Advances in p-adic and Non-Archimedean Analysis

CONTEMPORARY MATHEMATICS 508 Advances in p -adic and Non-Archimedean Analysis Tenth International Conference June 30–Ju...

Author: Martin Berz | Khodr Shamseddine (eds.) 2010

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CONTEMPORARY MATHEMATICS 508

Advances in p -adic and Non-Archimedean Analysis Tenth International Conference June 30–July 3, 2008 Michigan State University East Lansing, Michigan

Martin Berz Khodr Shamseddine Editors

American Mathematical Society

Advances in p-adic and Non-Archimedean Analysis

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CONTEMPORARY MATHEMATICS 508

Advances in p-adic and Non-Archimedean Analysis Tenth International Conference June 30–July 3, 2008 Michigan State University East Lansing, Michigan

Martin Berz Khodr Shamseddine Editors

American Mathematical Society Providence, Rhode Island

Editorial Board Dennis DeTurck, managing editor George Andrews

Abel Klein

Martin J. Strauss

2000 Mathematics Subject Classification. Primary 46S10, 11S80, 12J25, 16W30, 46G10, 32P05, 11D88, 30G06, 47B37.

Library of Congress Cataloging-in-Publication Data International Conference on p-Adic and Non-Archimedean Analysis (10th : 2008 : Michigan State University) Advances in p-adic and non-Archimedean analysis : Tenth International Conference on p-Adic and Non-Archimedean Analysis, June 30–July 3, 2008, Michigan State University, East Lansing, Michigan / Martin Berz, Khodr Shamseddine, editors. p. cm. — (Contemporary mathematics ; v. 508) Includes bibliographical references. ISBN 978-0-8218-4740-4 (alk. paper) 1. p-adic analysis—Congresses. 2. Topological fields—Congresses. I. Berz, M. II. Shamseddine, Khodr, 1966– III. Title. QA241I5848 2008 512.55—dc22 2009042367

Copying and reprinting. Material in this book may be reproduced by any means for educational and scientific purposes without fee or permission with the exception of reproduction by services that collect fees for delivery of documents and provided that the customary acknowledgment of the source is given. This consent does not extend to other kinds of copying for general distribution, for advertising or promotional purposes, or for resale. Requests for permission for commercial use of material should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294, USA. Requests can also be made by e-mail to [email protected]. Excluded from these provisions is material in articles for which the author holds copyright. In such cases, requests for permission to use or reprint should be addressed directly to the author(s). (Copyright ownership is indicated in the notice in the lower right-hand corner of the first page of each article.) c 2010 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Copyright of individual articles may revert to the public domain 28 years after publication. Contact the AMS for copyright status of individual articles. Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines

established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ 10 9 8 7 6 5 4 3 2 1

15 14 13 12 11 10

Contents Preface

vii

Strict topologies on spaces of vector-valued continuous functions over non-Archimedean field Jos´ e Aguayo, Samuel Navarro, and Miguel Nova

1

Some subalgebras of the algebra of bounded linear operators of the one variable Tate algebra Bertin Diarra

13

The ultrametric corona problem Alain Escassut and Nicolas Ma¨ınetti

35

Vector-valued p-adic measures Athanasios K. Katsaras

47

On the Clifford algebra of orthomodular spaces over Krull valued fields Hans A. Keller and Herminia Ochsenius

73

Divergence and convergence of conjugacies in non-Archimedean dynamics Karl-Olof Lindahl

89

A criterion for the invertibility of Lipschitz operators on type separating spaces H´ ector M. Moreno 111 On monomial dynamical systems on the p-adic n-torus Marcus Nilsson and Robert Nyqvist

121

On the value group and norms of a Form Hilbert space Herminia Ochsenius and Elena Olivos

133

Compact perturbations of Fredholm operators on Norm Hilbert spaces over Krull valued fields Herminia Ochsenius and Wim H. Schikhof

147

Applications of the p-adic Nevanlinna theory to problems of uniqueness Jacqueline Ojeda

161

Tensor products of p-adic locally convex spaces having the strongest locally convex topology Cristina P´ erez-Garc´ıa and Wim H. Schikhof

181

Tensor products of p-adic measures Chrysostomos G. Petalas and Athanasios K. Katsaras

187

v

vi

CONTENTS

p-adic arithmetic coding Anatoly Rodionov and Sergey Volkov

201

Analysis on the Levi-Civita field, a brief overview Khodr Shamseddine and Martin Berz

215

Criteria for non-repelling fixed points Per-Anders Svensson

239

A p-adic q-deformation of the Weyl algebra, for q a pN -th root of unity Fana Tangara

253

Preface The Tenth International Conference on p-adic and non-Archimedean Analysis took place at Michigan State University in East Lansing from June 30 to July 3, 2008. It follows other recent meetings of the series in Concepcion/Chile, ClermontFerrand/France, Nijmegen/Netherlands, and Ioannina/Greece. Further information on this meeting and the series can be found at www.bt.pa.msu.edu/NA08/. This volume contains a selection of papers presented at the meeting, and provides a cross section of some of the recent advances in the field. In July 2008, the participants of our conference were saddened to learn about the death of Nicole De Grande-De Kimpe, who has been a leading personality in p-adic analysis for over 30 years. She also contributed in the organization of the very first International Conference on p-adic Functional Analysis in Laredo, with colleagues from Santander, in the year 1990. We gratefully acknowledge the help and support from the international organizing committee with the selection of speakers for the meeting, the refereeing of the papers, and lots of valuable comments and good advice. Current members of the committee are Jesus Araujo, Martin Berz, Alain Escassut, José Aguayo Garrido, Athanasios Katsaras, Herminia Ochsenius, Wim Schikhof, and Khodr Shamseddine. Financial and logistic support was appreciated from Michigan State University, in particular the Department of Physics and Astronomy and the Department of Mathematics, as well as the US Department of Energy. We are also very thankful for the help of the local organizers José Aguayo Garrido, Kyoko Makino, Alexander Wittig, and Brenda Wenzlick, who helped assure smooth operation of all parts of the meeting. We are now looking forward to the next meeting in the series, which will be organized by Alain Escassut and take place at Université Blaise Pascal in Clermont-Ferrand in July of 2010. Martin Berz Khodr Shamseddine

vii


Contemporary Mathematics Volume 508, 2010

Strict topologies on spaces of vector-valued continuous functions over non-Archimedean field José Aguayo, Samuel Navarro, and Miguel Nova Abstract. Locally convex topologies βP on Cb (X, E) are introduced and developed, where X is a zero-dimensional topological space, E a locally Kconvex space and K is a non-Archimedean valued field. Among other things, different strict topologies are compared, bounded subsets of Cb (X, E) are characterized, polarity property of these strict topologies and gDF property are studied. The paper finishes with the proof of a theorem as the same type of the Ascoli’s Theorem.

1. Introduction Continuous functions and measures on topological spaces has been fundamental in the development of the integration theory. A number of locally convex topologies on spaces of continuous functions have been studied to clarify this relationship. These topologies are often called strict topologies. They enable the powerful duality theory of locally convex spaces to be profitably applied to the topological measure theory. It was R. C. Buck [3] who coined the term “strict topology” in the 1950’s, but the pre-history of the subject goes back to the familiar Riesz Representation Theorem. J. Prolla [10] was the first researcher who took this real or complex development to the non-Archimedean case. He defined the strict topologies in the space of all continuous and bounded functions from a locally compact and zero-dimesional topological space to a locally convex vector space E over a non-Archimedean valued field K. A. K. Katsaras [4, 5, 6, 7] , who also worked in the classical case, followed developing this interesting theme, introducing a variety of such topologies. One of these topologies is a particular case of the strict topologies introduced in the present paper. In this paper we introduce and study the locally convex topologies βP on Cb (X, E), where X is a zero-dimensional topological space, E a locally K-convex 2000 Mathematics Subject Classification. Primary 46S10, 46E10, 46E27, 46A70, 46G10; Secondary 46A03, 54C35. Key words and phrases. Non-Archimedean structures, strict topologies, zero-dimensional spaces, space of non-Archimedean measures. Research partially supported by Proyecto DIUC No. 205.013.024-1.0, UDEC, and by Proyecto DICYT, No. 040833NH, USACH. c 2001 c 2010 American Mathematical enter name of copyrightSociety holder

1

2

´ AGUAYO, SAMUEL NAVARRO, AND MIGUEL NOVA JOSE

space and K is a non-Archimedean valued field. Among other things, the authors define and compare different strict topologies, characterize bounded subsets of Cb (X, E) and study the polarity and gDF properties of such topologies. Finally, the authors prove a theorem of the same type as the Ascoli’s Theorem. 2. Preliminaries and notation Throughout this paper, X is a zero-dimensional Hausdorff topological space, (K, |·|) is a non-Archimedean, non-trivial valued complete field and E is a Hausdorff locally K-convex space. We will denote by VT (0) the collection of all neighborhoods of 0 of E and by cs (E) the collection of all continuous semi-norms on E. Now, if p ∈ cs (E) , we will also denote by Bp (e, ) the set of all v ∈ E such that p (e − v) ≤ . If A is a subset of X, f is a function from X to E and p is a semi-norm on E, then we define f A,p and f p by f A,p = sup {p (f (x)) : x ∈ A} f p = f X,p . If E = K, then f A = sup {|f (x)| : x ∈ A} f = f X . It is clear that all of them are non-Archimedean semi-norms on Cb (X, E), the space of all continuous and bounded functions from X into E. We denote by K (X) the set of all compact subsets of X and designate by B (X) the collection of all directed families of K (X) whose union is X. Let P1 , P2 ∈ B (X) ; we say that P1 is a refinement of P2 if for any K1 ∈ P1 , there exists K2 ∈ P2 such that K1 ⊂ K2 . For a given P ∈ B (X) , a function v : X → K is said to be P −vanished at infinite if it is bounded and for each > 0 there exists K ∈ P such that vX\K < . We denote by B (P ) the set of all functions on X which are P −vanished at infinite. A semi-norm p on a K-vector space F is a polar semi-norm if p = sup {|x∗ | : x∗ ∈ F ∗ , |x∗ | ≤ p} . A locally convex space F is polar if its topology is generated by a family of polar semi-norms. A locally convex space E is quasi-complete if each bounded and closed set is complete. 3. Topologies on Cb (X, E) The space Cb (X, E) has a natural locally convex topology,called the uniform topology Tu , and which is generated by the family of semi-norms ·p : p ∈ cs (E) . For P ∈ B (X) , the family of semi-norms ·K,p : K ∈ P, p ∈ cs (E) defines the locally convex topology denoted by TP . Now, for a fixed directed family P of K (X) , we define the following semi-norm f v,p = sup {|v (x)| p (f (x)) : x ∈ X} where p is a continuous semi-norm on E, v is on P −vanished function at infinite, K ∈ P and f ∈ Cb (X, E). It is clear that ·v,p is a semi-norm on Cb (X, E) and we

STRICT TOPOLOGIES

3

will topology generated by the family of semi-norm denote by βP the locally convex ·v,p : v ∈ B (P ) , p ∈ cs (E) . It is easy to see that TP is coarser than βP , since for any K ∈ P, v = XK is on P −vanished functions at infinite and ·v,p = ·K,p . Proposition 1. If P1 is a refinement of P2 , then (1) TP2 is finer than TP1 and βP2 is finer than βP1 . (2) Tu is finer than βP , for any directed family P of K (X) . (3) βP = TP if and only if each union of a countable family of P is contained in some element of P. (4) βP = Tu if and only if X ∈ P. (5) βP is a Hausdorff locally convex topology. Proof. (1.)It is derived from the definitions. (2.) Since B1 = f ∈ Cb (X, E) : f v,p ≤ 1 is a βP −neighborhood of 0 , where v ∈ B (P ) and p ∈ cs (E) , and it contains the Tu −neighborhood of 0, B2 = 1 f ∈ Cb (X, E) : f p ≤ 1+v , then it is enough to prove that Tu is finer than βP . (3.) Suppose that βP = TP and let {Kn } be a sequence of elements of P. We have to prove that there exists some K ∈ P such ∪Kn ⊂ K. that ∞ For λ ∈ K, with 0 < |λ| < 1, we define v = n=1 λn XKn . We claim that v ∈ B (P ) . In fact, ∞ n λ XKn (x) ≤ max {|λn XKn (x)| : n ∈ N} ≤ |λ| , |v (x)| = n=1

which says v is bounded. On the other hand, if > 0 is given, then there exists n N ∈ N such that for n ≥ N, |λ| < . Since P is directed, there exists K ∈ P such that ∪N i=1 Ki ⊂ K. Now, if x ∈ X K, then ∞ n |v (x)| = λ XKn (x) ≤ max {|λn | : n ≥ N + 1} < , n=N +1

that is, v is P −vanished at infinite. By the assumption, the identity linear operator from (Cb (X, E) , TP ) onto (Cb (X, E) , βP ) is continuous, hence for a given p ∈ cs (E) and for the above v we can take a positive number M, K ∈ P and q ∈ cs (E) such that (3.1)

f v,p ≤ M f K,q , ∀f ∈ Cb (X, E) . ∪∞ i=1 Kn

We claim that ⊂ K. In fact, if x0 ∈ ∪∞ i=1 Kn K, then there exists a clopen subset U of X that contains x0 and U is contained in X K. Choose e ∈ E with p (e) = 0 and take f0 = XU ⊗ e ∈ Cb (X, E) . Note that ∞ λn XKn (x0 ) vU ≥ |v (x0 )| = n=1

= max {|λn XKn (x0 )| : n ∈ N} ≥ |λ|n > 0 which implies f0 v,p ≥ p (e) vU > 0.

4


This is a contradiction, since f0 K,q = sup {q (f0 (x)) : x ∈ K} = 0. Conversely, suppose that each union of a countable family of P is contained in some element of P. We must prove that βP ⊆ TP . Let B1 = f ∈ Cb (X, E) : f v,p ≤ 1 be a βP −neighborhood of 0. Since v ∈ B (P ), for each n ∈ N, there exists a compact Kn such that vXKn < n1 . By the assumption, there exists K ∈ P 1 which is a containing ∪∞ i=1 Kn . We define B2 = f ∈ Cb (X, E) : f K,p ≤ 1+v TP −neighborhood of 0. We claim that B2 ⊆ B1 ; in fact, for a f ∈ B2 and x ∈ X, we have v ≤ 1 if x ∈ K, |v (x)| p (f (x)) ≤ v f K,p ≤ 1 + v and |v (x)| p (f (x)) ≤ vXK f p ≤ vXKn f p 1 < f p , ∀n ∈ N, if x ∈ /K n that is, |v (·)| p (f (·)) = 0, ∀x ∈ X K. Thus, f v,p = sup {|v (x)| p (f (x)) : x ∈ X} ≤ v f K,p ≤ 1 and then f ∈ B1 . It is enough to say βP ⊆ TP , and then βP = TP . (4.) Suppose βP = Tu and prove X ∈ P. Thus, the identity linear operator from (Cb (X, E) , βP ) onto (Cb (X, E) , Tu ) is continuous, hence for a given p ∈ cs (E) there exist M > 0 q ∈ cs (E) and v ∈ B (P ) such that f p ≤ M f v,q ; ∀f ∈ Cb (X, E) . Note that if e ∈ E is such that p (e) > 0, then q (e) > 0. Now, since v ∈ B (P ) , there exists K ∈ P such that vXK
1 and for each x ∈ X choose λx ∈ K such that |λx | ≤ |v (x)| ≤ |λλx | . Define Φ : X → K by Φ (x) = λx and note that Φ = sup {|Φ (x)| : x ∈ X} = sup {|λx | : x ∈ X} ≤ v < ∞ and if > 0 is given, then there exists a compact subset K of X such that vXK < . Thus, since ΦXK ≤ vXK , we conclude that Φ ∈ B (P ) . On the other hand, f Φ,p ≤ f v,p ≤ |λ| f Φ,p ; ∀f ∈ Cb (X, E) , that is, ·Φ,p ≤ ·v,p ≤ |λ| ·Φ,p . We claim that ·Φ,p is a polar semi-norm in Cb (X, E) , i. e., ∗ ·Φ,p = sup |Ψ (·)| : Ψ ∈ Cb (X, E) , |Ψ| ≤ ·Φ,p . Let f ∈ Cb (X, E) such that f Φ,p > α > 0; hence there exists x0 ∈ X with |Φ (x0 )| p (f (x0 )) > α. Since p is polar, there exists e∗ ∈ E ∗ with |e∗ | ≤ p such that |Φ (x0 )| |e∗ (f (x0 ))| > α. Now, we define Λ : Cb (X, E) → K by Λ (g) = Φ (x0 ) e∗ (g (x0 )) . It is clear that Λ is a linear functional and α < |Λ (f )| = |Φ (x0 )| |e∗ (f (x0 ))| ≤ |Φ (x0 )| p (f (x0 )) ≤ f Φ,p . Thus, if we take α = f Φ,p −

1 n

for a big n, then

1 |Λ (f )| > f Φ,p − ⇒ |Λ (f )| ≥ f Φ,p n ∗ ⇒ sup |Ψ (f )| : Ψ ∈ Cb (X, E) , |Ψ| ≤ ·Φ,p ≥ f Φ,p . It is enough to have

·Φ,p = sup |Ψ (·)| : Ψ ∈ Cb (X, E)∗ , |Ψ| ≤ ·Φ,p .

The following lemma will give us tools to characterize the βP − bounded sets of Cb (X, E). Lemma 3.2. Let B be a subset of E X = {f : X → E : f function} . Then, B is uniformly bounded if and only if for any P ∈ B (X) , v ∈ B (P ) and p ∈ cs (E) , the set f v,p : f ∈ B is bounded in R.

6


Proof. Suppose that B is uniformly bounded, i. e., for any p ∈ cs (E) , ∃M > 0 such that B ⊂ f ∈ E X : f p ≤ M . Take P ∈ B (X), v ∈ B (P ) and p ∈ cs (E). Thus, if f ∈ B, then f v,p = sup {|v (x)| p (f (x)) : x ∈ X} ≤ M v . Conversely, if B is not uniformly bounded, then there exists p ∈ cs (E) , a sequence (fn ) in B and a sequence (xn ) in X such that p (fn (xn )) ≥ |λ|

2n

, for some λ ∈ K with |λ| > 1

We consider the function v=

∞

λ−n X{xn }

n=1

which is 0 if x = xn and λ−n , if x = xn ; also, v is bounded and vanishes at infinite. The last assertion comes from the following fact: for a given > 0, there exists N ∈ N such that |λ−n | < if n ≥ N, and if we choose the compact subset K = {x1 , x2 , · · · , xN } , then ∞ λ−n X{xn } (x) : x ∈ X K vXK = sup {|v (x)|} = sup n=1 ∞ k ≤ λ−n = lim λ−n k→∞ n=N +1 n=N +1 = lim max λ−n : n = N + 1, · · · , k = λ−(N +1) < , k→∞

that is v ∈ B (P ) , where P is the family of all finite subsets of X. Now, by the above sequence, we have fn v,p = sup {|v (x)| p (fn (x)) : x ∈ X} ≥ |v (xn )| p (fn (xn )) ≥ |λ| which means that

−n

f v,p : f ∈ B

|λ|

2n

n

= |λ| → ∞

is not bounded in R. This is a contradiction to the assumption.

Theorem 3.3. A subset of Cb (X, E) is βP − bounded if and only if it is uniformly bounded. Proof. If B ⊂ Cb (X, E) is βP − bounded, then B ⊂ f ∈ Cb (X, E) : f v,p ≤ 1 , that is f v,p : f ∈ B is bounded in R. Thus, by the above lemma, B is uniformly bounded. The converse is obvious since the uniform topology is finer than βP topology. The next theorem will relate the topology βP with the topology TP . Theorem 3.4. The topologies βP and TP coincide on each uniformly bounded subset of Cb (X, E) .

7

STRICT TOPOLOGIES

Proof. Let B be a uniformly bounded subsets of Cb (X, E) ; hence there exists M > 0 such that sup f p : f ∈ B ≤ M Since TP is coarser than βP , it is enough to prove that βP is coarser than TP on B. Take p ∈ cs (E) and v ∈ B (P ) and consider the βP −neighborhood of 0 V = f ∈ Cb (X, E) : f v,p ≤ 1 . Now, since v is P −vanished at infinite, there exists K ∈ P such that vXK ≤ 1/M. Thus, if f ∈ B, then f v,p = sup {|v (x)| p (f (x)) : x ∈ X}

≤ max sup |v (x)| p (f (x)) , sup |v (x)| p (f (x)) x∈K x∈XK

≤ max sup |v (x)| p (f (x)) , 1 ≤ max f K,p v , 1 . x∈K

Therefore, if

W =

f ∈ Cb (X, E) : f K,p

is a TP −neighborhood of 0, then we claim that

1 ≤ v + 1

W ∩ B ⊂ V. In fact, 1 ∧ f ∈B v + 1 ⇒ f v,p ≤ max f K,p v , 1

1 v , 1 ≤ 1 ≤ max v + 1 ⇒f ∈V

f ∈ W ∩ B ⇒ f K,p ≤

4. gDF spaces and the strict topologies Definition 4.1. A locally convex space (E, T ) is a gDF space if i. there exists a countable fundamental system of bounded sets of E, and ii. T is the finest locally convex topology on E which coincides with T on each element of the fundamental system. Definition 4.2. An increasing sequence {An } of bounded sets of E is said to be absorbent if for any x ∈ E there exists n ∈ N and λ ∈ K, λ = 0 such that x ∈ λAn . The next lemma can be found in [9] : Lemma 4.3. Let {An } be an absorbent sequence of K-convex subset of E. Then, the collection {∩∞ n=1 (An + Un ) : Un ∈ VT (0) } is a basis of neighborhoods of 0 for the finest locally convex topology which agrees with T on each An .


8

Theorem 4.4. Suppose that E is a non-Archimedean Banach space. Then, for any P ∈ B (X) , (Cb (X, E) , βP ) is a gDF space. Proof. Let B be the unit ball of Cb (X, E) and let {δn } be a sequence of K such that |δn | ∞. The family {Bn = δn B : n ∈ N} is a fundamental system of βP −bounded subsets of Cb (X, E). By Th. 2, βP and TP agree on each uniformly bounded subset of Cb (X, E) , therefore if we prove that βP is the finest locally convex topology with this property, then we are done. Let us denote by γ the finest locally convex topology which agrees with TP . Since βP and TP agree on each uniformly bounded subset, we have that βP ≤ γ. Now, by the previous lemma, a typical 0-neighbourhood of γ is of the form U = ∩∞ n=1 (Un + Bn ) , where Un is a TP −neighborhood of 0. We claim that U is a βP −neighborhood of 0; in fact, take a sequence (ρn ) on K such that |δn−1 | < |ρn | ≤ |δn | and a βP −neighborhood Wp (Kn , ρn ) of 0, with Kn ∈ P and p = · . Take f ∈ Wp (Kn , ρn ) and n ∈ N; since f (∪ni=1 Ki ) is compact in E, there exists xj ∈ ∪ni=1 Ki , j = 1, · · · , k, such that f (∪ni=1 Ki ) ⊂ ∪kj=1 B (f (xj ) , |δn−1 |) . Note that

Cn = f −1 ∪kj=1 B (f (xj ) , |δn−1 |)

is clopen and ∪ni=1 Ki ⊂ Cn . It is clear that f = XX\Cn f + XCn f,

XX\Cn f ∈ Un and XCn f ∈ Bn .

Since n ∈ N was arbitrary, we conclude that f ∈ U. From this, we conclude that γ ≤ βP . 5. Compactoid and the Ascoli theorem We start this section with the definition of compactoid subsets of a locally convex space. Definition 5.1. A subset C of a locally convex space E is said to be a compactoid if for a given neighborhood U of 0 in E there exists a finite subset F of E such that C ⊂ co (F ) + U, where co (F ) denote the absolutely convex hull of F. Of course, every finite subset of E is a compactoid; if C is contained in a compactoid, then C is also a compactoid; and every compactoid in a locally convex space is bounded. The next definition is topological concept: Definition 5.2. Let X be a zero-dimensional topological space. We will say that X is said to be a P −space if f : X → K is continuous whenever f|K is continuous in K, for every K ∈ P .

STRICT TOPOLOGIES

9

Remark 5.3. Note that if P = K (X) , then X is a k0 −space (see [12] , p. 273). If P is co-final, that is, for every compact K of X there exists a compact C ∈ P such that K ⊂ C, then k0 −space and P −space coincide. Theorem 5.4. Suppose that X is a P −space. Then, (Cb (X) , βP ) is quasicomplete. Proof. It suffices that each bounded and βP −Cauchy net is βP −convergent. Let {fα } be a bounded and βP −Cauchy net. For a fixed x ∈ X, we define Hx : Cb (X) → K by Hx (f ) = f (x) . We claim that Hx is a βP −continuous linear functional; in fact, if > 0 is given, choose v ∈ B (P ) such that |v (x)| ≥ 1 (it is possible, since x ∈ K for some K ∈ P and XK ∈ B (P )). It is easy to see that f ∈ V = {f ∈ Cb (X) : f v ≤ } ⇒ |f (x)| ≤ . From this, {fα (x)} is Cauchy in K and then it is convergent. Since x ∈ X is arbitrary, we can define f (x) = lim fα (x) . α

In order to prove that f is continuous, we will prove that f|K is continuous for every K ∈ P. Let K ∈ P, x0 ∈ K and a net {xγ } in K be such that xγ → x0 , x0 ∈ X. It is clear that v = XK ∈ B (P ) . Thus, for a given > 0 there exists α0 such that fα − fβ v ≤ α, β ≥ α0 in particular, |fα (x) − fβ (x)| = |v (x)| |fα (x) − fβ (x)| ≤ fα − fβ v ≤ α, β ≥ α0 ∀x ∈ K and then Now, since fα0

|fα (x) − f (x)| ≤ , α ≥ α0 ∀x ∈ K is continuous, there exists γ0 such that |fα0 (xγ ) − fα0 (x0 )| ≤ γ ≥ γ0 .

So, |f (xγ ) − f (x0 )| ≤ max {|f (xγ ) − fα0 (xγ )| , |fα0 (xγ ) − fα0 (x0 )| , |fα0 (x0 ) − f (x0 )|} ≤ . Thus, since K ∈ P and x0 ∈ K are arbitrary, we conclude that f|K is continuous and f is continuous. Therefore, f ∈ Cb (X) . The following proposition was proved in [6] Proposition 3. Let F be a Hausdorff polar space and let G denote its dual equipped with the topology of uniform convergence on compactoid subsets of F . If F is quasi-complete, then G = F. Theorem 5.5. Let X be a P −space. Then, a subset F ⊂ Cb (X) is a βP − compactoid if and only if the following properties are satisfied, (1) R = supx∈X; f ∈F |f (x)| < ∞. (2) F is equicontinuous.


10

Proof. Suppose that F is a βP −compactoid. Then condition 1 holds immediately, since βP −compactoid implies βP −bounded or equivalently, uniformly bounded. In order to prove that F is equicontinuous, let us take F = (Cb (X) , βP ) and G = (Cb (X) , βP ) as in the previous proposition. By Prop. 2, F is polar and by Th. 4, F is quasi-complete. For a fixed x ∈ X, the linear functional δx (f ) = f (x) is βP − continuous. Then we can define Ψ : X → (G, σ (G, F )) by Ψ (x) = δx . Ψ is continuous, since if K ∈ P, then Ψ (K) ⊂ V 0 , where V = {f ∈ Cb (X) : f v ≤ 1} and v = XK . Even more, Ψ (K) is equicontinuous in F. Now, since on equicontinuous sets of the dual of a locally convex space the weak topology coincides with the topology of uniform convergence on compactoids (see [11] , lemma 10.6), we have that Ψ : K → Ψ (K) is continuous, where Ψ (K) is provided with the topology of uniform convergence on compactoids. Thus, since X is a P − space and F|K is equicontinuous for any K ∈ P, we conclude that F is equicontinuous. Conversely, let W be a βP −neighborhood of 0. Thus, there exist v ∈ B (P ) and > 0 such that V = {f ∈ Cb (X) : f v ≤ } ⊂ W. In order to prove that F is a βP −compactoid, we need to find a finite subset {f1 , f2 , · · · , fn } such that F ⊂ co (f1 , f2 , · · · , fn ) + W, where co (f1 , f2 , · · · , fn ) denotes the absolutely convex hull of {f1 , f2 , · · · , fn } . Let λ ∈ K be such that |λ| > max {1, v , R} . For

|λ|2

> 0, there exists K ∈ P such that |v (x)|
n. Since for any non negative integer n, the polynomial Πn has degree n, one sees that this sequence of polynomials is a linear basis of the K-vector space of polynomials K[X], with (Πj )0≤j≤n a basis of the space of polynomials of degree ≤ n. n βj (n)Πn (X), one sets βj (n) = 0, Hence for any integer n ≥ 0, one has X n = j=0

whenever j > n. The numbers βj (n) are the Stirling numbers of second kind. For any fixed integer n, the triangular matrix (αj ())0≤j,≤n ∈ M atn (Z) has only the number 1 on its principal diagonal. Hence its determinant is equal 1 and it is invertible in M atn (Z), with inverse the triangular matrix (βj ())0≤j,≤n which therefore has its cœfficients βj () in the ring Z, with βn,n = 1. It follows that ∀j, n ≥ 0, one has max(|αj (n)|, |βj (n)|) ≤ 1. As a consequence one obtains the following lemma : Lemma 1.1. The sequence of Pochhammer polynomials (Πn )n≥0 is an orthonormal basis of K{X}. We shall see later that there is a similar orthonormal basis of K{X} attached 1 to any element q of K such that |q − 1| p−1 , whenever K is an extension of the field of p-adic numbers Qp .

SUBALGEBRAS OF BOUNDED LINEAR OPERATORS ONalgebra TATE ALGEBRA Subalgebras of bounded linear operators on Tate

15 3

One aim of this paper is to give, on one hand, the description of the set of all continuous linear endomorphisms of K{X} that commute with the translation operators with elements of Λ and on the other hand that of the continuous linear endomorphisms that commute with the homothety operators defined by the elements of Λ. A second aim concerns the study of some commutation relations between specific continuous linear operators of K{X} wich give rise to non-commutative subalgebras of the algebra L(K{X}) which are models of the Weyl algebra as well quantum Weyl algebra and quantum plane algebra. The closure of these algebras in the Banach algebra L(K{X}) of the continuous linear endomorphisms of K{X} are shown to be their completion and an orthogonal basis of them are constructed. 2. The algebra of continuous linear operators of K{X} commuting with translation operators 2.1. The translation operators . −•− Let a ∈ Λ, then for any restricted formal power series f (X) = an X n ∈ K{X}, the series f (a) =

n≥0

an an converges in K. In fact one defines by this way

n≥0

an analytic function on the closed unit disc (= the valuation ring ) Λ with values in K. Moreover, since an (X + a)n = |an | → 0, one defines the translation operator by a by setting τa (f )(X) = f (X + a) = an (X + a)n ∈ K{X}. n≥0 n i+j i j i j an ai+j aX = aX = Notice that f (X + a) = i i i+j=n i,j n≥0 i+j i bj (a)X j , where the sequence bj (a) = ai+j a converges to 0, because i

j≥0

i≥0

|bj (a)| ≤ sup |ai+j | and lim sup |ai+j | = 0. j→+∞ i≥0

i≥0

Furthermore τa (f ) = sup |bj (a)| ≤ sup sup |ai+j | = f . j≥0

j≥0 i≥0

However, τ−a (f ) ≤ f , hence τa (f ) = f and one obtains an isometric linear operator τa on K{X}. Since τ1 (f ) belongs to K{X}, the finite difference operator ∆ = τ1 − id is a well defined continuous linear operator on K{X} with norm equal 1. For the Pochhammer polynomials, one has ∆(Π0 ) = 0 and for any integer n ≥ 1, one verifies that ∆(Πn ) = nΠn−1 . Hence ∆j (Πn ) = n · · · (n − j + 1)Πn−j , 0 ≤ j ≤ n, ∆n (Πn ) = n!Π0 and j ∆ (Πn ) = 0, ∀j > n. bn Πn be the expansion of the restricted power series f with respect Let f = n≥0

to the Pochhammmer basis. For any integer j ≥ 0, on has ∆j (f ) = j + 1)bn Πn−j . Therefore ∆j (f )(0) = j!bj and f =

∆n (f )(0) Πn n!

n≥0

n≥j

n · · · (n −

16 4

BERTIN Bertin DIARRA Diarra

− • •− Let D be the operator of derivation on K{X} such that for f (X) = n an X ∈ K{X}, one has D(f )(X) = nan X n−1 . As usual, for any inn≥1 j D (f ) Dj (f ) n+j n = (X) = an+j teger j ≥ 0, one has X . Hence j! j! j n≥0 j n+j |an+j | ≤ sup |an+j |. It follows that lim D (f ) = 0 and the sesup j→+∞ j j! n≥0 n≥0 Dj (f ) converges in K{X}. ries exp(D)(f ) = j! n≥0

j≥0

Dj as defined j! above are continuous on K{X} with norms equal to 1, see for instance [10]. Let us also remind, that on the algebra L(K{X}), there is a structure of locally convex topology called the strong operator topology that is defined by the family of semi-norms {u(f ), f ∈ K{X}}. −&− For a ∈ Λ and f = an X n ∈ K{X}, one has Let us notice that it is readily seen that the linear operators

n≥0 Di (f ) i + j (X) = exp(aD)(f ). Hence, one has τa (f ) = Xj = ai+j ai ai i! i i,j i≥0 ai D i the strong convergence expansion of series of operators and exp(aD) = τa i!

i≥0

( Taylor formula ). −&&−

Since f =

∆n (f )(0) Πn , for any a ∈ Λ, one has n!

n≥0

Πn (a) ∆ (f )(0) . Furthermore, for any other b ∈ Λ, one has f (a) = n! n≥0 a τa (f )(b) = τb (f )(a) = f (a + b) = ∆n (f )(b). Moreover, one has in K{X} n n≥0 a the expansion of convergent series τa (f ) = ∆n (f ); that means that in the n n

n≥0

strong topology, one has the expansion a ∆n = (id + ∆)a . τa = n n≥0

2.2. The algebra of operators that commute with the translation operators . Let Endcom (K{X}) be the set of the continuous linear endomorphisms u of K{X} such that u ◦ τa = τa ◦ u, ∀a ∈ Λ. The elements of Endcom (K{X}) are said to be the shift-invariant operators or composition operators or difference operators of K{X}. It is readily seen that Endcom (K{X}) is a closed unitary subalgebra of the Banach algebra L(K{X}).


17 5

The counterpart of this algebra for the space of p-adic continuous functions on the ring of p-adic integers has been studied by L. Van Hamme in [16], see also [18] and concerns p-adic umbral calculus which, with the aid of Theorem 2.1 below, can also be performed on the space K{X} to. −†− According to the fact that one has for any a ∈ Λ, the expansion in a the strong topology τa = ∆n = (id + ∆)a , one sees that u belongs to n n≥0

Endcom (K{X}) if and only if u commutes with ∆ = τ1 − id. f (X + a) − f (a) = D(f )(X). For any f ∈ K{X}, one has lim a→0 a a (−1)n−1 1 a = , one obtains ∆n (f (X)) and lim Since f (X + a) = a→0 a n n n n≥0 (−1)n−1 D(f )(X) = ∆n (f )(X) = log(id + ∆)(f )(X), that is, in the strong n n≥1 (−1)n−1 ∆n = log(id + ∆). topology one has the expansion D = n n≥1

−††− As a consequence u ∈ Endcom (K{X}) ⇐⇒ u commutes with D. Theorem 2.1. Any shift-invariant operator u ∈ Endcom (K{X}) can be written as a unique strongly convergent series of one of the forms : Dn , where αn = u(X n )(0) ∈ K. Furthermore u = αn −(i)− u = n! n≥0

sup |αn |. n≥0

−(ii)−

u=

n≥0

cn

∆n , where cn = u(Πn )(0). Moreover u = sup |cn |. n! n≥0

Proof :

Dn (f )(a) X n . Let n! n≥0 Dn (f )(a) u(X n ). It u ∈ Endcom (K{X}), then τa ◦ u(f )(X) = u ◦ τa (f )(X) = n! n≥0 Dn (f )(a) n follows that τa ◦ u(f )(0) = u(f )(a) = u(X )(0), ∀a ∈ Λ and u(f ) = n! n≥0 n D Dn (f ) n . Since sup |u(X n )(0)|f . u(X )(0) n! = 1, one has u(f ) ≤ n≥0 n! n≥0 m m n m u(X )(0) X m−n . It follows But for any integer m ≥ 0, one has u(X ) = n n=0 m n m |u(X )(0)| ≥ |u(X m )(0)| and in conclusion u = that u(X ) = max 0≤n≤m n sup |u(X n )(0)| − • − For a ∈ Λ and f ∈ K{X}, one has τa (f )(X) =

n≥0

18 6


− • •− f (X + b) =

Since f (a + b) = n≥0

∆n (f )(b)

a , for all a, b ∈ Λ, one sees that n

n≥0 ∆n (f )(b) X ∆n (f )(b) = Πn (X). n n! n≥0

In the same way, as above, if u ∈ Endcom (K{X}), one has ∆n (f ) u(Πn )(0) and u = sup |u(Πn )(0)|, because, as for the derivau(f ) = n! n≥0 n≥0 n ∆ tive, one can prove that n! = 1. Remark 1 : -(1)- One deduces from Theorem 2.1 that Endcom (K{X}) is a unitary commutative Banach algebra isometrically isomorphic to the subalgebra K! < T >= {S = Tn αn ∈ K[[T ]] / sup |αn | < +∞} of the formal Hurwitz algebra. n! n≥0 n≥0

-(2)- Consider the continuous Banach algebra morphism c : K{X} −→ K{X}⊗K{X} (= topologically tensor product) such that c(X) = X ⊗ 1 + 1 ⊗ X. Hence, one defines on K{X} a coproduct for which K{X} becomes a Banach coalgebra with counit σ(f ) = f (0). n −†− Notice that c(X n ) = X i ⊗ X j and i i+j=n n c(Πn ) = Πi ⊗ Π j . i i+j=n In other words, the bases (X n )n≥0 and (Πn )n≥0 are binomial divided powers sequences. −††− Let us say that a continuous linear endomorphism u of K{X} is a (left) comodule endomophism of K{X} if c ◦ u = (idH ⊗ u) ◦ c. It can be easily proved that the space of continuous comodule endomorphisms of K{X} coincides with the algebra Endcom (K{X}). This point of view gives an unified proof of the parts (i) and (ii) of Theorem 2.1. -(3)- If the field K is of residue characteristic zero, that is stated above can be reproduced without any modifications. On the other hand if K is of characteritic p = 0, defining the higher derivations

(resp. higher shift-invariant operators) by setting D[j] (X n ) = nj X n−j resp. ∆j Dj

resp. by ∆[j] (Πn ) = nj Πn−j , one has analogue results by replacing j! j!

D[j] resp. ∆[j] . 3. Restricted power series q-base for K{X} It is well known that the radius of convergence of the formal logarithm power (−1)n−1) xn (resp. series (resp. the formal exponential series) log(1 + x) = n n≥1


exp(x) =

19 7

1 xn ) in a valued complete field K extension of the p-adic numbers n!

n≥0

1

field Qp is equal 1 (resp. |p| p−1 ). Let q ∈ K be such that |q −1| for any x in the p-adic integers ring Zp , the < 1. Then (q − 1)n x expansion q x = Πn (x) defines a continuous function = (q − 1)n n! n n≥0

n≥0 1

on Zp . Furhermore, if |q − 1| < |p| p−1 , then one has formal power series q X =

|q − 1|n = 0 and the n→+∞ |n!| lim

(q − 1)n Πn (X) belongs to K{X}, with q X = 1. n!

n≥0

Hence we have an analytic function defined on Λ. Let us notice that, with the same condition on q, one also has q X = exp(X logp (q)), where logp is the p-adic logarithm function and | logp (q)| = |q − 1|. q X−a − 1 Consider for a ∈ Λ, the element [X − a]q = of K{X}. For a = 0, q−1 n | logp (q)|n 1 (logp (q)) 1 one has [X]q = X n and [X]q = sup = q−1 n! |q − 1| n≥1 |n!| n≥1

|q − 1|n−1 sup = 1. |n!| n≥1 Since τ−a is an isometry,one has [X− a]q = τ−a [X]q = [X]q . logp (q) 1 (logp (q))n n −1 X+ X of K{X}, For the element [X]q − X = q−1 q−1 n! n≥2 logp (q) −1 |q − 1|n−1 − 1 , sup ≤ γp (q) = |q−1||p| p−1 < one has [X]q −X = max q−1 |n!| n≥2 1. Moreover for any a ∈ Λ, one has [X − a]q − (X − a) = [X]q − X ≤ γp (q). n−1 − • − Put Ψn,q (X) = [X − j]q . j=0

Considering the Pochhammer polynomials Πn (X) =

n−1

(X − j), one sees that

j=0

Ψn,q (X) − Πn (X) =

n−1

X · · · (X − j + 1)([X − j]q − (X − j))[X − n + j + 1]q · · · [X −

j=0

n + 1]q . Hence Ψn,q (X) − Πn (X) ≤

max

0≤j≤n−1

[X − j]q − (X − j) ≤ γp (q) =

−1 p−1

< 1. |q − 1||p| Applying a well known theorem on perturbation of orthonormal basis (cf. [11], [1], [2] ) one obtains the following proposition proved in [1] 1

Proposition 3.1. Let q ∈ K be such that |q − 1| < |p| p−1 . n−1 [X − j]q is an orThen the sequence of formal power series Ψn,q (X) = thonormal basis of K{X}.

j=0

20 8


We shall now give in this context a q-expansion for the shift-invariant operators of K{X}. − • − Let us consider the q-symbolic powers of the operator τ1 defined by (n) (n) setting (τ1 − id)(n) = (τ1 − id) · · · (τ1 − q (n−1) .id) = ∆q . One has ∆q ≤ 1. qn − 1 Let us remind that the q-integers and q-factorials are given by [n]q = q−1 n−1 Ψn,q (X) are analogue [n − j]q . The formal power series Cn,q (X) = and [n]q ! = [n]q ! j=0 of the binomial polynomials. One verifies directly, or using computations on the Cn,q done in [1] and [13], that ⎧ (j) ⎪ ⎨ ∆q Ψ (X) = n q −j(n−j) q jX Ψ ∀0 ≤ j ≤ n n,q n−j,q (X), [j]q ! j q ⎪ ⎩ (j) ∆q Ψn,q (X) = 0, ∀j ≥ n + 1 n [n]q ! are the Gauss q-binomials cœfficients. Here the = j q [n − j]q ![j]q ! Let f = cn Ψn,q be the expansion of the element f of K{X} in the basis n≥0 (j)

(Ψn,q )n≥0 . One has ∆q (f )(X) =

≥0

cj+

[j + ]q ! −j jX q q Ψ,q (X). []q !

(j) ∆q(n) (f )(0) ∆q (f )(0) = cj and f = Ψn,q . Hence [j]q ! [n]q ! n≥0

(j)

∆q Ψn,q (X), one sees that − • − With the above formulas giving [j] q! ∆(j) ∆(j)

q n q n Ψn,q = =⇒ = 1 =⇒ ∆q(j) = |[j]q !| = |j!| . = sup [j]q ! j q [j]q ! n≥j j q (j)

− • •− Also from the above expansion of ∆q (f )(X), one deduces that (j) ∆(j) (f ) ∆q (f ) q = 0. ≤ sup |cj+ | =⇒ lim j→+∞ [j]q ! [j]q ! ≥0 It follows that if (γn )n≥0 is a bounded sequence in K, then for any element f ∆q(n) (f ) γn converges in K{X}. One defines by the of K{X}, the series w(f ) = [n]q ! n≥0

way a continuous linear endomorphism w of K{X} such that w = sup |γn |. n≥0

Let a ∈ Λ, one has τa (f )(X) = f (X + a) =

n≥0

(n) ∆q (f )(a)

[n]q !

Ψn,q (X). As in

Theorem 2.1, one obtains q-expansions of the shift-invariant operators. Theorem 3.2. Any shift-invariant operator u ∈ Endcom (K{X}) can be ex ∆q(n) , with γn = u(Ψn,q )(0) γn panded as a unique strongly convergent series u = [n]q ! and u = sup |γn |. n≥0

n≥0


21 9

4. The Jackson q-derivative and the q-difference operators In the first two subsections of this section, we assume only that K is an ultrametric complete valued field and we suppose that q is a unit of K that is not a root of unit ( for instance, if K is an extension of Qp any element q of K such that 1 |q − 1| < |p| p−1 is not a root of unit). 4.1. The Jackson q-derivative . The Jackson q-derivative Dq defined on the ring of polynomials can be extended in a natural way on K{X} by setting for any formal power series f : Dq (f )(X) = f (qX) − f (X) . For any integer n ≥ 0, one has Dq (X n ) = [n]q X n−1 , where [n]q qX − X is the n-th q-integer; hence for f (X) = an X n ∈ K{X}, one has Dq (f )(X) =

n≥0

[n]q an X

n−1

. Furthermore Dq (f ) = sup |[n]q ||an | ≤ f and since Dq (X) = 1, n≥1

n≥1

one sees that Dq = 1. For integers 0 ≤ j ≤ n, one has Dqj (X n ) = [n]q · · · [n − j + 1]q X n−j and n Dqj (f ) j n a X n−j and Dq (X ) = 0, ∀j ≥ n + 1. Hence one obtains = j q n [j]q ! n≥j Dj (f ) n q |a | ≤ sup |an |. = sup [j]q ! n≥j j q n n≥j Dj (f ) Dqn (f )(0) q Xn Therefore, lim = 0. Hence on one hand, one has f = j→+∞ [j]q ! [n]q ! n≥0

and on the other hand, for any bounded sequence (δn )n≥0 ⊂ K, the series ω(f ) = Dqn (f ) δn converges in K{X} and one defines by this way a continuous linear [n]q ! n≥0

endomorphism ω of K{X} with norm ω = sup |δn |. n≥0 Dqj (X n ) n n n−j = Noticing that X , with = 0, whenever j ≥ n + 1, one j q j q [j]q ! Dj q n obtains = 1. = sup [j]q ! n≥j j q Proposition 4.1. The subalgebra Dq of L(K{X}) of the continuous linear endomorphisms of K{X} that commute with the Jackson derivative Dq is closed and Dqn any element ω of Dq is a unique strong sum ω = , with ω = sup |δn |, δn [n]q ! n≥0 n≥0

and δn = ω(X n )(0). Proof :

Let cq be the linear map from K{X} into the topological tensor Dqn (f ) ⊗ X n ( a coproduct). Setting product K{X}⊗K{X} such that cq (f ) = [n]q ! n≥0

σ(f ) = f (0), one can show that (id ⊗ σ) ◦ cq (f ) = f = (σ ⊗ id) ◦ cq (f ). Moreover ω ∈ Dq if and only if cq ◦ ω = (id ⊗ ω) ◦ cq . Hence, if ω commutes with

22 10


Dq , one has ω(f ) = (id ⊗ σ) ◦ cq ◦ ω(f ) = (id ⊗ σ)(id ⊗ ω) ◦ cq (f ) = Dqn (f ) Dqn (f ) ω(X n )(0) ⊗ σ ◦ ω(X n ) = . [n]q ! [n]q ! n≥0

n≥0

4.2. An analogue of Jackson derivative 1 . We assume here that Qp ⊂ K and that q ∈ K is such that |q − 1| < |p| p−1 . f (X + 1) − f (X) For f ∈ K{X} let us put ∇q (f )(X) = . Hence, one defines q X (q − 1) a continuous linear endomorphism ∇q of K{X}, called the second kind of Jackson 1 1 f and since ∇q (X) = X , one sees derivative. One has ∇q (f ) ≤ |q − 1| q (q − 1) 1 that ∇q = . |q − 1| We have defined previously the restricted power series Ψ0,q (X) = 1 and n ≥ n−1 q X−j − 1 and showed that they form 1, Ψn,q (X) = [X − j]q , with [X − j]q = q−1 j=0 an orthonormal basis of K{X}. 1 (1) ∆(1) (f ). Since ∆q (Ψn,q ) = [n]q q −(n−1) q X Ψn−1,q , Notice that ∇q (f ) = X q (q − 1) q [n]q Ψn−1,q . Then, for 0 ≤ j ≤ n, one verone obtains ∇q (Ψn,q ) = n−1 q (q − 1) j(j+1) ∇jq (Ψn,q ) ∇nq (Ψn,q ) q 2 −nj n = = ifies by induction that Ψ , with n−j,q [j]q ! (q − 1)j j q [n]q ! q

n(n+1) −n2 2

∇jq (Ψn,q ) = 0, ∀j ≥ n + 1. Ψ ; from which one deduces that 0,q (q − 1)n q! [j] ∇j 1 1 q n sup . Furthermore, one has = = [j]q ! |q − 1|j n≥j j q |q − 1|j dn Ψn,q is an element of K{X}, for any integer j ≥ 0, one has If f =

n≥0 j(j+1) ∇j (f ) ∇jq (f ) q 2 −nj n 1 q = dn Ψn−j,q . It follows that sup |d |, ≤ j j [j]q ! |q − 1|j n≥j n [j]q ! (q − 1) q n≥j

with sup |dn | → 0, when j → +∞. n≥j

∇nq (f )(0) ∇nq (f )(0) −n(n−1) =q 2 . n [n]q ! (q − 1) · · · (q − 1) |γj | < +∞. For Let (γj )j≥0 be a squence in K such that M = sup |q − 1|j j≥0 ∇j (f ) q dn Ψn,q , one has |γj | · any restricted formal power series f = ≤ [j]q ! −††−

Notice that dn = (q−1)n q

−n(n−1) 2

n≥0

|γj | sup |dn | ≤ M · sup |dn | → 0, when j → +∞. Hence the series (f ) = |q − 1|j n≥j n≥j


j≥0

γj

23 11

∇jq (f ) converges in K{X}. Thus one definies a continuous linear endomor[j]q !

phism of K{X} with norm = sup j≥0

|γj | . As in 4.1, one can prove the |q − 1|j

following proposition: Proposition 4.2. The subalgebra Cq of L(K{X}) of the continuous linear endomorphisms of K{X} that commute with the second kind of Jackson derivative ∇nq ∇q is closed and any element of Cq is a unique strong sum = γn with [n]q ! n(n−1) |γn | and γn = (q − 1)n q 2 (Ψn,q )(0). = sup n |q − 1| n≥0

n≥0

4.3. The q-difference operators . − • − Let us consider for a ∈ Λ and f ∈ K{X} the formal power series ha (f )(X) = f (aX). It is clear that ha (f )(X) is a restricted power series and one obtains a continuous algebra endomorphism ha of K{X}, called a dilatation or a homothety operator, such that ha (1) = 1 and ha = 1. If |a| = 1, one sees that ha is a bijective isometry with reciprocical h−1 a = ha−1 . − • •− Let as above q ∈ K be such that |q| = 1, and that is not a root of unit. For any a ∈ Λ, one has the commutation rule Dq ◦ ha = aha ◦ Dq . Consider the q-monomials (the q-symbolic powers of X) (0)

(n)

(X − 1)q = 1, (X − 1)q = (X − 1) · · · (X − q n−1 ), whenever n ≥ 1. One has the well known relations (see for instance [8]) ⎧ n (n−j)(n−j−1) n ⎪ (n) ⎪ 2 ⎪ (−1)n−j q Xj (X − 1)q = ⎪ ⎨ j q j=0 an X n ∈ K{X}, Let f = n n ⎪ ⎪ (j) n n≥ ⎪ (X − 1)q . ⎪ ⎩X = j q j=0

n n (X − 1)q(j) = j q j=0 n≥0 n n (j) (j) an (X − 1) = a (X − 1) = bj (X − 1)q(j) , with bj = n q q j q j q n≥0 j≥0 j≥0 n≥j j≥0 n a and |bj | ≤ sup |an | → 0, as j → +∞. j q n n≥j n≥j n (n) However, (X − 1)q = sup q = 1. Hence f = bj (X − 1)q(j) ≤ j 0≤j≤n

one has f =

an

j≥0

sup |bj | ≤ sup sup |an | = f . It follows that f = sup |bj | and one has the j≥0

j≥0 n≥j

j≥0

following proposition :

Proposition 4.3. The family K{X}.

(j)

(X − 1)q

is an orthonormal basis of j≥0

24 12

BERTIN Bertin DIARRA Diarra (j)

−††− Let (Dq )j≥0 be the elementary q-difference operators defined by (0) (j) setting Dq = id, Dq = (hq − id)(j) = (hq − id) · · · (hq − q (j−1) id), j ≥ 1. j−1 (j) − • − For the integers n and j ≥ 0, one has Dq (X n ) = (q n − q ) · X n = =0 (j)

(q n − 1)(j) · X n , with Dq (X n ) = 0, if j ≥ n + 1. For 0 ≤ j ≤ n, one has (q n − 1)(j) n = [ ]q . Since (X n )n≥0 is an orthonormal basis of K{X}, one sees that j (q j − 1)(j) (j) Dq = sup |(q n − 1)(j) | = |(q j − 1)(j) |. n≥0 (n)

− • •−

Consider the sequence of polynomials Qn (X) =

(X − 1)q

. It is well (n) (q n − 1)q (j) known that Dq (Qn (X)) = q −j(n−j) X j Qn−j (X), ∀0 ≤ j ≤ n. (n) (j) Hence Dq (Qn (X)) = X n and Dq (Qn (X)) = 0, ∀j ≥ n + 1. From what, one deduces that (j) (n) Dq ((X − 1)q ) n (j) (n) (n−j) Dq ((X − 1)q ) = X j (X − 1)q , ∀ 0 ≤ j ≤ n, and = j q (q j − 1)(j) 0, ∀j ≥ n + 1. Let H be the subset of L(K{X}) of the operators U that commute with the operators of dilatation ha , a ∈ Λ, i.e. U ◦ ha = ha ◦ U, ∀a ∈ Λ, such operators will be called multiplicative shift-invariant. It is readily seen that H is a closed subalgebra of L(K{X}). Furthermore, one has : Theorem 4.4. Any element U of the closed subalgebra H of L(K{X}) can be (n) Dq written in a unique form of a strongly convergent series U = βn n , (q − 1)(n) n≥0 with U = sup |βn | and βn = U ((X − 1)(n) )(1). Moreover H is a commutative n≥0

algebra. (n)

Proof : It is obvious that any elementary q-difference operator Dq = (n) (hq − id)q is multiplicative shift-invariant. Let f = bn (X − 1)(n) ∈ K{X}, n≥0

one has (j) D(j) (f ) n Dq (f ) q = bn X j (X − 1)q(n−j) and j ≤ sup |b | → 0, when j q (q − 1)(j) n≥j n (q j − 1)(j) n≥j j → +∞. Therefore, for any bounded sequence (βj )j≥0 ⊂ K, the series (j) Dq (f ) βj j converges in K{X}. By the way, one obtains a continuU (f ) = (q − 1)(j) j≥0 ous linear endomomorphism of K{X} that commutes with the multiplicative shift operators ha , a ∈ Λ. Moreover U = sup |βj |. j≥0 (n) However, for any f = bn (X − 1) ∈ K{X} and any integer n ≥ 0, one has n≥0 (n) Dq (f ) (1) (q j − 1)(n)

= bn . It follows that for a ∈ Λ, one has


ha (f ) =

25 13

Dq(n) (f )(a) (X − 1)(n) . Therefore, if U is a continuous linear endon − 1)(n) (q n≥0

morphism of K{X} that commutes with the multiplicative shift operators ha , Dq(n) (f )(a) U ((X − 1)(n) ). one has U (f )(aX) = ha ◦ U (f )(X) = U ◦ ha (f ) = n − 1)(n) (q n≥0 and U (f )(a) =

(n)

U ((X − 1)(n) )(1)

n≥0

Dq (f )(a) One then concludes that U (f ) = (q n − 1)(n)

(n)

U ((X − 1)(n) )(1)

n≥0

Dq (f ) . (q n − 1)(n)

Remark 2 : -(1)- Let us consider the algebra homomorphism c1 : K{X} −→ K{X}⊗K{X} such that c1 (X) = X ⊗ X, then again one has on K{X} a structure of Banach coalgebra with coproduct c1 and counit σ1 defined by σ1 (f ) = f (1). One sees that the subalgebra H of the elements of L(K{X}) which are multiplicative shift-invariant is equal to the algebra Endcom1 (K{X}) of the (left) comodule endomorphisms of the Banach coalgebra ( K{X}, c1 , σ1 ), i.e. the continuous linear endomorphisms U of K{X} such that c1 ◦ U = (id ⊗ U ) ◦ c1 . -(2)- Notice also that if K is of residue characteritic a prime number p and if q ∈ K, is not a root of unit such that there exists an integer > 0 with |q − 1| < 1, then (cf. [6]) the closed subgroup Vq of the group of units of Λ is a compact group and any restricted power series f ∈ K{X} defines a continuous function from Vq into K. This gives an explanation to the fact that any element U of H = Endcom1 (K{X}) can be expanded with the aid of the elementary q-différence (n) operators Dq .

5. Commutation relations for some continuous linear operators on K{X} 5.1. The two variables Weyl algebra . We assume here that the field K is of characteristic 0. Let Z be the operator of multiplication by X on K{X} and let D be the usual derivation of formal power series. For f ∈ K{X}, one has : D ◦ Z(f ) = D(Xf ) = D(X)f + XD(f ) = f + XD(f ), i.e. D ◦ Z − Z ◦ D = id (C1 ). Let W1 be the subalgebra of L(K{X}) generated Z and D. From the commutation rule, one deduces that any element u of W1 can be expanded in the form Dj λi,j Z i . Obviously, one has u ≤ sup |λij |. Furthermore u = j! i,j u(1)

(i,j)∈[0,m]×[0,n] m D0 (1) = λi,0 Z i 0! i=0

=

m i=0

λi,0 X i . Hence u(1) = max |λi,0 | ≤ u. 0≤i≤m

26 14


Let us suppose by induction that

sider vj =

max (i,)∈[0,m]×[0,j−1]

λi Z i

(i,)∈[0,m]×[0,j−1] j

One has uj (X ) =

|λi, | ≤ u and let us con-

D and uj = u − vj . !

λi Z

iD

(i,)∈[0,m]×[j,n]

(X j ) λij X i . = ! i=0 m

Therefore uj (X j ) = max |λij | ≤ max(u(X j ), vj (Xj )) 0≤i≤m |λi | ≤ u. ≤ max u, max (i,)∈[0,m]×[0,j−1] Dj |λi,j | and Z i is Hence we have proved that u = sup j! (i,j)∈N×N (i,j)∈[0,m]×[0,n] an orthonormal family in L(K{X}). Which implies that it is a base of the vector space W1 . −††− As a consequence, the algebra W1 is isomorphic to the Weyl algebra A1 with two generators, that is A1 = K < x, y > /I < x, y >, where I < x, y > is the two-sided ideal generated by xy − yx − 1 in the free algebra K < x, y > with the two generators x, y.

1 be the closure of W1 in L(K{X}). Theorem 5.1. Let W j 1 is a Banach subalgebra of L(K{X}) which admits Z i D Then W j! (i,j)∈N×N as an orthonormal base. 1 can be expanded as a unique convergent sum u = In other words any u ∈ W j D , lim λij = 0, with u = sup |λij |. λij Z i j! i,j (i,j)∈N×N (i,j)∈N×N

1 is a completion of W1 with respect the operator norm. Moreover W

of Proof : Let B1 be the subspace of L(K{X}) the operators u of the form j j D iD λij Z is an orthonormal u = , lim λij = 0. Since Z i j! i,j j! (i,j)∈N×N (i,j)∈N×N

family, one sees that B1 is isometrically isomorphic to the Banach space c0 (N×N, K) of the families (λij )(i,j)∈N×N such that lim λij = 0. Hence B1 is a closed subspace i,j

of L(K{X}). It is easily seen that W1 is a dense subspace of B1 . Hence one has 1 = B1 and W 1 is a completion of W1 . W Remark 3 : The closure MX of the subalgebra of L(K{X}) generated by the operator Z of multiplictaion by X is isometrically isomorphic to K{X}. i One has MX = {S = λi Z / lim |λi | = 0}. i→+∞

i≥0

1 = {u = Moreover W

j≥0

Sj (Z)

Dj / Sj ∈ K{X}, lim Sj = 0}. j→+∞ j!


Put K! {D} = {w =

j≥0

βj

Dj / j!

27 15

lim |βj | = 0}. This space is a closed subal-

j→+∞

1 . gebra of W The algebras MX and K! {D} are integral rings and one can prove that the 1 . same is true for W Also, W1 as a Banach space is isometrically isomorphic to the topological tensor ! {D}. product K{X}⊗K

5.2. The algebra W1∆ . Let f ∈ K{X}, one has τ1 ◦ Z(f )(X) = τ1 (Xf (X)) = (X + 1)f (X + 1) = Xτ1 f (X) + f (X + 1) =⇒ τ1 ◦ Z − Z ◦ τ1 = τ1 =⇒ ∆ ◦ Z − Z ◦ ∆ = id + ∆ (C2 ). Since ∆Z = id + ∆ + Z∆, on a ∆Z 2 = Z + ∆Z + Z∆Z = id + ∆ + 2Z + 2Z∆ + 2 Z ∆ = id + 2Z + (id + 2Z + Z 2 )∆ = id + 2Z + (id + Z)2 ∆. Inductively, one sees that ∆Z n = Pn (Z)+(id+Z)n ∆, with Pn (Z) = ZPn−1 (Z)+ (id + Z)n−1 . n−1 Z n − (id + Z)n From what, one deduces that Pn (Z) = Z j (id+Z)n−1−j = = Z − id − Z j=0 n−1 n n n Zj. (id + Z) − Z = j j=0 More generally, one has ∆i Z n =

i

Pi,n, (Z)∆ , where the Pi,n, are polyno-

=0

mials with cœfficients in K. As a consequence, one obtains that any element u of the subalgebra W1∆ of L(K{X}) generated by Z and ∆ can be written as a linear expansion u = mj n ∆j ∆j λij Z i Pj (Z) , where Pj (T ) = λij T i is a polynomial with = j! j! j=0 i=0 f inite i,j

cœfficients in K. At a first step one has u(1) = P0 (Z)(1) = P0 (X) = P0 (X) =

sup |λi,0 | ≤ u. 0≤i≤m0

And in an inductive process, one proves as above that

m0

λi,0 and u(1) =

i=0

∆j is an Zi j! (i,j)∈N×N

orthonormal family in L(K{X}). As a consequence on one hand, one has that this family is a linear basis of the K-vector space W1∆ . ∆ ∆ On the other hand it is an orthonormal base of the closure W 1 of W1 in j ∆ ∆ , with λij Z i L(K{X}) : any element v of W 1 is a convergent sum v = j! (i,j)∈N×N

lim λij = 0 and v = sup |λij |. i,j

i,j

Summing up, one has :

28 16


Remark 4 : For this kind of Weyl algebra generated by the operator Z of multiplication by X and the difference operator ∆, one can state and prove results analogue to Theorem 5.1 and Remark 3. 5.3. The quantum Weyl algebras Wq . Let q be a unit of K that we assume is not a root of unit. For the Jackson derivative Dq and for any restricted power series f , one has qXf (qX) − Xf (X) f (qX) − f (X) . Since qZ ◦ Dq f (X) = qX , one Dq ◦ Z(f ) = qX − X qX − X obtains Dq ◦ Z(f )(X) − qZ ◦ Dq f (X) = f (X). In other words, one has Dq ◦ Z − qZ ◦ Dq = id (C3 ). One sees by induction that Dq ◦ Z n = [n]q Z n−1 + q n Z n ◦ Dq . In the sequel, as previously done, we shall often omit the sign ◦ of composition of operators. Let Wq be the subalgebra of L(K{X}) generated by Z and Dq . One deduces from the relations Dq ◦Z n = [n]q Z n−1 +q n Z n ◦Dq that any element n Dqj Dqj v of Wq can be written as a linear expansion v = = , αij Z i Pj (Z) [j]q ! [j]q ! i,j j=0 where Pj is a polynomial with cœfficients in K and degree mj . Dqj (X n ) n = Let us remind that X n−j , ∀0 ≤ j ≤ n and Dqj (X n ) = 0, ∀j ≥ j q [j]q ! n + 1. n Dqj (1) = P0 (Z)(1) = P0 (X). Hence Pj (Z) One has v(1) = [j]q ! j=0 v(1) = P0 (X) =

sup |αi,0 | ≤ v.

0≤i≤m0

One sees by induction process as in 5.1 that v = sup |αij |. i,j j Dq This means that Z i is an orthonormal family and hence is [j]q ! (i,j)∈N×N

linearly independent. −††− The algebra Wq is isomorphic to the Weyl q-algebra Aq = K < x, y > /Iq < x, y >, where Iq < x, y > is the two-sided ideal genarated by xy − qyx − 1 in the free algebra K < x, y > with free generators x and y. q in L(K{X}) of the subalgebra Wq is a compleTheorem 5.2. The closure W tion of this model ofthe quantum Weyl algebra in two variables. j D q More precisely, Z i is an orthonormal basis of this completion: [j]q ! (i,j)N×N

q , can be expanded as a convergent sum v = any element v of W

Sj (Z)

j≥0

with Sj ∈ K{X} and lim Sj = 0. j→+∞

Proof : j D q Zi [j]q !

It follows as previous analogue results from the fact that

(i,j)N×N

is an orthonormal family L(K{X}).

Dqj , [j]q !


29 17

q is isometrically isomophic to the -(1)- The Banach algebra W Dqj ! {Dq }, where K! {Dq } = {v = ∈ γj topological tensor product K{X}⊗K [j]q ! Remark 5:

L(K{X}) /

j≥0

lim |γj | = 0}.

j→+∞

Dqj Dqj and v2 = are two elements γ2,j [j]q ! [j]q ! j≥0 j≥0 n Dqn . Hence K! {Dq } is an integral ring. of K! {Dq }, one has v1 v2 = k q [n]q ! -(2)- Notice that if v1 =

γ1,j

n≥0 k+j=n

q is also an integral ring. One can show that W 1

5.4. The quantum Weyl algebras Wc,q for |q − 1| < |p| p−1 . Assume that the valued field K is an extension of Qp . Let Zq be the operator of multiplication in K{X} by the restricted power series q X and let ∇q the second kind of the Jackson q-derivative defined by setting f (X + 1) − f (X) ∇q (f )(X) = ( cf. 4.2 ). q X (q − 1) One verifies that ∇q ◦ Zq (f )(X) − qZq ∇q (f )(X) = f (X). That is : ∇q ◦ Zq − qZq ◦ ∇q = id

(C4 ).

Let Wc,q be the subalgebra of L(K{X}) generated by ∇q and Zq . According to the relation (C4 ) the algebra Wc,q is a surjective image of the quantum Weyl algebra Aq by the morphism of algebras sending x onto ∇q and y onto Zq . One then sees that the linear span of the Zqi ∇jq is equal to Wc,q . Let K[q X ] be the subalgebra of K{X} generated by the restricted power series q . The sequence of formal power series (q iX )i≥0 is a linear basis of K[q X ]. Another linear basis of K[q X ] is given by the sequence of restricted power series (Ψi,q (X))i≥0 , i−1 q X−j − 1 . [X − ]q and [X − ]q = with Ψi,q (X) = q−1 X

=0

−•−

Let βi,q (Zq ) =

i−1

q − Zq − id , this operator is the operator of multiq−1

=0

plication by Ψi,q (X). Lemma 5.3. The family

∇jq βi,q (Zq ) [j]q !

is an orthogonal family in (i,j)∈N×N

L(K{X}) and therefore is a linear basis of the vector space Wc,q . Proof : One processes as in previous similar situations. n ∇jq ∇jq = , αij βi,q (Zq ) Qj (Zq ) Indeed let w = [j]q ! [j]q ! i,j j=0 mj i=0

αij βi,q (Zq ).

where

Qj (Zq ) =

30 18

BERTIN Bertin DIARRA Diarra m0

m0

αi,0 Ψi,q (X) and w(1) = ∇j q sup |αi,0 | ≤ w. And by induction, one proves that w = max Qj (X) 0≤j≤n [j]q ! 0≤i≤m0 ∇j q = max max |αij | . 0≤j≤n 0≤i≤mj [j]q ! ∇jq This means that βi,q (Zq ) is an orthogonal family in L(K{X}) [j]q ! One has w(1) = Q0 (Zq )(1) =

αi,0 βi,q (Zq )(1) =

i=0

i=0

(i,j)∈N×N

and it is a linear basis of Wc,q . One then sees that the algebra Wc,q . is isomorphic to the quantum Weyl algebra Aq . m αi Ψi,q (X) ∈ K[q X ], then one has −††− It is readily seen that if Q = Q(Zq ) =

m

i=0

αi βi,q (Zq ). Since Q(Zq )(1) =

i=0

obtains Q(Zq ) = sup |αi |.

m

αi βi,q (Zq )(1) =

i=0

m

αi Ψi,q (X), one

i=0

0≤i≤m

Notice that if the element f of K{X} is such that f = 1, then the operator mf of multiplication by f is an isometry and this because the norm on K{X} is multiplicative. As a consequence each operator βi,q (Zq ), beeing the operator of multiplication by Ψi,q (X), is an isometry. − • − Moreover, since the algebra K[q X ] is dense in K{X} and since for any element f of K{X} the norm of the operator mf is mf = f , setting Mq (Zq ) the closed subalgebra of the operators of multiplication by the elements of K{X}, one sees that the algebras Mq (Zq ) and K{X} are isometrically isomorphic. We have an analogue of Theorem 5.2. Theorem 5.4. The closure W c,q in L(K{X}) of the algebra Wc,q is a completion of this other model Weyl algebra in two variables. of the quantum ∇jq is an orthogonal basis of this compleMore precisely, βi,q (Zq ) [j]q ! (i,j)∈N×N

tion. Any element w of W c,q is a unique convergent sum ∇j q j aij βi,q (Zq )∇q , with w = sup |aij | w= . [j]q ! (i,j)∈N×N (i,j)∈N×N Also w = Tj (Zq )∇jq , where Tj = aij Ψi,q is an element of K{X} with j≥0

i≥0

corresponding operator of multiplication ∇j q aij βi,q (Zq ) and w = sup Tj Tj (Zq ) = . [j]q j≥0 i≥0

N. B. One has, as in Remark 4, that the Banach space W c,q is isometrically ! {∇q }, where K! {∇q } = isomorphic to the topological tensor product K{X}⊗K ∇jq |γj | w= γj =0 . ∈ L(K{X}) / lim j j→+∞ |q − 1| [j]q ! j≥0


31 19

The Banach space W c,q can be isometrically identified with the Banach space ∇jq Tj . : Tj ∈ K{X} / lim K{X}! {∇q } = Tj · j→+∞ |q − 1|j [j]q ! j≥0

6. The ultrametric quantum plane algebras As for the quantum Weyl algebra, we have different models of the quantum plane algebra as operator algebras on the Tate algebra K{X}. 6.1. The quantum plane algebras Pq . In this subsection K may be any complete ultrametric valued field and q a unit in K that is not a root of unit. Let us remind that Z and hq are the operators on the Tate algebra K{X} such that Z(f )(X) = X · f (X) and hq (f )(X) = f (qX). One has hq ◦ Z(f )(X) = Z(f )(qX) = qXf (qX) = qZ ◦ hq (f )(X). Hence, one has the commutation relation : hq ◦ Z = qZ ◦ hq (C5 ). The quantum plane algebra is by definition the algebra Kq [x, y] = K < x, y > /Jq < x, y >, where Jq < x, y > is the two-sided ideal generated by xy − qyx in the free algebra K < x, y >. Let Pq be the subalgebra of L(K{X}) generated by the operators Z and hq . It is well known that the family (xi y j )i≥0,j≥0 is linear basis of Kq [x, y] = K < x, y > /Jq < x, y >, hence one obtains a surjective algebra homomorphism of Kq [x, y] onto Pq which sends x onto hq and y onto Z. Let us also remind that one has the sequence of q-difference operators (0) (j) (j) Dq = id, Dq = (hq − id)(hq − qid) · · · (hq − q j−1 id), j ≥ 1 such that Dq = (n) |(q j − 1)(j) |. On the other hand, (X − 1)q )n≥0 is an orthonormal basis of K{X}, (j) (n) where (X − 1)(n) = (X − 1) · · · (X − q n−1 ). Moreover Dq (X − 1)q ) = (q j − n (n−j) (j) (n) , ∀ 0 ≤ j ≤ n and Dq ((X − 1)q ) = 0, ∀j ≥ n + 1. 1)(j) [ ]q X j (X − 1)q j

(j) Lemma 6.1. The family of operators Z i Dq is an orthogonal family (i,j)∈N×N

in L(K{X}) and is a linear basis of the K-vector space Pq . Proof As we have repetetively done above one gives the proof by induction. Set mj n u= βij Z i Dq(j) = Pj (Z)Dq(j) , where Pj (X) = βij X i ∈ K[X]. j=0

i,j;f inite

Considering u − one obtains

u−

j−1

P (Z)Dq() =

=0 j−1

i=0

P (Z)Dq()

n

P (Z)Dq() ,

=j

(j) (j) (j) (X − 1)q ) = Pj (X)Dq ((X − 1)q )) = (q j −

=0 (j)

1)(j) Pj (X). Hence Dq Pj (X) = |(q j − 1)(j) |Pj (X) ≤ max(u, max P (X)Dq() ) = u. 0≤≤j−1

32 20


One concludes by induction that u = max Pj (X)Dq(j) = max |βij |Dq(j) ). i,j 0≤j≤n

(j) The families Z i hjq (i,j)∈N×N and Z i Dq space Pq .

(j) Since Z i Dq

span each other the vector

(i,j)∈N×N

is an orthogonal family, it is linearly independent and

(i,j)∈N×N

thus is a basis of Pq . Let us notice that a consequence is that the algbras Kq [x, y] and Pq are iso morphic. Futhermore Z i hjq (i,j)∈N×N is also a linear basis of Pq . q be the closure of Pq in L(K{X}). Theorem 6.2. Let P i (j) q . The family (Z Dq )(i,j)∈N×N is an orthogonal basis of P Moreover any element Pq can be written in the form of convergent sum u = Sj (Z)Dq(j) , with Sj (X) ∈ K{X}, lim Sj Dq(j) = 0 and u = sup Sj j→+∞

j≥0 j

j≥0

|(q − 1)(j) |. N.B. ( ) γj Dq(j) : γj ∈ K, The subspace K{Dq } = w = j≥0

lim |γj |Dq(j) = 0

j→+∞

of L(K{X}) is a equal to the closure of K[hq ] in L(K{X}) and is a complete commutative Banach algebra. ( ) q and K{X}⊗K{D As above the Banach spaces P q } are isometrically isomorphic. Question : The Ore extension K{X}, y, hq , 0 of K{X} is isomorphic to the algebra Opq of the continuous linear operators u of K{X} of the form n u= Sj (X)hjq , Sj ∈ K{X}. This algebra is an integral ring, it is the algebra of j=0

the q-difference operators. q which is a completion of Opq an integral ring ? Is the Banach algebra P Linked to the theory of q-difference equations, it should be interesting to study q and naturally that of the elements of the other the spectrum of the elements of P algebras of operators we have encountered until now. −1

6.2. The quantum plane algebras Pc,q , for |q − 1| < |p| p−1 . We assume in this subsection that K is an extension of the p-adic field Qp and −1 that q ∈ K is such that |q − 1| < |p| p−1 For any restricted formal power series f , one has τ1 ◦ Zq (f )(X) = Zq (f )(X + 1) = q X+1 Xf (X + 1) = qZq ◦ τ1 (f )(X) =⇒ τ1 ◦ Zq = qZq ◦ τ1 (C6 ). Let Pc,q be the subalgebra of L(K{X}) generated by Zq and τ1 . As above sending x onto τ1 and y onto Zq one defines a surjective algebra endomorphism of Kq [x, y] onto Pc,q .


Any element u of Pc,q is a finite sum u =

33 21

αij Zqi τ1j .

i,j

Let us remind (see 5.4) that if Q is an element of K[q X ] then the operator of m γi βi,q (Zq ), where βi,q is the operator multiplication by Q is given by Q(Zq ) = i=0

of multiplication by the basic formal power series Ψi,q . (n) (n) Let us consider ∆q = (τ1 − 1)q . Quite as many results stated above we have:

(j) −•− The family βi,q (Zq )∆q is an orthogonal family in L(K{X}). (i,j)∈N×N

Hence it is a linear basis of Pc,q and Pc,q is isomorphic to the quantum plane algebra Kq [x, y]. As a consequence, one also has

(j) is an orthogonal basis of the − • •− The same family βi,q (Zq )∆q (i,j)∈N×N

closure P c,q in L(K{X}) of the operator model Pc,q of the quantum plane algebra Kq [x, y]. Moreover, any element u of P c,q can be expanded as a unique convergent (j) sum βi,q (Zq )∆q = Sj (Zq )∆q(j) , where Sj ∈ K{X} and Sj (Zq ) is (i,j)∈N×N

j≥0

the operator of multiplication by the restricted power series Sj . u = sup Sj ∆q(j) .

Furthermore

j≥0

τ1 (f )(X) − f (X) , one sees that q X (q − 1) τ1 = id + (q − 1)Zq ◦ ∇q belongs to the algebra Wc,q generated by Zq and ∇q . Hence Pc,q ⊂ Wc,q . − • • • − Since ∇q (f )(X) =

Theorem 6.3. One has W c,q = Pc,q . Proof We have seen that the closed algebra Mq (Zq ) in L(K{X}) generated by Zq coincides with the set of operators defined by multiplication by the elements of K{X} and is isometrically isomorpphic to K{X}. But q −X belongs to K{X}, hence the associated multiplication operator mq−X = Zq−1 belongs to Mq (Zq ). It is the reciprocical linear isomorphism of Zq . Since Mq (Zq ) ⊂ P c,q and the operator ∇q = mq −X (τ1 − id) belongs to Pc,q , one sees that the algebra Wc,q is contained in Pc,q . It follows that W c,q = Pc,q . References [1] K. Conrad, A q-analogue of Mahler Expansion, Advances in Mathematics 153, (2000), 185-230. [2] T. Diagana, Towards a theory of some unbounded linear operators on p-adic Hilbert spaces and applications, Annales Math´ ematiques Blaise Pascal, vol. 12, n◦ 1 (2005) 205-222 [3] B. Diarra, Bases de Mahler et autres, S´ eminaires d’Analyse - Universit´ e Blaise Pascal (199495) Expos´ e 16 - MR, 98e:46093. [4] B. Diarra, Complete ultrametric Hopf algebras which are free Banach spaces, in p-adic functional analysis, edited by W.H. Schikhof, C. Perez-Garcia, J. K¸ akol, Lecture notes in pure and applied mathematics, vol. 192, Marcel Dekker Inc., New York (1997), 61-80. [5] B. Diarra, The continuous coalgebra endomorphisms of C(Zp , K), Bull. Belg. Math. Soc. supplement - December 2002, 63-79.

34 22


[6] B. Diarra, Ultrametric q-calculus, in Ultrametric functional analysis, edited by B. Diarra, A. Escassut, A.K. Katsaras, L. Narici, Contemporary Mathematics, vol. 384, AMS (2005), 63-78. [7] B. Diarra and F. Tangara, The p-adic quantum plane algebras and quantum Weyl algebra, to appear [8] V. Kac and P. Cheung, Quantum Calculus - Universitext, Springer 2001. [9] A. Kochubei, p-Adic commutation relations, Journal of Physics A: Math. Gene., 29 (1996), 6375-6378. [10] A. Robert, Le th´ eor` eme des accroissements finis p-adique, Annales Math´ ematiques Blaise Pascal, Volume 2, N◦ 1 (1995), 245-258. [11] A.C.M. van Rooij, Non-archimedean analysis, Marcel Dekker Inc., New-York (1978). [12] G-C. Rota, Finite operator calculus, Academic Press, New York (1975). [13] F. Tangara, Bases orthonormales et calcul ombral en analyse p-adique, Th` ese de l’Université Blaise Pascal, Clermont-Ferrand (2006). [14] F. Tangara, The p-adic quantum Weyl algebra, to appear . [15] L. Van Hamme, Jackson’s Interpolation formula in p-adic analysis, Proceedings of the Conference on p-adic analysis, Report 7806, Nijmegen, June 1978, 119-125. [16] L. Van Hamme, Continuous operators which commute with translations, on the space of continuous functions on Zp , In ”p-adic functional analysis”, edited by J.M. Bayod, N. De GrandeDe Kimpe and J. Mart´ınez-Maurica, Marcel Dekker, New-York (1991), 75-88. [17] A. Verdoodt, The use of operators for the construction of normal bases for the space of continuous functions on Vq , Bull. Belg. Soc. 1 (1994), 685-699. [18] A. Verdoodt, Bases and operators for the space of continuous functions defined on subset of Zp , Thesis, Vrije Universiteit Brussel, September 1995. [19] A. Verdoodt, p-adic q-umbral calculus, Journal of Mathematical Analysis and Applications, 198 (1996), 166-177. ´matiques, UMR 6620 - Universit´ Laboratoire de Mathe e Blaise Pascal, Complexe Scientifique des C´ ezeaux- 63 177 Aubi` ere Cedex, France E-mail address: [email protected]


The ultrametric corona problem Alain Escassut and Nicolas Ma¨ınetti Abstract. Let K be a complete ultrametric algebraically closed field and let A be the Banach K-algebra of bounded analytic functions in the ”open” unit disk D of K provided with the Gauss norm. Let M ult(A, . ) be the set of continuous multiplicative semi-norms of A provided with the topology of simple convergence, let M ultm (A, . ) be the subset of the φ ∈ M ult(A, . ) whose kernel is a maximal ideal and let M ulta (A, . ) be the subset of the φ ∈ M ult(A, . ) whose kernel is a maximal ideal of codimension 1. For every maximal ideal M, there exist ultrafilters U on D such that M is the set of functions f ∈ A vanishing along U and there exists at least one ψ ∈ M ultm (A, . ) of kernel M equal to a limit of |f (x)| along U. Certain ultrafilters define a unique ψ ∈ M ultm (A, . ). If every maximal ideal is the kernel of only one ψ ∈ M ultm (A, . ), then M ulta (A, . ) is dense in M ultm (A, . ). This is the case when K is strongly valued or spherically complete. Given a continuous multiplicative norm ψ on A other than the Gauss norm, ψ is defined by a circular filter on D of diameter r < 1. If K is of characteristic zero, the algebra A admits proper closed prime ideals that are neither zero nor maximal ideals.

1. Introduction Let K be an algebraically closed field complete with respect to an ultrametric absolute value | . |. Given a ∈ K and r > 0, we denote by d(a, r) the disk {x ∈ K | |x − a| ≤ r}, by d(a, r− ) the disk {x ∈ K | |x − a| < r}, by C(a, r) the circle {x ∈ K | |x − a| = r} and set D = d(0, 1− ). Let A be the K-algebra of bounded power series converging in D which is complete with respect to the Gauss ∞ norm defined as an xn = sup |an |: we know that this norm actually is the n=1

n∈IN

norm of uniform convergence on D [6], [16]. In [19] the Corona problem was considered in a similar way as it is on the field Cl [3], [15]: the author asked the question whether the set of maximal ideals of A defined by the points of D (which are well known to be of the form (x − a)A) is dense in the whole set of maximal ideals with respect to the so-called ”Gelfand Topology”. In fact, this makes no sense because the maximal ideals which are not of 1991 Mathematics Subject Classification. 2J25, 46S10. Key words and phrases. ultrametric analysis, analytic functons, corona problem, multiplicative semi-norms. c Mathematical 0000 (copyright Society holder) c 2010 American

1 35

36 2

ALAIN ESCASSUT AND NICOLAS MAÏNETTI

the form (x − a)A are of infinite codimension [11]. Consequently, a Corona problem should be defined in a different way. Given a commutative K-algebra B with unity, provided with a K-algebra norm . , the set of continuous multiplicative K-algebra semi-norms of B was studied in many works [1], [5], [6], [9] and is usually denoted by M ult(B, . ) [5], [6], [9]. For each φ ∈ M ult(B, . ), we denote by Ker(φ) the closed prime ideal of the f ∈ B such that φ(f ) = 0. The set of the φ ∈ M ult(B, . ) such that Ker(φ) is a maximal ideal is denoted by M ultm (B, . ), the set of the φ ∈ M ult(B, . ) such that Ker(φ) is a maximal ideal of codimension 1 is denoted by M ulta (B, . ) and here, the set of the continuous multiplicative norms of A will be denoted by M ult0 (B, . ). 1 We know that sup{φ(f ) | φ ∈ M ult(B, . )} = limn→∞ (f n ) n ∀f ∈ B [7], [8], [13], [14]. On the other hand, M ult(B, . ) is provided with the topology of simple convergence and is compact in this topology. We know that for every M ∈ M ax(B), there exists at least one φ ∈ M ultm (B, . ) such that Ker(φ) = M but in certain cases, there exist infinitely many φ ∈ M ultm (B, . ) such that Ker(φ) = M [4], [7], [8]. A maximal ideal M of B is said to be univalent if there is only one φ ∈ M ultm (B, . ) such that Ker(φ) = M and the algebra B is said to be multbijective if every maximal ideal is univalent (so, non-multbijective commutative Banach K-algebras with unity do exist). Thus, the ultrametric Corona problem may be viewed at two levels: 1) Is M ulta (A, . ) dense in M ultm (A, . ) (with respect to the topology of simple convergence)? 2) Is M ulta (A, . ) dense in M ult(A, . ) (with respect to the same topology )? Actually, this way to set the Corona problem on an ultrametric field is not really different from the original problem once considered on Cl because on a commutative C-Banach l algebra with unity, all continuous multiplicative semi-norms are known to be of the form |χ| where χ is a character of A. Thus the Corona problem was equivalent to show that the set of multiplicative semi-norms defined by points of the open disk was dense inside the whole set of continuous multiplicative semi-norms, with respect to the topology of simple convergence. Remark: Given a filter G, if for every f ∈ A, |f (x)| admits a limit ϕG (f ) along G, the function ϕG obviously belongs to M ult(A, . ). Moreover, it clearly lies in the closure of M ulta (A, . ). Consequently, if we can prove that every element of M ultm (A, . ) is of the form ϕG , with G a certain filter on D, Question 1) is solved. And similarly, if we could prove that every element of M ult(A, . ) is of the form ϕG , Question 2) would be solved. Studying such problems first requires to know the nature of continuous multiplicative semi-norms on A. Definitions and notation: Let a ∈ D and let R ∈]0, 1]. Given r, s ∈ IR such that 0 < r < s we set Γ(a, r, s) = {x ∈ K |r < |x − a| < s}. We call circular filter of center a and diameter R on D the filter F which admits as a generating system the family of sets Γ(α, r , r ) ∩ D with α ∈ d(a, R), r
0. j∈IN

n∈IN n=j

An ultrafilter U is said to be regular if it is thinner than a regular sequence. Thus, by definition, a regular ultrafilter is a coroner ultrafilter. Two coroner ultrafilters F, G are said to be contiguous if for every subsets F ∈ F, G ∈ G of D the distance from F to G is null. Let f ∈ A. Recall that f is said to be quasi-invertible if it factorizes in A in the form f = P g with P ∈ K[x] and g an invertible element of A. On K[x], circular filters on K are known to characterize multiplicative seminorms by associating to each circular filter F the multiplicative semi-norm ϕF defined as ϕF (f ) = limF |f (x)| [5], [6], [7], [12]. We know that every f ∈ A is an analytic element in each disk d(a, r) whenever a ∈ D, r ∈]0, 1[ [6]. Consequently, by classical results [6], several properties of polynomials have continuation to A. Remark: A regular sequence is an idempotent polar sequence which and is not a T -polar sequence [18]. However, a polar sequence which and is not a T -polar sequence is not necessarily a regular sequence. The paper is aimed at showing that in at least when the field K satisfies certain hypotheses, A is multbijective and therefore M ulta (A, . ) is dense in M ultm (A, . ). On the other hand, we will examine relations between closed prime ideals, maximal ideals and multiplicative semi-norms.


38 4

2. Basic results Describing problems and results requires to state a lot of basic results (some of them are well known by specialists). Theorems 1, 2, 3 are immediate: Theorem 1:

For every f ∈ A, f = lim |f (x)|.

Theorem 2:

Every element of A is uniformly continuous in D.

Y

Corollary 2.1: J (U1 ) = J (U2 ).

Let U1 , U2 be two coroner contiguous ultrafilters on D. Then

Remark: An interesting question is whether two coroner ultrafilters U1 and U2 such that J (U1 ) = J (U2 ) are contiguous. The problem is partially solved in Theorem 24. Theorem 3: Let U be an ultrafilter on D. For every f ∈ A, |f (x)| admits a limit ϕU (f ) along U. Moreover, the mapping ϕU from A to IR+ belongs to M ult(A, . ) and Ker(ϕU ) = J (U). Given two contiguous ultrafilters U1 , U2 on D, ϕU1 = ϕU2 . Theorem 4: For every circular filter F secant with D, of diameter r < 1, ϕF has continuation to an element of M ult(A, . ). Theorem 5 also is classical [9]: Theorem 5: An element f ∈ A is not quasi-invertible if and only if it has infinitely many zeros. If f is not quasi-invertible, its set of zeros is a coroner sequence (an ). Theorems 6, 7 and 9, 10, 11, 12 may be found in [6]. Theorem 6: Let I be an ideal of A. The following two statements are equivalent: i) I is generated by a polynomial whose zeros lie in D, ii) I contains a quasi-invertible element. Theorem 7: Let M be a principal maximal ideal of A. Then there exists a ∈ D such that M = (x − a)A. Concerning maximal ideals, Theorem 8 is proved in [9]. Theorem 8: finite type.

Let M be a non-principal maximal ideal of A. Then M is not of

Corollary 8.1: The mapping from D to M ax(A) associating to each point a of D the maximal ideal (x − a)A is a bijection from D onto the set of principal maximal ideals. Theorem 9:

Non-principal maximal ideals of A are of infinite codimension. [9]

Theorem 10: Let M be a maximal ideal of A. The following statements are equivalent: i) M is of finite type, ii) M is principal, iii) there exists a ∈ D such that M = (x − a)A iv) M is of codimension 1.

39 5

THE ULTRAMETRIC CORONA PROBLEM

Corollary 10.1: An element φ of M ult(A, . ) belongs to M ulta (A, . ) if and only if there exists a ∈ D such that φ(f ) = |f (a)|, ∀f ∈ A. In order to understand the way to prove the main theorems, it is indispensable to define sets D(h, ). Notation:

Let h ∈ A, > 0. We set D(h, ) = {x ∈ D | |h(x)| ≤ }.

Theorem 11: Let P be a prime non-principal ideal of A. Let r ∈]0, 1[ and ∈ ]0, 1[. There exists h ∈ P such that D(h, ) ⊂ Γ(0, r, 1). Theorem 12: Let M be a non-principal maximal ideal of A and let ψ ∈ M ultm (A, . ) satisfy Ker(ψ) = M. Every quasi-invertible element f ∈ A satisfies ψ(f ) = lim |f (x)| = f . W

Theorem 13 is shown in [19] and is a Bezout-like theorem: Theorem 13: Let f1 , ..., fq ∈ A satisfy fj < 1 ∀j = 1, ..., q and q gj fj = 1 inf{ max (|fj (x)|) x ∈ D} = ω > 0. There exist g1 , ..., gq ∈ A s.t. j=1,...,q

j=1

and max gj < ω −2 . j=1,...,q

Corollary 13.1: Let I be an ideal of A different from A. The family of sets D(f, ), f ∈ I, > 0, makes a system of generators of a filter on D. Notation: Let I be an ideal of A different from A. We will denote by GI the filter generated by the sets D(f, ), f ∈ I, > 0. By definition, GI is minimal, with respect to the relation of thinness, among the filters H such that limH f (x) = 0 ∀f ∈ I. As a corollary of Theorem 11 and Corollary 13.1, we have Corollary 13.2 Corollary 13.2: Let P be a non-principal prime ideal of A. Then GP is coroner. Corollary 13.3: Let M be a non-principal maximal ideal of A. Then GM is coroner and M = J (GM ). Moreover, for every ultrafilter U thinner than GM , then J (U) = M. Corollary 13.4: Let M be a non-principal maximal ideal of A and let U be an ultrafilter thinner than GM . Then ϕU belongs to the closure of M ulta (A, . ) in M ultm (A, . ). Corollary 13.5: Let M be a univalent non-principal maximal ideal of A and let φ ∈ M ultm (A, . ) satisfy Ker(φ) = M. Then φ is of the form φ(f ) = lim |f (x)| U

with U a coroner ultrafilter such that J (U) = M. Moreover, φ belongs to the closure of M ulta (A, . ) in M ultm (A, . ). Corollary 13.6: Suppose A is multbijective. Then every multiplicative semi-norm φ ∈ M ultm (A, . ) \ M ulta (A, . ) is of the form φ(f ) = lim |f (x)| with U a U

coroner ultrafilter such that J (U) = M. Moreover, M ulta (A, . ) is dense in M ultm (A, . ).


40 6

Definition: The field K is said to be strongly valued if at least one of the following sets is not countable: the set of values of K = { |x| | x ∈ K} and the residue class field of K. Let us recall the following Theorem [5], [7]: Theorem 14: Suppose K is strongly valued. Every commutative K-Banach algebra is multbijective. Corollary 14.1: Suppose K is strongly valued. Then every multiplicative seminorm φ ∈ M ultm (A, . ) \ M ulta (A, . ) is of the form φ(f ) = lim |f (x)| with U

U a coroner ultrafilter such that J (U) = M. Moreover, M ulta (A, . ) is dense in M ultm (A, . ). 3. Multbijectivity in a spherically complete field Theorem 18 is proved with help of Propositions 15,16,17. The hypothesis K spherically complete is essential in Proposition 17 because if the field K is not spherically complete, we can’t factorize h as done in Proposition 16. Proposition 15: Let (B, . ) be a commutative ultrametric K-Banach algebra with unity. Suppose there exist f ∈ B , φ, ψ ∈ M ult(B, . ) such that ψ(f ) < φ(f ), sp(f ) ∩ Γ(0, ψ(f ), φ(f )) = ∅ and there exists ∈]0, φ(f ) − ψ(f )[ satisfying further (f − a)−1 ≤ M ∀a ∈ Γ(0, ψ(f ), φ(f ) − ). Then there exists γ ∈ B such that ψ(γ) = 1, φ(γ) = 0. Proposition 16: Let M be a non-principal maximal ideal of A and let U be an ultrafilter on D such that M = J (U). Let f ∈ A \ M be not invertible in A and let g ∈ A, h ∈ M such that f g = 1 + h. Let λ = ϕU (f ), let ∈]0, min(λ, 1)[ and let Λ = {x ∈ D |f (x)g(x)| − 1|∞ < , | |f (x)| − λ|∞ < }. Suppose that there exist a function h ∈ A admitting for zeroes in D the zeroes of h in D \ Λ and a function h ∈ A admitting for zeroes the zeroes of h in Λ, each h. Then |h(x)| has a strictly positive lower counting multiplicities, so that h = h bound in Λ and h belongs to M. Moreover, there exists ω ∈]0, λ[ such that ω ≤ inf{max(|f (x)|, | h(x)|) x ∈ D}. Further, for every a ∈ d(0, (λ−)), we have ω ≤ inf{max(|f (x)−a|, | h(x)|) x ∈ D}. Proposition 17: Suppose K is spherically complete. Let M be a non-principal maximal ideal of A and let U be an ultrafilter on D such that M = J (U). Let f ∈ A \ M satisfy f < 1, let λ = ϕU (f ) and let ∈]0, λ[. There exists ω > 0 such that, for every a ∈ d(0, λ − ), there exists ga ∈ A satisfying (f − a)ga − 1 ∈ M and ga ≤ ω −2 . Now, we can conclude when K is spherically complete: Theorem 18:

If K is spherically complete, then A is multbijective. The proof of

Theorem 18 consists of assuming that there exists ψ, φ ∈ M ultm (A, . ) satisfying ψ(P ) = φ(P ) = P ∀P ∈ K[x] Ker(ψ) = Ker(φ) and ψ(f ) < φ(f ) for certain f ∈ A. Using the ultrametric holomorphic functional calculus and Propositions 15, 16, 17 we can construct a function g satisfying ψ(g) = 0, φ(g) > 0, a contradiction. Corollary 18.1: If K is spherically complete, then for every φ ∈ M ultm (A, . ) there exists a coroner ultrafilter U such that φ = ϕU .

41 7


Corollary 18.2: M ultm (A, . ).

If K is spherically complete, then M ulta (A, . ) is dense in

4. Regular maximal ideals Regular coroner ultrafilters give a very nice representation of certain maximal ideals. We will first recall links with bounded sequences [19]. Definitions and notation: Let B(IN, K) be the K-Banach algebra of bounded sequences of K provided with the usual laws of K-algebra and with the usual norm defined as (an )n∈IN = sup{|an | | n ∈ IN}. For every ultrafilter G on IN we will denote by Θ(G) the ideal of B(IN, K) consisting of sequences (an ) such that lim an = 0. G

Let S = (an )n∈IN be a coroner sequence. We will denote by Σ(S) the set of ultrafilters thinner than S, by I(S) the ideal of the f ∈ A such that f (an ) = 0 ∀n ∈ IN. Recall that an ultrafilter G is said to be principal if F is a singleton. F ∈G

Theorem 19 is classical: Theorem 19: Θ is a bijection from the set of ultrafilters on IN onto M ax(B(IN, K)). The restriction of Θ to the subset of non-principal ultrafilters on IN is a bijection from this set onto the set of non-principal maximal ideals of B(IN, K). Moreover, a maximal ideal of B(IN, K) is principal if and only it is of codimension 1. Theorem 20: Let M be a non-principal maximal ideal of B(IN, K) and let U = Θ−1 (M ). Let θ be the canonical surjection from B(IN, K) onto the field L = B(IN, K) . Let . M be the K-algebra quotient norm of L. Then, every sequence M (an )n∈IN ∈ B(IN, K) satisfies (an )M = lim |an |. U

Corollary 20.1: of

B(IN, K) is multbijective. Let M be a principal maximal ideal

B(IN, K). The K-Banach algebra quotient norm of the field tive.

B(IN, K) is multiplicaM

Definitions and notation: Let S = (an )n∈IN be a coroner sequence. We will denote by I(S) the ideal of the f ∈ A such that f (an ) = 0 ∀n ∈ IN. We will denote ∞ a n xn by TS the mapping from A into B(IN, K) which associates to each f (x) = n=0

the sequence (f (an )n∈IN ). Remark: Given a regular maximal ideal M = J (U) where U is thinner than a regular sequence S, then M contains I(S). From [19], (4.6), we have the following theorem: Theorem 21: Let S be a coroner sequence. Then TS is surjective on B(IN, K) if and only if the sequence S is regular.

42 8


Theorem 22: Let S be a regular sequence and let M be a maximal ideal of A. The following two statements are equivalent: i) I(S) ⊂ M ii) There exists an ultrafilter U thinner than S such that M = J (U). Moreover, the mapping Ψ which associates to each ultrafilter U thinner than S the ideal J (U) is a bijection from Σ(S) onto the set of maximal ideals of A containing I(S). By Theorems 21 and 22 we have Corollary 22.1 Corollary 22.1: Let S be a regular sequence. For every maximal ideal M conA B(IN, K) taining I(S), the field is isomorphic to . M TS (M) Let S be a regular sequence. For every maximal ideal M of A A is equivalent to an containing I(S), the K-Banach algebra quotient norm of M absolute value extending this of K.

Theorem 23:

Corollary 23.1: ideal.

For every regular ultrafilter U, J (U) is a univalent maximal

Remark: Consider a maximal ideal M of A and suppose that there exists φ ∈ M ultm (A, . ) which does not lie in the closure of M ulta (A, . ) (which obviously implies that K is not spherically complete). Then M is not univalent A and therefore the K-Banach algebra quotient norm of the field is not equivalent M to its norm . si . Theorem 24: Let U1 , U2 be two regular ultrafilters. Then J (U1 ) = J (U2 ) if and only if U1 and U2 are contiguous. Corollary 24.1: The relation R on regular ultrafilters defined as U1 RU2 if U1 and U2 are contiguous is an equivalence relation whose classes are in bijection with the set of regular maximal ideals of A. Remark: In the general case, Relation R is not an equivalence relation on a set of filters in K. Example: consider a sequence (an )n∈IN such that |an | < |an+1 | ∀n, put bn = a2n , cn = a2n+1 and consider the filters F, G, H associated to these sequences, respectively. Clearly, both G and H are contiguous to F but G is not contiguous to H. 5. Differential ideals of A In order to study more carefully certain multiplicative semi-norms and closed prime ideals, we have to notice a basic theorem: Notation: Let F be a field, let R be a commutative F -algebra with unity and let D be a derivation on R. Given an ideal J of R, we will denote by J the set of f ∈ R such that D (n) (f ) ∈ J ∀n ∈ IN. Let ψ ∈ M ult(A, . ). We set Subker(ψ) = Ker(ψ).


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Theorem 25: Let F be a field, let R be a commutative F -algebra with unity and let D be a derivation on R. Let J be an ideal of R. Then J is an ideal of R and

= J. Moreover, if F is of characteristic 0 and if J is prime, then so is J. (J) Since f ≤ f ∀f ∈ A, we can derive Corollary 25.1: Corollary 25.1: Suppose K is of characteristic zero. Let P be a prime ideal of is a prime ideal of A such that ( = P. Moreover, if P is closed, then A. Then P P) so is P. Corollary 25.2: Suppose K is of characteristic zero. Let ψ ∈ M ult(A, . ). Then Subker(ψ) is a prime closed ideal . Theorem 26 shows that Subker(ψ) is not equal to Ker(ψ) in the general ∞ case. Recall that, given a strictly increasing sequence (rn ) of limit 1 such that n=0 > 0, it is always possible to construct a function f ∈ A having exactly one zero (of order 1) in the circle C(0, rn ) and no other zero. However, due to Lazard’s problem, if K is not spherically complete, we can’t control the place of these zeroes. With such an element f , it is easily proved that f has no zero in the class of C(0, rn ). Theorem 26 is proved by using this property. Theorem 26: There exist regular maximal ideals M of A and f ∈ M, having a sequence of zeroes of order 1 and no other zeroes, such that f ∈ / M. In the particular case when K is spherically complete, we can get a more general statement: Theorem 27: Suppose K is spherically complete and let M be a regular maximal ideal of A. There exists f ∈ M, having a sequence of zeroes of order 1 and no other zeroes, such that f ∈ / M. Remark: When K is not spherically complete, Theorem 26 is less general than Theorem 27 because of Lazard’s problem on zeroes of an analytic function [17]. Now we may notice that when the field is of characteristic 2, it is easy to show

is not prime. Indeed, by Theorem that for certain maximal ideals M of A, M 26, there exists a coroner maximal ideal M and f ∈ M such that f ∈ / M. Then

Now consider g = f 2 . Then g = 2f f = 0 hence f does not belong to M. g (n) ∈ M ∀n ∈ IN. If K is of characteristic 3, we can also construct a similar but less simple counter-example. Let ψ ∈ M ult(A, . ) be such that ψ(P ) = ϕF (P ) ∀P ∈ K[x], with F a circular filter on D of diameter r < 1. In [10] it is shown that ψ has a unique continuation to A defined in the same way as ψ(f ) = ϕF (f ) ∀f ∈ A. Theorem 28: Let ψ ∈ M ult(A, . ) satisfy ψ(P ) = ϕF (P ) ∀P ∈ K[x], with F a circular filter on D of diameter r < 1. Then ψ(f ) = ϕF (f ) ∀f ∈ A. Consequently, all multiplicative norms whose restrictions to polynomials are not the Gauss norm . are defined by the circular filters on D of diameter < 1 and therefore the problem of characterizing continuous multiplicative norms of A only concerns the various continuations of the Gauss norm to A.


44 10

Theorem 29 shows lets us characterize all continuous multiplicative norms of A Theorem 29: Let ψ ∈ M ult(A, . ) be coroner. Then Subker(ψ) is not null. Moreover, if K is spherically complete, then, for every f ∈ A such that ψ(f ) < f , there exists g ∈ Subker(ψ) admitting no zero which is not a zero of f (the zero of f eventually having a smaller order). When K is spherically complete, we can adjust the order of zeroes in order that they have slowly increasing orders (in the wide sense), so that the new function we make remains bounded. But any derivation will admit the zeroes of f after certain rank. When the field is not spherically complete, we try to follow a similar way by using Theorem 25.5 [6] in order to obtain a function having some more zeroes than we ask but ”the work ” of the additional zeroes remains bounded. Corollary 29.1:

Let ψ ∈ M ult(A, . ) be coroner. Then ψ is not a norm.

Corollary 29.2: Let ψ ∈ M ult(A, . ) be a norm. If ψ is not . , there exists a circular filter F on D, of diameter r < 1, such that ψ = ϕF . Corollary 29.3: Let ψ ∈ M ult(A, . ) be a norm. If ψ is not . , there exists a circular filter F on D, of diameter r < 1, such that ψ = ϕF . On the other hand, each coroner maximal ideal is the kernel of some coroner continuous multiplicative semi-norm of A. Consequently:

is not null. Corollary 29.4: Let M be a coroner maximal ideal of A. Then M Concerning the Corona Problem, we may notice this: Corollary 29.5:

M ult0 (A, . ) is included in the closure of M ulta (A, . ).

On the other hand, using Theorems 27 and 29, we can prove Theorem 30: Theorem 30: Let K be spherically complete and M be a regular coroner maximal

is neither null nor equal to M. ideal. Then M Corollary 30.1: Suppose K is of characteristic zero. Then A admits prime closed ideals that are neither null nor maximal ideals. Moreover, if K is spherically complete, then every regular coroner maximal ideal M of A contains a prime closed

that is neither null nor equal to M. ideal M Remark: The prime closed ideal we construct, which is neither null nor maximal, does not seem to be the kernel of an element of M ult(A, . ). Recall that in [2] an example of a Banach-K-algebra of analytic elements with no divisors of zero, admitting no continuous multiplicative norm, was constructed. Now, suppose K is spherically complete. If M ult(A, . ) only consists of M ult0 (A, . ) and M ultm (A, . ), then M ulta (A, . ) is dense in M ult(A, . ). Actually, this situation seems much likely.


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References [1] V. Berkovich, Spectral Theory and Analytic Geometry over Non-archimedean Fields. AMS Surveys and Monographs 33, (1990). [2] K. Boussaf, and A. Escassut, Absolute values on algebras of analytic elements Annales Math´ ematiques Blaise Pascal 2, n2, p.15-23 (1995). [3] L. Carleson, Interpolation by bounded analytic functions and the corona problem. Annals of Math. 76, p. 547-559 (1962). [4] A. Escassut. Spectre maximal d’une algèbre de Krasner, Colloquium Mathematicum, XXXVIII, fasc. 2, p. 339-357, (1978). [5] A. Escassut . The ultrametric spectral theory Periodica Mathematica Hungarica, Vol.11, (1), p7-60, (1980). [6] A. Escassut. Analytic Elements in p-adic Analysis World Scientific Publishing Inc., Singpore (1995). [7] A. Escassut. Ultrametric Banach Algebra. World Scientific Publishing Inc., Singapore (2003). [8] A. Escassut and N. Ma¨ınetti. Spectral semi-norm of a p-adic Banach algebra Bulletin of the Belgian Mathematical Society, Simon Stevin, vol 8, p.79-61, (1998). [9] A. Escassut and N. Ma¨ınetti. On Ideals of the Algebra of p-adic Bounded Analytic Functions on a Disk Bulletin of the Belgian Mathematical Society, Special issue for the Proceedings of the 9-th International Conference on p-adic Functional Analysis. [10] A. Escassut and N. Ma¨ınetti,. About the ultrametric Corona problem Bulletin des Sciences Math´ ematiques 132, p. 382-394 (2008) [11] A. Escassut. Ultrametric Corona problem and spherically complete field To appear in Proceedings of the Edinburgh Mathematical Society. [12] G. Garandel. Les semi-normes multiplicatives sur les alg` ebres d’´ el´ ements analytiques au sens de Krasner. Indag. Math., 37, n4, p.327-341, (1975). [13] B. Guennebaud. Alg` ebres localement convexes sur les corps valu´ es. Bull. Sci. Math. 91, p.7596, (1967). [14] B. Guennebaud,. Sur une notion de spectre pour les algèbres normées ultram´ etriques th` ese Universit´ e de Poitiers, (1973). [15] K. Hoffman. Banach Spaces of Analytic Functions. Prentice-Hall Inc. (1962). [16] M. Krasner,. Prolongement analytique uniforme et multiforme dans les corps valu´ es complets. Les tendances g´ eom´ etriques en alg` ebre et th´ eorie des nombres, Clermont-Ferrand, p.94-141 (1964). Centre National de la Recherche Scientifique (1966), (Colloques internationaux du C.N.R.S. Paris, 143). [17] M. Lazard. Les z´ eros des fonctions analytiques sur un corps valué complet. IHES, Publications Math´ ematiques n14, p.47-75 ( 1962). [18] M.C. Sarmant and A. Escassut,. T-suites idempotentes. Bull. Sci. Math. 106, p.289-303, (1982). [19] M. Van Der Put. The Non-Archimedean Corona Problem. Table Ronde Anal. non Archimedienne, Bull. Soc. Math. M´ emoire 39-40, p. 287-317 (1974). Laboratoire de math´ ematiques UMR 6620, Universit´ e Blaise Pascal, F-63177 Aubi` ere, France E-mail address: [email protected] LAIC, EA 2146, IUT, Campus des C´ ezeaux, Universit´ e d’Auvergne F-63170 Aubi` ere, Frabce E-mail address: [email protected]



Vector-valued p-adic measures Athanasios K. Katsaras Abstract. For a separating algebra R of subsets of a set X and E a Hausdorff non-Archimedean locally convex space, we study the space M (R, E) of all E-valued bounded finitely-additive measures on R as well as its subspaces Mσ (R, E) and Mτ (R, E) of all σ-additive and all τ -additive members, respectively. We also study integrals of scalar-valued functions on X with respect to members of M (R, E). We show that, if X is a Hausdorff zero-dimensional topological space, Cb (X) the space of all bounded continuous scalar-valued functions on X and K(X) the algebra of all clopen subsets of X, then, in case E is complete, Mτ (K(X), E) is algebraically isomporphic to the space of all linear maps from Cb (X) to E which are continuous with respect to the strict topology βo .

1. Preliminaries Throughout this paper, K stands for a complete non-Archimedean valued field whose valuation is non-trivial. By a seminorm, on a vector space E over K, we mean a non-Archimedean seminorm. Also by a localy convex space we will mean a nonArchimedean locally convex space over K, (see [13] and [14]). For E a locally convex space, we denote by cs(E) the collection of all continuous seminorms on E and by E the topological dual of E. For a zero-dimensional Hausdorff topological space X, βo X is the Banachewski compactification of X, Cb (X) the space of all bounded continuous K-valued functions on X and Crc (X) the space of all f ∈ Cb (X) whose range is relatively compact. Every f ∈ Crc (X) has a continuous extension f βo to all of βo X. For f ∈ KX and A ⊂ X, we define f A = sup{|f (x)| : x ∈ A}

and f = f X .

βo X

we will denote the closure of A in βo X. By A Next we will recall the definition of the strict topology β on Cb (X) which was given in [5]. Let Ω be the family of all compact subsets of βo X which are disjoint from X. For Z ∈ Ω, let CZ be the set of all h ∈ Crc (X) for which hβo vanishes on Z. We denote by βZ the locally convex topology on Cb (X) generated by the seminorms ph , h ∈ CZ , where ph (f ) = hf . The inductive limit of the topologies βZ , Z ∈ Ω, 2000 Mathematics Subject Classification. 46S10, 46G10. Key words and phrases. non-Archimedean fields, zero-dimensional spaces, p-adic measures, locally convex spaces. c Mathematical 0000 (copyright Society holder) c 2010 American

1 47

48 2

ATHANASIOS K. KATSARAS

is the strict topology β. As it is shown in [7], Theorem 2.2, an absolutely convex subset W of Cb (X) is a βZ -neighborhood of zero iff, for each r > 0, there exist a βo X disjoint from Z, and > 0 such that clopen subset A of X, with A {f ∈ Cb (X) : f A ≤ , f ≤ r} ⊂ W. Monna and Springer initiated in [12] non-Archimedean integration. In [13] and [14], van Rooij and Schikhof developed a non-Archimedean integration theory for scalar valued measures. Some results on measures with values in Banach spaces were given in [1] and [2]. In this paper we will study measures with values in a locally convex space as well as integrals of scalar valued functions with respect to such measures. 2. Vector Measures Let R be a separating algebra of subsets of a non-empty set X, i.e. R is a family of subsets of X with the following properties : (1) X ∈ R and, if A, B ∈ R, then A ∪ B, A ∩ B, A \ B are also in R. (2) If x, y are distinct elements of X, then there exists a member of R containing x but not y. We will call the members of R measurable sets. Clearly R is a base for a Hausdorff zero-dimensional topology τR on X. For a net (Vδ ) of subsets of X we will write Vδ ↓ ∅ if it is decreasing and Vδ = ∅. Similarly we will write Vn ↓ ∅ for a sequence (Vn ) of sets which decreases to the empty set. Let now E be a Hausdorff locally convex space. We denote by M (R, E) the space of all bounded finitely-additive measures m : R → E. For m ∈ M (R, E) and p ∈ cs(E), we define mp : R → R,

mp (A) = sup{p(m(V )) : V ∈ R, V ⊂ A}

and mp = mp (X). We also define Nm,p : X → R,

Nm,p (x) = inf{mp (V ) : x ∈ V ∈ R}.

An element m of M (R, E) is called σ-additive if m(Vn ) → 0 if Vn ↓ ∅ and τ -additive if m(Vδ ) → 0 if Vδ ↓ ∅. Let Mσ (R, E) (resp. Mτ (R, E)) be the space of all σ-additive (resp. τ -additive) members of M (R, E). The proof of the following Theorem is analogous to the one given in [3] in the case of scalar measures. Theorem 2.1. Let m ∈ M (R, E). Then (1) m is τ -additive iff, for all p ∈ cs(E), we have that mp (Vδ ) → 0 when Vδ ↓ ∅. (2) m is σ-additive iff, for all p ∈ cs(E), we have that mp (Vn ) → 0 when Vn ↓ ∅. Theorem 2.2. Let m∈ Mτ (R, E) and let (Vi )i∈I ⊂ R . If p ∈ cs(E), then for each V ∈ R contained in i∈I Vi , we have that mp (V ) ≤ supi mp (Vi ). c Proof : For each finite subset S of I, let WS = i∈S Vi . Then WS ↓ ∅. V If mp (V ) > 0, there exists a finite subset S of I such that mp (V WSc ) < mp (V ). Now

VECTOR-VALUED p-ADIC MEASURES

mp (V )

49 3

= max{mp (V ∩ WS ), mp (V ∩ WSc )} = mp (V ∩ WS ) ≤ mp (WS ) = max mp (Vi ). i∈S

Corollary 2.3. Let m ∈ Mτ (R, E), p ∈ cs(E) and V ∈ R. Then mp (V ) = sup Nm,p (x). x∈V

Proof : Clearly mp (V ) ≥ α = supx∈V Nm,p (x). On the other hand, if > 0, then for each x ∈ V there exists a measurable set Vx , with x ∈ Vx ⊂ V , such that mp (Vx ) < Nm,p (x) + ≤ α + . Since V = x∈V Vx , we have that mp (V ) ≤ sup mp (Vx ) ≤ α + , x∈V

and the result follows as > 0 was arbitrary. Theorem 2.4. Let m ∈ M σ (R, E) and let (Vn ) be a sequence of measurable sets. If V ∈ R is contained in Vn , then mp (V ) ≤ supn mp (Vn ). n Proof : Let Wn = k=1 Vk . Suppose that mp (V ) > 0. Since V Wnc ↓ ∅, there exists an n such that mp (V ∩ Wnc ) < mp (V ). Now mp (V ) = max{mp (V ∩ Wnc ), mp (V ∩ Wn )} = mp (V ∩ Wn ) ≤ mp (Wn ) = max mp (Vk ). 1≤k≤n

Theorem 2.5. If m ∈ M (R, E) and p ∈ cs(E), then Nm,p is upper semicontinuous. Proof : Let α > 0 and V = {x : Nm,p (x) < α}. For x ∈ V , there exists a measurable set A containing x and such that mp (A) < α. Now x ∈ A ⊂ V and so V is open. Theorem 2.6. Let m ∈ Mτ (R, E), p ∈ cs(E) and > 0. Then the set Xp, = {x : Nm,p (x) ≥ } is τR -compact. Proof : Let (Vi )i∈I be a family of measurable sets covering Xp, = Y . Since Nm,p isupper semicontinuous, the set Y is closed. For each finite subset S of I, let WS = i∈S Vi . Consider the family F of all measurable sets of the form [WS ∪ V ]c , where V is a measurable set disjoint from Y and S a finite subset of I. Then F is downwards directed and F = ∅. Since m is τ -additive, there are S and V such that mp ([WS ∪ V ]c ) < . But then [WS ∪ V ]c ⊂ Y c and thus Y ⊂ WS . Definition 2.7. A subset G of X is said to be a support set of an m ∈ M (R, E) if m(V ) = 0 for each measurable set V disjoint from G. Theorem 2.8. Let m ∈ Mτ (R, E). Then the set {x : Nm,p (x) > 0} supp(m) = p∈cs(E)

is the smallest of all closed support sets of m.

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Proof : If V is a measurable set disjoint from supp(m), then for each p ∈ cs(E) we have p(m(V )) ≤ mp (V ) = sup Nm,p (x) = 0, x∈V

which proves that supp(m) is a support set of m since E is Hausdorff. On the other hand, let F be a closed support set of m. Given x ∈ F c , there exists V ∈ R with x ∈ V ⊂ F c . Now, for each p ∈ cs(E) and y ∈ V , we have that Nm,p (y) ≤ mp V ) = 0 and so the set B= {x : Nm,p (x) = 0} p∈cs(E)

does not intersect V , which implies that x ∈ / B = supp(m). Thus supp(m) ⊂ F and the result follows. 3. A Universal Measure Let R be a separating algebra of subsets of X and let S(R) be the vector space of all K-valued R-simple functions on X. Let χ : R → S(R), A → χA . Let E be a Hausdorff locally convex space. Every m ∈ M (R, E) induces a linear map n n m ˆ : S(R) → E, m ˆ λk χVk = λk m(Vk ). k=1

k=1

On S(R) we consider the locally convex topologies φ, φσ , φτ defined as follows : (1) φ is the weakest locally convex topology for which, for each Hausdorff locally convex space E and each m ∈ M (R, E), the map m ˆ : S(R) → E is continuous. (2) φσ is the weakest locally convex topology for which, for each Hausdorff locally convex space E and each m ∈ Mσ (R, E), the map m ˆ : S(R) → E is continuous. (3) φτ is the weakest locally convex topology for which, for each Hausdorff locally convex space E and each m ∈ Mτ (R, E), the map m ˆ : S(R) → E is continuous. Clearly φτ ⊂ φσ ⊂ φ. Lemma 3.1. The topology φτ is Hausdorff. Proof: Every x ∈ X defines a τ -additive measure mx : R → K,

mx (A) = χA (x).

Let g ∈ S(R), g = 0 and let g(x) = 0. Let 0 < < |g(x)|. The set {h ∈ S(R) : |m ˆ x (h)| = |h(x)| < } is a φτ -neighborhood of zero not containing g. Theorem 3.2. If F = (S(R), ρ), where ρ = φ, φσ or φτ , then χ : R → F is a member of M (R, F ), Mσ (R, F ) or Mτ (R, F ), respectively. Proof : Assume that F = (S(R), φτ ). Clearly χ is finitely additive. Let E be a Hausdorff locally convex space and let m ∈ Mτ (R, E), p ∈ cs(E). Let W = {s ∈ E : p(s) ≤ 1}.

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Since m ∈ Mτ (R, E), there exists λ ∈ K such that m(R) ⊂ λW . If D = {g ∈ S(R) : m(g) ˆ ∈ W }, then χ(R) ⊂ λD, which proves that χ : R → F is bounded. If (Vδ ) is a net of measurable sets with Vδ ↓ ∅, then m(Vδ ) → 0, and so m(Vδ ) ∈ W eventually, which implies that χVδ ∈ D eventually. Thus χ ∈ Mτ (R, F ). The proofs for the cases of φ and φσ are analogous. Theorem 3.3. Let E be a Hausdorff locally convex space. Then : (1) The map m → m, ˆ from M (R, E) to the space L ((S(R), φ), E), of all continuous linear maps from (S(R), φ) to E, is an algebraic isomorphism. (2) The map m → m, ˆ from Mσ (R, E) to the space L((S(R), φσ ), E), is an algebraic isomorphism. (3) The map m → m, ˆ from Mτ (R, E) to the space L((S(R), φτ ), E), is an algebraic isomorphism. Proof : (1) By the definition of φ, each m ˆ is continuous. On the other hand, let u : (S(R, ), φ) → E be a continuous linear map and take m = u ◦ χ. Then m ∈ M (R, E) and m ˆ = u. The proofs of (2) and (3) are analogous. Since, for every Hausdorff locally convex space E, every measure m : R → E is of the form m = u ◦ χ, for some φ-continuous linear map u from S(R) to E, we will refer to the measure χ : R → (S(R), φ) as a universal measure. Taking K in place of E and identifying each scalar measure µ on R with the corresponding linear functional µ ˆ, we get the following Theorem 3.4. The spaces M (R) = M (R, K), Mσ (R) and Mτ (R) are algebraically isomorphic with the spaces (S(R), φ), (S(R), φσ ) and (S(R), φτ ) , respectively. Theorem 3.5. On the space S(R), the topology φ is coarser than the topology τu of uniform convergence. Proof : Let E be a Hausdorff locally convex space and let m ∈ M (R, E). It suffices to show that m ˆ : (S(R), τu ) → E is continuous. Indeed, let p ∈ cs(E). There exists r > 0 such that p(m(A)) ≤ r for all A ∈ R. Now, for V = {g ∈ S(R) : g ≤ 1/r}, we have that p(m(g)) ˆ ≤ 1 for all g ∈ V . Indeed, let g ∈ V , g = A1 , . . . , An are pairwise disjoint sets . Then |λk | ≤ 1/r and so p(m(g)) ˆ = p(

n k=1

n k=1

λk χAk , where

λk m(Ak )) ≤ max |λk | · p(m(Ak )) ≤ 1. k

This completes the proof. Theorem 3.6. φ is the finest of all Hausdorff locally convex topologies ρ on S(R) such that, for F = (S(R), ρ), the map χ : R → F is in M (R, F ). Analogous results hold for φσ and φτ . Proof : Let ρ be a Hausdorff locally convex topology on S(R) such that χ : R → (S(R), ρ) is a bounded finitely additive measure. By the definition of φ , the linear map χ ˆ : (S(R), φ) → (S(R), ρ)

52 6


is continuous. Since χ ˆ is the identity map, it follows that φ is finer that ρ. Thus the result holds for φ. Analogous are the proofs for φσ and φτ . Corollary 3.7. On S(R) the topology φ coincides with the topology τu of uniform convergence. Let σ = σ(M (R), S(R)). For a σ-bounded subset H of M (R), we denote by Hσ the set H equipped with the topology induced by σ. Let Cb (Hσ ) be the space of all bounded continuous K-valued functions on Hσ endowed with the sup norm topology. For A ∈ R, the function m → m(A), m ∈ H, is σ-continuous. Also this function is bounded because H is σ-bounded. Hence we get a map µ = µH : R → Cb (Hσ ),

< µ(A), m >= m(A).

Theorem 3.8. For a subset H of M (R), the following are equivalent : (1) H is φ-equicontinuous. (2) H is σ-bounded and the map µ = µH : R → F = Cb (Hσ ) is in M (R, F ). Proof : (1) ⇒ (2). Since H is φ-equicontinuous, it is σ-bounded. Clearly µ is finitely additive. We need to show that µ(R) is a norm bounded subset of Cb (Hσ ). Indeed, let V be a φ-neighborhood of zero in S(R) such that H ⊂ V o . Since χ : R → (S(R), φ) is a bounded measure, there exists a non-zero element λ of K such that χA ∈ λV for all A ∈ R. Thus, for A ∈ R and m ∈ H, we have that |m(A)| ≤ |λ| and hence µ(A) ≤ |λ|. Thus, supA∈R µ(A) ≤ |λ|, which proves that µ ∈ M (R, F ). (2) ⇒ (1). Since µ : R → F = Cb (Hσ ) is a bounded finitely-additive measure, it follows that µ ˆ : (S(R), φ) → F is continuous. Thus, there exists a φ-neighborhood V of zero such that ˆ µ(g) ≤ 1 for all g ∈ V . Then H ⊂ V o and the result follows. Theorem 3.9. For a subset H of Mσ (R), the following are equivalent : (1) H is φσ -equicontinuous. (2) H is σ-bounded and the map µ = µH : R → Cb (Hσ ) is a σ-additive measure. (3) H is σ-bounded and uniformly σ-additive. (4) supm∈H m < ∞ and H is uniformly σ-additive. Proof : (1) ⇒ (2). Since φσ ⊂ φ, it follows that H is φ-equicontinuous and thus (by the preceding Theorem) µ : R → Cb (Hσ ) is a bounded finitely-additive measure. We need to show that µ is σ-additive. So let (Vn ) be a sequence of measurable sets which decreases to the empty set. Since H is φσ -equicontinuous, there exists a φσ -neighborhood V of zero in S(R) such that H ⊂ V o . Let λ = 0. As χ : R → (S(R), φσ ) is a σ-additive measure, there exists no such that χVn ∈ λV , for all n ≥ no . Thus, for n ≥ no and m ∈ H, we have |m(Vn )| ≤ |λ| and thus |µ(An | ≤ |λ|, which proves that µ is σ-additive. (2) ⇒ (3). Let Vn ↓ ∅. Since µ(Vn ) → 0 in Cb (Hσ ), given > 0, there exists no such that |µ(Vn )| ≤ for all n ≥ no . Thus, for n ≥ no , we have that |m(Vn )| ≤ for all m ∈ H, which proves that H is uniformly σ-additive. (3) ⇒ (2). It is trivial. (2) ⇒ (1). Since µ = µH : R → Cb (Hσ ) is a σ-additive measure, the map µ ˆ : (S(R), φσ ) → F is continuous. Hence, there exists a φσ -neighborhood V of zero such that ˆ µ(g) ≤ 1 for all g ∈ V . But then H ⊂ V o . (1) ⇒ (4). Since φσ is coarser than the topology τu of uniform convergence, it


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follows that H is τu -equicontinuous and hence supm∈H m < ∞. Also H is uniformly σ-additive since (1) implies (3). This clearly completes the proof. The proof of the next Theorem is analogous to the one of the preceding Theorem. Theorem 3.10. For a subset H of Mτ (R), the following are equivalent : (1) H is φτ -equicontinuous. (2) H is σ-bounded and the map µ = µH : R → Cb (Hσ ) is a τ -additive measure. (3) H is σ-bounded and uniformly τ -additive. (4) supm∈H m < ∞ and H is uniformly τ -additive. Theorem 3.11. φτ is the weakest of all locally convex topologies ρ on S(R) such that, for each non-Archimedean Banach space E and each m ∈ Mτ (R, E), the map m ˆ : (S(R), ρ) → E is continuous. Proof : Let τo be the weakest of all locally convex topologies ρ on S(R) having the property mentioned in the Theorem. Clearly τo is coarser than φτ . On the other hand, let W be a polar φτ -neighborhood of zero and let H be the polar of W in Mτ (R). By the preceding Theorem, µ = µH : R → E = Cb (Hσ ) is a τ -additive measure. If V is the unit ball of E, then (ˆ µ)−1 (V ) ia a τo -neighborhood of zero. −1 o Since (ˆ µ) (V ) ⊂ H = W , the result clearly follows. 4. Integration Throughout the rest of the paper we will assume that E is a complete Hausdorff locally convex space (unless it is stated otherwise ) and R a separating algebra of subsets of a non-empty set X. Let m ∈ M (R, E) and A ∈ R. Let DA be the family of all α = {A1 , A2 , . . . , An ; x1 , x2 , . . . , xn }, where {A1 , A2 , . . . , An } is a finite Rpartition of A and xi ∈ Ai . We make DA into a directed set by defining α1 ≥ α2 iff the partition of A in α1 is a refinement of the one in α2 . For α = {A1 , A2 , . . . , An ; x1 , x2 , . . . , xn } ∈ DA and f ∈ KX , we define ωα (f, m) =

n

f (xk )m(Ak ).

k=1

over A and If the limα ωα (f, m) exists in E, we will say that f is m-integrable denote this limit by A f dm. For A = X, we write simply f dm. It is easy to see that, if f is m-integrable over X, then it is m-integrable over every measurable subset A and A f dm = f χA dm. If f is bounded on A, then

f dm ≤ f A · mp (A) p A

for every p ∈ cs(E). Using an argument analogous to the one used in [6], Theorem 2.1 for scalar-valued measures, we get the following Theorem 4.1. If m ∈ M (R, E), then an f ∈ KX is m-integrable iff, for each p ∈ cs(E) and each > 0, there exists an R-partition {A1 , A2 , . . . , An } of X such

54 8


that |f (x) − f (y)| · mp (Ai ) ≤ , for all i, if the x, y are in Ai . Moreover in this case, if xi ∈ Ai , then

n f (xi )m(Ai ) ≤ . p f dm − i=1

Theorem 4.2. Let m ∈ M (R, E) and let f ∈ KX be m-integrable. Then : (1) f is continuous at every x in the set {x : Nm,p (x) = 0}. D= p∈cs(E)

(2) For each p ∈ cs(E), there exists a measurable set A, with mp (Ac ) = 0, such that f is bounded on A. Proof : (1). Suppose that Nm,p (x) = d > 0 and let > 0. There exists an Rpartition {A1 , A2 , . . . , An } of X such that |f (x) − f (y)| · mp (Ai ) ≤ d, if x, y ∈ Ai . If x ∈ Ai , then |f (y) − f (x)| ≤ for all y ∈ Ai . (2). Let {A1 , A2 , . . . , An } be an R-partition of X such that |f (x) − f (y)| · mp (Ai ) ≤ 1, if x, y ∈ Ai . Let A = {Ai : mp (Ai ) > 0}. It follows easily that f is bounded on A and that mp (Ac ) = 0. Theorem 4.3. Let m ∈ M (R, E). If f, g ∈ KX are m-integrable, then h = f g is also m-integrable. Proof : Let p ∈ cs(E) and > 0. There are measurable sets A, B such that mp (Ac ) = mp (B c ) = 0 and f, g are bounded on A, B, respectively. Let D = A ∩ B. d > f D , gD . Now there exists an R-partition {A1 , A2 , . . . , An } of X, which is a refinement of {D, Dc }, such that |f (x) − f (y)| · mp (Ai ) < /d and

|g(x) − g(y)| · mp (Ai ) < /d

if x, y ∈ Ai . Let now x, y ∈ Ai . If Ai ⊂ Dc , then |h(x) − h(y)| · mp (Ai ) = 0. For Ai ⊂ D, we have that |h(x) − h(y)| = |[f (x) − f (y)]g(x) + f (y)[g(x) − g(y)]| ≤ max{d · |f (x) − f (y)|, d · |g(x) − g(y)|} and so |h(x) − h(y)| · mp (Ai ) < . This completes the proof in view of Theorem 4.1. Let now m ∈ M (R, E) and let g ∈ KX be m-integrable. Define

g dm. mg : R → E, mg (A) = A

Clearly mg is finitely-additive. Also, mg is bounded. In fact, let p ∈ cs(E). There exists a measurable set B such that mp (B c ) = 0 and g is bounded on B. Let d = gB . Let A ∈ R, W1 = A ∩ B, W2 = A ∩ B c . Since g is m-integrable, there exists an R-partition {V1 , V2 , . . . , Vn } of A, which is a refinement of {W1 , W2 } such that |g(x) − g(y)| · mp (Vi ) < 1 if x, y ∈ Vi . Let xi ∈ Vi . Then

n g dm − g(xi )m(Vi ) < 1. p A

k=1


55 9

If Vi ⊂ W1 , then p (g(xi )m(Vi )) ≤ d · mp (X), while for Vi ⊂ W2 we have that p (g(xi )m(Vi )) = 0. Thus

p g dm ≤ max{1, d · mp (X)}. A

This proves that mg is bounded and hence mg ∈ M (R, E). X X Theorem 4.4. Let m ∈ M (R, E) and let g ∈ K be m-integrable. If f ∈ K is m-integrable, then f is mg -integrable and f dmg = f g dm.

Proof : Let p ∈ cs(E). There exists a measurable set D, with mp (Dc ) = 0, such that f, g are bounded on D. Let d > max{f D , gD }. If V is a measurable set contained in Dc , then p(mg (V )) = 0. This follows from the fact that, for A ⊂ V we have that p(g(x)m(A)) = 0. Let now > 0 be given. There exists an R-partition {V1 , V2 , . . . , VN } of X , which is a refinement of {D, Dc }, such that |f (x) − f (y)| · mp (Vi ) < /d, and |g(x) − g(y)| · mp (Vi ) < /d n if x, y ∈ Vi . We may assume that i=1 Vi = D. For A ∈ R, A ⊂ Vi ⊂ D, we have

g dm ≤ gA · mp (A) ≤ d · mp (Vi ), p A

and hence (mg )p (Vi ) ≤ d · mp (Vi ). Thus, for x, y ∈ Vi ⊂ D, we have |f (x) − f (y)| · (mg )p (Vi ) ≤ d · |f (x) − f (y)| · mp (Vi ) ≤ . The same inequality holds when Vi ⊂ Dc . This proves that f is mg -integrable. If xk ∈ Vk , then

n f (xk )mg (Vk ) ≤ . p f dmg − k=1

Since, for x, y ∈ Vk ⊂ D, we have |g(x) − g(y)| · mp (Vk ) ≤ /d, it follows that p (mg (Vk ) − g(xk )m(Vk )) ≤ /d. For x, y ∈ Vk ⊂ D, we have |f (x)g(x)−f (y)g(y)|·mp (Vk )≤mp (Vk )·max{|g(x)|·|f (x)−f (y)|, |f (y)|·|g(x)−g(y)|}≤. Since mp (Vk ) = 0 if Vk ⊂ Dc , we get that

n g(xk )f (xk )m(Vk ) ≤ . p gf dm − k=1

Also, for 1 ≤ k ≤ n, we have p (f (xk )g(xk )m(Vk ) − f (xk )mg (Vk )) ≤ . It follows that

p gf dm − f dmg ≤ . This, being true for all > 0, and the fact that E is Hausdorff, imply that

gf dm = f dmg , which completes the proof. Theorem 4.5. Let m ∈ M (R, E), p ∈ cs(E) and x ∈ X. If g ∈ KX is m-integrable, then Nmg ,p (x) = |g(x)| · Nm,p (x).

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Proof : Let > 0. There exists an R-partition {V1 , V2 , · · · , Vn } of X such that |g(y) − g(z)| · mp (Vi ) ≤ if y, z ∈ Vi . Claim I : If V is a measurable subset of Vi containing x, then, for each A ⊂ V , we have p(mg (A)) ≤ max{, |g(x)| · mp (V )} = θ. Indeed, if x ∈ A, then for each y ∈ A we have that |g(x) − g(y)| · mp (A) ≤ , which implies that p(mg (A) − g(x)m(A)) ≤ and so p(mg (A)) ≤ max{, |g(x)| · p(m(A))} ≤ θ. In case x ∈ V \A, we get in the same way that p(mg (V \A)) ≤ θ. Also p(mg (V )) ≤ θ, since x ∈ V . Thus p(mg (A)) = p(mg (V ) − mg (V \ A)) ≤ θ, and the claim follows. Claim II. If W is a measurable subset of Vi containing x, then for each measurable set A ⊂ W , we have that |g(x)| · p(m(A)) ≤ max{, (mg )p (W )} = d. Indeed, if x ∈ A ⊂ W , then p(mg (A) − g(x)m(A)) ≤ and so |g(x)| · p(m(A)) ≤ max{, p(mg (A))} ≤ d. If x ∈ W \ A, then |g(x)| · p(m(W \ A)) ≤ d. Also g(x)| · p(m(W )) ≤ d, and so again |g(x)| · p(m(A)) ≤ d, which proves the claim. Now there are measurable subsets V, W of Vi containing x such that mp (V ) < Nm,p (x) + ,

and (mg )p (W ) < + Nmg ,p (x).

By claim I, we have Nmg ,p (x) ≤ (mg )p (V )

≤ max{, |g(x)| · mp (V )} ≤ max{, |g(x)|[ + Nm,p (x)]}.

Taking → 0, we get that Nmg ,p (x) ≤ |g(x)| · Nm,p (x). Also |g(x)| · Nm,p (x) ≤ |g(x)| · mp (W ) ≤ max{, (mg )p (W )} < + Nmg ,p (x). Taking → 0, we get that |g(x)| · Nm,p (x) ≤ Nmg ,p (x), which completes the proof. Theorem 4.6. Let m ∈ M (R, E) and let g ∈ KX be m-integrable. If m is τ -additive (resp. σ-additive ), then mg is τ -additive (resp. σ-additive ). Proof : Assume that m is τ -additive and let Vδ ↓ ∅ and p ∈ cs(E) There exists an A ∈ R such that mp (Ac ) = 0 and f A = d < ∞. Given > 0, there exists a δo such that mp (Vδ ) < /d if δ ≥ δo . For a measurable set V disjoint from A, we have p(mg (V )) = 0. Thus, for δ ≥ δo , we have p(mg (Vδ )) = p(mg (Vδ ∩ A)) ≤ gVδ ∩A · mp (Vδ ∩ A) ≤ d · mp (Vδ ) < . This proves that mg is τ -additive. The proof for the σ-additive case is analogous.


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The proof of the next Theorem is analogous to the one given in [8], Theorem 2.16, for scalar-valued measures. Theorem 4.7. Let m ∈ M (R, E). For a subset Z of X, the following are equivalent: (1) χZ is m-integrable. (2) For each p ∈ cs(E) and each > 0, there are measurable sets V, W such that V ⊂ Z ⊂ W and mp (W \ V ) < . For m ∈ M (R, E), let Rm be the family of all A ⊂ X such that χA is mintegrable. Using the preceding Theorem, we show easily that Rm is a separating algebra of subsets of X which contains R. Define

m : Rm → E, m(A) = χA dm. The proofs of the next two Theorems are analogous to the corresponding ones for scalar valued measures (see [8], Lemma 2.18, Theorems 2.22, 2.23, 2.24, 2.26 and Corollary 2.25 ). Theorem 4.8. (1) For A ∈ R and p ∈ cs(E), we have mp (A) = mp (A). (2) m is σ-additive iff m is σ-additive. (3) m is τ -additive iff m is τ -additive. (4) Nm,p = Nm,p . (5) Rm = Rm . X Theorem 4.9. (1) If f ∈ K is m-integrable, then f is also m-integrable and f dm = f dm. (2) If f ∈ KX is m-integrable and bounded, then f is also m-integrable.

For m ∈ M (R, E), p ∈ cs(E) and A ⊂ X, we define m∗p (A) = inf{mp (W ) : A ⊂ W ∈ R}. Theorem 4.10. For all A ∈ Rm we have m∗p (A) = m ¯ p (A). Proof : If A ⊂ W ∈ R, then m ¯ p (A) ≤ m ¯ p (W ) = mp (W ). It follows from this that m∗p (A) ≥ m ¯ p (A). On the other hand, given > 0, there are sets V, W in R such that V ⊂ A ⊂ W and mp (W \ V ) < . Now ¯ p (W ) = max{m ¯ p (A), m ¯ p (W \ A)} m∗p (A) ≤ mp (W ) = m ≤ max{m ¯ p (A), m ¯ p (W \ V )} ≤ max{m ¯ p (A), }. ¯ p (A) and the Theorem follows. Taking → 0, we get that m∗p (A) ≤ m Theorem 4.11. A subset A of X is in Rm iff for all p ∈ cs(E) we have inf{m∗p (A V ) : V ∈ R} = 0. Proof : Assume that A ∈ Rm and let > 0. There are V, W in R such that V ⊂ A ⊂ W and mp (W \ V ) < . Now m∗p (A V ) ≤ mp (W \ V ) < . Conversely, assume that the condition is satisfied and let > 0. There exist V, W in R such that A V ⊂ W and mp (W ) < . Let V1 = V ∩ W c and W1 = V ∪ W . Then V1 ⊂ A ⊂ W1 and mp (W1 \ V1 ) = mp (W ) < . This proves that A ∈ Rm and the result follows.

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Lemma 4.12. If m ∈ Mτ (R, E), then every τR -clopen set A is in Rm . Proof : Let p ∈ cs(E) and > 0. Consider the collection F of all R-measurable sets of the form W \ V , where V, W ∈ R and V ⊂ A ⊂ W . Then F ↓ ∅. As m is τ -additive, there exists an W \ V ∈ F such that mp (W \ V ) < , which proves that A ∈ Rm . Theorem 4.13. Let m ∈ Mτ (R, E) and f ∈ KX . If f is bounded and τR continuous, then f is m-integrable (and hence m-integrable ). Proof : Without loss of generality, we may assume that f ≤ 1. Let p ∈ cs(E) and > 0. The set Y = {x : Nm,p (x) ≥ } is τR -compact. Choose 0 < 1 < such that 1 .mp (X) < . There are x1 , x2 , · · · , xn ∈ Y such that the sets Ak = {x : p(f (x) − f (xk )) ≤ 1 },

k = 1, · · · , n.

are pairwise disjoint and cover Y . Each Ak is τR -clopen and hence it is a member of Rm . Let nVk , Wk ∈ R be such that Vk ⊂ Ak ⊂ Wk and mp (Wk \ Vk ) < . Let Vn+1 = ( k=1 Vk )c . Then Vn+1 is disjoint from Y . As m is τ -additive, we have that mp (Vn+1 ) = supx∈Vn+1 Nm,p (x) ≤ . If now x, y ∈ Vi , i ≤ n, then |f (x) − f (y)| · mp (Vi ) ≤ 1 . · mp (X) < . Also, if x, y ∈ Vn+1 , then |f (x) − f (y)| · mp (Vn+1 ) ≤ . This proves that f is m-integrable. Theorem 4.14. Let m ∈ Mτ (R, E). For a subset A of X, the following are equivalent : (1) A ∈ Rm . (2) A is τRm -clopen. Proof : Clearly (1) ⇒ (2). On the other hand let A be τRm -clopen. Since m is τ -additive, it follows (by Theorem 4.11) that χA is m-integrable and hence χA is m-integrable (by Theorem 4.9), which means that A ∈ Rm . Theorem 4.15. Let m ∈ Mτ (R, E) and consider on X the topology τR . Then the map

um : Cb (X) → E, um (f ) = f dm = f dm is β-continuous. Also, every β-continuous linear map form u = um for some m ∈ Mτ (R, E).

u : Cb (X) → E is of the

Proof : Let p ∈ cs(E) and G ∈ Ω. We need to show that the set V = {f ∈ Cb (X) : p(um (f )) ≤ 1} is a βG -neighborhood of zero. Indeed, let r > 0. There exists a decreasing net (Vδ ) βo X of τR -clopen sets with δ Vδ = G. Since Vδ ∈ Rm and m is τ -additive, there exists a δ such that mp (Vδ ) < 1/r. Now V1 = {f ∈ Cb (X) : f ≤ r,

f Vδc ≤ 1/mp } ⊂ V.

In fact, let f ∈ V1 and set h = f χVδ , g = f χVδc . Then

g dm = p g dm ≤ 1. p h dm = p h dm ≤ 1 and p


59 13

Thus p( f dm) ≤ 1, which shows that V1 ⊂ V . Since the closure of Vδc in βo X is disjoint from G, this proves that V is a βG -neighborhood of zero. This, being true for every G ∈ Ω, implies that V is a β-neighborhood of zero and so um is β-continuous. Conversely let u : (Cb (X), β) → E be continuous. Since β is coarser than the topology of uniform comnvergence, for each p ∈ cs(E), there exists a non-zero λ ∈ K such that {f ∈ Cb (X) : f ≤ |λ|} ⊂ {f : p(u(f )) ≤ 1}. Let K(X) be the algebra of al τR -clopen subsets of X. Define µ : K(X) → E,

µ(A) = u(χA ).

Clearly µ is finitely-additive. Also, since |λχA | ≤ |λ|, it follows that p(µ(A)) ≤ |λ|−1 , and so µ is bounded. If (Vδ ) is a net of clopen sets which decreases to the empty set, then χVδ → 0 with respect to the topology β and so µ(Vδ ) → 0. Thus µ ∈ Mτ (K(X), E). The restriction m = µ|R is in Mτ (R, E). The subspace F of Cb (X) spanned by the functions χA , A ∈ K(X), is β-dense in Cb (X). Since u and um are both β-continuous and they coincide in F , it follows that u = um on Cb (X). This completes the proof. Theorem 4.16. Let X be a zero-dimensional Hausdorff topological space and E a Hausdorff locally convex space. Then a linear map u : Cb (X) → E is β-continuous iff it is βo -continuous. ˆ be the completion of E and let K(X) be the algebra of all Proof : Let E ˆ is clopen subsets of X. Suppose that u is β-continuous. Then u : (Cb (X), β) → E ˆ continuous. In view of the preceding Theorem, there exists an m ∈ Mτ (K(X), E) such that u(f ) = f dm for all f ∈ Cb (X). Let p ∈ cs(E) and V = {f : p(u(f )) ≤ 1}. We need to show that V is a βo -neighborhood of zero. By [4], Theorem 2.8, it suffices to show that, for each r > 0, there exists a compact subset Y of X and > 0 such that V1 = {f ∈ Cb (X) : f ≤ r, f Y ≤ } ⊂ V. Choose > 0 such that ·mp (X) ≤ 1 and r· ≤ 1. The set Xp, = {x : Nm,p (x) ≥ } is compact. In the definition of V1 take as Y the set Xp, . Let f ∈ V1 and A = {x : |f (x)| ≤ }. Then mp (Ac ) = supx∈Ac Nm,p (x) ≤ . Now

f dm ≤ · mp (X) ≤ 1, and p f dm ≤ r · mp (Ac ) ≤ 1. p Ac

A

Thus V1 ⊂ V and the result follows. Theorem 4.17. Let R be a separating algebra of subsets of a set X and consider on X the topology τR . Then φτ coincides with the topology induced on S(R) by βo and by the topology induced by β. Proof :

If (Vδ ) is a net of measurable subsets of X which decreases to the β

empty set, then χVδ ↓ 0 and so χVδ → 0. Thus χ : R → (S(R), β) is a τ -additive measure. In view of Theorem 3.6 , it follows that φτ is finer than the topology induced on S(R) by β. On the other hand, let E be a Hausdorff locally convex

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ˆ be its completion. If m ∈ Mτ (R, E), then m ∈ Mτ (R, E). ˆ The space and let E map

ˆ u : Cb (X) → E, u(f ) = f dm, ˆ is continˆ = u|S(R) , it follows that m ˆ : (S(R), βo ) → E is βo -continuous. Since m uous and hence m ˆ : (S(R), βo ) → E is continuous. This implies that φτ is coarser than the topology induced on S(R) by βo and the result follows. Lemma 4.18. Let Z be a vector space over K , D a subspace of Z and τ1 , τ2 Hausdorff locally convex topologies on Z which induce the same topology on D and for both of which D is dense in Z. If τ2 is finer than τ1 , then τ1 and τ2 coincide on Z. ˆ be its completion. The identity map T : Proof: Let G = (Z, τ2 ) and let G ˆ is clearly continuous. Let S = T |D . Since τ1 and τ2 induce the same (Z, τ2 ) → G ˆ is continuous. As D is τ1 -dense in Z, topology on D, it follows that S : (D, τ1 ) → G ˆ ˆ Now Sˆ : (Z, τ2 ) → G ˆ is there exists a unique continuous extension S : (Z, τ1 ) → G. ˆ ˆ continuous. Since S = T on D and D is τ2 -dense in Z, it follows that S = T on Z. Thus ˆ T = Sˆ : (Z, τ1 ) → G is continuous, which clearly implies that τ1 is finer than τ2 and the Lemma follows. Theorem 4.19. For any zero-dimensional Hausdorff topological space X, the topologies β and βo coincide on Cb (X). Proof : Let K(X) be the algebra of all clopen subsets of X. Since S(K(X)) is β-dense in Cb (X), the result follows from Theorem 4.15 and the preceding Lemma. Theorem 4.20. Let ∆ be the family of all pairs (m, p) for which there exists a Hausdorff locally convex space E such that p ∈ cs(E) and m ∈ Mτ (R, E). To each δ = (m, p) ∈ ∆ corresponds the non-Archimedean seminorm · Nm,p on S(R). Then φτ coincides with the locally convex topology ρ generated by these seminorms. Proof : LetE be a Hausdorff locally convex space, m ∈ Mτ (R, E) and p ∈ cs(E). If g = nk=1 αk χAk ∈ S(R), then n αk m(Ak ) ≤ max |αk | · p(m(Ak )) ≤ gNm,p . p(m(g)) ˆ =p k=1

k

Thus m ˆ : S(R), ρ) → E is continuous and so φτ is coarser than ρ. On the other hand, let (m, p) ∈ ∆ and V = {g ∈ S(R) : p(m(g)) ˆ ≤ 1}. Since φτ is locally solid, there exists a solid φτ -neighborhood V1 of zero contained in V . Now V1 ⊂ {g : gNm,p ≤ 1}. In fact, assume that, for some g = nk=1 αk χAk ∈ V1 , we have that gNm,p > 1. There exists an x in some Ak such that |g(x)| · Nm,p (x) = |αk | · Nm,p (x) > 1. There is a measurable set A contained in Ak such that |αk | · p(m(A)) > 1. If h = αk χA , then |h| ≤ |g| and so h ∈ V1 , which is a contradiction since p(m(h)) ˆ > 1. This contradiction shows that V1 ⊂ {g : gNm,p ≤ 1}. Thus ρ is coarser than φτ and the result follows.


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5. (VR)-Integrals Throughout this section, R will be a separating algebra of subsets of a set X, E a complete Hausdorff locally convex space and m ∈ Mτ (R, E). For p ∈ cs(E), and f ∈ KX , let f Nm,p = supx∈X |f (x)| · Nm,p (x). Let Gm be the space of all f ∈ KX for which f Nm,p < ∞, for each p ∈ cs(E). Each .Nm,p is a non-Archimedean seminorm on Gm . We will consider on Gm the locally convex topology generated by these seminorms. Lemma 5.1. If g = nk=1 αk χAk ∈ S(R), then n αk m(Ak ) ≤ gNm,p . p k=1

Proof : We first observe that gNm,p ≤ g · mp (X) < ∞. If g = α · χA , where α ∈ K and A ∈ R, then p(α · m(A)) ≤ |α| · mp (A) = |α| · sup Nm,p (x) x∈A

sup |g(x)| · Nm,p (x) = gNm,p .

=

x∈X

In the general case, we may assume that the sets Ak , k = 1, . . . , n, are pairwise disjoint. Then n αk · m(Ak ) ≤ max |αk | · mp (Ak ) = max sup |g(x)| · Nm,p (x) = gNm,p . p k=1

k

k

x∈Ak

Lemma 5.2. If we consider on S(R) the topology induced by the topology of Gm , then

ω : S(R) → E, ω(g) = g dm is a continuous linear map. Proof : It follows from the preceding Lemma. Let now S(R) be the closure of S(R) in Gm and let ω : S(R) → E be the unique continuous extension of ω. Definition 5.3. A function f ∈ KX is said to be (VR)-integrable with respect to m if it belongs to S(R). In this case, ω(f ) is called the (VR)-integral of f , with respect to m, and will be denoted by (V R) f dm. We will denote by L(m) the space S(R). Theorem 5.4. If f is (VR)-integrable, then, for each p ∈ cs(E), we have

p (V R) f dm ≤ f Nm,p . Proof : There exists a ne (gδ ) in S(R) such that gδ → f in S(R). Then

(V R) f dm = lim gδ dm, and gδ Nm,p → f Nm,p . δ

Since p

gδ dm

≤ gδ Nm,p ,

62 16


the result follows. Theorem 5.5. The space Gm is complete and hence L(m) is also complete. Proof : Let (fδ ) be a Cauchy net in Gm and let A= {x : Nm,p (x) > 0}. p∈cs(E)

Let x ∈ A and choose p ∈ cs(E) such that Nm,p (x) = d > 0. Given > 0, there exists a δo such that fδ − fδ Nm,p < d if δ, δ ≥ δo . Now, for δ, δ ≥ δo , we have |fδ (x) − fδ (x)| < . This proves that the net (fδ (x)) is Cauchy in K. Define f (x) = limδ fδ (x), if x ∈ A and f (x) arbitrarily if x ∈ / A. We will show that f ∈ Gm and that fδ → f . Indeed, given p ∈ cs(E) and > 0, there exists δo such that |fδ (x) − fδ (x)| · Nm,p (x) < for all x and all δ, δ ≥ δo . Let now δ ≥ δo be fixed. If x ∈ A, then taking the limits on δ , we get that |fδ (x) − f (x)| · Nm,p (x) ≤ . The same inequality also holds when x∈ / A. Thus, for all δ ≥ δo , we have supx∈X |fδ (x) − f (x)| · Nm,p (x) ≤ . It follows from this that , for all x ∈ X, we have |f (x)| · Nm,p (x) ≤ max{, fδo Nm,p } which proves that f ∈ Gm . Also, f − fδ Nm,p ≤ for δ ≥ δo . Hence fδ → f and the proof is complete. Theorem 5.6. For a subset A of X, the following are equivalent: (1) χA is (VR)-integrable. (2) For each p ∈ cs(E) and each > 0, there exists V ∈ R such that Nm,p < on A V . (3) For each p ∈ cs(E) and each > 0, there exists V ∈ R such that V ∩ Xp, = A ∩ Xp, . Proof : (1) ⇔ (2). The proof is analogous to the one given in [14], Lemma 7.3 for scalar valued measures. (2) ⇔ (3). It follows from the fact that, for V ∈ R, ˜ m be the family of all subsets A of X for which χA is (VR)-integrable Let now R ˜ m is a separating algebra of subsets with respect to m. It is easy to see that R ˜m of X which contains R. Let τR˜ m be the zero dimensional topology having R as a basis. In view of Theorem 2.5, for all p ∈ cs(E) and all > 0, the set ˜ m iff , for all p ∈ cs(E) and Xp, = {x : Nm,p (x) ≥ } is τR -compact. Since A ∈ R all > 0 , there exists V ∈ R such that V ∩ Xp, = A ∩ Xp, , it follows that Xp, is τR˜ m -compact. Also, since τR is Hausdorff, τR and τR˜ m induce the same topology on Xp, . Now we define

˜ m → E, m(A) m ˜ :R ˜ = (V R) χA dm. A

Clearly m ˜ is finitely-additive. Also m ˜ is bounded since, for each p ∈ cs(E), we have p(m(A)) ˜ ≤ sup Nm,p (x) ≤ mp (X). x∈A

˜ m , E). Thus m ˜ ∈ M (R


63 17

Lemma 5.7. If V ∈ R, then mp (V ) = m ˜ p (V ). Proof : It is clear that mp (V ) ≤ m ˜ p (V ). Suppose that m ˜ p (V ) > θ > 0. There ˜ m , A ⊂ V, p(m(A)) exists A ∈ R ˜ > θ. Since p(m(A)) ˜ ≤ supx∈A Nm,p (x), there exists x ∈ A such that Nm,p (x) > θ and so mp (V ) ≥ Nm,p (x) > θ. This proves that mp (V ) ≥ m ˜ p (V ) and the Lemma follows. Lemma 5.8. Nm,p = Nm,p ˜ . Proof : Since mp (V ) = m ˜ p (V ) for V ∈ R, it follows that Nm,p ≥ Nm,p ˜ . ˜m (x). Let x ∈ A ∈ R Assume that there exists an x such that Nm,p (x) > θ > Nm,p ˜ be such that m ˜ p (A) < θ. Let Y = Xp,θ and let V ∈ R be such that V ∩ Y = A ∩ Y . Since x ∈ A ∩ Y , we have that x ∈ V and so mp (V ) ≥ Nm,p (x) > θ. Let D ∈ R, D ⊂ V be such that p(m(D)) > θ. Now p(m(D ˜ ∩ A)) ≤ m ˜ p (A) < θ and hence p(m(D)) = p(m(D ˜ ∩ Ac )) ≤ sup Nm,p (y). y∈D\A

But, for y ∈ D \ A, we have that Nm,p (y) < θ since D ⊂ V and A ∩ Y = V ∩ Y . Thus θ < p(m(D)) ≤ θ, a contradiction. This completes the proof. ˜ m iff A is τ ˜ -clopen. Lemma 5.9. For A ⊂ X, we have A ∈ R Rm ˜ m is τ ˜ -clopen. On the other hand let A be Proof : Clearly every A ∈ R Rm τR˜ m -clopen and let p ∈ cs(E), > 0. Since τR and τR˜ m induce the same topology induced by τR . on Xp, , the set G = A Xp, is clopen in Xp, for the topology For each x ∈ G, there exists an Ax ∈ R such that x ∈ Ax Xp, ⊂ G. As G is τR -compact, there are x1 , x2 , . . . , xn ∈ G such that G=

n

Axk ∩ Xp, = V ∩ Xp, ,

k=1

where V = follows.

n

k=1 Axk

˜ m and the result ∈ R. In view of Theorem 5.6, A is in R

˜ m , E). Theorem 5.10. m ˜ ∈ Mτ ( R ˜ m which decreases to the empty set and let Proof : Let A be a family in R p ∈ cs(E), > 0 , Y = Xp, . For each A in A, there exists B ∈ R such that B ∩ Y = A ∩ Y . Let B = {B ∈ R : ∃A ∈ A, A ∩ Y = B ∩ Y }. It is easy to see that B ↓ ∅. Since m ∈ Mτ (R, E), there exists B ∈ B such that mp (B) < . Let A ∈ A be such that A ∩ Y = B ∩ Y . If x ∈ A, then x ∈ / Y and so ˜ m is contained in A, then Nm,p )x) < . If G ∈ R p(m(G)) ˜ ≤ sup Nm,p (x) ≤ x∈G

˜ m , E). and so m ˜ p (A) ≤ . This proves that limA∈A m ˜ p (A) = 0 and so m ˜ ∈ Mτ ( R ˜ m ), then for each p ∈ cs(E) and each > 0, there Lemma 5.11. If g ∈ S(R exists an h ∈ S(R) such that h − gNm,p ≤ .

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Proof : Assume that g = 0 and let A1 , A2 , . . . , An be pairwise disjoint members ˜ m and non-zero scalars α1 , α2 , . . . , αn such that g = n αk χA . Let r = of R k k=1 maxk |αk |. For each k, there exists a Bk ∈ R such that Nm,p < /r on Ak Bk . Since αk χAk − αk χBk Nm,p ≤ |αk | · sup Nm,p (x) ≤ , x∈Ak Bk

it follows that h − gNm,p ≤ . Using Lemmas 5.7 and 5.11, we get the following Theorem 5.12. A function f ∈ KX is (VR)-integrable with respect to m iff it is (VR)-integrable with respect to m. ˜ Moreover

(V R) f dm = (V R) f dm. ˜ Theorem 5.13. If f ∈ KX is m-integrable with respect to m, then it is also (VR)-integrable and

f dm = (V R) f dm. Proof : Let p ∈ cs(E) and > 0. There exists a R-partition {V1 , V2 , . . . , Vn } of X n such that |f (x) − f (y)| · mp (Vi ) < if x, y ∈ Vi . Let xk ∈ Vk and g = k=1 f (xk )χVk . For x ∈ Vk , we have |f (x) − g(x)| · Nm,p (x) = |f (x) − f (xk )| · Nm,p (x) ≤ |f (x) − f (xk )| · mp (Vk ) < .

This proves that f is (VR)-integrable. Also p f dm − g dm ≤ and

p (V R) f dm − g dm = p (V R) (f − g) dm ≤ f − g|Nm,p ≤ . Thus

p

f dm − (V R)

Since E is Hausdorff, it follows that plete.

f dm

f dm = (V R)

≤ .

f dm and the proof is com-

Theorem 5.14. Let Y be a zero-dimensional topological space and f : X → Y . Then f is τR˜ m -continuous iff, for each p ∈ cs(E) and each > 0, the restriction of f to Xp, is τR - continuous. Proof : Since τR and τR˜ m induce the same topology on Xp, , the necessity of the condition is clear. On the other hand, assume that the condition is satisfied and let Z be a clopen subset of Y . We need to show that f −1 (A) is τR˜ m -clopen, ˜ m . Let p ∈ cs(E) and > 0. The restriction h of or equivalently that f −1 (A) ∈ R f to Xp, is τR -continuous. Thus G = f −1 (A) ∩ Xp, = h−1 (A) is clopen in Xp, for the topology induced by τR . For each x ∈ G, there exists Vx ∈ R such that x ∈ V τR -compact, there are x1 , x2 , . . . , xn in G such that Xp, ⊂ G. Since G is x n n G = k=1 Vxk ∩ Xp, . If V = k=1 Vxk ∈ R, then V ∩ Xp, = f −1 (A) ∩ Xp, . In ˜ m and we are done. view of Lemma 5.9, we get that f −1 (A) ∈ R


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Theorem 5.15. Let m ∈ Mτ (R, E) and f ∈ KX . Then, f is (VR)-integrable iff : a) f is τR˜ m -continuous. b) For each p ∈ cs(E) and each > 0, the set D = {x : |f (x)| · Nm,p (x) ≥ } is τR˜ m -compact. Proof : Assume that f is (VR)-integrable and let p ∈ cs(E) and > 0. There exists a sequence (gn ) in S(R) such that f − gn Nm,p → 0. For x ∈ Xp, , we have that |f (x) − gn (x)| ≤ 1/ · f − gn Nm,p → 0 uniformly. Since each gn is τR -continuous , it follows that f |Xp, is τR -continuous and so f is τR˜ m -continuous. Also, given > 0, there exists a g ∈ S(R) such pairwise disjoint members of R and that f − gNm,p < . Let {V1 , V2 , . . . , Vn } be n α1 , α2 , . . . , αn non-zero scalars such that g = k=1 αk χVk . Now D = {x : |g(x)| · Nm,p (x) ≥ } =

n

[Vk ∩ {x : Nm,p (x) ≥ /|αk |}],

k=1

and so D is τR˜ m -compact. Moreover D = {x : |f (x)| · Nm,p (x) ≥ }. Conversely, assume that the conditions (a), (b) are satisfied. Let p ∈ cs(E), > 0 and D = {x : |f (x)| · Nm,p (x) ≥ }. ˜ m such that For each x ∈ D, there exists an Ax ∈ R x ∈ Ax ⊂ {y : |f (y) − f (x)| < /mp (X)}.

˜ m -compactness of D, there are y1 , y2 , . . . , yn ∈ D such that D ⊂ n Ay . By the R k k=1 ˜ m such that D ⊂ N Vj Now, there are pairwise disjoint sets V1 , V2 , . . . , VN in R j=1 N and each Vj is contained in some Ayk . Let xj ∈ Vj , g = j=1 f (xj )χVj . If x ∈ Vj , then |f (x) − g(x)| · Nm,p (x) = |f (x) − f (xj )| · Nm,p (x) ≤ mp · /mp = , N / D, which implies that while, for x ∈ / j=1 Vj we have g(x) = 0 and x ∈ |f (x) − g(x)| · Nm,p (x) = |f (x)| · Nm,p (x) ≤ . This proves that f is (VR)-integrable with respect to m ˜ and hence it is (VR)integrable with respect to m. This completes the proof. 6. The Measure m ˜f In this section we will assume that E is a complete Hausdorff locally convex space, R a separating algebra of subsets of a set X and m ∈ Mτ (R, E). Let f ∈ KX be (VR)-integrable with respect to m and define

˜ ˜ f (A) = (V R) f dm = (V R) χA f dm. m ˜ f : Rm → E, m A

Then, for each p ∈ cs(E), we have p(m ˜ f (A)) ≤ sup |f (x)| · Nm,p (x) ≤ f Nm,p , x∈A

and so m ˜ f is bounded and clearly finitely-additive. Also m ˜ f is τ -additive. Indeed, ˜ m which decreases to the empty set and let p ∈ cs(E), > 0. let (Aδ ) be a net in R

66 20


There exists a g ∈ S(R) such that f − gNm,p < . If g = V1 , V2 , . . . , Vn are pairwise disjoint members of R, then n αk m(V ˜ k ∩ Aδ ). m ˜ g (Aδ ) =

n

k=1 αk χVk ,

where

k=1

˜ is τ -additive, there exists δo such that p(m ˜ g (Aδ )) < if Since Aδ ∩ Vk ↓ ∅ and m δ ≥ δo . Also p(m ˜ f −g (Aδ )) ≤ f − gNm,p < . ˜ m , E). ˜ f (Aδ )) < , which proves that m ˜ f ∈ Mτ ( R Thus, for δ ≥ δo , we have that p(m Lemma 6.1. If g ∈ S(R), then Nm ˜ g,p (x) = |g(x)| · Nm,p (x). n Proof : Let g = k=1 αk χVk , where {V1 , V2 , . . . , Vn } is an R-partition of X. ˜ m is contained in Vk , then Let x ∈ Vk and h = αk χVk . If A ∈ R

m ˜ g (A) = m ˜ h (A) = αk · (V R) χA dm = g(x)m(A). ˜ Thus Nm ˜ g,p (x) = |g(x)| · Nm,p ˜ (x) = |g(x)| · Nm,p (x). Lemma 6.2. Let f, g ∈ KX be (VR)-integrable with respect to m. Then for each ˜ m , we have V ∈R ˜ g )p (V )| ≤ f − gNm,p . |(m ˜ f )p (V ) − (m Proof : Assume (say) that (m ˜ f )p (V ) − (m ˜ g )p (V ) ≥ 0. Given > 0, there ˜ m contained in V such that (m exists A ∈ R ˜ f )p (V ) < p(m ˜ f (A)) + . Now 0 ≤ (m ˜ f )p (V ) − (m ˜ g )p (V )

< + p(m ˜ f (A)) − p(m ˜ g (A)) ˜ g (A)) ≤ + p(m ˜ f (A) − m = + p(m ˜ f −g (A)) ≤ + f − gNm,p

and the Lemma follows taking → 0. Lemma 6.3. Let f, g ∈ KX be (VR)-integrable with respect to m. Then |Nm ˜ f ,p (x) − Nm ˜ g ,p (x)| ≤ f − gNm,p . ˜ Proof : Suppose (say) that 0 ≤ Nm ˜ f ,p (x) − Nm ˜ g ,p (x) and choose a V ∈ Rm containing x such that (m ˜ g (V ) < Nm ˜ g ,p (x) + . Now 0 ≤ Nm ˜ f )p (V ) − [(m ˜ g )p (V ) − ] ≤ + f − gNm,p . ˜ f ,p (x) − Nm ˜ g ,p (x) ≤ (m Taking → 0, the Lemma follows. Theorem 6.4. If f ∈ KX is (VR)-integrable with respect to m, then Nm ˜ f ,p (x) = |f (x)| · Nm,p (x). Proof : Given > 0, there exists a g ∈ S(R) such that f − gNm,p < . By Lemma 6.1, we have Nm ˜ g,p (x) = |g(x)| · Nm,p (x). Also | |g(x)| · Nm,p (x) − |f (x)| · Nm,p (x) | ≤ |g(x) − f (x)| · Nm,p (x) < . Thus |Nm ˜ f ,p−|f (x)|·Nm,p (x)|≤|Nm ˜ f ,p (x)−Nm ˜ g ,p (x)|+| |g(x)|·Nm,p (x)−|f (x)|·Nm,p (x) |≤2. As > 0 was arbitrary, the Theorem follows.


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Lemma 6.5. If f ∈ KX is (VR)-integrable with respect to m and h ∈ S(R), then hf is (VR)-integrable. Proof : Let > 0, p ∈ cs(E), d > h. Choose g ∈ S(R) such that g − f Nm,p < /d. Now gh ∈ S(R) and hf − ghNm,p < , which proves the Lemma. Theorem 6.6. Let f ∈ KX be (VR)-integrable with respect to m. If g ∈ KX is (VR)-integrable with respect to m ˜ f , then gf is (VR)-integrable with respect to m and

(V R) gf dm = (V R) g dm ˜ f. ˜ m ) be such that g−hN Proof : Given p ∈ cs(E) and > 0, let h ∈ S(R < . m ˜ f ,p ˜ m , then Let d > h and choose f1 ∈ S(R) such that f − f1 Nm,p < /d. If V ∈ R

χV dm ˜f = m ˜ f (V ) = (V R) χV f dm and so h dm ˜ f = (V R) hf dm. Now

p (V R) g dm ˜ f − h dm ˜ f ≤ g − hNm˜ f ,p < . If f2 = f − f1 , then hf2 Nm,p ≤

and g − h)f Nm,p = g − hNm˜ f ,p ≤ .

It follows that gf − hf1 Nm,p ≤ . Since hf1 is (VR)-integrable with respect to m, we get that gf is (VR)-integrable with respect to m. Also,

p (V R) f g dm − (V R) hf dm ≤ gf − hf Nm,p ≤ . It follows that

p (V R) f g dm − (V R) g dm ˜ f ≤ ,

which clearly completes the proof. Theorem 6.7. Let f, g ∈ KX be (VR)-integrable with respect to m. If g is bounded, then : (1) g is (VR)-integrable with respect to m ˜ f. (2) gf is (VR)-integrable with respect to m. (3) (V R) gf dm = (V R) g dm ˜ f. The same result holds if we assume that f is bounded. Proof : Assume that g is bounded. In view of the preceding Theorem, we only need to prove (1). By Theorem 5.15, g is τR˜ m -continuous. As g was assumed to be bounded, we get that g is integrable with respet to m ˜ f , which implies that it is (VR)-integrable with respect to the same measure (by Theorem 5.13). Thus (1) holds. In case f is bounded, let d > f and choose h ∈ S(R) such that g − hNm,p < /d. Now g − hNm˜ f ,p = (g − h)f Nm,p < , and so the result follows. Theorem 6.8. Let f ∈ KX be (VR)-integrable with respect to m and let g ∈ KX be m-integrable. Then :

68 22


(1) g is (VR)-integrable with respect to m ˜ f. (2) gf is (VR)-integrable with respect to m. (3) (V R) gf dm = (V R) g dm ˜ f. Proof : Let p ∈ cs(E) and > 0. Since g is m-integrable, there exists a V ∈ R, with mp (V c ) = 0, such that gV = d < ∞. Let g1 = gχV . By the ˜ m ) such that g1 − hN < . For preceding Theorem, there exists an h ∈ S(R m ˜ f ,p c x ∈ V , we have |g(x) − h(x)| · Nm ˜ f,p (x) = |f (x)(g(x) − h(x))| · Nm,p (x) = 0. Thus g − hNm˜ f ,p ≤ . This proves (1) and the result follows. 7. The Completion of (S(R), φτ ) In this section, R will be a separating algebra of subsets of a non-empty set X. We will equip X with the topology τR . As in [ 9 ], we will denote by X (k) the set X equipped with the zero-dimensional topology which has as a base the family of all subsets A of X such that A ∩ Y is clopen in Y for each compact subset Y of X. We will prove that (Cb (X (k) ), βo ) coincides with the completion Fˆ of F = (S(R), φτ ). As F is a polar Hausdorff space, its completion is the space of all linear functionals on F = Mτ (R) which are σ(F , F )-continuous on φτ -equicontinuous subsets of Mτ (R) (see [10 ]). The topology of Fˆ is the one of uniform convergence on the φτ -equicontinuous subsets of Mτ (R). Since φτ is the topology induced on S(R) by βo and since βo and the topology τu of uniform convergence have the same bounded sets, it follows that the strong topology on F is the topology given by the norm m → m. Theorem 7.1. The completion Fˆ of F is an algebraic subspace of the second dual F . The topology of Fˆ is coarser than the topology induced on Fˆ by the norm topology of F . Proof : Let u be a linear functional on Mτ (R) which is σ(F , F )-continuous on φτ -equicontinuous subsets of Mτ (R). Then u is norm-continuous. Indeed, let (mn ) be a sequence in Mτ (R) with mn → 0. The set H = {mn : n ∈ N} is uniformly τ -additive. In fact, let (Vδ ) be a net in R which decreases to the empty set and let > 0. Choose no such that mn < if n > no . If δo is such that |mn |(Vδ ) < for all δ ≥ δo and all n = 1, 2, . . . , no , then |m|(Vδ ) < for all m ∈ H and all δ ≥ δo . In view of Theorem 3.10, H is φτ -equicontinuous. As g dmn → 0 for all g ∈ S(R), it follows that u(mn ) → 0 and so u ∈ F . The last assertion is a consequence of the fact that every φτ equicontinuous subset of Mτ (R) is norm bounded. Let K(X) be the algebra of all τR -clopen subsets of X. Define m ˜ : K(X) → ˜ ∈ Mτ (K(X)). K, m(A) ˜ = χA dm. Then m Lemma 7.2. If H is a uniformly τ -additive subset of Mτ (R), then the set ˜ = {m H ˜ : m ∈ H} is a uniformly τ -additive subset of Mτ (K(X)). Proof : Let (Vδ ) be a net in K(X) which decreases to the empty set. Consider the family F of all A ∈ R which contain some Vδ . Let A1 , A2 ∈ F and let δ1 , δ2 be such that Vδi ⊂ Ai , for i = 1, 2. If δ ≥ δ1 , δ2 , then Vδ ⊂ A = A1 ∩ A2 , which proves that F is downwards directed. Also, F = ∅. Indeed, let x ∈ X and


69 23

c choose Vδ not containig x. There exists a B ∈ S(R such that x ∈ B ⊂ Vδ . Now c Vδ ⊂ A = B and x ∈ / A, which proves that F = ∅. As H is uniformly τ -additive, there exists A ∈ F with |m|(A) < for all m ∈ H. If Vδ is contained in A, then ˜ = |m|(A) < , for all m ∈ H, and the Lemma follows. |m|(V ˜ δ ) ≤ |m|(A)

Theorem 7.3. (Cb (X), βo ) is a topological subspace of Fˆ . Proof : Let f ∈ Cb (X). Without loss of generality we may assume that f ≤ 1. For each m ∈ Mτ (R), the integral f dm exists. Thus f may be considered as a linear functional on Mτ (R) = F . Let H be an absolutely convex φτ -equicontinuous subset of Mτ (R) andlet (mδ ) be a net in H which is σ(F , F )convergent to zero. We will show that f dmδ → 0. As H is φτ -equicontinuous, ˜ is we have that d = supm∈H m < ∞. By the preceding Lemma, the set H a norm-bounded uniformly τ -additive subset of Mτ (K(X)). By [4], Theorem 3.6, given > 0, there exists a compact subset Y of X such that |m|(V ) = |m|(V ˜ ) 0 such that < 1/n and · mp < 1/n. The set Z = Y {x : Nm,p (x) ≥ } is compact. For each y ∈ Z, there exists Vy ∈ S(R) containing y and such that Vy ∩ Z ⊂ {z : |f (z) − f (y)| < }. By the compactness of Z, there are pairwise disjoint W1 , W2 , . . . , WN in R covering Z and such that each Wi is contained in some Vy . Choose zk ∈ Wk and take N gγ = k=1 f (zk )χWk . Then gγ ∈ B. If x ∈ Y , then |f (x) − gγ (x)| ≤ < 1/n and N so f − gγ Y ≤ 1/n. Also, if x ∈ W = k=1 Wk , then |f (x) − gγ (x)| · Nm,p (x) ≤ · mp < 1/n, while for x ∈ / W we have that Nm,p (x) ≤ < 1/n. Thus f − gγ Nm,p ≤ 1/n, which proves our claim. Now the net (gγ ) is in B and converges to f with respect to the topology of uniform convergence on compact subsets of X and so (gγ ) is βo -convergent to f , which implies that u(f ) = lim u(gγ ). On the other hand, (gγ ) is contained in Gm and converges to f in the topology of Gm Thus

u(f ) = lim u(gγ ) = lim gγ dm = (V R) f dm. This completes the proof. Theorem 7.6. Let X be a zero-dimensional Hausdorff space and let ∆ be the family of all pairs (m, p) for which there exists a Hausdorff locally convex space E such that p ∈ cs(E) and m ∈ Mτ (K(X), E), where K(X) is the algebra of all clopen subsets of X. Then the topologies β and βo on Cb (X) coincide with the locally convex topology ρ generated by the seminorms · Nm,p , (m, p) ∈ ∆. Proof : As it is shown in the proof of the preceding Theorem, the space F = S(K(X)) is ρ-dense in Cb (X). Also F is dense in Cb (X) for the topologies β and βo . In view of Theorems 4.15, 4.18 and 4.19, the topologies βo , β and ρ coincide on F . Also, ρ is coarser than βo . Indeed, let (m, p) ∈ ∆ and V = {f ∈ Cb (X) : f Nm,p ≤ 1}. Let r > 0 and choose 0 < < 1/r such that · mp (X) < 1. The set Y = {x : Nm,p (X) ≥ } is compact. Moreover V1 = {f ∈ Cb (X) : f ≤ r, f Y ≤ } is contained in V . In fact, let f ∈ V1 . If x ∈ Y , then |f (x)|·Nm,p (x) ≤ ·mp (X) ≤ 1, while for x ∈ / Y we have |f (x)| · Nm,p (x) ≤ r ≤ 1. Thus f Nm,p ≤ 1, i.e. f ∈ V . This, being true for each r > 0, implies that V is a βo -neighborhood of zero. Now the result follows from Lemma 4.17.


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References [1] J. Aguayo, Vector measures and integral operators, in : Ultrametric Functional Analysis, Cont. Math., vol. 384(2005), 1-13. [2] J. Aguayo and T. E. Gilsdorf, Non-Archimedean vector measures and integral operators, in: p-adic Functional Analysis, Lecture Notes in Pure and Appplied Mathematics, vol 222, Marcel Dekker, New York (2001), 1-11. [3] J. Aguayo, A. K. Katsaras and S. Navarrto, On the dual space for the strict topology β1 and the space M (X) in function spaces, in : Ultrametric Functional Analysis, Cont. Math., vol. 384(2005), 15-37. [4] A. K. Katsaras, The strict topology in non-Archimedean vector-valued function spaces, Proc. Kon. Ned. Akad. Wet. A 87 (2) (1984), 189-201. [5] A. K. Katsaras, Strict topologies in non-Archimedean function spaces, Intern. J. Math. and Math. Sci., 7 (1), (1984), 23-33. [6] A. K. Katsaras, Separable measures and strict topologies on spaces of non-Archimedean valued functions, in : P-adic Numbers in Number Theory, Analytic Geometry and Functional Analysis, edided by S. Caenepeel, Bull. Belgian Math., (2002), 117-139. [7] A. K. Katsaras, Strict topologies and vector measures on non-Archimedean spaces, Cont. Math. vol. 319 (2003), 109-129. [8] A. K. Katsaras, Non-Archimedean integration and strict topologies, Cont. Math., vol. 384 (2005), 111-144 [9] A. K. Katsaras, P-Adic Measures and p-adic spaces of continuous functions, Note di Mat. (to appear in 2009). [10] A. K. Katsaras, The non-Archimedean Grothendieck’s completeness theorem, Bull. Inst. Math. Acad. Sinica 19(1991), 351-354. [11] A. K. Katsaras, P-adic spaces of continuous functions I, Ann. Math. Blaise Pascal 15 (2008), 109-133. [12] A. F. Monna and T. A. Springer, Integration non-archimedienne, Indag. Math. 25, no 4(1963), 634-653. [13] W. H. Schikhof, Locally convex spaces over non-spherically complete fields I, II, Bull. Soc. Math. Belg., Ser. B, 38 (1986), 187-224. [14] A. C. M. van Rooij, Non-Archimedean Functional Analysis, New York and Bassel, Marcel Dekker, 1978. [15] A. C. M. van Rooij and W. H. Schikhof, Non-Archimedean Integration Theory, Indag. Math., 31(1969), 190-199. Department of Mathematics, University of Ioannina, 45110 Ioannina, Greece E-mail address: [email protected]



On the Clifford algebra of orthomodular spaces over Krull valued fields Hans A. Keller and Herminia Ochsenius Abstract. Non-Archimedean orthomodular spaces are counterparts of the classical Hilbert spaces over R or C with which they share the basic property expressed by the Projection theorem. They are all constructed over fields with a non-Archimedean valuation of infinite rank. The geometry of these spaces diverges sharply from Euclidean geometry, which entails that there appear remarkable new properties in the orthogonal groups. In the paper we focus on a prominent example of an orthomodular space E. We show that the peculiar non-Archimedean features allow the construction of an algebra C˜ which is a topological analogue of the classical Clifford algebra. C˜ is obtained by a completion process and its inner automorphisms provide a representation of isometries T : E → E as the composition of an involution with an infinite product of reflections.

1. Introduction Let E be a vector space over a field K with char(K) = 2, endowed with a non-degenerate symmetric bilinear form . , . : E × E → K. Then E is called an orthomodular space if the Projection Theorem is valid: for every orthogonally closed subspace U of E there is an orthogonal projection from E onto U . In infinite dimension this is an extremely strong condition. Classical examples are the real or complex Hilbert spaces. Non-classical examples were discovered in 1980. These new spaces are all constructed as sequence spaces over certain non-Archimedeanly valued, complete fields, the valuations always being of infinite rank. With respect to their analytic and topological properties, they are very similar to their classical counterparts. In particular, the inner product . , . induces a non-Archimedean norm, the space E is complete in the norm topology, and a linear subspace U is topologically closed in E if and only if it is orthogonally closed. However, there arise striking differences of geometric kind. Indeed, in a Hilbert space H each straight line contains a unit vector, hence any two lines are isometric, which means 2000 Mathematics Subject Classification. Primary 46C99, Secondary 15A66. Key words and phrases. valuation, non-Archimedean norm, orthogonal group, Clifford algebra. Research supported by HSLU, Technik & Architektur, Projekt 07-17, and by Fondecyt, Proyecto No 1080194. c Mathematical 0000 (copyright Society holder) c 2010 American

1 73

74 2

HANS A. KELLER AND HERMINIA OCHSENIUS

that all directions in H are equivalent and H is perfectly homogeneous. A nonArchimedean orthomodular space E, in turn, is utmost inhomogeneous inasmuch as two orthogonal lines never contain vectors of the same length. In a previous work (see [6]) we showed that the peculiar geometry of a nonArchimedean orthomodular space E has a strong impact on the group Ω = Ω(E) of all isometries E → E. We showed that Ω contains no elements of finite order besides the obvious involutions. Moreover, we singled out a lattice L consisting of normal subgroups of Ω which is isomorphic to the Boolean algebra of a countably infinite set. Such a lattice has no analogue in Hilbert spaces. The smallest element Ω0 of L can be described as the group of all infinitesimal perturbation of the identity on E. Continuing these previous studies we now have a close look at the group Ω0 and we interrelate it with Clifford algebras. In the classical theory of orthogonal groups the prominant role of Clifford algebras is well-known. If (V, q) is a finite dimensional quadratic spaces then the Clifford algebra C(q) allows to represent isometries of V by inner automophisms of C(q) and consequently to define basic invariants. For a quadratic space (V, q) of infinite dimension, however, the power of the associated Clifford algebra C(q) seems to fade. The reason is that by means of C(q) we can capture only isometries which are products of finitely many reflections, and in infinite dimension these isometries amount only to a small portion of the whole orthogonal group. It is tempting to try to combine the algebraic part with topological methods. In the present paper we show that this idea can be carried out successfully for nonArchimedean orthomodular spaces. To this end we examine a prominant example (E, , ) of such a space. We introduce the notion of a norm-topological Clifford ˜ algebra C(E). As its algebraic analogue, it can be defined by a universal property. ˜ We set up a completion process that actually produces C(E). Our main result ˜ states that by means of inner automorphisms of C(E) we obtain all of Ω0 . Thus every element of Ω0 , i.e. every infinitesimal perturbations, can be represented as infinite product of reflections In sections 2 and 3 we review the construction of the orthomodular space (E, . , . ) and its group of isometries. In section 4 we give a summary on Clifford algebras. ˜ In the main section 5 we describe the construction of C(E) and we establish its interrelation with Ω0 . 2. Construction of an orthomodular space In this section we present the standard construction of orthomodular spaces. For details we refer to [1] and [5]. 2.1. The base field. For i = 1, 2, . . . let Gi = gi be an infinite cyclic group ordered by powers of the generator gi . That is, gir ≥ gim iff r ≥ m. Let G be the direct sum of G1 , G2 , . . . , ordered antilexicographically. Thus G consists of all sequences ∞ g = (g1r1 , . . . , giri , . . . ) ∈ Gi i=1

for which supp(g) := {i ∈ N : ri = 0} is finite, and if k := max supp(g) then g > 1 in the ordering of G iff rk > 0. For n ∈ N we put (1)

gn := (1, . . . , 1, gn , 1, . . . ) ,

ON THE CLIFFORD ALGEBRA OF ORTHOMODULAR SPACES OVER KRULL...

75 3

so that {gn : n ∈ N} is a base of G. In the terminology of [9] this group belongs to the family Γω . Next, we define recursively fields of rational functions by ◦ ◦ ◦

F0 := R, Fn := Fn−1 (Xn ) and weput ∞ F∞ := n=0 Fn .

for n > 1,

There is a uniquely determined Krull valuation | . | : F∞ → G ∪ {0} for which ◦ ◦

| . | is trivial on F0 | Xn | = gn for n = 1, 2, . . . .

To finish the construction of the base field we define K to be the completion of F∞ by means of Cauchy sequences. Notice that the field K (with the extended valuation) is far from being spherically complete. Indeed, this can be seen easily by comparing ˜ the maximal completion of F∞ . It can be described as the field of all K with K, maps f : G → R with well-ordered support supp(f ) := {g ∈ G : f (g) = 0}. Then ˜ is the smallest spherically complete field containing (K, | . |) as a subfield. K ˜ is K significantly larger than K. Indeed, if we define the sequence a0 , a1 , . . . by 1 if n = m! for some m ≥ 0, an := 0 otherwise, then for any fixed r = 1, 2, . . . the series ∞ 1 1 1 am ak =1+ + 2 + 6 + · · · + m! + . . . Xrk! Xr Xr Xr Xr

k=0

˜ but this series (and many others of similar obviously represents an element of K, shape) does not correspond to an element in K, as can be seen by a straightforward argument. Let us point out that the field K can as well be conceived as a non-Archimedeanly ordered, complete field. To this end we order the fields Fn ∼ = Fn−1 (Xn ) recursively by powers of the variable Xn . That is, a polynomial p(Xn ) = a0 + a1 Xn + a2 Xn2 + · · · + as Xns ∈ Fn

where ai ∈ Fn−1 , as = 0,

is positive in the ordering of Fn iff as > 0 in Fn−1 . This induces an ordering on F∞ which can be extended in a unique way to K. This ordering ≤ on K is compatible with the valuation, i.e. 0 < ξ ≤ η ⇒ |ξ| ≤ |η|

for all ξ, η ∈ K.

It follows, in particular, that the order topology coincides with the topology given by the valuation. 2.2. The space. Let E be the vector space of all sequences x = (ξi )i∈N ∈ K N0 for which the series ∞ ξi2 Xi i=1

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converges in the valuation topology. Operations in E are componentwise. Define an inner product . , . : E × E → K by x, y :=

∞

ξi ηi Xi

for x = (ξi )i , y = (ηi )i ∈ E.

i=1

The symmetric bilinear form . , . is positive-definite in the ordering introduced above. As usual we say that x, y ∈ E are orthogonal, x ⊥ y, if x , y = 0, and for a subspace U ⊂ E we define its orthogonal by U ⊥ := {x ∈ E : x ⊥ u for all u ∈ U }. Next, we introduce a non-Archimedean norm . on E. Of course, we should like to have x 2 = | x, x|. So we first have to care about the group of values for the norm we have in mind. Let D be the divisible hull of G.√Consider the subgroups H := {h ∈ D : h2 ∈ G} √ 2 and J := {h ∈ D : h = 1}. Let G √ := H/J. Then G contains G as a subgroup. By√ construction, x2 ∈ G for √ all x ∈ G. The ordering on G can readily be extended to √G by the rule x ≤ y in G iff x2 ≤ y2 in G. For every g ∈ G there is a unique √ x ∈ G for which x2 = g; we write x = g. Now consider the assignment x → x :=

√ | x , x | ∈ G.

Since . , . is positive-definite it follows immediately that . satisfies the requirements of a non-Archimedean norm. The most basic properties of the space (E, . , . ) thus constructed are summarized in the following result. Theorem 2.1. (1) E is complete in the norm topology. (2) A subspace U ⊆ E is topologically closed if and only if it is orthogonally ¯ ⇔ U = U ⊥⊥ . closed, thus U = U (3) The Projection Theorem is valid in (E, . , . ): U ⊆ E, U = U ⊥⊥

⇒

E = U ⊕ U ⊥.

For a proof refer to [1]. Remark 1: (a) Property (3) implies that the ortho-lattice L(E) := {U ⊆ E : U = U ⊥⊥ } satisfies the orthomodular law. This explains the terminology. (b) For finite dimensional spaces the Projection Theorem is simply equivalent to anisotropy. In infinite dimension, however, it is a very strong requirement. All examples known up to present (besides the classical Hilbert spaces) are built by modifications of the above procedure, thus they all rely essentially on topologies. The spaces obtained in this way are often called “form Hilbert space” (see [7]) in order, firstly, to distinguish them from spaces which have no inner product but instead a relation of ortogonality defined in terms of the norm and, secondly, in order to contrast them with (at the moment hypothetical) quadratic spaces which satisfy the Projection theorem but are obtained by essentially different methods.


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2.3. The canonical base. For n = 1, 2, . . . we put en := (0, . . . , 0, 1, 0, . . . , ) where 1 is in position n. Then en ⊥ em for n = m, and en , en = Xn , thus √ en = |Xn | = gn . The set {e1 , e2 , . . . , en , . . . } is a continuous base, called the canonical base, of (E, , ). That is, every vector x ∈ E can be expressed uniquely as x=

∞

ξi ei

i=1

where convergence is in the norm topology. We shall need the following simple facts. Lemma 2.2. Let n, m ∈ N with m > n. (a) Let h ∈ G. If h2 gm < gn then h2 gm < 1/gn . (b) Let λ ∈ K. If λem < en then λem < 1/ en = 1/ |Xn |. Proof: (a) Write h = (h1 , h2 , . . . ) ∈ G. Then (2)

h2 gm = (h21 , . . . , h2m−1 , h2m · gm , h2m+1 , . . . ) < (1, . . . , 1, gn , 1 . . . ) = gn Put k := max{i ∈ N : hi = 1}. Since m > n, and by the definition of the antilexicographic order,(2) is possible only when either k > m and h2k < 1 or k = m and h2m gm < 1. In both cases the crucial component is at the right of place n, so it is clear that the inequality (2) remains valid when the term gn on the right hand side is replaced by g−1 n . (b) This follows immediately from (a).

2.4. Algebraic types. In the study of non-classical orthomodular spaces the concept of types turned out to be a most powerful device. It will play a vital role in our reasonings in section 5 below. Intuitively, the type of an element ξ ∈ K reflects its “order of magnitude modulo squares”. For the present purpose we only introduce algebraic types. Let us mention that there is also a well developed concept of topological types, see [7]. By the algebraic type t(g) of an element g ∈ G we mean the image of g under the epimorphism G → G/G2 . The type of a scalar 0 = α ∈ K is t(α) := t(| α |). For example, the type of Xn is the class of squares in G represented by the element gn given by (1). In particular, t(Xn ) = t(Xm ) for n = m. It is clear that t(λ2 α) = t(α) for all λ = 0. Finally, we define the type of a vector x = 0 by t(x) := t( x , x ). Theorem 2.3. Let (E, . ) be the space constructed in 2.1 and 2.2. (1) Let x, y ∈ E. If x ⊥ y then t(x) = t(y), hence in particular x = y . (2) In any two maximal orthogonal families {ui : i ∈ N} and {vi : i ∈ N} of vectors in E there occur exactly the same types, thus there is a bijection σ : N → N such that t(ui ) = t(vσ(i) ) for all i. The property (2) is an instance of the more general “type condition” developed in [1]. This condition is a key for the study of non-Archimedean orthomodular spaces.

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Remark 2.4. The prototypical orthomodular space constructed in [1] contains a vector e0 with norm e0 = 1. Thre is no such vector in the space (E, . ) we are dealing with here. This slight modification was introduced in order to avoid some complications which require a more detailed study. 3. The group of isometries 3.1. Definitions and examples. Definition A linear operator T : E → E is called an isometry if T (x), T (x) = x, x for all x ∈ E. Clearly an isometry T : E → E preserves the norm of vectors, T (x) = x , thus T is an isomorphism by the norm in the sense of [8]. The converse is not true, as can be shown by simple counterexamples. Definition The group Ω(E) of all isometries T : E → E is called the orthogonal group of the space (E, . , . ). There are a host of isometries. Let us point out the following ones. (a) Each vector u = 0 of E gives rise to an isometry Su : E → E by Su (x) := x − 2

x, u · u, u, u

called the reflection with respect to the hyperplane Hu := K(u)⊥ . Notice that Su (u) = −u and Su (x) = x for x ∈ Hu . It follows that Su is an involution, Su ◦ Su = Su . (b) Let U be a closed linear subspace of E. Then E = U ⊕ U ⊥ by the orthomodular law. Therefore U gives rise to an involution QU ∈ Ω defined by QU (x) = x for x ∈ U, QU (x) = −x for x ∈ U ⊥ . Conversely, let Q ∈ Ω be an involution. Then U := {x ∈ E | Q(x) = x} is a norm-closed subspace of E, hence E = U ⊕ U ⊥ by Theorem 2.1. Clearly U ⊥ is invariant under Q. Let x ∈ U ⊥ and consider y := x + Q(x). Then Q(y) = y, which implies that y ∈ U , thus y = 0 and Q(x) = −x. Hence Q = QU . This means that there is a one-to-one relation between the closed subspaces of E and the involutions in Ω. Remark 2: In contrast to the classical situation of finite dimensional quadratic spaces, there are a lot of isometries which cannot be written as a (finite) product of reflections. It is a challenging problem to describe the subgroup Ωinv of Ω(E) generated by all the involutions of E. We conjecture that Ωinv is a proper subgroup of the whole group Ω. However, a proof is still missing. 3.2. Outstanding normal subgroups. A classical Hilbert space H is perfectly homogeneous. The orthomodular space E under consideration, in contrast, is extremely inhomogeneous, as is put into evidence by Theorem 2.3. In ([6]) we showed that the peculiar geometry of E has a strong impact on the orthogonal group. Indeed, there arise some remarkable normal subgroups which have no counterparts in the classical situation. To define them we must refer to the residue field of K. In fact there is an infinite sequence of residue fields associated to K because the value group is of infinite rank, but for the present purpose we need only the first one.


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To the valued field (K, | . |) there belongs a valuation ring A := {ξ ∈ K : | ξ | ≤ 1} ˆ = A/J. with maximal ideal J := {ξ ∈ K : | ξ | < 1} and therefore a residue field K ˆ ∼ It is readily seen that K R. Let π : A → R be the canonical map. = Now we have the following crucial result. Lemma 3.1. (See [6]) Let T : E → E be an isometry and consider its expression it in the standard base, T (ek ) =

∞

τik ei

where

τik =

i=1

T (ek ) , ei . ei , ei

Then for all k = 1, 2 . . . we have π(τkk ) ∈ {1, −1}. Thus the diagonal elements in the matrix of T are congruent to +1 or −1 modulo infinitesimals. It follows that T (ek ) , ek Λk := T ∈ Ω : π =1 ek , ek is a subgroup of Ω, in fact a subgroup of index 2 and therefore normal in Ω. Next we define

ΛM := Λk for M ⊆ N. k∈M

A detailed study of these normal subgroups ΛM and the lattice they generate can be found in ([6]). Let us now look at ∞

Ω0 := ΛN =

Λk .

k=1

If T ∈ Ω0 then every T (ek ) (for k ∈ N) is congruent to +ek modulo the set of all “infinitesimal” vectors, i.e. all vectors x with x2 ∈ J. Thus Ω0 may be considered as the group of infinitesimal perturbations of the identity I : E → E. Let us point out that Ω0 is a very large subgroup of Ω, in fact, it is of uncountable cardinality. Remark 4: The definition of Ω0 refers to the canonical base {ei : i ∈ N } of the space (E, . ). However, the group Ω0 does not depend on the choice of the base. Indeed, suppose that {ei : i ∈ N } is another continuous base of E. By Theorem 2.3 we may suppose, after renumbering, that t(ei ) = t(ei ) for all i ∈ N0 . Express the isometry T in terms of the new base, say

T (ek ) =

∞

τik ei .

i=1

Then a straightforward computation shows that again π(τkk ) ∈ {−1, 1} and in fact π(τkk ) = π(τkk ) for all k = 1, 2, . . . . 4. The Clifford algebra in finite dimension In this section we review some essential facts of the classical theory of Clifford algebras.

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4.1. Definition and the universal property. Consider a finite-dimensional vector space V over a field F with char(F ) = 2. Recall that a quadratic form q on V is a map q : V → F such that (i) q(ξ · v) = ξ 2 · q(v) for all v ∈ V, ξ ∈ F , (ii) the map Φ : V ×V → F , defined by Φ(u, v) := 12 ·(q(u + v) − q(u) − q(v)), is a symmetric bilinear form. We always suppose that Φ is non-degenerate. The Clifford algebra C(q) = (C(q), +, ◦, 1C ) of the quadratic form q is characterized by the following properties. (CL1) C(q) contains V as a linear subspace. (CL2) C(q) is generated as an algebra by the elements v ∈ V . (CL3) v ◦ v = q(v) · 1C for all v ∈ V . By polarization it follows that (CL4) u ◦ v + v ◦ u = 2 Φ(u, v) · 1C for all u, v ∈ V . In particular, u ⊥ v if and only if u ◦ v = −v ◦ u The Clifford algebra has the following universal property. (UP) If (A, +, ∗) is any algebra over F which contains V as a subspace and such that v∗v = q(v)·1A for all v ∈ V then there exists a unique homomorphism of algebras f : C(q) → A such that f (v) = v for all v ∈ V , that is, the diagram below commutes. id

-

V

id

A

f ? C(q)

It is clear that the Clifford algebra is determined uniquely (up to isomorphism) by the universal property. The existence must be established by some construction. This can be achieved conveniently by means of the tensor algebra. Recall that the tensor algebra of the vector space V consists of the set T (V ) := F ⊕ V ⊕ (V ⊗ V ) ⊕ (V ⊗ V ⊗ V ) ⊕ . . . together with the addition + and the multiplication ⊗ defined in the obvious way. Consider the two-sided ideal J = J (q) ⊂ T (V ) generated by {v ⊗ v − q(v) · 1T (V ) : v ∈ V }. Let C(q) = T (V )/J be the quotient algebra. Now T (V ) contains V as a subspace. The canonical map T (V ) → T (V )/J is injective on V , hence C(q) contains an isomorphic copy of V , which is usually identified with V . The definition of J entails that v ◦ v − q(v) · 1 = 0 in C(q). In order to verify (UP) we observe that the identity id : V → A extends, by the universal property of the tensor algebra, to a homomorphism of algebras f˜ : T (V ) → A. For v ∈ V we have f˜(v ⊗ v − q(v) · 1T (V ) ) = f˜(v) ∗ f˜(v) − q(v) · 1A = v ∗ v − q(v) · 1A = 0. Thus f˜ vanishes on the ideal J and therefore induces a homomorphism f : C(q) → A, as required.


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4.2. Basic properties of C(q). For details we refer to [2]. Let n := dim V and assume again the the bilinear form Φ associated to q is non-degenerate. (1) If n = 1 then dim C(q) = 2 and C(q) is either a field or a direct sum of two copies of F . (2) If n = 2 then dim C(q) = 4 and C(q) is a quaternion algebra. (3) If n is even then C(q) is a central simple algebra. If n is odd then either C(q) is a simple algebra with a two-dimensional center which is a field, or C(q) is a direct sum of two isomorphic central simple algebras. (4) The Clifford algebra C(q) has dimension 2n . 4.3. Isometries of (V, q) and inner automorphisms of C(q). . Consider any vector 0 = u ∈ V ⊂ C. If u is anisotropic, i.e. q(u) = 0, then 1 u ◦ u = q(u) · 1C = 0. Therefore u is invertible in C, in fact u−1 = q(u) · u. Now let x ∈ V . Using (CL4) we find 1 1 · (u ◦ x ◦ u) = · (u ◦ (2 Φ(u, x) · 1C − u ◦ x)) u−1 ◦ x ◦ u = q(u) q(u) 2 Φ(u, x) · u − x. u−1 ◦ x ◦ u = q(u) We see that the conjugation x → u−1 ◦ x ◦ u of C leaves the subspace V invariant, and if Su : V → V is the reflection with respect to the hyperplane F (u)⊥ then u−1 ◦ x ◦ u = −Su (x) for all x ∈ V. Now assume that u1 , . . . , uk ∈ V are anisotropic. Let S1 , S2 , . . . , Sk : V → V be the corresponding reflections of V and consider T := S1 ◦ S2 ◦ · · · ◦ Sk ∈ Ω(V ). Then for all x ∈ V we have T (x) = (−1)k · a−1 ◦ x ◦ a

where a := uk ◦ uk−1 ◦ · · · ◦ u1 ∈ C

By the Cartan-Dieudonne Theorem every isometry T ∈ Ω(V ) can be written as a product of at most n = dim V reflections. If T is a rotation, that is, if det T = +1, then T is a product of an even number of reflections, and we see that T is induced by an inner automorphism of C(q). Assume that n = dim V is odd. Then −Id is a rotation. Writing an arbitrary reflection S as S = (−Id) ◦ (−S) we conclude that every reflection is induced by an inner automorphism of C(q). To summarize, we have the following result, which is the basis for many application of the Clifford algebra. Theorem 4.1. Let (V, q) be a non-degenerate quadratic space over F with dim V = n. (1) If n is odd then every isometry T : V → V is induced by an inner automorphism of C(q). (2) If n is even then an isometry T : V → V is induced by an inner automorphism of C(q) if and only if T is a rotation. 5. The norm-Clifford algebra C˜ Given a quadratic space (V, q) of infinite dimension we can define the Clifford algebra C(q) by the universal property (UP) and establish its existence by means of the tensor algebra as we did before. Anew it turns out that if an isometry T : V → V can represented as a product of an even number of reflections then T

82 10


is induced by an inner automorphism of C(q). However, this construction is not very useful because the whole orthogonal group Ω(V ) is large and the subgroup generated by all reflections covers only a small portion of Ω. With the aim to get a better knowledge on Ω(E) it is natural to try to introduce topologies and to define infinite products of reflections. Unfortunately, it seems that such a thing is out of reach in general when there is no additional information on the space V and the field F . In turn, in the case of the orthomodular space (E, , ) of section 2 the idea can be carried out successfully, as we are going to show now. ˜ Let E0 be the linear subspace of E spanned by the 5.1. Construction of C. canonical base {ek : k ∈ N} (see section 2.3) and let C0 = C(E0 ) be the Clifford algebra (in the algebraic sense) of E0 with the quadratic form q defined by the restriction of the form . . An algebraic base of C0 can be given explicitly. Indeed, let Q be the collection of all finite, strictly increasing sequences of elements from N0 . For Q ∈ Q, say Q = (k1 , k2 , . . . , kr ) where 1 ≤ k1 < k2 < · · · < kr we put eQ := ek1 ◦ ek2 ◦ · · · ◦ ekr In the case Q = ∅ we put eQ := 1. Then the set {eQ : Q ∈ Q} is an algebraic base for the Clifford algebra C0 (see, for example, [2], p.230 ; the arguments given there for finite dimensional spaces carry over without changes). Thus every element a ∈ C0 can be expressed uniquely as a=

where ξQ ∈ K.

ξQ eQ

Q∈Q

Next we define a map ψ from {eQ : Q ∈ Q} into the group (3)

ψ(eQ ) := ek1 · ek2 · · · ekr

√ G of norms by

for Q = (k1 , k2 , . . . , kr ).

Thus ψ(eQ )2 = ek1 2 · ek2 2 · · · ekr 2 = gk1 · gk2 · · · gkr = (1, . . . , 1, gk1 , 1, . . . , 1, gk2 , 1, . . . , 1, gkr , 1, . . . ) , from which it follows that (4)

Q, R ∈ Q, Q = R

⇒

ψ(eQ ) = ψ(eR ).

We introduce a non-Archimedean norm on C0 by (5) a= ξQ eQ → a := max{|ξQ | · ψ(eQ ) : Q ∈ Q}. Q∈Q

Let us have a closer look at the multiplication. Consider Q, R ∈ Q, say Q = (k1 , . . . , kr ), R = ( 1 , . . . , s ), so that (6) eQ = ek1 · · · ekr = | Xk1 | · · · | Xkr |, eR = e1 · · · es = | X1 | · · · | Xs |. (7) Recall that by the basic rules (CL3) and (CL4) we have ek ◦ e = −e ◦ ek for k = and ek ◦ ek = ek , ek · 1C = Xk · 1C . Applying this repeatedly we find Xi · eP (8) eQ ◦ eR = (ek1 ◦ · · · ◦ ekr ) ◦ (e1 ◦ · · · ◦ es ) = ± i


83 11

where the product is over all indices i in {k1 , . . . , kr } ∩ { 1 , . . . , s } and P is the ascending sequence consisting of all indices i which belong either to {k1 , . . . , kr } or to { 1 , . . . , s } but not to both. ¿From (6), (7), and (8) we infer eQ ◦ eR = eQ · eR .

(9) Now consider a=

ξQ eQ , b =

Q∈Q

ηR eR ∈ C0 .

R∈Q

By (4) and (5) there exist uniquely determined elements Q0 , R0 ∈ Q such that a = |ξQ0 | · eQ0

and

b = |ηR0 | · eR0 .

Using (9) we deduce that a ◦ b = a· b

for all a, b ∈ C0 .

Thus in particular, (C0 , . ) is a topological algebra. Now let C˜ be the completion of C0 and extend the norm on C0 to a norm . : C˜ → √ G ∪ {0} by continuity. Then clearly ˜ (10) a ˜ ◦ ˜b = a ˜ · ˜b for all a ˜, ˜b ∈ C. Notice that E0 is topologically dense in the space E, thus C˜ contains E as a linear subspace. Definition: The algebra C˜ is called the norm-Clifford algebra associated to the orthomodular space E. Remark 5: (10) entails that C˜ has no divisors of zero. 5.2. The main result. We are now able to establish the interrelation between C˜ and the group of isometries. We need the following technical result. Lemma 5.1. Let R : E → E be an isometry belonging to Ω0 . Suppose that, for some n ∈ N, we have R(ek ) = ek for all k ≤ n − 1. Then there exists an element c ∈ C˜ such that (a) c−1 ◦ ek ◦ c = ek for 1 ≤ k ≤ n − 1 (b) c−1 ◦ en ◦ c = R(en ). (c) c = 1 + c∗ with c∗ < |X1n | . (d) The isometry x → c−1 ◦ x ◦ c induced on E by c belongs to Ω0 . Proof: Put b := en ◦ (en + R(en )). We first show that (a) and (b) are satisfied for b in place of c. Notice that R(en ) = −en as R ∈ Ω0 , thus b = 0. The vectors ek with k ≤ n − 1 are orthogonal to both en and en + R(en ), so b−1 ◦ ek ◦ b = ek . Thus (a) is satisfied. To verify (b) we compute b−1 ◦ en ◦ b = (en + R(en ))−1 ◦ e−1 n ◦ en ◦ en ◦ (en + R(en )) −1

= (en + R(en ))

◦ en ◦ (en + R(en ))

= −Su (en ) where Su is the reflection with respect to u := en + R(en ). Now by definition Su (en ) = en − 2

en , u ·u u, u

84 12


Since R is an isometry we find en , en + en , R(en ) en , u = u, u en , en + 2en , R(en ) + R(en ), R(en ) 1 en , en + en , R(en ) = . = en , en + 2en , R(en ) + en , en 2 Hence

1 · (en + R(en )) = −R(en ) 2 −1 and consequently b ◦ en ◦ b = R(en ), as claimed. Clearly the conditions (a), (b) remain valid when b is replaced by a multiple λ · b with 0 = λ ∈ K. We show that (c) can be satisfied when the scaling factor λ ∈ K is chosen properly. To this end we decompose Su (en ) = en − 2 ·

R(en ) = η · en + w

with w ⊥ en .

Recall that R ∈ Ω0 , which means that π(η) = 1 (where π is the epimorphism from ˆ ∼ the valuation ring A onto the residue field K = R). Thus | η| = 1 and |1 − η| < 1. Hence η en = |η| · en = en . Moreover, since w ⊥ en we have w = en by Theorem 2.3. Since R(en ) = η en + w = en we conclude that w < en . Now R(en ) ⊥ ek for k = 0, . . . n − 1. Thus w=

∞

ξi ei

i=n+1

and for some m ≥ n + 1 we have w = max{ ξi ei : i ≥ n + 1} = ξm em It follows that w = ξm em < en , from which we deduce by Lemma 2.2 that 1 . w < |Xn | We scale the above element b by c :=

1 (1+η)·Xn ,

thus we put

1 · en ◦ (en + R(en )) (1 + η) · Xn

Then 1 · en ◦ (en + η en + w) (1 + η) · Xn 1 · [(1 + η)(en ◦ en ) + en ◦ w] = (1 + η) · Xn 1 · en ◦ w =1+ (1 + η) · Xn

c=

and for c∗ := c∗ =

1 (1+η)·Xn

· en ◦ w we find

1 1 1 1 · en · w < · | Xn | · , = | 1 + η| · | Xn | 1 · | Xn | | Xn | | Xn |

as claimed. It remains to prove that the isometry P : E → E, defined by x → c−1 ◦ x ◦ c, belongs to Ω0 . Let U := span{en , R(en )} = span{en , w}. Then P (x) = x for


85 13

all x ∈ U ⊥ because P is a product of two reflections with respect to the vectors en , en + R(en ) ∈ U . This means that we have only to examine the action of P on the plane U (cf. Remark 4). Write P (en ) = ηen + w,

P (w) = ξen + κw.

We want to show that π(κ) = +1. We know that | η| = 1 and w < en . Moreover, ξen ⊥ κw, hence ξen = κw by Theorem 2.3,(1). From w = R(w) = max{ ξen , κw} we deduce that ξen < w = κw, thus |ξ| · en < w < en , hence | ξ| < 1. Hence π(ξ) = 0. The restriction of the isometry P to U is a rotation, and the determinant, computed in the base {en , w}, is D = η κ−ξ. Hence 1 = π(D) = π(η) π(κ)−π(ξ) = +1 · π(κ) − 0, from which we obtain π(κ) = +1, as required. 2 Now we state the main result Theorem 5.2. Let (E, , ) be the orthomodular space constructed in section 2 and C˜ the corresponding norm Clifford algebra. Every isometry T : E → E which ˜ belongs to the subgroup Ω0 is induced by an inner automorphism of C. Proof : Step 1 We construct recursively elements c1 , c2 , c3 , . . . in C˜ along with a sequence of isometries R1 , R2 , R3 , . . . with the properties (a) For all n ≥ 1 we have (11)

c−1 n ◦ ei ◦ cn = ei

(12)

c−1 n

for 1 ≤ i < n

◦ en ◦ cn = Rn (en )

(b) The isometry E → E induced by the inner automorphism x → c−1 n ◦ x ◦ cn belongs to the subgroup Ω0 . (c) cn = 1 + c∗n with c∗n < | X1n | . To start the construction we apply Lemma 5.1 with n = 1 and R1 := T in place of R. This yields c1 . In the general recursive step we define Rn by (13)

Rn (x) := cn−1 ◦ Rn−1 (x) ◦ c−1 n−1 .

Then Rn is in Ω0 by virtue of (b). We obtain an element cn satisfying (a), (b) and (c) by applying Lemma 5.1 with Rn in place of R. Step 2 Put an := cn ◦ an−1 (= cn ◦ cn−1 ◦ · · · ◦ c1 ). We claim that for all n = 1, 2, . . . we have (14)

a−1 n ◦ ek ◦ an = T (ek )

for all

k ≤ n.

We prove (14) by induction on n. In the case n = 1 the claim is obvious, so suppose n ≥ 2. Consider first k < n. Then −1 −1 −1 a−1 n ◦ ek ◦ an = an−1 ◦ cn ◦ ek ◦ cn−1 ◦ an−1 = an−1 ◦ ek ◦ an−1 = T (ek )

which establishes (14) in the present case. Second, to verify (14) in the remaining case k = n we observe that Rn (x) = an−1 ◦ T (x) ◦ a−1 n−1 , as can be deduced readily from (13). Hence a−1 n−1 ◦ Rn (x) ◦ an−1 = T (x)

86 14


Using (12) we obtain −1 −1 −1 a−1 n ◦ en ◦ an = an−1 ◦ cn ◦ en ◦ cn−1 ◦ an−1 = an−1 ◦ Rn (en ) ◦ an−1 = T (en ).

Step 3 We claim that a1 , a2 , a3 , . . . is a Cauchy sequence. In fact, property (d) entails that ck = 1 for all k, hence an = 1 for all n. Moreover, an+1 − an = (1 + c∗n+1 ) ◦ an − an = c∗n+1 ◦ an = c∗n+1 · an 1 = c∗n+1 < | Xn | from which the claim follows. Step 4 We put a := lim an . n→∞

Then a = limn→∞ an = 1, so in particular a = 0. For the inner automorphism x → a−1 ◦ x ◦ a we obtain, by using (14), a−1 ◦ ek ◦ a = lim a−1 n ◦ ek ◦ an = T (ek ), n→∞

hence T is induced by the inner automorphism of C˜ defined by a. The proof is complete. 2 Combined with the results of [6] we obtain the following representation of isometries. Corollary 5.3. Every isometry T : E → E can be represented as a product of an involution composed with an isometry induced by an inner automorphism of ˜ C. References [1] Gross, H. and K¨ unzi, U.M. On a class of orthomodular quadratic spaces. L’ Enseignement Math., 31 (1985), 187-212. [2] N. Jacobson, Basic Algebra II, W.H.Freeman and Co., San Francisco, 1980. [3] H. Keller and H. Ochsenius, A Spectral Theorem for Matrices over Fields of Power Series, Ann. Math. Blaise Pascal, Vol. 2, No. 1 (1995), 169-179. [4] H. Keller and H. Ochsenius, On the Geometry of Orthomodular Spaces over Fields of Power Series, Int. Journal of Theor. Physics, Vol. 37, No. 1 (1998), 85-92. [5] H. Keller and H. Ochsenius, Bounded operator on non-archimedian orthomodular spaces, Math. Slovaca 45 (1995), 413-434. [6] H. Keller and H. Ochsenius, The Orthogonal group of a Form Hilbert Space , Bulletin of the Belgian Mathematical Society. Vol 14, Nr. 5, (2007), 937-946. [7] H. Ochsenius, Hilbert-like spaces over Krull Valued Fields. Contemporary Mathematics Vol. 319, (2003), 227-238, 2003. [8] H. Ochsenius ans W. Schikhof, Linear homomorphisms of non-classical Hilbert spaces, Indag. Math., N.S. 10 , (1999), 601 - 613. [9] E. Olivos, A family of totally ordered groups with special properties. Ann. Math. Blaise Pascal, Vol. 12, (1995), 79-90. [10] P. Ribenboim, Th´ eorie des Valuations, Les Presses de l’Universit´ e de Montreal, 1964. [11] O.F.G. Schilling, The Theory of Valuations, Amer. Math. Soc. Surveys, Providence, 1950.


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Hochschule Luzern, Technik & Architektur, CH-6048 Horw, Switzerland E-mail address: [email protected] ´ticas, Pontificia Universidad Cato ´ lica de Chile, Casilla 306, Facultad de Matema Correo 22, Chile E-mail address: [email protected]



Divergence and convergence of conjugacies in non-Archimedean dynamics Karl-Olof Lindahl Abstract. We continue the study in [21] of the linearizability near an indifferent fixed point of a power series f , defined over a field of prime characteristic p. It is known since the work of Herman and Yoccoz [13] in 1981 that Siegel’s linearization theorem [27] is true also for non-Archimedean fields. However, they also showed that the condition in Siegel’s theorem is ‘usually’ not satisfied over fields of prime characteristic. Indeed, as proven in [21], there exist power series f such that the associated conjugacy function diverges. We prove that if the degrees of the monomials of a power series f are divisible by p, then f is analytically linearizable. We find a lower (sometimes the best) bound of the size of the corresponding linearization disc. In the cases where we find the exact size of the linearization disc, we show, using the Weierstrass degree of the conjugacy, that f has an indifferent periodic point on the boundary. We also give a class of polynomials containing a monomial of degree prime to p, such that the conjugacy diverges.

The study of complex dynamical systems of iterated analytic functions begins with the description of the local behavior near fixed points, see [9, 3, 24]. Recall that, given a complete valued field K, a power series f ∈ K[[x]] of the form f (z) = λz + a2 z 2 + a3 z 3 . . . ,

|λ| = 1,

is said to be analytically linearizable at the indifferent fixed point at the origin if there is a convergent power series solution g to the following form of the Schr¨ oder functional equation(SFE) (0.1)

g ◦ f (x) = λg(x),

λ = f (0),

which conjugates f to its linear part. The coefficients of the formal solution g of (0.1) must satisfy a recurrence relation of the form bk =

1 Ck (b1 , . . . , bk−1 ). λ(1 − λk−1 )

If λ is close to a root of unity, the convergence of g then generates a delicate problem of small divisors. In 1942 Siegel proved in his celebrated paper [27] that 2000 Mathematics Subject Classification. Primary 32P05, 32H50, 37F50, 11R58. Key words and phrases. dynamical system, linearization, non-Archimedean field. This is a revised version of the preprint 04015 March 2004, MSI, V¨ axj¨ o University, Sweden. c Mathematical 0000 (copyright Society holder) c 2010 American

1 89

90 2

KARL-OLOF LINDAHL

the condition (0.2)

|1 − λn | ≥ Cn−β

for some real numbers C, β > 0,

on λ is sufficient for convergence in the complex field case. Later, Brjuno [8] proved that the weaker condition ∞ (0.3) − 2−k log inf |1 − λn | < +∞, k=0

1≤n≤2k+1 −1

is sufficient. In fact, for quadratic polynomials, the Brjuno condition is not only sufficient but also necessary as shown by Yoccoz [30]. Meanwhile, since the work of Herman and Yoccoz in 1981 [13], there has been an increasing interest in the non-Archimedean (or ultrametric) analogue of complex dynamics, see e.g. [1, 2, 4, 5, 6, 14, 15, 16, 18, 19, 21, 22, 23, 25, 26, 28]. Herman and Yoccoz proved that Siegel’s theorem, and its multi-dimensional counter part due to Zehnder [31], is true also for non-Archimedean fields. Moreover, they also showed that in dimension one, the Siegel condition is always satisfied over nonArchimedean fields of characteristic zero (e.g. p-adic fields). On the other hand they also showed that in the multi-dimensional p-adic case there exist multipliers that does not satisfy the corresponding Siegel condition, such that the conjugacy diverges. See Viegue’s thesis [29] for recent results on the multi-dimensional p-adic case. However, for complete valued fields of prime characteristic p > 0, which are necessarily non-Archimedean, there is a problem of small divisors also in dimension one. The Siegel condition, and even the weaker Brjuno condition, is only satisfied if λ is trivial, that is, that |1 − λn | = 1 ∀n ≥ 1. If λ is non-trivial (e.g. in locally compact fields all λ are non-trivial), then λ generates a problem of small divisors. One might therefore conjecture, as Herman [12], that for a locally compact, complete valued field of prime characteristics, the formal conjugacy ‘usually’ diverges, even for polynomials of one variable. Indeed, it was proven in [21] that in characteristic p > 0, like in complex dynamics, the formal solution may diverge also in the onedimensional case. On the other hand, in [21] it was also proven that the conjugacy may still converge due to considerable cancellation of small divisor terms. The main theorem of [21] stated that quadratic polynomials with non-trivial multipliers are linearizable if and only if the characteristic of the ground field char K = 2. In the present paper we present a new class of polynomials that yield divergence. We also note that the conjugacy converges for all power series f ∈ K[[x]], whoose monomials are all of degree divisible by char K = p. Furthermore, in case of of convergence, we estimate the radius of convergence for the corresponding semidisc, i.e. the maximal disc V such that the semi-conjugacy (0.1) holds for all x ∈ V , and the linearization disc 1 ∆, i.e. the maximal disc U , about the origin, such that the full conjugacy g ◦ f ◦ g −1 (x) = λx, holds for all x ∈ U . We also give sufficient conditions, related to the Weierstrass degree of the conjugacy, there being a periodic point on the boundary of the linearization disc. The first non-Archimedean results in this direction were obtained by Arrowsmith and Vivaldi [2] who showed that p-adic power functions may have indifferent periodic points on the boundary. In fact, we prove the following theorem, see Lemma 5.1. 1Here we use the term ‘linearization disc’ rather than ‘Siegel disc’, because in nonArchimedean dynamics the Siegel disc is often refered to as the larger maximal disc on which f is one-to-one.

DIVERGENCE AND CONVERGENCE OF CONJUGACIES IN NON-ARCHIMEDEAN...

91 3

Theorem 0.1. Let K be a complete algebraically closed non-Archimedean field. Let f ∈ K[[x]] have a linearization disc ∆ about an indifferent fixed point. Suppose that ∆ is rational open, and that the radius of the corresponding semi-disc of f is strictly greater than that of ∆, then f has an indifferent periodic point on the boundary of ∆. This theorem and other results of the present paper, stated below, support the idea that the presence of indifferent periodic points on the boundary of a linearization disc about an indifferent fixed point is typical in the non-Archimedean setting. For a more thorough treatment of the problem and its relation to earlier works on non-Archimedean and complex dynamics the reader is referred to [21]. Estimates of p-adic linearization discs were obtained in [20, 22]. 1. Summary of results 1.1. Divergence and convergence. In the present paper we find a new class of polynomials that yield divergence. Theorem 1.1. Let char K = p > 0 and let λ ∈ K, |λ| = 1. Then, polynomials of the form f (x) = λx + ap+1 xp+1 ∈ K[x], ap+1 = 0, are not analytically linearizable at the fixed point at the origin if |1 − λ| < 1. On the other hand we also prove convergence for all power series f whose monomials are all of degree divisible by charK = p. Theorem 1.2. Let char K = p > 0 and let λ ∈ K, |λ| = 1, but not a root of unity. Then, convergent power series of the form f (x) = λx + a i xi , p|i

are linearizable at the origin. These results indicate that the convergence depends mutually on the powers of the monomials of f and the characteristic p of K, ‘good’ powers for convergence being those divisible by p, ‘bad’ powers being those prime to p. However, the blend of prime and co-prime powers may sometimes yield convergence, sometimes not, at least for non-polynomial power series as shown in [21]. However, there might be a sharp distinction for polynomials: Open problem 1.3. Let K be of characteristic p > 0. Is there a polynomial of the form f (x) = λx+O(x2 ) ∈ K[x], with λ not a root of unity satisfying |1−λn | < 1 for some n ≥ 1, and containing a monomial of degree prime to p, such that the formal conjugacy g converges? 1.2. Estimates of linearization discs and periodic points. Let K be a field of prime characteristic p. Let λ ∈ K, not a root of unity, be such that the integer (1.1)

m = m(λ) = min{n ∈ Z : n ≥ 1, |1 − λn | < 1},

exists. The case in which such an m does not exist was treated in [21]; it was shown that if |1 − λn | = 1 for all n ≥ 1, then the linearization disc of a power series f (x) = λx + a2 x2 + a3 x3 + . . . ,

92 4

KARL-OLOF LINDAHL

is either the closed or open disc of radius 1/a where a = supi≥2 |ai |1/(i−1) . Note that, by Lemma 2.2 below, m is not divisible by p. Given λ and hence m, the integer k is defined by (1.2)

k = k (λ) = min{k ∈ Z : k ≥ 1, p|k, m|k − 1}.

Let a > 0 be a real number. We will associate with the pair (λ, a), the family of power series ⎧ ⎫ ⎨ ⎬ p ai xi ∈ K[[x]] : a = sup |ai |1/(i−1) , (1.3) Fλ,a (K) = λx + ⎩ ⎭ i≥2 p|i

and the real numbers 1

(1.4)

ρ = ρ(λ, a) =

|1 − λm | mp , a

and 1

(1.5)

|1 − λm | k −1 , σ = σ(λ, a) = a

respectively. p As stated in Theorem 1.2, power series in the family Fλ,a (K) are linearzable at the origin. Given f , the corresponding conjugacy function g, will be defined as the unique power series solution to the Schr¨ oder functional equation (0.1), with g(0) = 0 and g (0) = 1. In Section 4 we use the ansatz of a power series solution to the Schr¨ oder functional equation, to obtain estimates of the coefficients of g, and its radius of convergence. Moreover, applying a result of Benedetto (Proposition 2.5 below), we estimate the radius of convergence for the inverse g −1 . The main result can be stated in the following way. p Theorem 1.4. Let f ∈ Fλ,a (K). Then, the semi-conjugacy g ◦ f (x) = λg(x) holds on the open disc Dρ (0). Moreover, the full conjugacy g ◦ f ◦ g −1 (x) = λx holds on Dσ (0).

Under further assumptions on f , the linearization disc may contain the larger disc Dρ (0). p (K) be of the form Theorem 1.5. Let f ∈ Fλ,a f (x) = λx + a i xi , i≥i0

for some integer i0 > k . Then, the full conjugacy g ◦ f ◦ g −1 (x) = λx holds on a disc larger than or equal to Dρ (0) or the closed disc Dρ (0), depending on whether g converges on Dρ (0) or not. Note that the estimate of the linearization disc in Theorem 1.2 is maximal

the completion of the algebraic closure of K, quadratic in the sense that in K, polynomials have a fixed point on the sphere Sσ (0) if m(λ) = 1, breaking the conjugacy there. In fact, the estimate is maximal in a broader sense, according to the following theorem.


93 5

p Theorem 1.6. Let f ∈ Fλ,a (K). Suppose a = |ak |1/(k −1) and |ai | < ai−1 for

we all i < k . Then, Dσ (0) is the linearization disc of f about the origin. In K

on have deg(g, Dσ (0)) = k . Furthermore, f has an indifferent periodic point in K κ the sphere Sσ (0) of period κ ≤ k , with multiplier λ .

Here, deg(g, D) denotes the Weierstrass degree of g on the disc D, as defined in Section 5. The Weierstrass degree is the same as the notion of degree as ’the number of pre-images of a given point, counting multiplicity’. Since we assume that

g(0) = 0 and g (0) = 1, deg(g, Dσ (0)) = k means that in the algebraic closure K, g maps the disc Dσ (0) onto itself exactly d-to-1, counting multiplicity. The result in Theorem 1.6 is based on Lemma 5.1 that shows that if there is a shift of the value of the Weierstrass degree from 1 to d > 1, of the conjugacy function on a sphere S, then there is an indifferent periodic point of period κ ≤ d, on the sphere S. 2. Preliminaries Throughout this paper K is a field of characteristic p > 0, complete with respect to a nontrivial absolute value |·|K . That is, |·|K is a multiplicative function from K to the nonnegative real numbers with |x|K = 0 precisely when x = 0, and nontrivial in the sense that it is not identically 1 on K ∗ , the set of all nonzero elements in K. If a field L is equipped with an absolute value, we say that L is a valued field. In fact, all valued fields of strictly positive characteristic are non-Archimedean. In what follows, we often use the shorter notation | · | instead of | · |K . Recall that a non-Archimedean (or ultrametric) field is a field K equipped with a non-trivial absolute value | · |, satisfying the following strong or ultrametric triangle inequality: (2.1)

|x + y| ≤ max[|x|, |y|],

for all x, y ∈ K.

One useful consequence of ultrametricity is that for any x, y ∈ K with |x| = |y|, the inequality (2.1) becomes an equality. In other words, if x, y ∈ K with |x| < |y|, then |x + y| = |y|. For a field K with absolute value | · | we define the value group as the image (2.2)

|K ∗ | = {|x| : x ∈ K ∗ }.

Note that |K ∗ | is a multiplicative subgroup of the positive real numbers. We will also consider the full image |K| = |K ∗ | ∪ {0}. The absolute value | · | is said to be discrete if the value group is cyclic, that is if there is is an element π ∈ K such that |K ∗ | = {|π|n : n ∈ Z}. The absolute value | · | can be extended to an absolute

the completion of the value on the algebraic closure of K. We shall denote by K

is algebraically closed algebraic closure of K with respect to | · |. The fact that K ∗

and that | · | is nontrivial forces the value group |K |K to be dense on the positive real line. In particular, | · |K cannot be discrete. Standard examples of non-Archimedean fields include the p-adics, see for example [17], and various function fields, see for example [10]. The p-adics include the p-adic integers and their extensions. These fields are all of characteristic zero. The most important function fields include fields of formal Laurent series over various fields. These can be of any characteristic.

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KARL-OLOF LINDAHL

Example 2.1. Let F be a field of characteristic p > 0, and fix 0 < < 1. Let K = F((T )) be the field of all formal Laurent series in variable T , and with coefficients in the field F. Then K is also of characteristic p. An element x ∈ K is of the form (2.3) x= xj T j , xj0 = 0, xj ∈ F, j≥j0

for some integer j0 ∈ Z. This field is complete with respect to the absolute value for which (2.4) | xj T j | = j0 . j≥j0

Note that j0 is the order of the zero (or negative the order of the pole) of x at T = 0. Let us also note that | · | is the trivial valuation on F, the subfield consisting of all constant power series in K. In this case |K ∗ | is discrete and consists of all nonzero

∗| integer powers of . The value group of the completion of the algebraic closure |K is not discrete and consists of all nonzero rational powers of . Moreover, K can be viewed as the completion of the field of rational functions over F with respect to the absolute value (2.4) (see, e.g. [10]). Given an element x ∈ K and real number r > 0 we denote by Dr (x) the open disc of radius r about x, by Dr (x) the closed disc, and by Sr (x) the sphere of radius r about x. If r ∈ |K ∗ | (that is if r is actually the absolute value of some nonzero element of K), we say that Dr (x), D r (x), and Sr (x) are rational. Note that Sr (x) is non-empty if and only if it is rational. If r ∈ / |K ∗ |, then we will call Dr (x) = Dr (x) an irrational disc. In particular, if a ∈ K and r = |a|s for some rational number s ∈ Q, then Dr (x) and Dr (x) are rational considered as discs in

However, they may be irrational considered as discs in K. Note that all discs K. are both open and closed as topological sets, because of ultrametricity. However, as we will see in Section 2.3 below, power series distinguish between rational open, rational closed, and irrational discs. Again by ultrametricity, any point of a disc can be considered its center. In other words, if b ∈ Dr (a), then Dr (a) = Dr (b); the analogous statement is also true for closed discs. In particular, if two discs have nonempty intersection, then they are concentric, and therefore one must contain the other. The open and closed unit discs, D1 (0) and D1 (0), respectively play a fundamental role in non-Archimedean analysis, because of their algebraic properties. In fact, due to the strong triangle inequality (2.1), D1 (0) is a ring and D1 (0) is the unique maximal ideal in D1 (0). The corresponding quotient field, k = D1 (0)/D1 (0) is called the residue field of K. The residue field k will also be of characteristic p. Hence we always have k ⊇ Fp . The absolute value | · | is trivial on k. For x ∈ D1 (0), we will denote by x the reduction of x modulo D1 (0). 2.1. The formal solution. The coefficients of the formal solution of (0.1) must satisfy the recurrence relation l! 1 k aα1 · ... · aα bl ( k ) λ(1 − λk−1 ) α1 ! · ... · αk ! 1 k−1

(2.5)

bk =

l=1


95 7

where α1 , α2 , . . . , αk are nonnegative integer solutions of ⎧ ⎨ α1 + ... + αk = l, α1 + 2α2 ... + kαk = k, (2.6) ⎩ 1 ≤ l ≤ k − 1. The convergence of g will depend mutually on the denominators |1 − λk−1 | and the factorial terms in (2.5). In view of Lemma 2.2, the denominator is small if k − 1 is divisible by m and a large power of the characteristic p. In fact, the conjugacy may diverge as in Theorem 3.6 below. On the other hand, the factorial term l! α1 ! · ... · αk ! is always an integer and hence of modulus zero or one, depending on whether it is divisible by p or not. Accordingly, factorial terms may extinguish small divisor terms as in Theorem 4.6 below. 2.2. Arithmetic of the multiplier. Let λ ∈ S1 (0), be an element in the unit sphere. The geometry of the unit sphere and the roots of unity in K was discussed in [21]. We are concerned with calculating the distance |1 − λn |,

for n = 1, 2, . . . .

Note that if x, y ∈ D1 (0), then |x − y| < 1 if and only if the reductions x, y belong to the same residue class. Consequently, (2.7)

|1 − λn | < 1

⇐⇒

n

λ − 1 = 0 in k.

Hence, the behavior of 1 − λn falls into one of two categories, depending on whether the reduction of λ is a root of unity or not. More precisely we have the following lemma that was proven in [21]. Lemma 2.2 (Lemma 3.2 [21]). Let char K = p > 0 and let k be the residue field of K. Let Γ(k) be the set of roots of unity in k. Suppose λ ∈ S1 (0). Then, (1) λ ∈ / Γ(k) ⇐⇒ |1 − λn | = 1 for all integers n ≥ 1. (2) If λ ∈ Γ(k), then the integer m = min{n ∈ Z : n ≥ 1, |1 − λn | < 1} exists. Moreover, p m and 1, if m n, (2.8) |1 − λn | = m pj |1 − λ | , if n = mapj , p a. Note that category 2 in Lemma 2.2 is always non-empty since k ⊇ Fp . Moreover, if k ⊆ Fp , then Γ(K) = k∗ and all λ ∈ S1 (0) falls into category 2. Consequently category 1 is empty in this case. This happens for example when K is locally compact, see e.g. [10]. Proposition 2.3. Let K be a non-Archimedean field with absolute value | · |. Then K is locally compact (w.r.t. | · |) if and only if all three of the following conditions are satisfied: (i) K is complete, (ii) | · | is discrete, and (iii) the residue field is finite. On the other hand if K is algebraically closed, then k is infinite and K cannot be locally compact. In this case k ⊇ Fp . We shall only consider the case in which λ belongs to category 2. The other case was treated in [21].

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KARL-OLOF LINDAHL

2.3. Mapping properties. Let K be a complete non-Archimedean field. Let h be a power series over K of the form h(x) =

∞

ck (x − α)k ,

ck ∈ K.

k=0

Then h converges on the open disc DRh (α) of radius Rh =

(2.9)

1 , lim sup |ck |1/k

and diverges outside the closed disc DRh (α). The power series h converges on the sphere SRh (α) if and only if lim |ck |Rhk = 0. k→∞

The following proposition is useful to estimate the size of a linearization disc, i.e. the maximal disc on which the full conjugacy, g ◦ f ◦ g −1 = λx, holds. 2.4 (Lemma 2.2 [5]). Let K be algebraically closed. Let h(x) = ∞Proposition k c (x − x ) be a nonzero power series over K which converges on a rational k 0 k=0 closed disc U = D R (x0 ), and let 0 < r ≤ R. Let V = Dr (x0 ) and V = Dr (x0 ). Then s

= max{|ck |r k : k ≥ 0},

d = max{k ≥ 0 : |ck |r k = s},

d

and

= min{k ≥ 0 : |ck |r = s} k

are all attained and finite. Furthermore, a. s ≥ |f (x0 )| · r. b. if 0 ∈ f (V ), then f maps V onto Ds (0) exactly d-to-1 (counting multiplicity). c. if 0 ∈ f (V ), then f maps V onto Ds (0) exactly d -to-1 (counting multiplicity). Benedetto’s proof uses the Weierstrass Preparation Theorem [7, 11, 17]. We will be interested in the special case in that c0 = x0 = 0. In this case, we have the following proposition. Proposition 2.5. Let K be an algebraically closed complete non-Archimedean ∞ field and let h(x) = k=1 ck xk be a power series over K. 1. Suppose that h converges on the rational closed disc DR (0). Let 0 < r ≤ R and suppose that |ck |r k ≤ |c1 |r for all k ≥ 2. Then h maps the open disc Dr (0) one-to-one onto D|c1 |r (0). Furthermore, if d = max{k ≥ 1 : |ck |r k = |c1 |r}, then, h maps the closed disc Dr (0) onto D|c1 |r (0) exactly d-to-1 (counting multiplicity). 2. Suppose that h converges on the rational open disc DR (0) (but not necessarily on the sphere SR (0)). Let 0 < r ≤ R and suppose that |ck |r k ≤ |c1 |r

for all k ≥ 2.

Then h maps Dr (0) one-to-one onto D|c1 |r (0).


97 9

Lemma 2.6. Let K be an algebraically closed complete non-Archimedean field. i Let f (x) = λx+ ∞ i=2 ai x ∈ K[[x]], |λ| = 1, be convergent on some non-empty disc. Then, the real number a = supi≥2 |ai |1/(i−1) exists and |ai | ≤ ai−1 for all i ≥ 2. Furthermore, Rf ≥ 1/a and f : D1/a (0) → D1/a (0) is bijective. If |ai | < ai−1 for all i ≥ 2 and f converges on the closed disc D1/a (0), then f : D1/a (0) → D1/a (0) is bijective. Finally, f cannot be bijective on a (rational) disc greater than D1/a (0). Proof. As f is convergent, we have sup |ai |1/i < ∞ and hence we have that sup |ai |1/(i−1) < ∞; 2

i/(i−1) 1/(i−1) 1/(i−1) 1/i 1/i sup |ai | ≤ sup |ai | + sup |ai | ≤ 1 + sup |ai | .

|ai |≤1

|ai |>1

1/(i−1) i−1

|ai |>1

. Moreover, Rf = (lim sup |ai |1/i )−1 ≥ 1/a. Clearly, |ai | ≤ supi≥2 |ai | That f : D1/a (0) → D1/a (0) is bijective follows from the second statement in Proposition 2.5. Remark 2.7. Proposition 2.5 and Lemma 2.6 also hold when K is not algebraically closed with the modification that the mappings h : Dr (0) → D|c1 |r (0) and f : D1/a (0) → D1/a (0) are one-to-one but not necessarily surjective; the analogous statement is also true for the closed discs. Remark 2.8. Note that the disc D1/a (0) in Lemma 2.6 may be irrational. Let

∗ | = {r : r ∈ Q} for some 0 < < 1 (as in Example K be a field such that |K 2.1). Let β be an irrational number and let pn /qn be the n-th convergent of the continued fraction expansion of β. Let the sequence {ai ∈ K}i≥2 satisfy p (1/) n , if i − 1 = qn and pn /qn < β, |ai | = 0, otherwise. Then,

sup |ai |1/(i−1) = (1/)β ∈ / |K|. On the other hand, if the sequence {ai } is such that maxi≥2 |ai |1/(i−1) exists, then

for any K. This is always the case for polynomials. a = maxi≥2 |ai |1/(i−1) ∈ |K| 3. Divergence

As proven in Section 4, power series of the form f (x) = λx + O(x2 ), with monomials of degree divisible by some nonnegative integer power of p are analytically linearizable at the origin. In the paper [21], it was proven that quadratic polynomials, f (x) = λx + a2 x2 , with |1 − λ| < 1, are not analytically linearizable at the origin. We will prove another result in this direction. Polynomials of the form f (x) = λx + ap+1 xp+1 , with |1 − λ| < 1, are not analytically linearizable in characteristic p > 0. The key result is Lemma 3.4 below. In that lemma we obtain the exact modulus of a subsequence of coefficients of the conjugacy function g. It turns out that this subsequence contains sufficiently many small divisor terms to yield divergence. Lemma 3.1. Let K be a complete valued field and let f be a power series of the form f (x) = λx + ai xi ∈ K[[x]], i≥i0

98 10

KARL-OLOF LINDAHL

where λ = 0 but not a root of unity, i0 ≥ 2 is an integer, and ai0 = 0. Then, the associated formal conjugacy function is of the form g(x) = x + bk xk , k≥i0

where bi0 = ai0 /λ(1 − λ

i0 −1

).

By definition, the formal conjugacy g is of the form g(x) = x + Proof. k b x . The left hand side of the Schröder functional equation (0.1) is of k k≥2 the form i i 0 −1 0 −1 bk (λx + ai0 xi0 + . . . )k + O(xi0 ) = bk (λx)k + O(xi0 ). g ◦ f (x) = k=1

k=1

For the right hand side we have λg(x) = λ

i 0 −1

bk xk + O(xi0 ).

k=1

Recall that λ = 0 is not a root of unity. Identification term by term yields that bk = 0 for 1 < k < i0 . Consequently, for k = i0 , the recursion formula (2.5) yields bi0 = b1 ai0 /λ(1 − λi0 −1 ). But by definition, b1 = 1. This completes the proof of the lemma.

Lemma 3.2. Let char K = p > 0. Let f (x) = λx + ap+1 xp+1 ∈ K[x], with |λ| = 1 but not a root of unity. Then, the formal solution g of the SFE (0.1) has coefficients bk of the form ⎤ ⎡ j−1 1 ip + 1 ip+1−(j−i) j−i ⎥ ⎢ bip+1 ap+1 ⎦ , λ (3.1) bjp+1 = ⎣ jp λ(1 − λ ) j−i j−1 i= p+1

for all integers j ≥ 1, b1 = 1, and bk = 0 otherwise. Proof. First note that the case k ≤ p + 1 follows by Lemma 3.1. Now assume that the lemma holds for all k ≤ (n − 1)p + 1 for some n ≥ 2. In particular, this means that bk = 0 for all (i − 1)p + 1 < k < ip + 1 where 1 ≤ i ≤ n − 1. We now consider the case (n − 1)p + 1 < k ≤ np + 1. We have regarding the left hand side of the SFE g ◦ f (x) =

np+1

bk (λx + ap+1 xp+1 )k + O(xnp+2 ),

k=1

and by hypothesis, (3.2) g ◦ f (x) =

n−1

bip+1 (λx + ap+1 xp+1 )ip+1 +

i=0

np+1

bk (λx + ap+1 )k + O(xnp+2 ).

k=(n−1)p+2

For all integers i ≥ 0, we have the binomial expansion ip+1 ip + 1 p+1 ip+1 (λx)ip+1− (ap+1 xp+1 ) . = (λx + ap+1 x ) =0


99 11

Consequently, with δ = min{ip + 1, n − i}. δ ip + 1 ip+1− p+1 ip+1 (3.3) (λx + ap+1 x ) = ap+1 x(i+)p+1 + O(xnp+2 ). λ =0

Also note that for k ≥ (n − 1)p + 2 we have (λx + ap+1 xp+1 )k = (λx)k + O(xnp+2 ).

(3.4)

Combining (3.2), (3.3), and (3.4) we obtain that g ◦ f (x) is of the form np+1 δ n−1 ip + 1 ip+1− (i+)p+1 bip+1 ap+1 x + bk (λx)k + O(xnp+2 ). λ i=0 =0

k=(n−1)p+2

The right hand side of the SFE is of the form λg(x) = λ

np+1

bk xk + O(xnp+2 ).

k=1

Note that (3.3) contains no powers of x in the closed interval [(n − 1)p + 2, np]. Consequently, for powers of x in this interval the SFE gives np

k

bk (λx) = λ

k=(n−1)p+2

np

bk xk ,

k=(n−1)p+2

and by identification term by term bk = 0,

for all (n − 1)p + 1 < k < np + 1.

In the remaining case the xnp+1 -term in (3.3) (if it exists) occur for = n − i. Such an exits if and only if n − i ≤ ip + 1, that is if i ≥ n−1 p+1 . Hence, the equation for the xnp+1 -term yields ⎤ ⎡ n−1 ip + 1 ip+1−(n−i) n−i ⎥ 1 ⎢ λ bnp+1 = bip+1 ap+1 ⎦ , ⎣ np λ(1 − λ ) n−i n−1 i= p+1

as required.

Lemma 3.3. For each summand in the recursion formula (3.1), the total power of ap+1 is j, i.e each bjp+1 is of the form (3.5)

bjp+1 = cjp+1 ajp+1 ,

where where cjp+1 is a sum of products with factors of the form a β 1 λ . (3.6) λ(1 − λα ) b Proof. First note that in view of (3.1) we have for j = 1 that the coefficient bp+1 = ap+1 /(λ(1 − λp )). Now assume that (3.5) holds for j ≤ n. For j = n + 1 and 1 ≤ i ≤ n, we then have regarding the right hand side of (3.1) that n+1−i n+1−i bip+1 ap+1 = cip+1 aip+1 ap+1 = cip+1 an+1 p+1 ,

as required.

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KARL-OLOF LINDAHL

Lemma 3.4. Let f (x) = λx+ap+1 xp+1 , |λ| = 1 but not a root of unity. Suppose |1 − λ| < 1. Then the formal solution g of the SFE (0.1) has coefficients bk of absolute value |ap+1 |j , (3.7) |bjp+1 | = j ip i=1 |1 − λ | for all integers j ≥ 1, b1 = 1, and bk = 0 otherwise. Proof. By Lemma 3.3, the total power of ap+1 in the products bip+1 aj−i p+1 of the right hand side of (3.1) is j. Also recall that |λ| = 1. Furthermore ip + 1 = 1, 1 for all nonnegative integers i. Now, since |1 − λ| < 1, it follows by induction over j that the b(j−1)p+1 -term in (3.1) is strictly greater than all the others and by ultrametricity we obtain j 1 (j − 1)p + 1 (j−1)p = |ap+1 | |bjp+1 | = a b λ . p+1 (j−1)p+1 j ip λ(1 − λjp ) 1 i=1 |1 − λ |

This completes the proof of Lemma 3.4.

Lemma 3.5. Let char K = p > 0 and λ ∈ K, |λ| = 1, but not a root of unity. Suppose |1 − λ| < 1, then N −1 p

(3.8)

N

|1 − λip | = |1 − λ|p

( p−1 p (N −1)+1)

,

i=1

for all integers N ≥ 1. Proof. First note that for each n = 1, . . . , N − 1, the number of elements in {1, 2, . . . , pN } that are divisible by pn but not by pn+1 , are given by the number pN /pn − pN /pn+1 . Since |1 − λ| < 1 we can apply Lemma 2.2 in the case m = 1 to obtain N −1 p

pN +

|1 − λip | = |1 − λ|

N −1 n=1

N

( ppn −

pN pn+1

)pn

N

= |1 − λ|p

(1+ p−1 p (N −1))

,

i=1

as required.

Theorem 3.6. Let char K = p > 0 and f (x) = λx+ap+1 xp+1 ∈ K[x]. Suppose |λ| = 1 and |1 − λ| < 1. Then f is not analytically linearizable at x = 0. Proof. Let j = pN −1 for some integer N ≥ 1. By Lemma 3.4, N −1

|ap+1 |p

|bpN +1 | = pN −1 i=1

|1 − λip |

.

But in view of Lemma 3.5 this means that N −1

|bpN +1 | = and since |1 − λ| < 1,

|ap+1 |p

N (1+ p−1 (N −1)) p

|1 − λ|p

N

lim |bpN +1 |1/(p

Consequently, lim sup |bk |

N →∞ 1/k

+1)

,

= ∞.

= ∞ so that the conjugacy diverges.

DIVERGENCE AND CONVERGENCE OF CONJUGACIES IN NON-ARCHIMEDEAN... 101 13

Further investigation has led us to believe that we have divergence also in the case m > 1 in Theorem 3.6, that is m > 1 is the smallest integer such that |1 − λm | < 1. However, the proof seems to be more complicated; we do not necessarily have a strictly dominating term in the right hand side of (3.1) for all j, and consequently, (3.7) may no longer be valid. Conjecture 3.7 (Generalization of Theorem 3.6). Let char K = p > 0 and f (x) = λx + ap+1 xp+1 ∈ K[x]. Suppose |λ| = 1 and that m ≥ 1 is the smallest integer such that |1 − λm | < 1. Then f is not analytically linearizable at x = 0. 4. Estimates of linearization discs In this section we consider power series in the family ⎫ ⎧ ⎬ ⎨ p (4.1) Fλ,a (K) = λx + ai xi ∈ K[[x]] : a = sup |ai |1/(i−1) , ⎭ ⎩ i≥2 p|i

p as defined in Section 1.2. Note that each f ∈ Fλ,a (K) is convergent on D1/a (0) and by Lemma 2.6 f : D1/a (0) → D1/a (0) is one-to-one. We will prove in Theorem p 4.6 that each each f ∈ Fλ,a (K) is linearizable at the fixed point at the origin. We also estimate the region of convergence for the corresponding conjugacy function g, and its inverse.

Remark 4.1. By the conjugacy relation g ◦ f ◦ g −1 (x) = λx, f must certainly be one-to-one on the linearization disc. Consequently, by Lemma 2.6, the full conjugacy relation cannot hold on a disc greater than D1/a (0). We begin by proving the following, simple but important fact. Given a power p series f ∈ Fλ,a (K), the conjugacy function g only contains monomials of degree divisible by some nonnegative integer power of p. More precisely, we have the following Lemma. p Lemma 4.2. Let char K = p > 0 and let f ∈ Fλ,a (K), where |λ| = 1 but not a root of unity. Then, the formal conjugacy g is of the form bk xk . (4.2) g(x) = x + p|k

and let fp (x) = (i,p)>1 ai xi . Let the conjugacy Proof. Let f ∈ ∞ g(x) = x + k=2 bk xk . We will prove by induction that if g is the formal solution of the Schröder functional equation, then bk = 0 for all k ≥ 2 such that p k. By definition b1 = 1 and by Lemma 3.1 bk = 0 for 1 < k < p. Assume that bk = 0 for all k > 1 such that (j − 1)p < k < jp, where 1 ≤ j ≤ N . Let hN be the polynomial p (K) Fλ,a

hN (x) = λx + fp (x) +

N

bip (λx + fp (x))ip

mod x(N +1)p .

i=1

Note that the terms (λx + fp (x))

ip

ip ip (λx)l (fp (x))ip−l , = l l=0

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KARL-OLOF LINDAHL

do only contain powers of x divisible by p. This follows because, in characteristic p, ipl = 0 if p l. Accordingly, hN (x) − λx contains only powers of x divisible by p. By hypothesis,

(N +1)p−1

g ◦ f (x) = hN (x) +

bk (λx + fp (x))k + O(x(N +1)p ).

k=N p+1

As noted above the monomials of fp are of degree greater than or equal to p. Consequently,

(N +1)p−1

g ◦ f (x) = hN (x) +

bk (λx)k + O(x(N +1)p ).

k=N p+1

Recall that, hN (x) − λx contains only powers of x divisible by p. Consequently, identification term by term with the right hand side λg(x) yields that bk = 0 for all N p < k < (N + 1)p as required. In the following we shall estimate the coefficients of the conjugacy (4.2). As noted in Section 2.1 these coefficients are given by the recursion formula (2.5). Our main result (Theorem 4.6) of this section is based on the estimate obtained in Lemma 4.4 below. In preparation, we recall the following definitions from Section 1.2. The integer m is defined by (4.3)

m = m(λ) = min{n ∈ Z : n ≥ 1, |1 − λn | < 1}.

Note that, by Lemma 2.2, m is not divisible by p. Given m, the integer k is defined by (4.4)

k = k (λ) = min{k ∈ Z : k ≥ 1, p|k, m|k − 1}.

Note the following lemma. Lemma 4.3. Let k ≥ 2 be an integer. Then, (k − k )/mp + 1 is the the number of positive integers l ≤ k that satisfies the two conditions p | l and m | l − 1. Proof. Let k be the smallest positive integer such that p | k and m | k − 1. Let Zm be the residue class modulo m. Also recall that by definition p m. Accordingly, k = j p where j is the unique solution in Zm to the congruence equation (4.5)

xp − 1 ≡ 0 mod m.

The integer solutions of (4.5) are thus of the form x = j + mn, where n runs over all the integers. It follows that an integer l satisfies the two conditions p | l and m | l − 1 if and only if it is of the form l = (j + mn)p = k + mpn, for some integer n. Given k ≥ 2, let t ≥ 0 be the largest integer such that k + (t − 1)mp ≤ k.


It follows that

t = (k − k )/mp + 1,

as required. Lemma 4.4. Let f ∈ conjugacy function satisfies

p Fλ,a (K).

|bk | ≤

(4.6)

Then, the coefficients of the corresponding ak−1

|1 − λm | (k−k )/mp+1

.

for all k ≥ 2. Proof. Recall that the coefficients |bk | are given by the recursion formula (2.5) where each factorial term l!/α1 ! · ... · αk ! is an integer and thus of modulus zero or one, depending on whether it is divisible by p or not. By Lemma 4.2, bk = 0 if k ≥ 2 and p k. If p | k, we have in view of Lemma 2.2 that 1, if m k − 1, (4.7) |1 − λk−1 | = |1 − λm |, if m | k − 1. Also recall that |ai | ≤ ai−1 . It follows by Lemma 4.3 that

|bk | ≤ |1 − λm |− (k−k )/mp+1 aα , for some integer α. In view of equation (2.6) we have k

(i − 1)αi = k − l.

i=2

Consequently, since |ai | ≤ ai−1 , we obtain k

(4.8)

|ai |αi ≤

i=2

k

a(i−1)αi = ak−l .

i=2

Now we use induction over k. By definition b1 = 1 and, according to the recursion formula (2.5), |b2 | ≤ |1 − λm |− (2−k )/mp+1 |a|. Suppose that

|bl | ≤ |1 − λm |− (l−k )/mp+1 al−1 for all l < k. Then

m − (k−k )/mp+1 l−1

|bk | ≤ |1 − λ |

a

max

k

|ai |

αi

,

i=2

and the lemma follows by the estimate (4.8).

The above estimate of |bk | is maximal in the sense that we may have equality in (4.6) for k = k as shown by the following example. Example 4.5. Let f be of the form

f (x) = λx + ak xk .

Then, a = |ak |1/(k −1) . Also note that by Lemma 3.1

bk = ak /λ(1 − λk −1 ), and consequently,

|bk | = ak −1 /|1 − λm |.

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Thus we have equality in (4.6) for k = k in this case. By the estimates in Lemma 4.4 and application of Proposition 2.5, the radius of convergence of the conjugacy function g and its inverse g −1 can now be estimated by 1

|1 − λm | mp ρ= , a

(4.9) and

1

|1 − λm | k −1 , a respectively. In other words, we have the following theorem.

(4.10)

σ=

p (K). Then, f is analytically linearizable at x = 0. Theorem 4.6. Let f ∈ Fλ,a The semi-conjugacy relation g ◦ f (x) = λg(x) holds on Dρ (0). Moreover, the full conjugacy g ◦ f ◦ g −1 (x) = λx holds on Dσ (0). The latter estimate is maximal in the sense that there exist examples of such f which have a periodic point on the sphere Sσ (0).

Remark 4.7. Note that g : Dσ (0) → Dσ (0) is bijective if we consider Dσ (0)

in the algebraic closure K. Proof. In view of Lemma 4.4, g converges on the open disc of radius 1/ lim sup |bk |1/k ≥ a−1 |1 − λm |1/mp = ρ. Given m ≥ 1, by definition k must be one of the numbers p,

2p,

...,

mp,

and accordingly, k − 1 < mp.

(4.11)

To estimate the radius of the maximal disc on which g is one-to-one we consider 1/(k−1) |b1 | σ0 = a−1 inf |1 − λm | (k−k )/mp+1 /(k−1) ≤ inf . k≥2 k≥2 |bk | We will show that the maximum value of the exponent δ(k) = (k − k )/mp + 1/(k − 1) is attained if and only if k = k . First note that δ(k) = 0 for all 2 ≤ k < k . In view of (4.11), we have for each integer n ≥ 1 that δ(k + nmp) −

n((k − 1) − mp) 1 = < 0. k − 1 (k − 1 + nmp)(k − 1)

Moreover,

δ(k) < δ(k + nmp), if mnp < k − k < (n + 1)mp. Hence, the maximum of δ is attained if and only if k = k so that 1

σ0 = a−1 |1 − λm | k −1 = σ. Note that in view of (4.11) σ < ρ so that g certainly converges on the closed disc Dσ (0). Also note that, in terms of δ, we have by (4.6) that |bk | ≤ ak−1 |1 − λm |−(k−1)δ(k) .


Consequently, according to the derived properties of δ, |bk |σ k ≤ a−1 |1 − λm | k −1 +(k−1)( k −1 −δ(k)) ≤ σ = |b1 |σ. 1

(4.12)

1

In view of Proposition 2.5 g : Dσ (0) → Dσ (0) is a bijection if we consider Dσ (0) in

Consequently, g : Dσ (0) → Dσ (0) is one-to-one if we consider Dσ (0) in K. K. Recall that by Lemma 2.6 f : D1/a (0) → D1/a (0) is one-to-one. Moreover, 1/a > ρ > σ. It follows that the semi-conjugacy and the full conjugacy holds on Dρ (0) and Dσ (0) respectively. That this estimate of σ is maximal follows from the following example. Let charK = 2 and f (x) = λx + a2 x2 ∈ K[x], where |1 − λ| < 1. Then m = 1 and k = p = 2 so that σ = |1 − λ|/|a2 |. But x ˆ = (1 − λ)/a2 is a fixed point of f , breaking the conjugacy on Sσ (0). This completes the proof of Theorem 4.6. If bk = 0, then we can extend the estimate of the full conjugacy to the disc Dρ (0). p (K). Suppose the coefficient bk of the conjugacy Lemma 4.8. Let f ∈ Fλ,a function g is equal to zero. Then, the full conjugacy g ◦ f ◦ g −1 (x) = λx holds on a disc larger than or equal to Dρ (0) or D ρ (0), depending on whether g converges on the closed disc Dρ (0) or not.

bk

Proof. Recall that k is the positive integer defined by (4.4). Assume that = 0. Let k > k be the integer k := min{k ∈ Z : k > k , bk = 0, p | k, m | k − 1}.

In the same way as in Lemma 4.4, |bk | ≤

ak−1 |1 − λm |(k−1)γ(k) ,

where

k ≥ k , (k − k )/mp + 1/(k − 1), 0, k < k . It follows from the proof of Lemma 4.3 that k = k + mpn, for n = 1. Hence, γ(k) =

k ≥ p + mp. Consequently, γ(k) −

k − k 1 1 mp − (k − 1) 1 ≤ + − = < 0, mp (k − 1)mp k − 1 mp (k − 1)mp

for k ≥ 2. Moreover, lim γ(k) = 1/mp

k→∞

so that sup γ(k) = 1/mp. Accordingly, max γ(k) does not exist. Also note that for ρ = a−1 |1 − λm |1/mp , we have |bk |ρk ≤ a−1 |1 − λm | mp +(k−1)( mp −γ(k)) . 1

1

Hence, according to the derived properties of γ, (4.13)

|bk |ρk < ρ = |b1 |ρ,

for k ≥ 2. It follows that g : Dρ (0) → Dρ (0) is one-to-one. If g converges on the closed disc Dρ (0), then, since we have strict inequality in (4.13), we also have that g : Dρ (0) → D ρ (0) is one-to-one.

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Recall that f : D1/a (0) → D1/a (0) is one-to-one and that 1/a > ρ. Consequently, the full conjugacy g ◦ f ◦ g −1 (x) = λx holds on a disc larger than or equal to Dρ (0) or Dρ (0), depending on whether g converges the closed disc Dρ (0) or not. Note that a sufficient condition that bk = 0 is that ai = 0 for all 2 ≤ i ≤ k . In fact, in view of Lemma 3.1 we and the previous lemma we have the following result. p Theorem 4.9. Let f ∈ Fλ,a (K) be of the form a i xi , f (x) = λx + i≥i0

for some integer i0 > k . Then, the full conjugacy g ◦ f ◦ g −1 (x) = λx holds on a disc larger than or equal to Dρ (0) or D ρ (0), depending on whether g converges on the closed disc Dρ (0) or not. 5. Linearization discs and periodic points In this section we give sufficient conditions under which the estimate σ in Theorem 4.6 is maximal in the sense that Dσ (0) is equal to the linearization disc. In other words, we give sufficient conditions under which Dσ (0) is the maximal disc U ⊂ K about the fixed point x = 0, such that the conjugacy g ◦ f ◦ g −1 (x) = λx holds for all x ∈ U . In theorem 4.6 we proved that the estimate σ is maximal in the sense that, in the the special case that f is quadratic and |1 − λ| < 1 (m = 1), there is a fixed

on the sphere Sσ (0), breaking the conjugacy there. point in the algebraic closure K Using the notion of Weierstrass degree of the conjugacy function, defined below, we will give sufficients conditions for the existence of an indifferent periodic point on the boundary Sσ (0) for more general f . The Weierstrass degree is defined as follows. Let K be an algebraically closed complete non-Archimedean field. Let U ⊂ K be a rational closed disc, and let h be a power series which converges on U . For any disc V ⊆ U , the Weierstrass degree or simply the degree deg(h, V ) of h on V is the number d (if V is closed) or d (if V is open) in Proposition 2.4. Note that if 0 ∈ h(V ), then the Weierstrass degree is the same as the notion of degree as ’the number of pre-images of a given point, counting multiplicity’. More information on the properties of the Weierstrass degree can be found in [5]. The following lemma shows that a shift of the value of Weierstrass degree from 1 to d > 1, of the conjugacy function on a sphere S, reveals the existence of an indifferent periodic point on the sphere S. Lemma 5.1. Let K be a complete non-Archimedean field. Let f be a linearizable power series of the form f (x) = λx + i≥2 ai xi ∈ K[[x]], such that |λ| = 1 and a = supi≥2 |ai |1/(i−1) . Let g be the corresponding conjugacy function. Let τ < 1/a.

Suppose deg(g, Dτ (0)) = 1 and deg(g, Dτ (0)) = d > 1 in the algebraic closure K.

Then, f has an indifferent periodic point in K on the sphere Sτ (0) of period κ ≤ d. In particular, Dτ (0) is the linearization disc of f about the fixed point at the origin. Proof. Let τ < 1/a, and suppose deg(g, Dτ (0)) = 1 and deg(g, Dτ (0)) = d >

∗ |. Hence, Sτ (0) is rational and non-empty in the 1. Note that by definition τ ∈ |K


The proof that there is a periodic point in K

on the sphere algebraic closure K. Sτ (0), goes as follows. Since the conjugacy g maps the closed disc Dτ (0) onto itself exactly d-to-1 (counting multiplicity), and the open disc Dτ (0) one-to-one onto x) = 0. In view of the itself, there exist at least one point x ˆ ∈ Sτ (0) such that g(ˆ Schröder functional equation |g(f (ˆ x)) − g(f ◦n (ˆ x))| = |g(ˆ x)||λ − λn | = 0,

(5.1)

for all n ≥ 1. Recall that 1/a > τ . By Lemma 2.6 f : Dτ (0) → Dτ (0) is bijective

The same is true for all the iterates f ◦n , n ≥ 1. Moreover f ◦n (0) = 0 and in K. consequently f ◦n can have no zeros on the sphere Sτ (0) for any n ≥ 1. In particular x) = 0 for all n ≥ 1. f ◦n (ˆ Let y = g(f (ˆ x)). Then y ∈ Dτ (0). The equation g(x) = y can have only d x) = solutions on Dτ (0) and we conclude from (5.1) that we must have that f ◦(κ+1) (ˆ

of period f (ˆ x) for some κ ≤ d. This shows the existence of a periodic point x ˆ∈K κ ≤ d. Finally, since f ◦κ is one-to-one on Dτ (0), it follows by the first statement of proposition 2.4 that |(f ◦κ ) (ˆ x)| = 1. This proves that x ˆ is indifferent. p (K). By Theorem 4.6, the Weierstrass Let us return to the case f ∈ Fλ,a degree of g on the open disc deg(g, Dσ (0)) = 1. In the following lemma we find a necessary and sufficient condition that the Weierstrass degree on the closed disc deg(g, Dσ (0)) > 1. Again, the integer k , defined by (4.4), plays a significant role. p

deg(g, Dσ (0)) > 1 if and only if Lemma 5.2. Let f ∈ Fλ,a (K). Then, in K, the coefficient, bk , of g satisfies

|bk | = ak −1 /|1 − λm |.

(5.2)

Moreover, if (5.2) holds, then deg(g, Dσ (0)) = k . Proof. By the estimate (4.6) we always have

|bk | ≤ ak −1 /|1 − λm |.

If |bk | = ak −1 /|1 − λm |. Then, we have equality in (4.12) if and only if k = 1 or

g maps the closed disc Dσ (0) k = k . Consequently, in the algebraic closure K, onto Dσ (0) exactly k -to-1 counting multiplicity. It follows that the Weierstrass degree deg(g, Dσ (0)) = k > 1. On the other hand, if |bk | < ak −1 /|1 − λm |, then we have equality in (4.12) if and only if k = 1. Consequently, deg(g, Dσ (0)) = 1 in this case. If f is of the form as in Example 4.5, then Lemma 5.2 applies and we have.

Theorem 5.3. Let f (x) = λx + ak xk , where ak = 0. Then, the linearization

we disc of f about the origin is equal to Dσ (0). In in the algebraic closure K,

have deg(g, Dσ (0)) = k . Moreover, f has an indifferent periodic point in K on the sphere Sσ (0) of period κ ≤ k , with multiplier λκ . Proof. It remains to prove that the multiplier of the periodic point is of the p form λκ . Because the degree of the nonlinear monomials of f ∈ Fλ,a (K) are all ◦n n

divisible by char K = p, the derivative (f ) (x) = λ for all x ∈ K and all n ≥ 1.

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It follows that x ˆ is an indifferent periodic point of period κ ≤ d, with multiplier λκ . In fact, this result can be generalized according to the following theorem.

p (K). Suppose a = |ak |1/(k −1) and |ai | < ai−1 for Theorem 5.4. Let f ∈ Fλ,a

we all i < k . Then, Dσ (0) is the linearization disc of f about the origin. In K

on have deg(g, Dσ (0)) = k . Furthermore, f has an indifferent periodic point in K the sphere Sσ (0) of period κ ≤ k , with multiplier λκ .

p Proof. Let f ∈ Fλ,a (K). Suppose |ak | = ak −1 and |ai | < ai−1 for all i < k . For k = k and l = 1 the equation (2.6) has the solution αk = 1, αj = 0 for j < k . Hence, for k = k , the recursion formula (2.5) contains the term

b1 ak /(1 − λk −1 ),

(5.3)

where b1 = 1, |ak | = ak −1 , and |1 − λk −1 | = |1 − λm |. The minimality of k and the assumption that |ai | < ai−1 for all i < k yields in view of Lemma 4.4 that the term (5.3) is strictly greater than all the other terms in the recursion formula (2.5). Hence, by ultrametricity, |bk | = ak −1 |1 − λm |−1 as required. p Corollary 5.5. Let f ∈ Fλ,a (K) be of the form ai xi , a = |ak |1/(k −1) > 0. f (x) = λx + i≥k

deg(g, Dσ (0)) = k . Then Dσ (0) is the linearization disc of f about the origin. In K,

Furthermore, f has an indifferent periodic point in K on the sphere Sσ (0) of period κ ≤ k , with multiplier λκ . Acknowledgements I would like to thank Prof. Andrei Yu. Khrennikov for fruitful discussions and for introducing me to the theory of p-adic dynamical systems and the problem on linearization. I thank Robert L. Benedetto for his hospitality during my visit in Amherst, for fruitful discussions, and consultancy on non-Archimedean analysis. References 1. D. K. Arrowsmith and F. Vivaldi, Some p-adic representations of the Smale horseshoe, Phys. Lett. A 176 (1993), 292–294. , Geometry of p-adic Siegel discs, Physica D 71 (1994), 222–236. 2. 3. A. F. Beardon, Iteration of rational functions, Springer-Verlag, Berlin Heidelberg New York, 1991. 4. R. Benedetto, Reduction dynamics and Julia sets of rational functions, J. Number Theory 86 (2001), 175–195. , Non-Archimedean holomorphic maps and the Ahlfors Islands theorem, Amer. J. 5. Math. 125 (2003), no. 3, 581–622. 6. J-P. B´ ezivin, Fractions rationnelles hyperboliques p-adiques, Acta Arith. 112 (2004), no. 2, 151–175. 7. S. Bosch, U. G¨ untzer, and R. Remmert, Non-archimedean analysis: A systematic approach to rigid analytic geometry, Springer-Verlag, Berlin, 1984. 8. A. D. Brjuno, Analytical form of differential equations, Trans. Moscow Math. Soc. 25,26 (1971,1972), 131–288,199–239. 9. L. Carleson and T. Gamelin, Complex dynamics, Springer-Verlag, Berlin Heidelberg New York, 1991.


10. J.W.S. Cassels, Local fields, Cambridge University Press, Cambridge, 1986. 11. J. Fresnel and M. van der Put, G´ eom´ etrie analytique rigide et applications, Birkh¨ auser, Boston, 1981. 12. M. Herman, Recent results and open questions on Siegel’s linearization theorem of complex analytic diffeomorphisms of Cn near a fixed point, Proc. VIII-th International Congress on Mathematical Physics 1986, World Scientific, 1987, pp. 138–184. 13. M. Herman and J.-C. Yoccoz, Generalizations of some theorems of small divisors to non archimedean fields, Geometric Dynamics, LNM, vol. 1007, Springer-Verlag, 1981, pp. 408– 447. 14. L. Hsia, Closure of periodic points over a non-archimedean field, J. London Math. Soc. 62 (2000), no. 2, 685–700. 15. A. Khrennikov, Small denominators in complex p-adic dynamics, Indag. Mathem. 12 (2001), no. 2, 177–188. 16. A. Khrennikov and M. Nilsson, On the number of cycles for p-adic dynamical systems, J. Number Theory 90 (2001), 255–264. 17. N. Koblitz, p-adic numbers, p-adic analysis, and zeta-functions, second ed., Springer-Verlag, New York, 1984. 18. H-C. Li, p-adic dynamical systems and formal groups, Compos. Math. 104 (1996), 41–54. , On heights of p-adic dynamical systems, Proc. Amer. Math. Soc. 130 (2002), no. 2, 19. 379–386. 20. K.-O. Lindahl, Estimates of linearization discs in p-adic dynamics with application to ergodicity, Preprint 04098, MSI, V¨ axj¨ o University, Sweden. , On Siegel’s linearization theorem for fields of prime characteristic, Nonlinearity 17 21. (2004), no. 3, 745–763. 22. , On the linearization of non-archimedean holomorphic functions near an indifferent fixed point, Ph.D. thesis, V¨ axj¨ o University, 2007. 23. J. Lubin, Non-archimedean dynamical systems, Compos. Math. 94 (1994), 321–346. 24. J. Milnor, Dynamics in One Complex Variable, 2nd ed., Vieweg, Braunschweig, 2000. 25. J. Rivera-Letelier, Dynamique des functionsrationelles sur des corps locaux, Ast´ erisque 287 (2003), 147–230. , Espace hyperbolique p-adique et dynamique des fonctions rationnelles., Compos. 26. Math. 138 (2003), no. 2, 199–231. 27. C. L. Siegel, Iteration of analytic functions, Ann. of Math. 43 (1942), 607–612. 28. S. De Smedt and A. Khrennikov, Dynamical systems and theory of numbers, Comment. Math. Univ. St. Pauli 46 (1997), no. 2, 117–132. 29. D. Viegue, Probl` emes de lin´ earisation dans des familles de germes analytiques, Ph.D. thesis, Universit´ e D’Orl´ eans, 2007. 30. J.-C. Yoccoz, Lin´ earisation des germes de diffomorphismes holomorphes de (C, 0), C. R. Acad. Sci. Paris S´ er. I Math. 306 (1988), no. 1, 55–58. 31. E. Zehnder, A simple proof of a generalization of a theorem by C.L. Siegel, Geometry and Topology (Berlin Heidelberg New York Tokyo) (J. Palis and Manfredo do Carmo, eds.), Lecture Notes in Mathematics, Springer-Verlag, 1976, Proceedings, Rio de Janeiro 1976, pp. 855– 866. ¨xjo ¨ University, 351 95, Va ¨xjo ¨, School of Mathematics and Systems Engineering Va Sweden E-mail address: [email protected]



A criterion for the invertibility of Lipschitz operators on type separating spaces Héctor M. Moreno Abstract. A Norm Hilbert space E with orthogonal base e1 , e2 , . . . is called a Type Separating space if the topological types of any two different basis vectors are also different. In this paper we study a criterion for the invertibility of a Lipschitz injective operator B in terms of a boundedness condition for the norms Be1 , Be2 . . .. It generalizes a result in [1]. From this we derive, following the work done in [6], a handy criterion in terms of the (infinite) matrix of B. We show applications for the case of the indecomposable selfadjoint Lipschitz operators studied in [5] and [8]

1. Preliminaries. In what follows G will be an abelian mutiplicative totally ordered group, we assume G = {1}. A subgroup H of G is said to be convex if ∀ g ∈ G ∀ h ∈ H h−1 ≤ g ≤ h ⇒ g ∈ H. A convex subgroup H is principal if there exists an h ∈ H such that H is the smallest convex subgroup of G containing h. Every proper convex subgroup is bounded above and below in G, and the set of all convex subgroups is linearly ordered by inclusion. K is a field with a Krull valuation | |, whose value group is G. That is, a surjective map | | : K → G ∪ {0} (where 0 is an element adjoined to G such that for all g ∈ G, 0 ≤ g, 0 · g = g · 0 for all g ∈ G), that satisfies (i) |x| = 0 if and only if x = 0, (ii) |xy| = |x||y| and (iii) |x + y| ≤ max{|x|, |y|}, for every x, y ∈ K. The order type of the set H := {H : H is a principal convex subgroup of G and H = {1}} is called the rank of the valuation. 1.1. G-modules. A linearly ordered set X is a G-module if it does not have a first element, and there exists an action (g, x) → gx of G onto X that (i) is increasing on both variables, and (ii) the orbit of x ∈ X is coinitial in X. 2000 Mathematics Subject Classification. Primary 46S10; Secundary 46H35. Key words and phrases. Norm Hilbert spaces, type separating spaces, Lipschitz operators. Research partially supported by Fondecyt Proyecto 1080194. c Mathematical 0000 (copyright Society holder) c 2010 American

1 111

´ HECTOR M. MORENO

112 2

It follows that for all x1 , x2 ∈ X and g ∈ G x1 < x2 ⇒ gx1 < gx2 . An important example of a G-module is G# (the completion of G by Dedekind cuts), with the action (g, s) → supG# {gx : x ∈ G, x ≤ s}. Definition 1.1. Let X be a G-module and s, t ∈ X. The algebraic type of s (denoted by Gs) is the orbit of s under the action of G on X. The topological type τ (s; t) of s with respect to t is a convex subgroup of G defined in the following way. If t ∈ Gs then τ (s; t) = {h ∈ G : ht = t}, if t ∈ / Gs then τ (s; t) is the largest among the convex subgroups H of G for which convX (Ht) ∩ Gs = ∅ where, for Z ⊂ X, convX (Z) = {x ∈ X : there exist z1 , z2 ∈ X with z1 ≤ x ≤ z2 }. A useful characterization of the topological type of an element is given by the following theorem. Theorem 1.2. The topological type τ (s; t) is the set of all h ∈ G satisfying (1) and (2) below. (1) If g ∈ G, t ≤ gs then ht ≤ gs. (2) If g ∈ G, t ≥ gs then ht ≥ gs. Proof. It follows directly from [2] 1.6.1 and 1.6.2.

Definition 1.3. Let X be a G-module and s1 , s2 , s3 , ... a sequence in X. (1) We say that s1 , s2 , ... satisfies the type condition if, for any sequence g1 , g2 , ... in G such that {g1 s1 , g2 s2 , ...} is bounded above in X it is true that limn gn sn = 0. (2) Let t ∈ X. We shall say that limn τ (sn ; t) = ∞ if for every proper convex subgroup H of G there exists n0 ∈ N such that H τ (sn ; t) for all n ≥ n0 . Theorem 1.4. Let G be a linearly ordered group, X a G-module and t ∈ X. For any sequence s1 , s2 , s3 , ... in X the following statements are equivalent: (1) s1 , s2 , ... satisfy the type condition. (2) limn τ (sn ; t) = ∞.

Proof. See [2] 1.6.6.

1.2. Banach spaces. Let X be a G-module to which we adjoin an element 0 = 0X such that 0X ≤ x, 0G x = 0G 0X = 0X for every x ∈ X. Let E be a K-vector space. An X-norm on E is a map : E → X ∪ {0} that satisfies the following properties: (i) x = 0 if and only if x = 0, (ii) λx = |λ| x, (iii) x + y ≤ max{x, y} for all x, y ∈ E, λ ∈ K. Then (E, ) will be called an X-normed space. Such an X-norm induces in a canonical way a topology with respect to which E is a Hausdorff topological vector space. Let K be a complete valued field. An X-normed K-vector space E is said to be complete or a Banach space if every Cauchy net in E converges in E. In addition E is said to be of countable type if it contains a countable set D whose linear hull is dense in E; that is to say span D = E. 1.3. Orthogonality. Let E be an X-normed space over the field K. Two subspaces D1 and D2 are orthogonal (D1 ⊥ D2 ) if for every d1 ∈ D1 , d2 ∈ D2 d1 + d2 = max{d1 , d2 }. A subspace D has an orthogonal complement in E if there exists a subspace S ⊥ D such that E = D + S.

A CRITERION FOR THE INVERTIBILITY OF LIPSCHITZ OPERATORS...

113 3

A set of vectors {ei : i ∈ I} of E is orthogonal if for any finite subset J ⊆ I y λj ∈ K λj ej = max λj ej . j∈J

j∈J

Clearly if x1 , x2 ∈ E are such that x1 ∈ / Gx2 , then x1 ⊥ x2 . A sequence of vectors e1 , e2 , ... in a Banach X-normed space E is an orthogonal ∞ base of E if for every x ∈ E there are unique λ1 , λ2 , ... ∈ K such that x = k=1 λi ei and ∞ x = λi ei = max λn en . n∈N

n=1

As in classical Hilbert spaces, if E is a Banach space of infinite dimension, then for an orthogonal sequence e1 , e2 , ... in E the following are equivalent: (1) e1 , e2 , ... is an orthogonal base. (2) en = 0 for all n ∈ N, and span{ei : i ∈ N} is dense in E. (See [2] 2.4.17). Definition 1.5. Let E be a Banach space of countable type and Σ the collection of all the algebraic types σ in the G-module E\{0} := {x : x ∈ E, x = 0}. A canonical (orthogonal) decomposition of E is an orthogonal direct sum E= Eσ σ∈Σ

where each Eσ is a closed subspace of E and Eσ \ {0} = σ. In [2] 3.4.5, the following theorem is proved. Theorem 1.6. Every Banach space E with orthogonalbase has a canonical decomposition. It is unique in the following sense. If E = σ∈Σ Eσ = σ∈Σ Fσ are two canonical decompositions, then for each σ ∈ Σ, Eσ and Fσ are isometrically isomorphic. 1.4. Norm-Hilbert spaces and type separating spaces. A Banach space E of countable type is a norm-Hilbert space (N.H.S.) if for every closed subspace D of E there exists a linear surjective projection P : E → D such that P (x) ≤ x for all x ∈ E. This is equivalent to the following, every closed subspace of E has an orthogonal complement. Proposition 1.7. Let E be a Banach X-normed space of countable type. Then the following are equivalent. (1) E is a Norm-Hilbert space. (2) Each orthogonal systen in E can be extended to an orthogonal base of E. (3) For each orthogonal base e1 , e2 , e3 , ... of E, the sequence n → en satisfies the type condition. Proof. See [3] 3.2.1, 3.2.3.

Within the class of N.H.S it has been recently defined a subclass where essentially it is asked that the topological types of the norms of the basis vectors should be all different. This concept generalizes an important property of the first known orthomodular space (H. Keller 1980). To be precise,

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Definition 1.8. Let E be a N.H.S. with orthogonal base {ei : i ∈ N}. We shall call E a type separating space if there exists t0 ∈ X such that if i = j τ (ei ; t0 ) = τ (ej ; t0 ). Remark 1.9. If E is a type separating space, then the algebraic types of any two different basis vectors ei and ej are also different. And since the base is orthogonal, for any x ∈ E, x = gek for some g ∈ G, k ∈ N. This implies that the algebraic type of x is equal to the algebraic type of ek . 2. The main results In what follows K is a complete valued field with an infinite rank valuation whose value group is G. We assume that the interval topology in G# satisfies the first axiom of countability and that G has a cofinal sequence (see [2] 1.4.4 for equivalent formulations). G is the union of a strictly increasing sequence of proper convex subgroups ([2] 4.3.1). E is a K-vector space, equipped with an X-norm, where X is a G-module. Definition 2.1. Let E,F be X-normed spaces. A linear operator A : E → F is Lipschitz (or bounded) if there exists a g ∈ G such that Ax ≤ gx for all x ∈ E. Lemma 2.2. Let E be an X-normed space, U a subspace of E and B : U → E a linear operator. Let V be a closed subspace of U with dim U/V < ∞. If the restriction of B to V is bounded, then B is bounded on U . Proof. By induction on n = dim U/V . Let n = 1, then U = K(w) ⊕ V with w ∈ U \ V , and there exists h ∈ G such that Bv ≤ hv for all v ∈ V . Let us assume that B is not bounded in U \ V , therefore for each g ∈ G there exists zg = αw + vg ∈ U , α ∈ K, α = 0, such that Bzg > gzg , without loss of generality g > h. Clearly if xg = α−1 zg := w + vxg then Bxg > gxg . Let {gi }n∈N be a cofinal sequence in G such that gi > h. For every i ∈ N we choose xi = w + vi with vi ∈ V and Bxi > gi xi . Since w is not an accumulation point of the closed set V , there exists an ∈ X such that xi > for i large enough. Then Bxi ≥ gi → ∞ when i → ∞, hence Bxi = Bw + Bvi = Bvi for i large enough. But vi ∈ V , therefore Bvi ≤ hvi which implies that vi → ∞ when i → ∞. Hence xi = w + vi = vi from some index i ∈ N on. In this way we obtain for those i Bxi = Bvi ≤ hvi = hxi but this contradicts the election of xi . If dim U/V = n + 1 we have that U = K(v1 ) ⊕ · · · ⊕ K(vn ) ⊕ K(vn+1 ) ⊕ V. By induction hypothesis B is bounded on n+1 i=2 K(vi )⊕V which is a closed subspace of U . With the same argument as in case n = 1 we obtain that B is bounded on U. Lemma 2.3. Let E be a Banach space and B : U → U a bijective linear operator between the subspaces U and U of E. Assume that there exists a closed


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subspace V ⊆ U with dim U/V < ∞ such that for all v ∈ V there exist g1 , g2 ∈ G with g1 v ≤ B(v) ≤ g2 v. Then there exist g1 , g2 ∈ G such that for all u ∈ U g1 u ≤ B(u) ≤ g2 u. Proof. Let V := B(V ). V is closed, therefore it is also complete. By hypothesis the restriction of B to V is a topological isomorphism. Therefore V is also complete, and hence closed. Since B is a bijection we see that dim U /V < ∞. Lemma 2.2 applied to B and B −1 ends the proof. Lemma 2.4. Let X be a G-module and s, t, t0 ∈ X such that g1 s ≤ t ≤ g2 s for some g1 , g2 ∈ G. Let H0 be the smallest convex subgroup containing g1 and g2 . If H0 ⊆ τ (s, t0 ) or H0 ⊆ τ (t, t0 ) then τ (s, t0 ) = τ (t, t0 ).

Proof. See [9] 4.3.

Now we come to the main theorem of this section. It was proven in [1] for a concrete space V (the space constructed by H. Keller), in it the norm is induced by a hermitean form, and therefore the notion of orthogonality is stronger in the sense that the orthogonal complement of any space is unique. The proof relied on that fact as well as other concrete aspects of V . Here we extend it to a class of spaces that includes V . Theorem 2.5. Let E be a type separating space with orthogonal base {ei : i ∈ N} and B : E → E an injective, bounded operator. If there exists g1 , g2 ∈ G such that g1 ei ≤ B(ei ) ≤ g2 ei for all i ∈ N, then B is surjective and B −1 , the algebraic inverse of B, is also bounded. Proof. The crucial fact to be proven is the surjectivity of B. By definition of type separating spaces there exists t0 ∈ X such that τ (ei ; t0 ) = τ (ej ; t0 ) whenever i = j, therefore by theorem 1.7, the sequence (en )n∈N satisfies the type condition. Let H be the smallest convex subgroup containing g1 and g2 , theorem 1.4 tells us that there exists m ∈ N for which H τ (ei ; t0 ) when i > m. Using now lemma 2.4 and the remark 1.9 we obtain that τ (ei ; t0 ) = τ (B(ei ); t0 ) for all i > m, which implies that τ (B(ei ); t0 ) = τ (B(ej ); t0 )

and Gei = GB(ei )

for all i = j such that i, j > m. The finite dimensional subspace W := span{ei : 1 ≤ i ≤ m} is closed. Define D := span{ei : i > m}, and clearly E = D + W . Moreover the sum is direct, since if 0 = x ∈ D ∩ W then for some k, l ∈ N, k ≤ m < l we would have Gx = Gek = Gel , a contradiction, since E is type separating. Therefore E = D ⊕ W and codimD = m. ∞ We shall prove that B(D) is also closed in E. Let x ∈ D, x = i=m+1 βi ei . By orthogonality and the fact that E is type separating x = max{βi ei : i > m} = βt et

´ HECTOR M. MORENO

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for some t ∈ N. As the elements of {B(ei ) : i > m} have different algebraic types, for some k ∈ N B(x) = B(βk ek ). Therefore g1 x = g1 βt et ≤ B(βt et ) ≤ B(βk ek ) ≤ g2 βk ek ≤ g2 βt et = g2 x, and we conclude that g1 x ≤ B(x) ≤ g2 x for all x ∈ D, and the restriction of B to D is a linear homeomorphism. Since E is a Banach space and D is closed, we deduce that B(D) is also closed. We have also proven that B(D) only contains vectors whose algebraic type belongs to {Gei : i ≥ m + 1}. We shall now estimate the codimension of B(D) in E. Pick an orthogonal complement W of B(D) in E. We contend that the algebraic type of any vector x ∈ W belongs to the set {Gei : 0 < i ≤ m}. If this were not the case, we could find an i0 > m such that x ∈ Gei0 . But then {B(ei ) : i > m} ∪ {x} ∪ {ei : 1 ≤ i ≤ m} is an orthogonal system in E and by 1.7 it can be extended to an orthogonal base E of the space. Then we would have a canonical decomposition E = σ∈Σ σ in which x, B(e ) ∈ E with σ = Ge , since x and e are orthogonal, dim Eσ ≥ 2. i σ i i 0 0 0 But E = i∈N ei is also a canonical decomposition of E and dim Eσ ≥ 2 > 1 = dimei0 , which contradicts theorem 1.6. Now W is closed, by [2] 3.4.1 it has an orthogonal base, therefore dim W ≤ m. Since B is injective m ≥ dim E/B(D) ≥ dim B(E)/B(D) = dim E/D = m. It follows that E = B(E). Now Lema 2.3 with U = E and V = D proves the theorem. There exists a subclass of N.H.S. in which the norm is induced by a hermitean form , , for that reason they are termed Form Hilbert spaces (F.H.S.). A remarkable fact that has appeared in some type separating F.H.S. is the existence of selfadjoint indecomposable operators, that is bounded linear maps A : E → E such that for all x, z ∈ E, Ax, z = x, Az, yet in sharp contrast to real or complex Hilbert spaces if for some closed subspace U of E A(U ) ⊆ U then U = {0} or U = E. It is easily proven (see [1]) that such an A must be injective, and that all operators C which commute with A are also injective. Therefore theorem 2.5 gives a criterion for the existence and boundedness of their inverses. Examples of such operators can be found in [1], [5] and [8]. In all cases an explicit description of their (infinite) matrices are given. Following the ideas of [6] we give now two theorems in which the boundedness condition of theorem 2.5 is translated in terms of the matrix of the operators. We need some preliminary definitions. Let E, F be X-normed spaces. An operator B : E → F is called strictly Lipschitz if there exists g ∈ G such that Bx < gx for every x = 0 in E. For a strictly Lipschitz operator B : E → F we define Γ∼ B := {g ∈ G : Bx < gx for every x = 0 ∈ E }


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and the strictly Lipschitz norm of B is B∼ := inf G# ∪{0} Γ∼ B . In [4] 2.2.4, it is proven that ∼ is a norm for the K-space of strictly Lipschitz operators. Definition 2.6. Let E be an X-normed vector space and e1 , e2 , ... an orthogonal base of E. For m, n ∈ N we define the operator Pmn : E → E as Pmn (ek ) = δkn em (k ∈ N). Notice that the matrix of Pmn has zero entries except for a 1 in the n’th column, m’th row. We use this to compare B with the set {amn Pmm : m ∈ N}. Lemma 2.7. For every m, n ∈ N let ‫ג‬mn = {g ∈ G : em < gen }, then Pmn ∼ =

inf

G# ∪{0}

‫ג‬mn .

Proof. See [4] 3.3.1. / ‫ג‬mn ⇐⇒ g < h for all h ∈ ‫ג‬mn . Clearly for any g ∈ G, gen ≤ em ⇐⇒ g ∈

Theorem 2.8. Let E be an X-normed Banach space with orthogonal base {ei : i ∈ N} and A : E → E a continuous linear operator with matrix ⎛ ⎞ a11 a12 a13 · · · ⎜ a21 a22 a23 · · · ⎟ ⎜ ⎟ ⎜ a31 a32 a33 · · · ⎟ . ⎝ ⎠ .. .. .. .. . . . . with respect to that base. Let k(n) := min{j ∈ N : Aen = aj n ej }. There exists g ∈ G such that gei ≤ Aei for all i ∈ N if and only if the sequence { ak(n)n Pk(n)n ∼ }n∈N is bounded below in G. Proof. Suppose that for some g ∈ G we have that gei ≤ Aei for all i ∈ N. Pick n ∈ N, then gen ≤ Aen = ak(n)n ek(n) , hence for g = g|ak(n)n |−1 we have that g en ≤ ek(n) . Then g < h for every h ∈ ‫ג‬k(n)n , and g ≤ Pk(n)n ∼ . Hence g is a lower bound for the sequence {ak(n)n Pk(n)n ∼ }n∈N . Suppose now that the sequence {ak(n)n Pk(n)n ∼ }n∈N is bounded below by some g ∈ G. We can assume, without loss of generality, that for all n ∈ N g < ak(n)n Pk(n)n ∼ . Then for any i ∈ N, g = g|ak(i)i |−1 < inf G# ∪{0} ‫ג‬k(i)i , hence g ∈ / ‫ג‬k(i)i and g ei ≤ ek (i) . Therefore gei ≤ ak(i)i ek(i) = Aei . Since i is chosen arbitrarily, the proof of the theorem is complete.

´ HECTOR M. MORENO

118 8

Let E be a type separating space with orthogonal base {ei : i ∈ N} and A ∈ Lip(E) with matrix ⎛ ⎞ a11 a12 a13 · · · ⎜ a21 a22 a23 · · · ⎟ ⎜ ⎟ ⎜ a31 a32 a33 · · · ⎟ ⎝ ⎠ .. .. .. .. . . . . with respect ⎛ a11 a12 ⎜ a21 a22 ⎜ ⎜ a31 a32 ⎝ .. .. . .

to that base. The ⎞ ⎛ a13 · · · ⎜ a23 · · · ⎟ ⎟ ⎜ =⎜ a33 · · · ⎟ ⎠ ⎝ .. .. . .

matrix descomposition of A ⎞ ⎛ 0 ··· a11 0 0 ⎜ a21 0 a22 0 ··· ⎟ ⎟ ⎜ +⎜ 0 0 a33 · · · ⎟ ⎠ ⎝ a31 .. .. .. .. .. . . . . .

a12 0 a32 .. .

a13 a23 0 .. .

··· ··· ··· .. .

⎞ ⎟ ⎟ ⎟ ⎠

gives rise to a descomposition A = D+S with D, S ∈ Lip(E), D a diagonal operator and S a nuclear operator(see [6] 3.3.2). Then, by [6] 3.1.2 we have that lim

m+n→∞, m=n

amn Pmn ∼ = 0.

If there exists g ∈ G for which gei ≤ Aei for every i ∈ N, then (see 2.8) there is a sequence (ak(n)n Pk(n)n ∼ )n bounded below in G. But the previous statement shows that no infinite subset of {amn Pmn ∼ : m = n} is bounded below in G. Hence, there exists N ∈ N such that the set {ann Pnn ∼ : n ≥ N } is bounded below in G which entails the following: Corollary 2.9. Let E be a type separating space and A ∈ Lip(E) with matrix (aij ), then there exists g ∈ G such that gei ≤ Aei for every i ∈ N if and only if for some N ∈ N the set {ann Pnn ∼ : n ≥ N } is bounded below in G. Corollary 2.10. Let E be a type separating space and A ∈ Lip(E) an injective operator with matrix (aij ), then A is invertible, and A−1 ∈ Lip(E) if and only if for some N ∈ N the set {ann Pnn ∼ : n ≥ N } is bounded below in G. Example 2.11. As in [1] let E be the orthomodular space constructed by Hans Keller which is type separating, and A : E → E the operator defined by ∞ 1 1 A(ei ) = (i ∈ N). k=0 Xk ek + 1 − Xi ei The matrix of A with respect to ⎛ 1 ⎜ X1 ⎜ 11 ⎜ ⎜ X12 ⎜ ⎝ X3 .. .

the standard basis {ei :∈ N} is ⎞ 1 1 1 ··· 1 1 1 ··· ⎟ X1 X1 ⎟ 1 1 1 ··· ⎟ X2 X2 ⎟. 1 1 1 ··· ⎟ X3 X3 ⎠ .. .. .. .. . . . .


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In [5], Carla Barrios described two classes of perturbations of A. Firstly the operators BQ,s : E → E defined by BQ,s (ei ) BQ,s (eqj ) BQ,s (es )

(i = q1 , q2 , ..., qp , s) 1 = A(eqj ) − es (j = 1, ..., p) Xs p 1 = A(es ) − eqj . X qj j=1 = A(ei )

with p, s ∈ N such that 1 < p < s and Q = {q1 , ..., qp } with q1 < q2 < ... < qp and qj ∈ {1, ..., s − 1} for j = 1, ..., p. Secondly the operators Bpqr : E → E defined by (i = p, q, r, r + 1) 1 A(ep ) − er Xr 1 1 A(eq ) − er − er+1 Xr Xr+1 1 1 A(er ) − ep − eq Xp Xq 1 A(er+1 ) − eq Xq

Bpqr (ei )

= A(ei )

Bpqr (ep )

=

Bpqr (eq )

=

Bpqr (er )

=

Bpqr (er+1 )

=

for p, q, r ∈ N and r ≥ 3. A, BQ,s and Bpqr are bounded selfadjoint indecomposable operators therefore injective(see [1], 4.3 and [5], 3.8). Moreover a direct computation shows that Aei = ei and since the value group G is quasidiscrete (see [6], 1.2.1) Pii ∼ = g0 :=

inf {g > 1 : g ∈ G} = 1

G# ∪{0}

for all i ∈ N. Therefore if (aij ), (bij ) and (cij ) are the matrices of A, BQ,s and Bpqr respectively in the base {ei : i ∈ N} of E, we observe that the sequences (aii Pii ∼ )i∈N = (g0 |aii |)i∈N (bii Pii ∼ )i∈N = (g0 |bii |)i∈N (cii Pii ∼ )i∈N = (g0 |cii |)i∈N are bounded below in the value group of the valuation. In fact, |aii | = 1 for all i ∈ N and BQ,s and Bpqr differ from A in a finite number of elements of {ei : i ∈ N}. Then, by Corollary 2.10 these operators are also invertible. Example 2.12. Let us again consider the K-vector space E and the operator A of the previous example. In [8] the operator algebra H is defined as H = {B : AB = BA and B(e0 ) =

∞ λi ei where {λi }i converges in K}. X i i=0

It is proven that every element B in H is a bounded selfadjoint injective operator which is completely described by the triple (η, λ, δ) with (1) λ = limi→∞ λi . (2) η is a null sequence, η = {ηi }i≥1 with ηi = λ − λi for every i ≥ 1. ∞ (3) δ = i=0 (λ − λi ).

´ HECTOR M. MORENO

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Lemma 2.13. Let B = (η, λ, δ) ∈ H. (1) If λ = δ we have that limm→∞ (2) If λ = δ we have that

B(em ) em

B(em ) em

= 0.

= |λ − δ| for all sufficient large m. 2

Proof. See [8], 3.6.

With the corollaries 2.9 and 2.10 it is easy to prove anew that this gives a criterion for invertibility. In fact, let B ∈ H. If δ = λ, then by the previous lemma there is no g ∈ Γ such that gei ≤ B(ei ) for every i ∈ N. By 2.5, B is not surjective, and therefore not invertible. Suppose now that δ = λ, by the previous lemma we have that em and B(em ) have the same algebraic type for all sufficiently large m. Therefore, if (βij ) is the matrix of B with respect to the orthogonal base {ei : i ∈ N} we have that B(em ) = |βmm |em and |βmm | = |λ − δ|2 for those m. Since Pii ∼ = g0 for all i ∈ N, by Corollary 2.10 we deduce that B is an invertible operator in Lip(E). References [1] H. Keller and H. Ochsenius, Bounded operators on a non-archimedean orthomodular spaces, Math. Slovaca 45 (1995), 4, 413-434. [2] H. Ochsenius and W. Schikhof, Banach spaces over fields with an infinite rank valuation, In p-Adic Functional Analysis, Lecture Notes in pure and applied mathematics 207, edited by J. Kakol, N. De Grande-De Kimpe and C. Perez-Garcia. Marcel Dekker (1999), 233-293. [3] H. Ochsenius and W. Schikhof, Norm Hilbert Spaces over Krull values fields, Indag. Math., N.S. 17 (2006), 1, 65-84. [4] H. Ochsenius and W. Schikhof, Lipschitz operators on Banach spaces over Krull valued fields. In Ultrametric Functional Analysis, Contemporary Mathematics 384, edited by B. Diarra, A. Escassut, A.K. Katsaras, L. Narici. A.M.S. (2005), 203-233. [5] Carla R. Barrios, Two families of Self-adjoint Indecomposable operators in a orthomodular space, Annales Math´ ematiques Blaise Pascal, Vol. 15, 2 (2008), 267-287. [6] H. Ochsenius and W. Schikhof, Matrix characterizations of Lipschitz operator on Banach spaces over Krull value fields, Bulletin of the Belgian Mathematical Society Simon Stevin, 14 (2007), 2, 193-212. [7] H. Moreno, Un teorema sobre invertibilidad de operadores en Type Separanting Spaces, Tesis de Magister en Ciencias Exactas(Matem´ aticas). Pontificia Universidad Cat´ olica de Chile (2006). [8] H. Keller and H. Ochsenius, An algebra of self-adjoint operators on a non-Archimedean orthomodular space. In p-adic Functional Analysis. Lecture notes in pure and applied mathematics 192, edited by W. Schikhof, C. Pérez Gar´ıa and J. Kakol. Marcel Dekker (1997), 253 - 264. [9] W. Schikhof and H. Ochsenius, Linear homeomorphisms of non-classical Hilbert spaces, Indag. Math., N.S. 10 (1999), 601 - 613. ´ticas, Pontificia Universidad Cato ´ lica de Chile, Casilla 306Facultad de Matema Correo 22, Santiago, Chile. E-mail address: [email protected]


On Monomial dynamical systems on the p-adic n-torus Marcus Nilsson and Robert Nyqvist Abstract. We first study monomial systems on the n-torus modulo a prime number p. We investigate the dynamics by using group structures and directed graphs. Under certain conditions the systems we get by lifting to the p-adic n-torus inherits much of the structure of the system modulo p. We also discuss the characters of the periodic points.

1. Introduction In this article we consider multidimensional discrete dynamical systems over finite fields and over the field of p-adic numbers. One-dimensional monomial dynamical systems over the p-adic numbers have been studied in for example [6], [9], [10], [8] and [11]. Monomial dynamical systems have also been used in the theory of cellular automata, see [2] and [4], and for Boolean networks [13]. In [12] and [3] multidimensional monomial systems over finite fields are investigated, especially systems that only have fixed points as periodic points. Multidimensional dynamical systems over the p-adics have been studied in [1]. 2. Structure of the phase space Definition 2.1. Let m be a positive integer and let Tn(m) = ((Z/mZ)∗ )n . We call Tn(m) the n-torus modulo m. By a monomial system modulo m we mean iterations of the monomial map f : Tn(m) → Tn(m) with f = (f1 , f2 , . . . , fn ) where l

l

fj (x1 , x2 , . . . , xn ) = x1j,1 x2j,2 · · · xlnj,n , for j = 1, 2, . . . , n and 0 ≤ lj,i for i, j = 1, 2 . . . n. We call L = (li,j ) the exponent matrix of the monomial system. Let a = (a1 , a2 , . . . , an ) ∈ Tn(m) and b = (b1 , b2 , . . . , bn ) ∈ Tn(m) The set Tn(m) is a group under the multiplication (a1 , a2 , . . . , an ) (b1 , b2 , . . . , bn ) = (a1 b1 , a2 b2 , . . . , an bn ) mod m. 1991 Mathematics Subject Classification. 11S82, 11F85. Key words and phrases. dynamical systems, polynomials, p-adic numbers. c Mathematical 0000 (copyright Society holder) c 2010 American

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MARCUS NILSSON AND ROBERT NYQVIST

−1 −1 The identity is (1, 1, . . . , 1) and the inverse of a is (a−1 1 , a2 , . . . , an ). The order n of a ∈ T(m) is ord(a) = lcm(ord(a1 ), ord(a2 ), . . . , ord(an )). We will first discribe the dynamics around the fixed point 1 = (1, 1, . . . , 1).

Definition 2.2. Let x ∈ Tn(m) . If f s (x) = x, for some positive integer s, then x is called a periodic point. The smallest such s is called the period of s, and s is said to be an s-periodic point. An element y that is not a periodic point but f t (y) is a periodic point for some positive integer t is called a preperiodic point. Remark 2.3. Since the Tn(m) is a finite set. The elements are either periodic or preperiodic. Definition 2.4. Let x be a periodic point. A preperiodic point y is said to be a preperiodic point to x if f k (y) = x for some integer k and that f j (y) is not a periodic point for any positive j less than k. The set of preperiodic points to x together with x is called the x-tree. As the following theorem shows, the 1-tree completely describes the preperiodic dynamics of f : Tn(m) → Tn(m) . We will need the following lemmas. Lemma 2.5. Let L be the exponent matrix of f . The exponent matrix of f r (the r-fold composition of f ) is Lr . Proof. Assume that the exponent matrix of f r−1 is A = (aj,i ) = Lr−1 . This means that a a fjr−1 (x) = x1 j,1 x2 j,2 · · · xanj,n , for 1 ≤ j ≤ n. (Note that fjr−1 denotes the jth component of f r−1 .) The jth component of f r is then fjr (x) = (f1r−1 (x))lj,1 (f2r−1 (x))lj,2 · · · (fnr−1 (x))lj,n l

= x1j,1 l

a1,1 +lj,2 a2,1 +...+lj,n an,1

· x2j,1

a1,2 +lj,2 a2,2 +...+lj,n an,2

.. . · xlnj,1 a1,n +lj,2 a2,n +...+lj,n an,n . It is clear that the exponent matrix of f r is LA = Lr since the exponent of xi in fjr (x) is lj,1 a1,i + lj,2 a2,i + . . . + lj,n an,i . Lemma 2.6. Let f be the monomial map described by the exponent matrix L. If x, y ∈ Tn(m) , then f s (x y) = f s (x) f s (y), for each positive integer s. Proof. We first prove by induction that fjs (x y) = fjs (x)fjs (y) for 1 ≤ j ≤ n and s ≥ 1. For s = 1 it is true since fj (x y) = (x1 y1 )lj,1 (x2 y2 )lj,2 · · · (xn yn )lj,n l

l

l

l

= x1j,1 y1j,1 x2j,2 y2j,2 · · · xlnj,n ynlj,n = fj (x)fj (y).

ONMONOMIAL MONOMIAL DYNAMICAL DYNAMICAL SYSTEMS SYSTEMS ON THE P ON P-ADIC -ADIC N N-TORUS -TORUS

123 3

If we assume that for s ≥ 2, fjs−1 (x y) = fjs−1 (x)fjs−1 (y) then fjs (x y) = (f1m−1 (x y))lj,1 (f2m−1 (x y))lj,2 · · · (fnm−1 (x y))lj,n = f1m−1 (x)lj,1 f1m−1 (y)lj,1 f2m−1 (x)lj,2 f2m−1 (y)lj,2 · · · fnm−1 (x)lj,n fnm−1 (y)lj,n = fjm (x)fjm (y). This gives us f s (x y) = (f1s (x y), f2s (x y), . . . , fns (x y)) = (f1s (x)f1s (y), f2s (x)f2s (y), . . . , fns (x)fns (y)) = f s (x) f s (y).

Theorem 2.7. The 1-tree is a subgroup of Tn(m) . We denote it by GA and call it the attractor group of the system. The set of periodic points, GP , is also a subgroup of Tn(m) . We also have Tn(m) /GA GP . Proof. We first prove that the GA is a subgroup of Tn(m) . Let x, y ∈ GA . Then there exists sx and sy such that f sx (x) = 1 and f sy (y) = 1. Set s = max(sx , sy ). From Lemma 2.6 it follows that f s (x y) = f s (x) f s (y) = 1 1 = 1, since sx ≤ s and sy ≤ s. Let x−1 be the inverse of x in Tn(m) .Then f sx (x−1 ) 1 = f sx (x−1 ) f sx (x) = f sx (1) = 1. This proves that GA is a subgroup of Tn(m) . Let us now prove that GP , the set of periodic points of f , is a subgroup of Tn(m) . Let x and y be periodic points of f . Let rx and ry be their least periods. Let t = lcm(tx , ty ). Then by Lemma 2.6 f t (x y) = f t (x) f t (y) = x y, since tx and ty divides t. In fact, the lcm(tx , ty ) is the least period of x y. We also have f sx (x−1 ) = x−1 f sx (x−1 ) x = x−1 f sx (x−1 ) f sx (x) = x−1 . Hence, GP is a subgroup. We now prove that Tn(m) /GA GP . Let u = (φ(m))n and let ψ(x) = f u (x). Then ψ is a homomorphism of Tn(m) such that Im ψ ⊆ GP since there are at most u elements in an orbit before it becomes periodic. We have left to prove that Im ψ = GP . Let y ∈ GP and assume that r is the period of y. For s ∈ Z+ satisfying u + s ≡ 0 (mod r) we have ψ(f s (y)) ≡ f u (f s (y)) ≡ f u+s (y) ≡ y

(mod m),

hence ψ is surjective. The rest follows from the fundamental theorem of homomorphisms. Theorem 2.8. The set of preperiodic points to any periodic point is isomorphic as a graph to the 1-tree.

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Proof. Let x ∈ GP and let r be the period of x. Let GA (x) be the set of preperiodic points to x and let Aj (x) ⊆ GA (x) denote the set of points that are mapped onto x after exactly j iterations. Hence we can write GA (x) as a disjoint union Aj (x). (1) GA (x) = j≥1

We will prove that there is a one-to-one correspondence between Aj (1) and Aj (x) for all j. Let a ∈ Aj (1) and let ψ(a) = f sj (x)a, where sj = r − j mod r. We will prove that ψ is bijective. From f j (ψ(a)) ≡ f j (f sj (x) a) ≡ f r (x) f j (a) ≡ x

(mod m)

and the fact that f k (ψ(a)) ≡ x (mod m) it follows that Im ψ ⊆ Aj (x). Assume that ψ(a) ≡ ψ(˜ a) (mod m) then a (mod m) f sj (x)a ≡ f sj (x)˜ ˜ (mod m). Hence, ψ is injective. and since (f sj (x), m) = 1 it follows that a ≡ a We will now prove that ψ is surjective. Let b ∈ Aj (x) and note that (f sj (x))−1 ≡ f sj (x−1 ). Then f j (f sj (x))−1 b)) ≡ f j (b) f r (x x−1 ) ≡ f j (b) f r (1) ≡ 1 (mod m), f k (f sj (x))−1 b) ≡ 1 (mod m) (k < j) and ψ(f sj (x))−1 b) ≡ b (mod m). Hence, ψ is surjective. By this we also have a one-to-one correspondence between GA (1) and GA (x). We need to show that this correspondence is also graph isomorphism. Let a ∈ Aj (1) and b ∈ Aj (x) such that ψ(a) ≡ b (mod m). Morover, let f (a ) ≡ a (mod m) and let b ≡ ψ(a ) (mod m). Note that a ∈ Aj+1 (1) and b ∈ Aj+1 (x). We have to show that f (b ) ≡ b (mod m). This follows from f (b ) ≡ f (f sj+1 (x)a ) ≡ f sj (x)f (a ) ≡ f sj (x)a ≡ b (mod m).

Let us now assume that m has a primitive root. The system of monomial equations y = f (x) mod m is then equivalent to the linear system t1 ≡ l1,1 s1 + l1,2 s2 + . . . + l1,n sn

(mod φ(m))

t2 ≡ l2,1 s1 + l2,2 s2 + . . . + l2,n sn

(mod φ(m))

.. . tn ≡ ln,1 s1 + ln,2 s2 + . . . + ln,n sn

(mod φ(m))

where tj = indr (yj ), sj = indr (xj ) and r is a primitive root modulo m. In terminology of dynamical systems this means that the dynamical system f : Tn(m) → Tn(m) is conjugated to the system g : (Z/φ(m)Z)n → (Z/φ(m)Z)n , where g(s) = Ls and the conjugacy is given by x → (indr (x1 ), . . . , indr (xn )).


125 5

3. Number of cycles The number of fixed points in Tn(m) is the number of (incongruent) solutions to the linear system (L − I)s ≡ 0 (mod φ(m)), where I is the identity matrix. Let us denote the number of r-periodic points by N (L, r, m). Let η(M, m) denote the number of solutions of M x ≡ 0 (mod m). We then have η(Lr − I, φ(m)) = N (L, d, m), d|r

since the solutions of (L − I)x ≡ 0 (mod φ(m)) are either r-periodic points or d-periodic points for a proper divisor d of r. By Möbius inversion we have µ(r/d)η(Ld − I, m). N (L, r, m) = r

d|r

If n = 1, L can be regarded as an integer and µ(r/d)(Ld − 1, φ(m)), N (L, r, p) = d|r

since the number of solutions to the linear congruence (Ld − 1)x ≡ 0 (mod φ(m)) is (Ld − 1, φ(m)). See [6] for details on the one dimensional case. If L is a diagonal matrix then the system consists of n independent monomial congruences ⎧ l y ≡ x11,1 (mod m), ⎪ ⎪ ⎪ 1 ⎪ ⎨y2 ≡ xl22,2 (mod m), .. ⎪ ⎪ . ⎪ ⎪ ⎩ l yn ≡ xnn,n (mod m). In this case we see that η(Lr − I, φ(m)) =

n

r (lj,j − 1, φ(m))

j=1

and N (L, r, p) =

µ(r/d)

d|r

n

d (lj,j − 1, φ(m)).

j=1

Example 3.1. Let us consider the dynamical system f (x, y, z) = (x3 y 3 z, xyz, x3 z), on T3(5) . The exponent matrix of f is ⎛

⎞ 3 3 1 L = ⎝1 1 1⎠ . 3 0 1

Since Ls ≡ 0 (mod 4) (φ(5) = 4) only has the solution (0, 0, 0) the only fixed point of f is 1 = (1, 1, 1). The 1-tree of can be found in Figure 3.1.

126 6


(2, 1, 3)

(3, 1, 2)

(4, 1, 4)

(1, 1, 1)

Figure 1. The 1-tree of f (x, y, z) = (x3 y 3 z, xyz, x3 z) in T3(5) . Figure 2. The component of f (x, y, z) = (x3 y 3 z, xyz, x3 z) containing the 3-cycle. Let us now see if there are any 3-periodic points to this system. For the exponent matrix of f 3 we have ⎞ ⎛ 2 1 2 L3 ≡ ⎝2 1 2⎠ (mod 4). 1 1 1 The system of linear congruences ⎧ ⎨ s1 + s2 + 2s3 ≡ 0 (mod 4) 2s1 + 2s3 ≡ 0 (mod 4) ⎩ s1 + s2 ≡ 0 (mod 4) has the solutions (0, 0, 0) (0, 0, 2) (2, 2, 0) (2, 2, 2). These solutions correspond to the points (1, 1, 1) (1, 1, 4) (4, 4, 1) (4, 4, 4) T3(5) .

in Since (1, 1, 1) is a fixed point there are three 3-periodic points (in a 3-cycle). In Figure 3.1 the component of phase space graph containing this 3-cycle is shown. 4. Generalizations to the p-adics In this section we will generalize some of the results from the other sections to the p-adic n-torus. Example 4.1. The system of monomial congruences x1 x22 ≡ 1 (mod 3k ), (2) x21 x2 ≡ 1 (mod 3k ), has the unique solution (1, 1) when k = 1. What happens if we consider the same system modulo 32 instead? Let x1 = 1 + 3a1 and x2 = 1 + 3b1 . If we substitute this into the system modulo 32 we get a1 + 2b1 ≡ 0 (mod 3) 1 + 3(a1 + 2b1 ) ≡ 1 (mod 9) ⇐⇒ . 1 + 3(2a1 + b1 ) ≡ 1 (mod 9) 2a1 + b1 ≡ 0 (mod 3)


127 7

This system has three solutions (a1 , b1 ) = (0, 0), (a1 , b1 ) = (1, 1) and(a1 , b1 ) = (2, 2) this gives us the following solutions to the system (2) (x1 , x2 ) = (1, 1), (x1 , x2 ) = (4, 4), (x1 , x2 ) = (7, 7). Observe that this solutions are congruent to (1, 1) modulo 3 from construction. We could not have other solutions modulo 9 since then there would be other solutions modulo 3. Theorem 4.2. Let h : Tnp → Tnp be the monomial map with exponent matrix L. Assume that p det(L − I). Then there is a one-to-one correspondence between the roots of h(x) = 1 on the n-torus modulo p and the p-adic n-torus. Proof. Let M = (mi,j )n×n = L − I. Let a = (a1 , a2 , . . . , an ) be a solution of hM (X) ≡ 1 (mod pk ). It is clear that if there is a solution x = (x1 , x2 , . . . , xn ) to this modulo pk+1 such that x ≡ a (mod pk ) then x1 = a1 + y1 pk x2 = a2 + y2 pk .. . xn = an + yn pk , where yj ∈ 0, 1, . . . , p − 1 for 1 ≤ j ≤ n. We will show that y1 , y2 , . . . , yn are uniquely determined. First, we observe that mi,j mi,j m −t m m −1 k mi,j (aj +p yj ) = (pk yj )t aj i,j ≡ aj i,j +mi,j yj aj i,j pk (mod pk+1 ). t t=0 Hence, hi (x) =

n

(aj + pk yj )mi,j ≡

j=0

Note that

n

mi,j

aj

mi,j

aj

n

i,s −1 mi,s pk am s

s=0

j=0 n

+

≡ 1 + bi pk

mi,j

aj

j=s

(mod pk+1 ),

j=0

for some bi ∈ {0, 1, . . . , p − 1} and that m k i,s −1 aj i,j ≡ a−1 am s s (1 + bi p )

(mod pk+1 ),

j=s

where

a−1 s

is the inverse of as modulo pk+1 . This gives us n k mi,s ys a−1 (mod pk+1 ). hi (x) ≡ 1 + bi pk + s p s=0

Hence hi (x) ≡ 1 (mod p

k+1

) if and only if n mi,s a−1 bi + s ys ≡ 0 (mod p). s=0

(mod pk+1 ).

128 8


The system of linear congruences ⎛ ⎛ ⎞ ⎞ ⎛ −1 ⎞ m1,1 m1,2 . . . m1,n a1 y1 b1 ⎜ m2,1 m2,2 . . . m2,n ⎟ ⎜ a−1 y2 ⎟ ⎜ b2 ⎟ ⎜ ⎜ ⎟ ⎟⎜ 2 ⎟ ⎜ .. .. .. ⎟ ⎜ .. ⎟ ≡ − ⎜ .. ⎟ ⎝ . ⎝ ⎝.⎠ ⎠ ⎠ . . . mn,1

mn,2

. . . mn,n

a−1 n yn

(mod p)

bn

−1

has a unique solution y = −(M b) a if and only if p det(M ). (0) (0) (0) By induction we have: Given a α(0) = (α1 , α2 , . . . , αn ) solution modulo p we get a solution α = (α1 , α2 , . . . , αn ) in Zp such that α ≡ α(0) (mod p). Remark 4.3. In fact this theorem is just a special case of a multidimensional variant of Hensel’s Lemma. 5. Character of periodic points In this section we will mention some results about the dynamics of an analytic function F over a closed ball U in a non-archimedean Banach space Kn , where K is a non-Archimedian field. The norm of x = (x1 , x2 , . . . , xn ) is given by

x = max |xi |, 1≤i≤n

where |.| is the non-Archimedean absolute value of K. Definition 5.1. Let a = (a1 , a2 , . . . , an ) ∈ Kn and let r be a positive real number. Let U = Ur (a) = {x ∈ Kn ; x − a ≤ r}. A function f : U → Kn such that n αi1 ,i2 ,...,in (xi − ai )ik k=1

and lim

i1 +i2 +···+in →∞

|αi1 ,i2 ,...,in |r i1 +i2 +···+in = 0

is called an analytic function on U . We will also recall some terminology for dynamical systems. We will in princip follow Aguayo et. al. in [1]. In the next section will apply these results to our monomial systems on the p-adic n-torus. Definition 5.2. Let a ∈ U be an r-periodic point of F (a fixed point of G = F r ). We say that a is an attractor if there is an open ball Bρ− (a) ⊆ U such that lims→∞ H(x) = a for each x ∈ Bρ− (a). The set ATT(a) = {x ∈ U ; lim Gs (x) = a}, s→∞

is called the basin of attraction of a (in U ). Definition 5.3. Let a ∈ U be an r-periodic point of F (a fixed point of G = F r ). We say that a is a repeller if there is an open ball Bρ− (a) ⊆ U such that

G(x) − a > x − a for all x ∈ Bρ− (a) and x = a. Definition 5.4. Let a ∈ U be an r-periodic point of F (a fixed point of G = F r ). We say that a is indifferent if there is an open ball Bρ− (a) ⊆ U such that Sρ (a) is invariant of G for ρ < ρ. The ball Bρ− (a) is called a Siegel disk with center a and the union of all Siegel disks with center a is called the Maximal Siegel disk. It is denoted by SI(a).


129 9

In the one dimensional case it is very easy to characterize the periodic points. A fixed point a of f is an attractor if |f (a)| < 1, it is indifferent if |f (a)| = 1 and it is a repeller if |f (a)| > 1. See for example [5] and [7] for details. In the multidimensional case it is not so easy. Instead of the derivative, the Jacobian matrix will play a large role. The norm of a matrix M = (mij ) is defined by

M = max |mi,j |. i,j

This norm is however not enough to determine the character of the periodic points. We will need som extra conditions on the Jacobian matrix. Definition 5.5. A matrix M is called a Single maximum in row and column matrix (a SMIRC-matrix) if there is a permutation σ of {1, . . . , n} such that |m1,σ(1) | = |m2,σ(2) | = . . . = |mn,σ(n) | = M and |mi,j | < M for each j = σ(i). The following proposition links the SMIRC property to the determinant. Proposition 5.6. Let M = 1. If M is a SMIRC-matrix then | det(M )| = 1. Proof. Let M = (mi,j )n×n . We will prove this by induction on n. First, consider the case when n = 2. For M to be a SMIRC-matrix it is necessary that one of the diagonals contain elements of S1 (0) and the other contain elements of B1− (0). Say that |m1,1 | = |m2,2 | = 1, |m1,2 | < 1 and |m2,1 | < 1. Then | det(M )| = |m1,1 m2,2 − m1,2 m2,1 | = max (|m1,1 m2,2 |, |m1,2 m2,1 |) = 1. Let us assume that the proposition is true for matrices of order n. Let M be a matrix of order n + 1. Let mh,1 be the maximum element in the first column. If we expand det M along the first column we get (3) det(M ) = mh,1 det(Mh,1 ) + mi,1 det(Mi,1 ) i=h

where Mi,1 is the n × n matrix we get if we delete row i and column 1 from M . It is clear that Mh,1 is a SMIRC matrix and that Mh,1 = 1. From the induction hypotesis we have |mh,1 det(Mh,1 )| = 1 and since |mi,1 | < 1 for i = h we have

| det(M )| = max |mh,1 det(Mh,1 )|, mi,1 det(Mi,1 ) = |mh,1 det(Mh,1 )| = 1. i=h

Example 5.7. It is clear that the converse of the above proposition does not hold. Consider for example

1 2 . M= 1 3 The matrix M is not SMIRC but | det(M )|p = 1. We will use the following theorems from [1]. Lemma 5.8. Let M be a SMIRC-matrix. Then

M x = M · x .

130 10


Lemma 5.9. Let F : U → U be an analytic function, and let a ∈ U . Then lim max max |cjα |r |α|−1 = 0,

r→∞ 1≤j≤n |α|≥2

where cjα are the coefficients of Fj around a. If JF = 0, r > 0 satisfies max max |cjα |r |α|−1 ≤ JF (a)

1≤j≤n |α|≥2

and Br (a) ⊆ U then

F (x) − F (u) ≤ JF (a) · x − u

(4) for x, u ∈ Br (a) and (5)

F (x) − F (u) = JF (a) · x − u

for x, u ∈ Br (a) and JF (a) is SMIRC. Theorem 5.10. Let F : U → U be an analytic function. Let a ∈ U be a fixed point. We then have (i) If Jf (a) < 1 then a is an attractor. In addition, if r > 0 satisfies (6)

max max |cjα |r |α|−1 < 1

1≤j≤n |α|≥2

and Br (a) ⊆ U then Br (a) ⊆ A(a). (ii) If Jf (a) = 1 and Jf (a) is a SMIRC-matrix, then a is a center of a Sigel disk. If r > 0 satisfies (6) and Br (a) ⊆ U then Br (a) ⊆ SI(a). (iii) If Jf (a) > 1 and Jf (a) is a SMIRC-matrix, then a is a repeller. 6. Character of the monomial periodic points Let L be the exponent matrix of the monomial map f = (f1 , f2 , . . . , fn ). Assume that (det(L − I), p) = 1. This will guarantee that we can lift the fixed points modulo p to the p-adics. Let a = (a1 , a2 , . . . , an ) be fixed point of f on the torus Tpn , then |ai | = 1 for 1 ≤ i ≤ n. We have ∂fi = max |li,j |.

Jf (a) = max 1≤i,j≤n ∂xj 1≤i,j≤n Note, that this is independent of a. Let g(x) = f r (x). From Lemma 2.5 it follows (r) that g has exponent matrix Lr = (li,j ). If a = (a1 , a2 , . . . , an ) is an r-periodic point of f then ∂gi = max |l(r) |p .

Jg (a) = max i,j 1≤i,j≤n ∂xj 1≤i,j≤n p Observe that the norm of the Jacobian matrix of f and g are independent of the periodic point. Proposition 6.1. Let L be a matrix over Zp . (i) If L < 1 then Lr < 1 for all r > 1. (ii) If L = 1 and L is SMIRC then Lr is SMIRC for r > 1.


131 11

r−1 Proof. (i): Let us assume that Lr−1 < 1, that is |li,j | < 1 for all i, j. Since n r−1 r li,j = li,k lk,j k=1 r we get |li,j | < 1 from the strong triangle inequality. Hence Lr < 1 and the rest follows by induction. (ii): Since L is SMIRC there is a permutation σ such that

|l1,σ(1) | = |l2,σ(2) | = . . . = |ln,σ(n) | = 1 and |li,j | < 1 if j = σ(i). We assume that Lr−1 is SMIRC. Hence, there is a permutation τ such that r−1 r−1 r−1 |l1,τ (1) | = |l2,τ (2) | = . . . = |ln,τ (n) | = 1 r−1 and |li,j | < 1 if j = τ (i). We are now going to prove that there is a permutation ρ such that r r r |l1,ρ(1) | = |l2,ρ(2) | = . . . = |ln,ρ(n) |=1 r | < 1 if j = ρ(i). We have that and |li,j r = li,j

n

r−1 li,k lk,j

k=1 r |li,j |

−1

r so < 1 if not τ (i) = σ (j). Hence |li,j | < 1 if j = σ(τ (i)). If j = σ(τ (i)) r | = 1, where then exactly one term in the sum has absolute value 1 and hence |li,ρ(i) ρ = σ ◦ τ . The Proposition now follows by induction.

Theorem 6.2. If L < 1 then all periodic points are attractors. Proof. Follows directly from Proposition 6.1 and Theorem 5.10.

Theorem 6.3. If L = 1 and L is SMIRC then all periodic points are indifferent (centers of Siegel disks). Proof. Follows directly from Proposition 6.1 and Theorem 5.10.

7. Discussion There are much more things to investigate about these systems. What are the sizes of the maximal Siegel discs the basins of attraction? What happen when

L = 1 and L is not SMIRC? Is it possible to fully characterize this case, or not? References [1] J. Aguayo, M. Saavedra, and M. Wallace, Attractor and repeller points for a several-variable analytic dynamical system in a non-archimeadean setting, Theoretical and Mathematical Physics 140 (2004), no. 2, 1175–1181. [2] R. Bartlett and M. Garzon, Monomial cellular automata, Complex Systems 7 (1993), 367– 388. [3] E. Delgado-Eckert, An algebraic and graph theoretical framework to study monomial dynamical systems over a finite field, arXiv:math.DS/0711.1230 (2006). [4] J. Kari, Theory of cellular automata: A survey, Theoretical computer science 334 (2005). [5] A. Yu. Khrennikov, Non-archimedean analysis: Quantum paradoxes, dynamical systems and biological models, Kluwer, Dordrecht, 1997. [6] A. Yu. Khrennikov and M. Nilsson, On the number of cycles of p-adic dynamical systems, Journal of Number Theory 90 (2001), no. 2, 255–264.

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[7] , p-adic deterministic and random dynamics, Kluwer Academic Publishers, 2004. [8] A. Yu. Khrennikov, M. Nilsson, and R. Nyqvist, The asymptotic number of periodic points of discrete polynomial p-adic dynamical system, Ultrametric functional analysis, seventh international conference on p-adic analysis, Contemporary Mathematics, vol. 319, Am. Math. Soc., 2003, pp. 159–166. [9] M. Nilsson, Cycles of monomial and perturbated monomial p-adic dynamical systems, Ann. Math. Blaise Pascal 7 (2000), no. 1, 37–63. , Distribution of cycles of monomial p-adic dynamical systems, p-adic functional anal[10] ysis (New York–Basel) (A. K. Katsaras, W.H. Schikhof, and L. van Hamme, eds.), Lecture notes in pure and applied mathematics, vol. 222, Marcel Dekker, 2001, pp. 233–142. , Fuzzy cycles of p-adic monomial dynamical systems, Far East J. Dynamical Systems [11] 5 (2003), no. 2, 149–173. [12] O. Colón Reyes, A. S. Jarrah, R. Laubenbacher, and B. Strumfels, Monomial dynamical systems over finite fields, arXiv:math.DS/0605439 (2006). [13] O. Colón Reyes, R. Laubenbacher, and B. Pareigis, Boolean monomial dynamical dystems, Annals of Combinatorics 8 (2004), 425–439. Växjö University, Sweden E-mail address: [email protected] Blekinge Institute of Technology, Sweden E-mail address: [email protected]


On the value group and norms of a Form Hilbert space Herminia Ochsenius and Elena Olivos Abstract. It is an essential feature of Hilbert spaces over R or C that their norm is induced by a Hermitean form. This also holds true in their counterparts, Form Hilbert spaces (FHS), constructed over an involutive field (K,∗ ), with a valuation of infinite rank. But in this case the norm does not take its √ values in the value group G but in the group G ⊇ G, and new questions arise. Our first set of results refer to the interplay between the Hermitean form , , the involution, the valuation | | and the norm |x, x|. Now, in the study of Lipschitz operators in a FHS there appear naturally classical inequalities √ such as Ax ≤ A · x. Here Ax and x belong to G while A is an element of the semigroup G# , the Dedekind completion of G, which does not √ √ in general contain G. The main questions concerns the density of G in G, √ and the continuity of G as a G module. We study these topics, firstly in a general setting of ordered, infinite rank groups, and then in the particular case √ of G.

1. Preliminaries We state first the necessary definitions and facts. Let (G, ·) be an abelian multiplicatively written totally ordered group. If G is isomorphic to a subgroup of (R+ , ·) we say that it has rank one. If that is not the case, we shall define the rank of G in the following way. Let H be the set of convex subgroups of G, that is those subgroups H such that for all h > 1 in H and g ∈ G, h−1 < g < h implies g ∈ H. Call such a subgroup H principal if there exists an 1 < h ∈ H such that H = n∈N (h−n , hn ). Now, the set of principal convex subgroups different from G is totally ordered by inclusion. The order type of this set is the rank of G. Another simple yet powerful clasification of totally ordered groups is obtained by considering the set of the elements greater than 1G . If this set has a minimal element we shall say that G is quasidiscrete; otherwise it is quasidense. It must be remarked that a quasidiscrete group may have quasidense subgroups (see [4]), 2000 Mathematics Subject Classification. Primary 46S10; Secondary 47A30. Key words and phrases. Krull valuation, Form Hilbert spaces, Hermitean form. Research Partially supported by Fondecyt 1080194, and by DIUFRO 108-0008. c Mathematical 0000 (copyright Society holder) c 2010 American

1 133

134 2

HERMINIA OCHSENIUS AND ELENA OLIVOS

it can also be proved that if G is quasidense, every nonempty interval of G has infinite elements (see [7]). An important source of non-trivial examples of infinite rank groups is given by a class of Hahn products defined as Γα . Their structure is the following. Let α be an ordinal. For each β < α, let Gβ be a totally ordered group of rank 1. The group Γα associated to the family {Gβ }β 1. But () does imply (V). Proposition 2.1. If there exists α ∈ K, such that |α∗ | = |α|, then neither () nor (C) can hold in any Hermitean K-space (E, , ). Proof. Suppose there exists α ∈ K such that |α∗ | < |α|. Let e ∈ E \ {0} and put u = αe, v = α∗ e. Then we have | u + v, u + v| = | (α + α∗ )e, (α + α∗ )e| = |α + α∗ |2 | e, e| = |α|2 | e, e| and | u, u| = | v, v| = |α||α∗ || e, e|. Hence, | u + v, u + v| > max{| u, u|, | v, v|} and () does not hold. In addition | u, v| = | αe, α∗ e| = |α e, eα| = |α|2 | e, e| > | u, u|. Therefore, we have | u, v|2 > | u, u|| v, v| and (C) is not true. The next example shows that there can be K-spaces where (P) is valid, even if (V) does not hold. Example 2.2. There exists a space (E, , ), such that it is not true that |α | = |α| for all α ∈ K, yet (P) is valid. ∗

Proof. We construct such a space. Let x1 , x2 , x3 be variables. For n = 1, 2, 3, we put F0 := Q, Fn := Fn−1 (xn ). To order Fn by powers of xn it is enough to s determine the positive polinomials in Fn−1 [xn ] ; hence we define p(xn ) = a i xn > i=0

0 if and only if as > 0 in Fn−1 . Let K := F3 with this non-archimedean order. The involutive automorphism ∗ on K is defined by x∗1 = x2 , x∗2 = x1 , x∗3 = x3 . Let Γ := g1 × g2 × g3 , with the antilexicographical ordering, and define a valuation | | : K → Γ by |q| = 1 for all q ∈ Q, |x1 | = (g1−1 , 1, 1), |x2 | = (1, g2−1 , 1), |x3 | = (1, 1, g3−1 ). Now let E be a K-two dimensional vector space generated by the vectors e and f , and , the Hermitean form defined by αe + βf, µe + δf = αµ∗ + βδ ∗ x3 . First, we prove a lemma. Lemma 1: ∀µ ∈ K |µµ∗ | = |x3 |. m µ1 , where µ1 = i=0 ai xi3 , µ2 = µ 2 n j j=0 bj x3 where ai , bj are elements of Q(x1 , x2 ). Then Proof. We may assume 0 = µ ∈ K, µ =

µµ∗ = (µ1 µ∗1 )(µ2 µ∗2 )−1 =

2m k=0

ck xk3

2n h=0

−1 dh xh3

ON THE VALUE GROUP AND NORMS OF A FORM HILBERT SPACE

137 5

where every ck , dh belong to Q(x1 , x2 ). By comparison of degrees we have that µµ∗ = x3 . Observe now that |µµ∗ | = |c2m ||x3 |2m (|d2n ||x3 |2n )−1 , so if |µµ∗ | = |x3 | we would have |x3 |2n+1−2m = |c2m ||d2n |−1 . But this is not possible, because in the right side we have an element of g1 × g2 . Note: In addition, this proposition shows that the form , is anisotropic. We can now prove that for each x, y ∈ E, x, y = 0 implies | x + y, x + y| = max{| x, x|, | y, y|}. We distinguish three cases: 1. Let x = αe, with α = 0, then x, y = 0 implies y = βf . Hence, | x + y, x + y| = |αα∗ + ββ ∗ x3 | = max{|αα∗ |, |ββ ∗ x3 |} = max{| x, x|, | y, y|}, since |αα∗ | = |ββ ∗ x3 |. 2. Let x = e + αf, y = e + βf . Then x, y = 0 if and only if β = −(α∗ x3 )−1 , which entails that | x, x| = |1 + αα∗ x3 | = max{1, |αα∗ x3 |} and | y, y| = |1 + ββ ∗ x3 | = max{1, |αα∗ x3 |−1 }. Therefore | x, x| = | y, y| and we have | x + y, x + y| = max{| x, x|, | y, y|}. 3. Let x = ae + bf with a = 0, b = 0 and y = ce + df a non-zero vector such that

x, y = 0. Write x = au with α = ba−1 , u = e + αf and also y = cv with β = dc−1 and v = e + βf . Since x, y = 0 if and only if u, v = 0, we have β = −(α∗ x3 )−1 . Therefore | x, x| = | au, au| = |aa∗ || u, u| and | y, y| = | cv, cv| = |cc∗ || v, v|. If |αα∗ x3 | < 1, then | u, u| = 1 and | v, v| = |αα∗ x3 |−1 . Thus | x, x| = |aa∗ | and | y, y| = |cc∗ ||αα∗ x3 |−1 , so | x, x| = | y, y| since Lemma 1, with µ = c(aα)−1 , gives a contradiction. A similar argument works in the case |αα∗ x3 | > 1 and therefore (P) holds true. We have established that if (V) does not hold in K then neither () nor (C) can be true in any Hermitean K-space (E, , ), but that (P) can be valid. Therefore in the absence of (V), (), (P) and (C) are no longer equivalent conditions. But the next proposition shows that if (C) is valid in (E, , ) then the form induces a non-archimedean norm on E and form orthogonality implies norm orthogonality. Proposition 2.3. Let (E, , ) be as stated at the start of this section. If (C) is true in E then so are () and (P). Proof. The implication (C)→() was proved in [1], we recall the argument for the reader’s convenience. Notice first that x, y∗ = y, x and (C) imply that | x, y∗ |2 = | y, x|2 ≤ | x, x|| y, y| ≤ max{| x, x|2 , | y, y|2 } hence | x, y∗ | ≤ max{| x, x|, | y, y|}. Now | x + y, x + y| = | x, x + x, y + x, y∗ + y, y| ≤ max{| x, x|, | x, y|, | x, y∗ |, | y, y|} = max{| x, x|, | y, y|} and the triangle inequality holds. On the other hand, the proof of ()→(P) given by Gross and K¨ unzi makes use of the conditions char(K) = 2 and |2| = 1. But it can be replaced by the following. Let x, y = 0. Then | x, x|2 = | x, x + y|2 ≤ | x, x|| x + y, x + y|. Hence, | x, x| ≤ | x + y, x + y|. Symmetrically, | y, y| ≤ | x + y, x + y|. That means

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| x + y, x + y| ≥ max{| x, x|, | y, y|} and since () is valid, we obtain the Pythagorean relation. We now turn to the following, more general, question. Is it possible that if (V) is not valid in K there exists a K-space (E, , ) where the Hermitean form defines a vector space topology on E? In the first place we observe that this is true in the above example for the topology where the family {Bk }k 1. Therefore, / U1G . we have u, v ∈ Uδ but u + v ∈

3. Density of groups of infinite rank In this section we consider two totally ordered abelian groups G G of arbitrary rank, and obtain conditions for G to be dense in G in the sense of the next definition. It will be shown that the convex subgroups of each one turn out to be relevant for this. Definition 3.1. Let A and B be two ordered sets. We shall say that B is dense in A if for every a ∈ A both supA {x ∈ B : x ≤ a} and inf A {x ∈ B : x ≥ a} exist and are equal to a. Clearly in arbitrary ordered sets one of the equalities does not imply the other. In fact, let us consider the sets A = {1 − n1 : n ≥ 2} ∪ {2 − n1 : n ≥ 2} ∪ {1} and B = {1 − n1 : n ≥ 2} ∪ {2 − n1 : n ≥ 2}. Then 1 = supA {x ∈ B : x ≤ 1}, but inf A {x ∈ B : x ≥ 1} = 32 > 1. The case of ordered groups is different.


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Proposition 3.2. Let G G , be two totally ordered abelian groups. Then for every x ∈ G , x = supG {g ∈ G : g ≤ x} if and only if for all u ∈ G , u = inf G {g ∈ G : g ≥ u}. Proof. Pick u ∈ G , by hypothesis u = supG {g ∈ G : g ≤ u}. Now let U = {g ∈ G : g ≥ u} and U −1 = {g ∈ G : g −1 ∈ U } = {g ∈ G : g ≤ u−1 }, again by our hypothesis, supG U −1 = u−1 . Let t be a lower bound of U , then we have that t ≤ g for every g ∈ U , hence t−1 ≥ g −1 for all g −1 ∈ U −1 , and t−1 is an upper bound of U −1 . Therefore u−1 ≤ t−1 , and t ≤ u, it follows that u = inf G U . 3.1. The convex subgroups of G and G . Let G G be two abelian multiplicative groups, totally ordered, of rank greater than 1 and let H := {H : H is a convex subgroup of G} and U := {U : U is a convex subgroup of G } be the sets of convex subgroups of G and G respectively. For any H ∈ H, we define the convex hull of H in G by H := {g ∈ G : ∃h1 , h2 ∈ H, h1 ≤ g ≤ h2 }. Consider now the maps ϕ : H → U, ϕ(H) = H and η : U → H, η(U ) = U ∩ G. A routine argument shows that both of them are well defined. Now we have the following. Proposition 3.3. i: Both maps preserve the order given by inclusion. ii: ϕ is an injective function; η is surjective. iii: η ◦ ϕ = IdH , but ϕ ◦ η(U ) ⊆ U . Proof. i. It is an immediate conclusion. ii. ϕ is an injective function. Let H1 , H2 ∈ H with H1 H2 . There exists h2 ∈ H2 \ H1 , hence h2 ∈ H2 . If it were the case that h2 ∈ H1 , there would exists h1 and h3 in H1 such that h1 ≤ h2 ≤ h3 , but then h2 ∈ H1 , a contradiction. In order to prove that η is surjective, let H be in H and therefore H ∈ U. We have that η(H) = H ∩ G = {g ∈ G : ∃h1 , h2 ∈ H1 h1 ≤ g ≤ h2 }. Hence H ∩ G is a subset of the convex subgroup H. This means that H ∩ G = H, and η is surjective. iii. The preceding argument shows that η ◦ ϕ(H) = H. Now, pick U ∈ U, and z ∈ ϕ ◦ η(U ) = U ∩ G. Then there exists u1 , u2 ∈ U ∩ G such that u1 ≤ z ≤ u2 , since U is a convex subgroup of G , we have that z ∈ U and ϕ ◦ η(U ) ⊆ U . Even in the case of two groups with the same finite rank, it can happen that ϕ is not surjective, nor is η injective, and therefore ϕ ◦ η = IdU as the following example shows. Example 3.4. For i = 1, 3 let Gi = Gi = (Q+ , ·); let G2 = {1} and G2 = (Q , ·). The group G = G1 × G2 × G3 is a subgroup of G = G1 × G2 × G3 . The order in both cases is antilexicographic. Now H1 = {(a, 1, 1) : a ∈ Q+ } is a convex subgroup of G. U1 = G1 × {1} × {1} is a convex subgroup of G . U2 = G1 × G2 × {1} is a convex subgroup of G . +

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Then a) η(U1 ) = U1 ∩ G = H1 = η(U2 ) and η is not injective, b) U2 is not the convex hull of any H ∈ H, hence ϕ is not surjective, c) ϕ ◦ η(U2 ) = ϕ(H1 ) = H1 = H1 = U1 U2 . Notice also that (U2 \ U1 ) ∩ G = φ. 3.2. Conditions for G to be dense in G . Proposition 3.5. Let G G . If G is dense in G then the map ϕ, defined as above, is a bijection between H and U. Therefore G and G have the same rank. Proof. Suppose that ϕ is not surjective, then there exists a convex subgroup U of G which is not the image of any convex subgroup of G. But U ∩ G is a convex subgroup of G and ϕ(U ∩ G) ⊆ U . Hence there is a u ∈ U \ G which is an upper bound of U ∩ G. Therefore for all g ∈ U ∩ G, gu ∈ / G is an upper bound of U ∩ G. Since gu < u if g < 1, G cannot be dense in G . The next lemma shows that a coset of a convex subgroup is an ‘interval’ of the group; its corollary relates this to density. Lemma 3.6. Let g ∈ G and H be a convex subgroup of G. If z in G is such that gh1 ≤ z ≤ gh2 for some h1 , h2 in H, then z ∈ gH. Proof. From gh1 ≤ z ≤ gh2 it follows that 1 ≤ zg −1 h−1 ≤ h−1 h2 . Since H is a convex subgroup, zg −1 h−1 ∈ H, hence zg −1 ∈ H and z = gh0 for some h0 ∈ H. Corollary 3.7. Let G G , and g ∈ G . If there exists a convex subgroup U of G such that g U ∩ G = φ, then for every g ∈ G, g < g implies that g < g u for all u ∈ U . Therefore G is not dense in G . Proof. If it were true that for some u ∈ U we have that g u < g then g u < g < g and by the previous lemma g ∈ g U , a contradiction. Our main theorem in this section states conditions both necessary and sufficient for G to be dense in G . Theorem 3.8. Let G G be two groups of the same rank ρ > 1. Let U∗ = {U ∈ U : U = {1}}. 1. If U∗ = {1} then G is dense in G if and only if for every g ∈ G and U ∈ U∗ we have that g U ∩ G = φ. 2. If U∗ = U0 = {1} then G is dense in G if and only if U0 ∩ G is dense in U0 and for every g ∈ G there is a g in G with g −1 g ∈ U0 . Proof. We will first prove [1]. (⇒) Suppose that there is a g ∈ G and U ∈ U∗ such that g U ∩ G = φ, then by the Corollary 3.7, G is not dense in G . (⇐) Let g ∈ G and A = {g ∈ G : g ≤ g }, for any z ∈ G , z ≤ g , let U be a convex subgroup of G that does not contain z(g )−1 . By hypothesis there exists z1 ∈ g U ∩ G and, without loss of generality, z1 ∈ A. Now z < z1 ≤ g , since if this


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were not true we would have, by the previous Lemma, that z ∈ z1 U . Therefore supG A = g and G is dense in G . The proof of [2]. (⇒) follows the same argument as in case [1]. (⇐) Let g ∈ G , by hypothesis there exists g ∈ G with g −1 g ∈ U0 . Take z ∈ G , z < g , by the Lemma we can assume, without loss of generality , that z ∈ g U0 = gU0 . Therefore g −1 z ∈ U0 , and g −1 z < g −1 g . Since U0 ∩ G is dense in U0 , there is a b ∈ G such that g −1 z < b < g −1 g . Hence z < bg < g , and since bg ∈ G we have that supG {g ∈ G : g ≤ g } = g . The condition G is dense in G is a strong one, it places clear restrictions on the groups, and in the set of their convex subgroups. Let G G , be totally ordered groups of the same rank. It is easy to see that if for some convex subgroup U0 of G we have that U0 = U0 ∩ G, then U = U ∩ G for any convex subgroup U of G such that U ⊆ U0 . In addition, U = U=U∩G U is also a convex subgroup of G that satisfies U = U ∩ G. Proposition 3.9. With the same notation as before, if G = G and U = {1}, then G is not dense in G Proof. Let z ∈ G \ G, a fortiori z ∈ / U. Since zu = g, with u ∈ U implies z ∈ G, then zU ∩ G = φ. We recall (see [9]) that if H is a convex subgroup of G, then G/H is totally ordered by the relation g1 H ≤ g2 H if and only if (g1 H = g2 H ∧ g1 < g2 ) ∨ (g1 g2−1 ∈ H). Proposition 3.10. If G is dense in G and U is a convex subgroup of G then G/(U ∩ G) is dense in G /U (with the canonical embedding of G/(U ∩ G) in G /U , g(U ∩ G) → gU for all g ∈ G). Proof. Let us suppose that G is dense in G and zU ∈ G /U . Define A := {g(U ∩ G) ∈ G/(U ∩ G) : gU ≤ zU }, clearly zU is an upper bound of A. Since G is dense in G , by Corollary 3.8, one has that zU ∩ G = φ. Hence, there exists g ∈ G such that gU = zU and obviously this is the maximum of A, and the proposition is proved. Proposition 3.11. Let G G . If G is dense in G then both groups are quasidense. Proof. Let us first suppose that G is quasidiscrete. Then there exists an element g0 ∈ G, the immediate successor of 1, which generates a first non-trivial convex subgroup in G, call it H0 . If there exists an element u ∈ G \ G such that 1 < u < g0 , then supG {g ∈ G : g ≤ u} = 1 = u. Hence G is not dense in G . If no such u exists, then g0 is the immediate successor of 1 in G , so H0 is the first non-trivial convex subgroup of G (if g0n < z < g0n+1 in G , then 1 < zg0−n < g0 , a contradiction). Hence we have that G and G share the convex subgroup H0 and by Proposition 3.10, G is not dense in G .

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Assume now that G is quasidense and G quasidiscrete. Let u0 ∈ G be the im/ G, since that would imply the existence of mediate successor of 1G . Clearly u0 ∈ infinite elements of G between 1 and u0 , which contradicts the fact that G ⊆ G . But then supG {g ∈ G : g ≤ u0 } = 1 and G is not dense in G . √ 4. The group G √ We start this section by giving the construction of G, we define an order compatible with the order of G, and study its convex subgroups.√ We prove next that it is a continuous G-module and discuss the density of G in G. Finally there are some remarks about norms. Let G be a multiplicative abelian group totally ordered by ≤, and G the divisible hull of G ([2] IV. 24). Define G0 = {h ∈ G : h2 ∈ G}, and ϕ: G0 → G be the epimorphism defined by ϕ(h) = h2 . Then Ker ϕ = {h ∈ G0 : h2 = 1} and, since G is totally ordered, Ker ϕ ∩ G = {1}. √ Definition 4.1. G := G0 / Ker ϕ. √ √ G is canonically embedded in G as the subgroup of squares of G. Additionally √ 2 ∼ G = G by the induced isomorphism ϕ(z ˆ Ker ϕ) = ϕ(z) = z . √ Lemma 4.2. Let h, k ∈ G. Then h2 = k2 if and only if h = k. Proof. Let h = a Ker ϕ, k = b Ker ϕ for some a, b ∈ G0 and assume h2 = k2 . Pick w ∈ a2 Ker ϕ ∩ G, then w ∈ G and for some z ∈ Ker ϕ, w = a2 z. But a2 ∈ G, therefore z ∈ G, and since Ker ϕ ∩ G = {1}, w = a2 . Therefore a2 Ker ϕ ∩ G = {a2 } and also b2 Ker ϕ ∩ G = {b2 }. It follows that a2 = b2 . As G is an ordered group a = b, therefore h = k. The other implication is trivial. √ 2 x = g will Notation: Letg ∈ G. The unique solution z ∈ G of the equation g h = gh and, setting be denoted by g. Then it is clear that for g, h ∈ G −1 h = g −1 , g −1 = g . Lemma 4.3. The order of G induces a total ordering in if and only if h2 ≤ k2 in G.

√

G by h ≤ k in

√ G

Proof. Clearly ≤ is a reflexive and transitive relation. By the previous √ lemma it is also antisymmetric. A direct computation shows that for any h,√ k, z ∈ G, h ≤ k implies hz ≤ kz. Since G is totally ordered, the same is true for G.

Example 4.4. Consider the classical group G = i∈N Gi , where each Gi is a √

cyclic group generated by gi . Then the group G = i∈N Hi , where each Hi is the cyclic group generated by hi such that h2i = gi . √ Example 4.5. If G = Γα with α an ordinal, then G is again a Γα -group associated to the family Gβ β 0. Let b ∈ Gβ \ Gβ and consider √ the element χ(β,b) ∈ Γα , the characteristic funtion where χ(β,b) (β) = b and it is / Γα for all g ∈ Γα , but this contradicts equal to one in other case. Then χ(β,b) g ∈ the Theorem 3.8. √ (⇐) On the other hand, if Gβ = Gβ for every β > 0, then any element z ∈ Γα

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can be written as the product z = h0 · g with h0 ∈ √ Theorem 3.8, Γα is dense in Γα .

√

H0 and g ∈ Γα . Again, by

√ The relation between G and the Dedekind completion of G shows yet another striking difference between the case of groups of rank one and the general case. From the theorem of density we obtain directly. √ √ Corollary 4.10. If G is not dense in G, then G is not contained in G# . Example 4.11. Consider the group Γω where Gβ = Q+ for all β < ω. Then √ √ the family { Q+ } is associated to Γω ( see [8]). √ √ √ The element z = (1, 2, 2, 1, . . .) belongs to Γω but it does not belong to the √ completion of Γω . Furthermore, the interval (zt0 , zs0 ) whose elements are √ {(a, 2, 2, 1, . . .) : a ∈ Q+ } does not contain any element of Γω . √ √ Proposition 4.12. G# ⊆ ( G)# and supG# H = sup(√G)# H = sup(√G)# H. √ Proof. The embedding g → g 2 allows us to consider G ⊆ G and therefore √ G# ⊆ ( G)# under the map x ∈ G# → sup√G# {g ∈ G : g ≤ x}. Moreover, by the √ G and the cofinality of correspondence between the convex subgroups of G and √ every convex subgroup H in√ H, the supremum (and infimum) of corresponding convex subgroups in G and G are identified. 4.2. √ Norms. We start with some propositions about norms in a√ quotient group of G. Let E be a Hermitean space with norm x = x, x ∈ G ∪ {0}. √ √ Proposition 4.13. Let H be a convex subgroup of G, then G/H ∼ = G/ H. √ √ √ √ √ Proof. Let T : G/H → G/ H be given by T ( gh) = g H. A direct verification shows that T is an isomorphism; in fact, an order isomorphism. √ convex subgroups, then Proposition √ √4.14. If H and H are corresponding √ H : E → G/ H defined by xH = x H is a norm on E that corresponds to the valuation | |H de K. Proof. Let K be a valued field | | : K → G ∪ {0} and H a convex subgroup of G. The map |α|H = |α|H = |α|H if α = 0 and |0|H = 0 is a valuation from K to G/H ∪ {0} that induceson K the same topology √ as | | (see √ [9]). √ We define the H on E by xH = T ( | x, x|H ) = | x, x| H in G/ H if x = 0 and 0H = 0. Clearly this is a norm on E with λxH = |λ|H xH . Proposition 4.15. The norms and H define the same topology in E. Proof. Denote by Bρ and BρH the ‘closed’ balls of center 0 and radius ρ in H . the topology given by and H respectively. Then for any δ ∈ G, B√δ ⊆ B√ δ And, with η < 1 ∈ G, BηH√δ ⊆ B√δ .


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We discuss now some facts about the norm of operators in the case of Form Hilbert spaces. Let E be a FHS over K, a√field with a Krull valuation with value group G. Then E is a Banach space with a G-norm and the norm of operators on E take their value on G# . Recall that L (E) is the set of continuous linear maps, Lip (E), the set of Lipschitz maps and Lip∼ (E), the set of strictly Lipschitz √ maps from E to E. Let us briefly summarize the central facts about them. √ As G is a group that contains G it is a faithful G-module (that is, for all z ∈ G, Stab (z) = {g ∈ G : gz = z} = {1}) we have that (see [4] 2.1.14 and 2.1.16) Lip∼ (E) = Lip (E) L(E). If G is quasidense, A∼ = A ([4] 2.2.11), but if G is quasidiscrete either A∼ = A or A∼ = g0 A where g0 = min{g ∈ G : g > 1} for all A ∈ Lip (E) ([4] 2.2.13). With respect to the classical inequalities, for A, B ∈ Lip (E), AB ≤ A∗B,by ([4] 2.2.16), but Ax ≤ A ∗ x must be proven. √ Proposition 4.16. The G-module ( G)# is continuous. Therefore for all x ∈ E, Ax ≤ A ∗ x Proof. It is well known (see [4]) that if G is quasidiscrete every G-module is continuous, therefore we only √ have to prove the proposition for the case of G quasidense. By Proposition 4.7, G (as a group) is quasidense if and only if G is quasidense and by [7], every open interval of a quasidense group contains infinite √ # points. We must show that r = inf √G# {gr : g ∈ G ∧ g > 1} for r ∈ G (see √ [4] 1.6.8). Let us first consider the case of r ∈ G. Define a := inf √G# {gr : g ∈ G ∧ g > 1}. The that 1 < r < a. Since √ case r = 1 follows from quasidensity of G. Suppose now √ G is quasidense, there exists g0 ∈ G such that 1 < r < g0 < a, hence 1 < r 2 < g0 < a2 en G. Multiplying by r −2 and g0−1 , we obtain 1 < g0 r −2 and 1 < g0−1 a2 , two elements of G strictly greater than 1, therefore a ≤ g0 r −2 r ∧ a ≤ g0−1 a2 r. Thus g0 = ar for each g0 between r 2 and a2 and that is a contradiction. Therefore a = r. The case r < 1 is analogous. √ √ # √ G) \ G. If r < a, there exists z ∈ G with Now we consider the case of√r ∈ ( √ r < z < a (by the density of G in ( G)# ). Hence, for every g ∈ G, g > 1, we have gr < gz, therefore inf (√G)# {gr : g ∈ G ∧ g > 1} ≤ inf (√G)# {gz : g ∈ G ∧ g > 1}. But this means that a ≤ z, a contradiction. The last assertion follows from [4] 2.2.17. √ Remark 4.17. Note the G-module G is also continuous, being a submodule √ of ( G)# . However it seems natural to consider yet another norm for operators

on a FHS. Let x √ : x = 0 . T : E → E be a Lipschitz map, we can set T sq = inf ( G)# ∪{0} T x √ This is well defined since, by definition, there exists g ∈ G√⊂ G such that T x ≤ gx. It is more precise in √ the sense that for any z ∈ G \ G, √ supG# {g ∈ G : √ √ g < z} < sup G# {w ∈ G : w < z} = z = inf G# {w ∈ G : w > z} < inf G# {g ∈ G : g > z}, hence T sq ≤ T . It is also clear from the definition of

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√ T x T sq , that in ( G)# it is true that T sq ≤ for any x = 0 in E. Therefore, x T sq x ≤ T x for all non zero vectors in E. The study of this new norm remains to be done. References [1] Gross, H. and K¨ unzi, U.M. On a class of orthomodular quadratic spaces. L’ Enseignement Math. 31 (1985), 187-212. [2] Fuchs, L. Partially ordered algebraic systems. Pergamon Press, 1963. [3] Ochsenius, H. and Schikhof, W.H. Banach spaces over fields with an infinite rank valuation. In p-Adic Functional Analysis, Lecture Notes in pure and applied mathematics 207, edited by J. Kakol, N. De Grande-De Kimpe and C. Perez-Garcia. Marcel Dekker (1999), 233-293. [4] Ochsenius, H. and Schikhof, W.H. Lipschitz operators on Banach spaces over Krull valued fields. Contemporary Mathematics. Volume 384, A.M.S. (2005), 203-233. [5] Ochsenius, H. and Schikhof, W.H. Norm Hilbert spaces over Krull valued fields. Indag. Mathem., N.S., 17(1), (2006), 65-84. [6] Olivos, E. A family of totally ordered groups with some special properties. Annales Mathematiques Blaise Pascal 12, (2005), 79-90. [7] Olivos, E.. Soto, H. and Mansilla, A., Metrizability of totally ordered groups of infinite rank and their completions. Bulletin of the Belgian Mathematical Society Simon Stevin. Vol 14 (2007), 969-977. [8] Olivos, E.. Soto, H. and Mansilla, A., A characterization of the Dedekind completion of a totally ordered group of infinite rank, to appear in Indag. Mathem. [9] Ribenboim, P. Th´ eorie des valuations. Les presses de l’Universit´ e de Montréal. Canada. 1968. ´ticas. Pontificia Universidad Cato ´ lica de Chile, Casilla 306, Facultad de Matema Correo 22, Santiago, Chile. E-mail address: [email protected] ´tica y Estad´ıstica. Universidad de la Frontera, Casilla Departamento de Matema 54-D, Temuco, Chile. E-mail address: [email protected]


Compact perturbations of Fredholm operators on Norm Hilbert spaces over Krull valued fields Herminia Ochsenius and Wim H. Schikhof Abstract. A continuous linear operator of a Banach space into itself is called Fredholm if its kernel and cokernel are finite-dimensional. The subject of compact perturbations of Fredholm operators on complex spaces is well-known, see e.g. [6]. For spaces over non-archimedean valued fields K of rank 1 (i.e. the range of the valuation is in [0, ∞)) the preservation of Fredholm operators under compact perturbations was proved in [1]. In this paper we allow K to have an arbitrary totally ordered abelian value group G (rather than a subgroup of (0, ∞)), but we restrict our study to Norm Hilbert Spaces (NHS) E over K i.e. each closed subspace admits a projection of norm ≤ 1. We prove the striking fact that the index of a Fredholm operator is 0. Further, we consider a natural class Φ(E) of so-called Lipschitz-Fredholm operators and prove that the operators A for which A + Φ(E) ⊂ Φ(E) form precisely the set of all nuclear operators. (An operator A is called nuclear if there exists a sequence A1 , A2 , . . . of continuous finite rank operators such that Ax − An x < gn x (x ∈ E \ {0}) for some sequence g1 , g2 , . . . in G, tending to 0. Here the strict inequality is essential!)

1. Preliminaries In this paper we need the concepts of an ordered group G, a Krull valuation on a field K and also the notion of a G-module, necessary to define norms on K-vector spaces. For a thorough treatment and examples, see [2], Ch. 1 or [3], but for the reader’s convienence we will recall some basics. Throughout G is an abelian multiplicatively written group with unit element 1, together with a linear ordering such that for all g1 , g2 , h ∈ G, g1 ≤ g2 implies hg1 ≤ hg2 . We assume G = {1}. Then G has no smallest or largest element and is torsion free. A subgroup H of G is called convex if for all h1 , h2 ∈ H, g ∈ G, h1 ≤ g ≤ h2 implies g ∈ H. Each proper convex subgroup is bounded from below and from above. 1991 Mathematics Subject Classification. Primary 46S10; Secondary 47A53. Key words and phrases. Fredholm operators, Norm Hilbert spaces, Krull valued fields. Research partially supported by Fondecyt 1080194. c Mathematical 0000 (copyright Society holder) c 2010 American

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HERMINIA OCHSENIUS AND WIM H. SCHIKHOF

Throughout we will assume that G is the union of a strictly increasing sequence of convex subgroups. For a typical example of such a group, see [2], 1.3.2. We augment G with an element 0 and extend the ordering and multiplication by declaring that 0 < g, 0 · g = g · 0 = 0 · 0 = 0 for all g ∈ G. A Krull valuation on a field K (with value group G) is a surjective map | | : K−→G ∪ {0} such that for all λ, µ ∈ K (i ) |λ| = 0 if and only if λ = 0 (ii ) |λ + µ| ≤ max(|λ|, |µ|) (iii ) |λµ| = |λ| |µ|. The map (λ, µ) −→ |λ−µ| behaves like an ultrametric and induces in a natural way a field topology on K. Thanks to our assumption on G this topology is metrizable ([2], 1.4.1). We assume from now on that K is complete i.e. that each Cauchy sequence converges. To be able to define normed spaces over K (see Section 2) we need the concept of a G-module. It is a linearly ordered set X together with an action of G on it, written (g, s) −→ gs, such that for all g, g1 , g2 ∈ G, s, s1 , s2 ∈ X we have (i ) (ii ) (iii ) (iv ) (v )

g1 (g2 s) = (g1 g2 )s 1·s=s g1 ≥ g2 =⇒ g1 s ≥ g2 s s1 ≥ s2 =⇒ gs1 ≥ gs2 for each ε ∈ X there is a h ∈ G such that hs < ε

Clearly G itself is a G-module in an obvious way. Another one is the Dedekind completion of G, from now on to be denoted G# . Indeed, the multiplication of G can be extended naturally to an action G × G# −→G# satisfying (i )–(v ) above, for details, see [2], 1.5.4. Let X be a G-module. For s ∈ X we define Stab(s) := {g ∈ G : gs = s}. It is a proper convex subgroup. X is called faithful if Stab(s) = {1} for all s ∈ X, almost faithful if there is a proper convex subgroup H of G such that Stab(s) ⊂ H for all s ∈ X. Obviously, the G-module G is faithful, but G# is not even almost faithful. In fact, for each proper convex subgroup H it is easily seen that Stab(sup H) = H (here, the supremum is taken in G# ). Finally, we recall the notion of topological type. Fix any s0 ∈ X. For s ∈ X, its topological type τ (s) is the collection of h ∈ G satisfying (i ) if g ∈ G, gs ≥ s0 then hs0 ≤ gs (ii ) if g ∈ G, gs ≤ s0 then hs0 ≥ gs. Then τ (s) is a proper convex subgroup of G. It depends only on the orbit of s i.e. τ (s) = τ (gs) for all g ∈ G. We say that a sequence s1 , s2 , . . . in X satisfies the type condition if τ (sn )−→∞ i.e. if for every proper convex subgroup H we have H ⊂ τ (sn ) for large n (For more background, see [2], Sec. 1.6). We quote the following from [5], 4.3. Lemma 1.1. Let X be a G-module, let s, t ∈ X. Suppose g1 s ≤ t ≤ g2 s for some g1 , g2 ∈ G. Let H be the smallest convex subgroup of G containing g1 , g2 , and suppose τ (s) ⊃ H. Then τ (s) = τ (t).

COMPACT PERTURBATIONS OF FREDHOLM OPERATORS ....

149 3

2. Operators on normed spaces We recall the fundamentals on normed (Banach) spaces and linear operators between them. For details and examples, see [2], Ch. 2. From now on K is a Krull valued field with the properties described in Section 1. To any G-module X we adjoin a zero element 0X satisfying 0X < s, 0 · s = 0 · 0X = 0X for all s ∈ X. However, from now on we will write 0 instead of 0X . Let E be a K-vector space, let X be a G-module. A norm (more precisely, an X-norm) is a map : E−→X ∪ {0} such that for all x, y ∈ E, λ ∈ K (i ) x = 0 if and only if x = 0 (ii ) λx = |λ| x (iii ) x + y ≤ max( x , y ). As E \ {0} = { x : x ∈ E, x = 0} is a G-submodule of X we often will assume that X = E \ {0}. The pair (E, ) is called an (X-)normed space, but often we write E instead of (E, ). The map (x, y) −→ x − y induces a vector topology on E in a natural way, for which it is metrizable ([2], 2.5). E is called a Banach space if each Cauchy sequence converges. Lemma 2.1. Finite-dimensional normed spaces are Banach. Linear maps between finite-dimensional normed spaces are continuous. Proof. [2], 2.3.4, 2.3.5. We now recall the concept of orthogonality. It is inspired by orthogonality in Hilbert spaces, but formulated in terms of the norm only. Let E be a normed space over K. Two vectors x, y ∈ E are called orthogonal if λx + µy = max( λx , µy ) for all λ, µ ∈ K. Two subspaces D1 , D2 of E are called orthogonal if, for each x ∈ D1 , y ∈ D2 , x, y are orthogonal. Then D1 ∩ D2 = {0}. If, in addition, D1 + D2 = E we say that D1 (D2 ) is an orthogonal complement of D2 (D1 ). A (finite or infinite) sequence e1 , e2 , . . . in E\{0} is called an orthogonal sequence if for each n, [en ] is orthogonal to [ei : i = n]; in other words if for all λ1 , λ2 , . . . ∈ K we have n λi ei = max λi ei i i=1

for all n. We recall the following Lemma 2.2. Let e1 , e2 , . . . be a sequence in E \ {0}, where E is a normed space. If n −→ τ ( en ) is injective then e1 , e2 , . . . is orthogonal. / G en . Proof. This follows easily from the fact that if m = n then em ∈ Let e1 , e2 , . . . be a (finite or infinite) sequence in a normed space E over K. We say that it is an orthogonal base of E if each x ∈ E has a unique expansion x = ∞ λ e , where λi ∈ K (then λi ei −→0) and, moreover, e1 , e2 , . . . is orthogonal. i i i=1

150 4


Let E, F be X-normed Banach spaces for some G-module X. We introduce the following sets of operators E−→F . Definition 2.3. Let L(E, F ) be the set of all continuous linear maps E−→F . Further, we set FR(E, F ) :=

{A ∈ L(E, F ) : dim AE < ∞}

Lip(E, F ) :=

{A ∈ L(E, F ) : there is a g ∈ G such that Ax ≤ g x for all x ∈ E} (‘Lipschitz operators’)

(‘finite rank operators’)

Lip∼ (E, F ) := {A ∈ L(E, F ) : there is a g ∈ G such that Ax < g x for all x ∈ E \ {0}} (‘strictly Lipschitz operators’) As customary we write L(E) := L(E, E), FR(E) := FR(E, E), Lip(E) := Lip(E, E), Lip∼ (E) := Lip∼ (E, E). Proposition 2.4. FR(E, F ) ⊂ Lip∼ (E, F ) ⊂ Lip(E, F ) ⊂ L(E, F ). Proof. See [3], 2.1.3 for the first inclusion. The other ones are obvious. Remark 2.5. One may wonder whether the last two inclusions might be strict. (a) It is known for quite some time that there exist spaces E for which Lip(E) = L(E), see [5], 4.2 for an example. In complex and rank 1 theory there is no difference between continuous and Lipschitz operators but it is mainly the Lipschitz property that is being used. With this in mind we consider ‘Lipschitz’ as the natural counterpart of ‘continuous’. We have another argument for doing so. From [2], 2.1.9 and [5], 3.2 it follows that for any Banach space (E, ) there exists an equivalent G# -norm such that each operator in L(E) is Lipschitz (E, )−→(E, ). Thus, the study of Lipschitz operators includes the study of continuous ones! (b) Less known is the fact that Lip∼ (E) may be = Lip(E)! This seems to be rather strange as, given an inequality Ax ≤ g x we could think of picking a g > g and conclude that Ax < g x . But this is false if (g )−1 g happens to be in Stab( x )! Let us ilustrate it with an example. Let H1 ⊂ H2 ⊂ · · · be a strictly increasing sequence of convex subgroups whose union is G, let sn := sup Hn , where the supremum is taken in G# . let E be the space of all sequences (λ1 , λ2 , . . .) in K for which |λn |sn is bounded and take on E the G# -norm (λ1 , λ2 , . . .) −→ sup |λn |sn n

The identity operator is clearly Lipschitz. If it were strictly Lipschitz there would be a g ∈ G such that sn < gsn for all n. Now g ∈ Hm for some m and it is easily seen that gsm = sm , a contradiction. In fact, the following was proved in [3], 2.1.16. See also 4.12. Theorem 2.6. Lip∼ (E) = Lip(E) if and only if E \ {0} is almost faithful. Still, it may seem that the introduction of Lip∼ (E) is rather arbitrary. However, we will see in 4.2 that strictly Lipschitz operators will appear naturally.


151 5

We say that a linear map A ∈ L(E, F ) is bi-Lipschitz if there are g1 , g2 ∈ G such that g1 x ≤ Ax ≤ g2 x for all x ∈ E, in other words, if A is injective, Lipschitz and A−1 : AE−→E is also Lipschitz. We now introduce natural G# -norms on Lip(E, F ) and Lip∼ (E, F ). Definition 2.7. Let E, F be X-normed Banach spaces for some G-module X. For A ∈ Lip(E, F ) put A := inf{g ∈ G : Ax ≤ g x for all x ∈ E}, ∼

and for A ∈ Lip (E, F ) we set A ∼ := inf{g ∈ G : Ax < g x for all x ∈ E \ {0}}, where the infima are taken in G# ∪ {0}. Proposition 2.8 ([3], 2.2.4). and ∼ are G# -norms on Lip(E, F ), Lip∼ (E, F ) respectively. Definition 2.9 ([3], 2.4.3). Let C(E, F ) be the closure of FR(E, F ) in Lip(E, F ) with respect to , and let C ∼ (E, F ) be the closure in Lip∼ (E, F ) with respect to ∼ . The elements of C(E, F ) are called compact, those of C ∼ (E, F ) are called nuclear. As ∼ is ≥ on Lip∼ (E, F ) we clearly have C ∼ (E, F ) ⊂ C(E, F ). See also 4.12. In this paper we will mainly be interested in C(E) := C(E, E) and C ∼ (E) := ∼ C (E, E). Let (E, ) be an X-normed Banach space for some G-module X. We shall, for each proper convex subgroup H, construct a G-module XH and a XH -norm · H on E such that X{1} = X, {1} = . This construction is new as far as we know. The formula s ∼ t if there exist g1 , g2 ∈ H such that g1 s ≤ t ≤ g2 s defines a equivalence relation ∼ on X. Put XH := X/ ∼, let ϕH : X−→XH be the natural map, and define a natural ordering on XH by v ≤ w if there exist s, t ∈ X with s ≤ t, ϕH (s) = v, ϕH (t) = w. Then ϕH is increasing and the formula gϕH (s) = ϕH (gs)

(g ∈ G, s ∈ X)

defines a G-module structure in XH . Now augment XH with a smallest element 0 as usual, and extend ϕH by putting ϕH (0) := 0. The map x −→ x H := ϕH ( x ) is easily seen to be an XH -norm on E. Clearly X{1} = X and {1} = . We shall write EH := (E, H ), but put E := E{1} . Theorem 2.10. Let E be an X-normed Banach space for some G-module X, let H be a proper convex subgroup of G. Then (i ) the identity E−→EH is a homeomorphism,

152 6


(ii ) for each closed subspace D of E, Lip(D, E) = Lip(DH , EH ) and Lip∼ (D, E) = Lip∼ (DH , EH ), (iii ) orthogonal sequences (bases) in E are orthogonal sequences (bases) in EH , (iv ) orthogonal subspaces (complements) in E are orthogonal subspaces (complements) in EH . Proof. (i ) Let ε ∈ X. Then {x ∈ E : x < ε} ⊃ {x ∈ EH : ϕH ( x ) < ϕH (ε)}. Conversely, let v ∈ XH ; choose ε ∈ X with ϕH (ε) = v. Then {x ∈ EH : ϕH ( x ) ≤ v} ⊃ {x ∈ H : x ≤ ε}, which proves (i ). (ii ) Let A ∈ Lip(D, E). Then Ax ≤ g x for some g ∈ G and all x ∈ D. By increasingness, Ax H = ϕH ( Ax ) ≤ ϕH (g x ) = g x H , so that A ∈ Lip(DH , EH ). Conversely, if A ∈ Lip(DH , EH ) then ϕH ( Ax ) ≤ ϕH (g x ) for some g ∈ G and all x ∈ DH . Choose g ∈ G, g > gh for all h ∈ H (This can be done as the coset gH is bounded in G). We claim that Ax ≤ g x for all x ∈ D. In fact, from ϕH ( Ax ) ≤ ϕH (g x ) of above it follows that there exist s, t ∈ X with s ≤ t, s ∼ Ax , t ∼ g x . Thus Ax ≤ h1 s ≤ h1 t ≤ h1 h2 g x ≤ g x for some h1 , h2 ∈ H and all x ∈ E. We conclude that Lip(D, E) = Lip(DH , EH ). Now suppose A ∈ Lip∼ (D, E). Then there is a g ∈ G such that Ax < g x for all x ∈ D \ {0}. Let g ∈ G be such that g > gh for all h ∈ H, as before. We claim that Ax H < g x H for all nonzero x ∈ DH . Suppose not. Then there is an x0 ∈ DH \{0} such that Ax0 H ≥ g x0 H . But Ax0 < g x0 implies, by increasingness of ϕH , Ax0 H ≤ g x0 H . We see that g x0 H ≤ Ax0 H ≤ g x0 H ≤ g x0 H so that we must have Ax0 H = g x0 H , i.e. Ax0 ∼ g x0 . So certainly there is a h ∈ H for which h g x0 ≤ Ax0 < g x0 so that h g x0 < g x0 and it follows that h g < g i.e. g < h−1 g, conflicting our choice of g. Thus, A ∈ Lip∼ (DH , EH ). Conversely, if A ∈ Lip∼ (DH , EH ) then Ax H < g x H for some g ∈ G and all x ∈ DH . Then by increasingness of ϕH we have Ax < g x and we are done. (iii ) Let e1 , e2 , . . . be an orthogonal sequence (base) of E. By (i ) and [2], 2.4.17 it suffices to prove that e1 , e2 , . . . are orthogonal in EH . So, let n ∈ N, λ1 , . . . , λn ∈ K. Then, using increasingness of ϕH , ni=1 λi ei H = n ϕH ( i=1 λi ei ) = ϕH (max1≤i≤n λi ei ) = max1≤i≤n ϕH ( λi ei ) = max1≤i≤n λi ei H . This proves (iii ). (iv ) Follows easily from (iii ) and the definitions. Remark 2.11. The reason behind the introduction of the norms H lies in the following fact, to be used in 3.5. Let D be a closed subspace of E, let A : D−→E be bi-Lipschitz. Then there is a proper convex subgroup H such that A is an isometry DH −→EH . (The proof is easy. Let g1 x ≤ Ax ≤ g2 x for some g1 , g2 ∈ G and all x ∈ D, take for H the smallest convex subgroup containing g1 , g2 ). We conclude this section by taking over the usual definition of a Fredholm operator.


153 7

Let E, F be X-normed Banach spaces for some G-module X. An operator T ∈ L(E, F ) is called Fredholm if Ker T and F/ Im T are finite-dimensional. Then the index χ(T ) of T is defined as χ(T ) := dim Ker T − dim F/ Im T. We need the following easy consequence of the Open Mapping Theorem ([2], 2.5.4). Theorem 2.12. Let E, F be as above and let T ∈ L(E, F ). If Im T has finite codimension then Im T is closed. Proof. (Classical, but included for convenience). First suppose that T is injective. Let D be an algebraic complement of Im T . By finite dimensionality D, with the norm inherited from F , is a Banach space, hence so is the product E × D. The map S : E × D−→F given by (x, d) → T x + d (x ∈ E, d ∈ D) is easily seen to be a continuous linear bijection. By the Open Mapping Theorem S is a homeomorphism, so that Im T = S(E × {0}) is closed. For the general case observe that E/ Ker T is a Banach space ([2], 2.5.1). We have the factorization T = T1 ◦ π, where π : E−→E/ Ker T is the quotient map and where T1 ∈ L(E/ Ker T, F ) is an injection, so we can apply the first part of the proof to T1 obtaining that Im T1 = Im T is closed. 3. Operators on Norm Hilbert spaces In this section we restrict our attention to a special class of Banach spaces, the so-called Norm Hilbert spaces (NHS). Definition 3.1. A Banach space E is called a Norm Hilbert space if for every closed subspace D of E there exists a linear surjective projection P : E−→D such that P x ≤ x for all x ∈ E. The NHS form an outstanding class of Banach spaces; they have many peculiar properties. We quote the ones we need and also prove some new results. For general background on NHS we refer to [4]. FROM NOW ON IN THIS PAPER E IS AN INFINITE-DIMENSIONAL NORM HILBERT SPACE Theorem 3.2 ([4], 3.2). (i ) E has an orthogonal base. (ii ) For each orthogonal sequence e1 , e2 , . . . in E the sequence e1 , e2 , . . . satisfies the type condition. (iii ) Each closed subspace of E has an orthogonal complement. (iv ) If F is a Banach space with an orthogonal base, and F is linearly homeomorphic to E then F is also a NHS. Corollary 3.3. For each proper convex subgroup H the space EH is a NHS. Proof. Use 2.10(i ), (iii ) to apply 3.2(iv ) above. We quote the following from [5], 3.6, 2.4. Theorem 3.4. Let D1 , D2 be closed subspaces of E, let D1C , D2C be orthogonal complements of D1 , D2 respectively. Then

154 8


(i ) if D1 and D2 are linearly homeomorphic then so are D1C and D2C , (ii ) if D1 and D2 are isometrically isomorphic then so are D1C and D2C . We now complete this result by proving the following Lipschitz version. Theorem 3.5. Let D1 , D2 be closed subspaces of E, let D1C , D2C be orthogonal complements of D1 , D2 respectively. Suppose there is a bi-Lipschitz map of D1 onto D2 . Then there is a bi-Lipschitz map of D1C onto D2C . Proof. Suppose we have a linear bijection A : D1 −→D2 for which g1 x ≤ Ax ≤ g2 x for some g1 , g2 ∈ G and all x ∈ D. Let H be the smallest convex subgroup of G containing g1 , g2 . Then, by Remark 2.11, A is an isometry of (D1 )H onto (D2 )H . By 2.10(iv ) (D1C )H and (D2C )H are orthogonal complements of (D1 )H , (D2 )H respectively with respect to H . Corollary 3.3 tells us that EH is a NHS so we can apply 3.4(ii ) to arrive at a linear surjective isometry (D1C )H −→(D2C )H . By 2.10(ii ) this map is in Lip(D1C , E) and its inverse is in Lip(D2C , E), so we have a bi-Lipschitz map of D1C onto D2C . The following consequences reveal the ‘rigidity’ of NHS. Corollary 3.6. Let D be a closed subspace of E. Then every linear homeomorphism (bi-Lipschitz map, isometry) A : D−→E can be extended to a linear : E−→E. homeomorphism (bi-Lipschitz map, isometry) A Proof. Let DC and (AD)C be orthogonal complements of D, AD respectively. By 3.4 and 3.5 there is a linear bijection B : DC −→(AD)C that is a homeomorphism : E−→E by the formula (bi-Lipschitz map, isometry). Define A + y) = Ax + By A(x

(x ∈ D, y ∈ DC ).

has the required properties. one easily checks that A Corollary 3.7. (i ) A linear homeomorphism of E onto a (closed) subspace is surjective. (ii ) A surjective continuous linear map E−→E is a homeomorphism. Proof. (i ) Let T be a linear homeomorphism E−→T E. By 3.4(i ) {0} = (T E)C for any orthogonal complement (T E)C of T E. Hence, T E = E. (ii ) Let T ∈ L(E) be surjective. Let (Ker T )C be an orthogonal complement of Ker T . By the Open Mapping Theorem T is a homeomorphism of (Ker T )C onto E. Then use again 3.4(i ) to conclude that Ker T = {0}.

More generally, if T ∈ L(E) and T E is closed we can choose orthogonal complements (Ker T )C , (T E)C of Ker T , T E respectively. This yields a factorization T = i ◦ T1 ◦ π


T

E = Ker T ⊕ (Ker T )C

π (Ker T )C

(T E)C ⊕ T E = E

i

T1

155 9

(∗)

TE

where π (resp. i) is the natural ‘second component’ projection (resp. injection) and where T1 is a linear homeomorphism. By 3.4(i ), Ker T and (T E)C are linearly homeomorphic. This leads to surprising results for Fredholm operators on NHS. Corollary 3.8. . (i ) Let T ∈ L(E) and let T E have finite codimension. Then T is Fredholm. (ii ) Let T ∈ L(E), let Ker T be finite-dimensional and let T E be closed. Then T is Fredholm. (iii ) Every Fredholm operator E−→E has zero index. Proof. In all cases (i ), (ii ), (iii ) T E is closed, so we can conclude that the spaces Ker T and (T E)C in (∗) are linearly homeomorphic. Thus, in case (i ) it follows that Ker T is finite-dimensional and in case (ii ) that T E is finite-codimensional. The linear homeomorphism between Ker T and (T E)C shows in case (iii ) that their dimensions are the same whence χ(T ) = 0. Remark 3.9. Operators satisfying the condition of 3.8(ii ) are usually called semi-Fredholm in complex and rank 1 theory. It is a curious fact that in our case these semi-Fredholm operators (and also the ‘dual’ ones of 3.8(i )) are automatically Fredholm! 4. Fredholm and nuclear operators in Norm Hilbert spaces According to the philosophy of 2.5(a) we ‘Lipschitzfy’ the notion of a Fredholm operator as follows. Definition 4.1. A Fredholm operator T : E−→E is called Lipschitz-Fredholm (L-Fredholm) if the map T1 in diagram (*) following 3.7 is bi-Lipschitz. The collection of all L-Fredholm operators E−→E is denoted by Φ(E). Perturbation theory for Φ(E) is dealing with the set of operators A : E−→E such that A + T ∈ Φ(E) whenever T ∈ Φ(E). Thus, we put (the symbol P standing for ‘perturbation’) P(E) := {A : E−→E : A + Φ(E) ⊂ Φ(E)}. Notice that P(E) ⊂ Lip(E), and that P(E) is a K-vector space. In [1] it was proved, for Banach spaces V over a complete valued field of rank 1, that the sum of a compact operator V −→V and a Fredholm operator V −→V is again Fredholm. Translated to our case we therefore could conjecture that C(E) ⊂ P(E). But the following Main Theorem of our paper shows that this conjecture is wrong (see 4.12).

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Theorem 4.2. P(E) = C ∼ (E). The proof runs in several steps. First we present a useful criterion for an operator to be L-Fredholm. Notice that bi-Lipschitz maps E−→E are automatically surjective (3.7). Lemma 4.3. Let T : E−→E. Then T ∈ Φ(E) if and only if T can be written as U + S, where U is bi-Lipschitz and S ∈ FR(E). Proof. We use the notations of diagram (*) following 3.7. Let T ∈ Φ(E). By 3.8 χ(T ) = 0 so Ker T and (T E)C have equal dimensions, and there is a bi-Lipschitz homeomorphism V : Ker T −→(T E)C (2.1, 2.3). Define U : E−→E by U (x + y) := V x + T y

(x ∈ Ker T, y ∈ (Ker T )C )

Clearly U is a bi-Lipschitz bijection, and since T = U on (Ker T )C , S := T − U is of finite rank. Conversely, let U : E−→E be bi-Lipschitz such that T −U is of finite rank. Then D := Ker(T −U ) is finite-codimensional. As T E ⊃ T D = U D and Ker T ∩D = {0}, T E is finite-codimensional and Ker T is finite-dimensional, so T is Fredholm. From T − U ∈ Lip∼ (E) (2.4) we obtain that T , hence T1 is Lipschitz. To prove that also T1−1 is Lipschitz, let U x ≥ g x for all x ∈ E, some g ∈ G. With D as above, let y ∈ T (D ∩ (Ker T )C ). Then T1−1 y ∈ (Ker T )C and since U = T on D we obtain y = T T1−1 y = U T1−1 y ≥ g T1−1 y , showing that T1−1 is Lipschitz on the closed finite-codimensional subspace T (D ∩ (Ker T )C ) of T ((Ker T )C ). But then it is also Lipschitz in the whole of T ((Ker T )C ) (consider an orthogonal complement V of T (D ∩ (Ker T )C ) in T ((Ker T )C ) and observe that T1−1 |V is Lipschitz (2.4)). Corollary 4.4. If T1 , T2 ∈ Φ(E) then T1 T2 ∈ Φ(E). Proof. By 4.3 we can write T1 = U1 + S1 , T2 = U2 + S2 , where U1 , U2 are biLipschitz and S1 , S2 ∈ FR(E). Then T1 T2 = U1 U2 + S1 U2 + U1 S2 + S1 S2 . Clearly, U1 U2 is bi-Lipschitz and S1 U2 + U1 S2 + S1 S2 ∈ FR(E) so, by 4.3, T1 T2 ∈ Φ(E). Lemma 4.5. FR(E) ⊂ P(E). Proof. Let A ∈ FR(E), let T ∈ Φ(E). By 4.3, T = U + S where U is bi-Lipschitz and S ∈ FR(E). Then A + T = U + S + A, and since S + A ∈ FR(E) we have by 4.3 that A ∈ P(E). Lemma 4.6. Let A ∈ P(E). Then, for each bi-Lipschitz U : E−→E we have U A ∈ P(E) and AU ∈ P(E). Proof. We only prove that U A ∈ P(E); the proof for AU is similar. Let T ∈ Φ(E). Then U −1 T ∈ Φ(E) by 4.4, so A + U −1 T ∈ Φ(E). Then, again by 4.4 U A + T = U (A + U −1 T ) ∈ Φ(E), showing that U A ∈ P(E). Now we arrive at the first half of Theorem 4.2. Proposition 4.7. C ∼ (E) ⊂ P(E).


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Proof. Let A ∈ C ∼ (E). There are A1 , A2 , . . . in FR(E) such that A − An ∼ −→0 (2.9). Let T ∈ Φ(E); we have to prove that T + A ∈ Φ(E). By 4.3 we have T = U + S where U is bi-Lipschitz and S is of finite rank, so by 4.5 and the fact that P(E) is a vector space it is enough to show that U + A ∈ Φ(E). In other words, we may assume that T itself is bi-Lipschitz. Writing T + A = T (I + T −1 A) and using 4.6 it suffices to prove that I + −1 T A ∈ Φ(E). It is easily seen that T −1 An , T −1 A are in Lip∼ (E) and that T −1 (An − A) ∼ −→0 so there exists an n ∈ N such that T −1 (An − A) < x for all non-zero x ∈ E implying that I − T −1 (An − A) is an isometry, hence surjective by 3.7(i ), so that it is in Φ(E). Now I + T −1 A = I − T −1 (An − A) + T −1 An , where T −1 An ∈ FR(E) ⊂ P(E). It follows that I − T −1 A ∈ Φ(E) and we are done. Next we consider the second half of Theorem 4.2. Lemma 4.8. Let E be a NHS with orthogonal base e1 , e2 , . . .. Let A ∈ L(E). Suppose for every g ∈ G we have Aen < g en for almost all n. Then A ∈ C ∼ (E). Proof. Let g ∈ G; we construct finite rank operators A1 , A2 , . . . ∈ L(E) such that (A − An )(x) < g x

(x ∈ E \ {0})

∼

for large n (It then follows that A ∈ C (E)). By assumption there is an m ∈ N such that Aen < g en for n > m. Let Pn be the standard projection of E onto [e1 , . . . , en ], and ∞put An := APn . Then An ∈ FR(E). Let x ∈ E \ {0} have the expansion x = i=1 ξi ei . Then for n > m we have ∞ (A − An )x = ξi Aei ≤ max ξi Aei = ξj Aej i>n i=n+1

for some j > n. If ξj = 0 we have (A − An )x = 0 < g x . Otherwise, (A − An )x ≤ ξj Aej < |ξj |g ej ≤ g maxi ξi ei = g x , and we are done. Lemma 4.9. Let E be a NHS with orthogonal base e1 , e2 , . . .. Let A ∈ Lip(E) \ C ∼ (E). Then there exists a closed infinite-dimensional subspace D of E such that A : D−→AD is bi-Lipschitz. Proof. By 4.8 there is a g1 ∈ G such that V1 := {n ∈ N : Aen ≥ g1 en } is infinite. Using the fact that A ∈ Lip(E) we find a g2 ∈ G such that g1 en ≤ Aen ≤ g2 en

(n ∈ V1 )

Since { en : n ∈ V1 } satisfies the type condition (3.2(ii )) there is an infinite subset V2 of V1 such that n, m ∈ V2 , n = m implies τ ( en ) = τ ( em ). Now let H be the smallest convex subgroup of G containing g1 and g2 . By deleting the finite set {n ∈ V2 : τ ( en ) ⊂ H} we arrive at an infinite subset V3 of N such that n, m ∈ V3 , n = m =⇒ τ ( en ) = τ ( em ) and τ ( en ) ⊃ H for all n ∈ V3 . Now apply 1.1 to conclude thar τ ( en ) = τ ( Aen ) for all n ∈ V3 . In particular we have τ ( Aen ) = τ ( Aem ) whenever n, m ∈ V3 , n = m implying by 2.2 that the Aen (n ∈ V3 ) are orthogonal. By a straightforward calculation we then arrive at g1 x ≤ Ax ≤ g2 x

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for all x ∈ D := [en : n ∈ V3 ] which proves the lemma. Lemma 4.10. Let E be a NHS. Then P(E) ⊂ C ∼ (E). Proof. Let A ∈ P(E), A ∈ / C ∼ (E); we arrive at a contradiction. Clearly we have A ∈ Lip(E) so by the previous lemma there is an infinite-dimensional closed subspace D of E for which A : D−→AD is bi-Lipschitz. By 3.5, A extends to a : E−→E. Then −A ∈ Φ(E), so that A − A ∈ Φ(E). But this is bi-Lipschitz A impossible as Ker(A − A) contains the infinite-dimensional subspace D. Proof of Theorem 4.2. Combine 4.7 and 4.10. Remark 4.11. In the beginning of this Section we claimed that our result P(E) = C ∼ (E) of 4.2 does not imply C(E) ⊂ P(E) (This last inclusion could be conjectured by analogy to the rank 1 theory). Thus, we will prove that, in general, C(E) and C ∼ (E) may differ. In fact, the example E given in 2.5(b) is easily seen to be a NHS for which E \ {0} is not almost faithful. By applying the next Theorem 4.12 we can conclude that, for this E, C ∼ (E) is strictly contained in C(E). Theorem 4.12. (Compare also 2.6) The following are equivalent. (α) C ∼ (E) = C(E). (β) E \ {0} is almost faithful. Proof. (β) =⇒ (α). We have (2.6) Lip(E) = Lip∼ (E) and it is not hard to see that and ∼ yield the same topologies so that, by definition C ∼ (E) = C(E). (α) =⇒ (β). Suppose E \ {0} is not almost faithful; we construct a P ∈ C(E) \ C ∼ (E) as follows. There are x1 , x2 , . . . ∈ E \ {0} such that Stab( x1 ) Stab( x2 ) · · · , n Stab( xn ) = G. For m = n we have xm ∈ / G xn so the sequence x1 , x2 , . . . is orthogonal. Put D := [x1 , x2 , . . .] and let P be an orthogonal projection of E onto D. We have P −I ∈ / Φ(E) as Ker(P − I) contains D. Thus, by 4.2, P ∈ / P(E) = C ∼ (E). To see that p ∈ C(E), let Pn be the standard projection D−→[x1 , . . . , xn ]. Then Pn P ∈ FR(E). Let g ∈ G; we prove P − Pn P ≤ g for large ∞n. In fact, there is an m such that g∈ Stab( xn ). Let x ∈ E; write P x = i=1 ξi xi . Then (P − ∞ Pn P )x = i=n+1 ξi xi = maxi>n ξi xi = g maxi>n ξi xi (as soon as n ≥ m) ≤ g maxi≥1 ξi xi = g P x ≤ g x , and we are done. References [1] J. Araujo, C. P´ erez-Garc´ıa and S. Vega. Preservation of the index of p-adic linear operators under compact perturbations. Compositio Math. 118, (1999), 291–303. [2] H. Ochsenius and W.H. Schikhof, Banach spaces over fields with an infinite rank valuation. Lecture Notes in pure and applied mathematics 207, edited by J. Kakol, N. De Grande-De Kimpe and C. P´ erez-Garc´ıa. Marcel Dekker (1999), 233–293. [3] H. Ochsenius and W.H. Schikhof, Lipschitz operators on Banach spaces over Krull valued fields. Ultrametric Functional Analysis, Contemporary Mathematics 319, edited by W.H. Schikhof, C. Pérez-Garc´ıa and A. Escassut. American Mathematical Society, (2003), 239– 249. [4] H.Ochsenius and W. H. Schikhof. Norm Hilbert spaces over Krull valued fields. Indagationes Mathematicae N.S. Vol 17(1). Pgs.65-84 (2006) [5] H. Ochsenius and W.H. Schikhof, Linear homeomorphisms of non-classical Hilbert spaces. Indag. Mathem., N.S., (1999), 601–613. [6] A.E. Taylor and D.C. Lay, Introduction to Functional Analysis, John Wiley & Sons, New York (1980).


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´ticas. Pontificia Universidad Cato ´ lica de Chile, Casilla 306, Facultad de Matema Correo 22, Santiago, Chile. E-mail address: [email protected] Department of Mathematics, Radboud University, Toernooiveld 525 ED Nijmegen, The Netherlands. E-mail address: w [email protected]



Applications of the p-adic Nevanlinna theory to problems of uniqueness Jacqueline Ojeda Abstract. In this paper we analyze conditions of uniqueness for meromorphic functions by applying the p-adic Nevanlinna Theory. We study meromorphic functions that share one finite nonzero value, counting or ignoring multiplicities. Let K be an ultrametric complete algebraically closed field of characteristic zero. We show that if two meromorphic functions of the form f n f and g n g share one constant value C.M. when n is a positive integer, there exists d ∈ K with dn+1 = 1, such that f = dg. For example, we obtain this when n ≥ 3 and f and g are entire functions; when n ≥ 4 and f and g are unbounded analytic functions inside an “open” disk; when n ≥ 11 and either f and g are meromorphic in K or f and g are “unbounded” meromorphic functions inside an “open” disk. Similarly, if f n f and g n g share one value I.M., we get to the same when n ≥ 8 and f and g are entire functions and when n ≥ 9 and f and g are “unbounded” analytic functions inside an “open” disk. We also prove Milloux’ Inequality in the field K which is useful in this last study.

1. Introduction 1.1. Definitions and Notations. Throughout this paper, K will denote an algebraically closed field of characteristic zero, complete for an ultrametric absolute value. Let a ∈ K and r > 0. We denote the “closed” disc and the “open” disc respectively by d(a, r) := {x ∈ K : |x − a| ≤ r} and d(a, r − ) := {x ∈ K : |x − a| < r}. We denote by A(K) the K-algebra of entire functions in K and by M(K) the set of meromorphic functions in K, i.e. the field of fractions of A(K). In the same way, we denote by A(d(a, r − )) the set of analytic functions in d(a, r − ), i.e. the K-algebra ∞ an (x − a)n converging in d(a, r − ), and by M(d(a, r − )) the set of of power series n=0

meromorphic functions inside d(a, r − ), i.e. the field of fractions of A(d(a, r − )). We denote by Ab (d(a, r − )) the K-subalgebra of A(d(a, r − )) consisting of the bounded 2000 Mathematics Subject Classification. 12J25, 30G06. Key words and phrases. meromorphic, Nevanlinna, ultrametric, sharing value, unicity. Research supported by a grant from the special program DLF 22 from the Chilean Government. c 2010 American Mathematical Society c 0000 (copyright holder)

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analytic functions f ∈ A(d(a, r − )) and by Mb (d(a, r − )) the field of fractions of Ab (d(a, r − )). Finally, we denote by Au (d(a, r − )) the set A(d(a, r − )) \ Ab (d(a, r − )) and by Mu (d(a, r − )) the set M(d(a, r − )) \ Mb (d(a, r − )). The paper is aimed at studying the following problem. Problem. Let n be a positive integer and let f, g ∈ M(K) be non-constant resp. f, g ∈ Mu (d(0, R− )) such that f n f and g n g “share” (C.M. or I.M.) one finite nonzero value a ∈ K . Does d ∈ K exists with dn+1 = 1, such that f = dg ? The question was studied in complex analysis in many papers concerning meromorphic functions or entire functions in C with various conclusions. In many cases the above conclusion holds. In some other cases the answer is that f g ≡ 1. See, for example, [5], [6], [8], [9] and [10]. Concerning A(K), we obtain always the same above conclusion since f g ≡ 1 is imposible. Moreover, we can also conclude when f, g ∈ Au (d(a, R− )) or when f, g ∈ Mu (d(0, R− )). 1.2. Nevanlinna Theory, Preliminary Results. We must now introduce some notation and results used in the Nevanlinna Theory. Let R > 0, for simplicity we will consider the open disc of center 0 and radius R. Let α ∈ d(0, R− ) and h ∈ M(d(0, R− )). If h has a zero of order n at γ, we put ωγ (h) = n. If h has a pole of order n at γ, we put ωγ (h) = −n and finally, if h(γ) = 0, ∞, we put ωγ (h) = 0. Let f ∈ M(d(0, R− )) be such that 0 is neither a zero nor a pole of f, and let r ∈]0, R[. We denote by Z(r, f ) the counting function of zeros of f in d(0, R− ) counting multiplities, we put Z(r, f ) := ωγ (f )(log r − log |γ|). Simωγ (f )>0 |γ|≤r − ilarly, we denote by Z(r, f ) the counting function of zeros of f in d(0, R ) ignoring multiplicities, we put Z(r, f ) := (log r − log |γ|). ωγ (f )>0 |γ|≤r

We shall also consider the counting functions of poles of f in d(0, R− ) 1 and N (r, f ) := counting or ignoring multiplicities. We put N (r, f ) := Z r, f 1 respectively. Z r, f The Nevanlinna Function T (r, f ) is defined by T (r, f ) := max Z(r, f ) + log |f (0)| , N (r, f ) . A. Boutabaa and A. Escassut in [3], A. Escassut in [4] and P. C. Hu and C. C. Yang in [7] give us results related to p-adic Nevanlinna Theory which we will use in the later proofs. Some of them are the following: Lemma 1. Let f ∈ A(K) (resp. f ∈ A(d(0, R− ))) be such that f has no zero at 0. Let r > 0 (resp. r ∈]0, R[), the functions T (r, f ) and Z(r, f ) are equivalent up to an additive constant.

APPLICATIONS OF THE p-ADIC NEVANLINNA THEORY TO PROBLEMS...

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Lemma 2. Let f, g ∈ M(K) (resp. f, g ∈ M(d(0, R− ))) be such that 0 is neither a zero nor a pole of f and g. Then, for r > 0 (resp. r ∈]0, R[), T (r, f −g) ≤ T (r, f ) + T (r, g) + O(1). Moreover, if f, g ∈ A(K) (resp. f, g ∈ A(d(0, R− ))), for r > 0 (resp. r ∈]0, R[), we have T (r, f g) ≤ T (r, f ) + T (r, g) + O(1) and T (r, f − g) ≤ max{T (r, f ), T (r, g)} + O(1). Lemma 3. Let f ∈ M(d(0, R− )) be such that 0 is neither a zero nor a pole of f . Then, f ∈ Mb (d(0, R− )) if and only if T (r, f ) is bounded in ]0, R[. Lemma 4. Let f ∈ M(K) (resp. f ∈ M(d(0, R− ))) be such that f is not identically zero. Suppose that 0 is neither a zero nor a pole of f and f . Then, for r > 0 (resp. r ∈]0, R[), N (r, f ) = N (r, f ) + N (r, f ) and Z(r, f ) ≤ Z(r, f ) + N (r, f ) − log r + O(1). Consequently T (r, f ) ≤ 2T (r, f ) + O(1). Moverover, if f ∈ A(K) (resp. f ∈ A(d(0, R− ))), for r > 0 (resp. r ∈]0, R[), we have T (r, f ) ≤ T (r, f ) + O(1). Let f ∈ M(d(0, R− )) be such that 0 is neither a zero nor a pole of f and let S be a finite subset of K. We denote by Z0S (r, f ) the counting function of zeros of f in d(0, R− ) which are not zeros of any f − s for s ∈ S : wα (f )(log r − log |α|) Z0S (r, f ) = s∈S, wα (f −s)=0, |α|≤r

Now we can state the ultrametric Nevanlinna second main Theorem in a basic form. Theorem 1. Let β1 , ..., βn ∈ K with n ≥ 2, and let f ∈ M(K) (resp. f ∈ M(d(0, R− ))). Let S = {β1 , ..., βn }. Assume that f, f , f − βj with 1 ≤ j ≤ n, have no zero and no pole at 0. Then, for r > 0 (resp. r ∈]0, R[), we have n (n − 1)T (r, f ) ≤ Z(r, f − βj ) + N (r, f ) − Z0S (r, f ) − log r + O(1). j=1

We now have to recall the notation given by A. Escassut in [4], useful for the ∞ proof of Lemma 9: Given f (x) = an xn in A(K) resp. in A(d(0, R− )) , for n=0 r > 0 resp. r ∈]0, R[ , we put |f |(r) :=

lim

|x|→r, |x|=r

|f (x)|.

This limit exists and | · | is an absolute value in A(K) resp. in A(d(0, R− )) . g ∈ M(K) resp. f ∈ M(d(0, R− )) by setting This may be continued to f = h |g|(r) |f |(r) = . |h|(r) As a Corollary of Lemma 2.1 [3] we obtain the following: Lemma 5. Let f ∈ M(K) resp. f ∈ M(d(0, R− )) be such that 0 is neither a zero nor a pole of f . Then log |f |(r) = Z(r, f ) − N (r, f ) + log |f (0)|.

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A classical and useful Theorem is the Nevanlinna Theorem on three small functions, which is described by P. C. Hu and C. C. Yang in [6]. Moreover, we include an analogous version for analytic functions and some interesting corollaries. Theorem 2. (Nevanlinna Theorem on three small functions) − Let f ∈ M(K) resp. f ∈ M(d(0, R and let α1 , α2 , α3 be )) be non-constant − distinct meromorphic functions in K resp. in M(d(0, R )) small with respect to f and such that 0 is neither a zero nor a pole of f , αi and f − ai (i = 1, 2, 3). Then T (r, f ) ≤

3

Z(r, f − αi ) − log r + S(r),

i=1

where S(r) = 4T (r, α1 ) + 4T (r, α2 ) + 5T (r, α3 ) + O(1). − Corollary 1. Let f ∈ M(K) resp. f ∈ M(d(0,−R )) be non-constant and let α be a meromorphic function in K resp. in d(0, R ) ) small with respect to f and such that 0 is neither a zero nor a pole of f , α and f − α. Then T (r, f ) ≤ Z(r, f ) + Z(r, f − α) + N (r, f ) − log r + S(r), where S(r) = 4T (r, α) + O(1). Proof. Consider α1 = −1, α2 =

1 and α3 = 0 in Theorem 2 and apply α−1

1 to prove the corollary. f −1 Theorem 3. Let f ∈ A(K) resp. f ∈ A(d(0, R− )) be non-constant and let α1 , α2 ∈ A(K) resp. α1 , α2 ∈ A(d(0, R− )) be distinct, small with respect to f and such that 0 is neither a zero nor a pole of f , αi and f − αi (i = 1, 2). Then this to ψ =

T (r, f ) ≤ Z(r, f − α1 ) + Z(r, f − α2 ) + S(r), where S(r) = 2T (r, α1 ) + 3T (r, α2 ) − log r + O(1). f − α1 . Without loss of generality we may assume that Φ f − α2 has no zero and no pole at 0. Let r > 0 resp. r ∈]0, R[ . Observe that Proof. Let Φ =

(1)

T (r, f ) ≤ T (r, f − α2 ) + T (r, α2 ) + O(1) α −α 1 2 1 + T r, + T (r, α2 ) + O(1) ≤ T r, f − α2 α2 − α1 α2 − α1 + 2T (r, α2 ) + T (r, α1 ) + O(1) ≤ T r, 1 + f − α2 = T (r, Φ) + 2T (r, α2 ) + T (r, α1 ) + O(1). Applying Theorem 1 to Φ, we have T (r, Φ) ≤ Z(r, Φ) + Z(r, Φ − 1) + N (r, Φ) − log r + O(1) ≤ Z(r, f − α1 ) + T (r, α1 − α2 ) + Z(r, f − α2 ) − log r + O(1)

(2)

≤ Z(r, f − α1 ) + Z(r, f − α2 ) + T (r, α1 ) + T (r, α2 ) − log r + O(1).

Therefore, from (1) and (2), T (r, f ) ≤ Z(r, f − α1 ) + Z(r, f − α2 ) + 2T (r, α1 ) + 3T (r, α2 ) − log r + O(1).


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We can easily deduce the following Corollaries 2 and 3 by remarking that S(r) only depends on the T (r, ui ) which are then bounded in ]0, R[. Corollary 2. Let f ∈ A(d(0,R−)) be transcendental and let α1 , α2 ∈ Ab (d(0,R−)) be distinct and such that 0 is not zero of f and f − αi (i = 1, 2). Then T (r, f ) ≤ Z(r, f − α1 ) + Z(r, f − α2 ) + S(r), where S(r) is bounded. Corollary 3. Let f ∈ A(d(0, R− )) be transcendental and let α ∈ Ab (d(0, R− )) such that 0 is not zero of f and f − α. Then T (r, f ) ≤ Z(r, f ) + Z(r, f − α) + S(r), where S(r) is bounded. We now have to show a p-adic version of a nice Theorem called Milloux’s Inequality, that has a fundamental role for dealing with value-sharing ignoring multiplicities of entire functions. It is necessary to mention that this has some variations with the classic inequality (see Theorem 3.2 [5]). Let f ∈ M(K) resp. f ∈ M(d(0, R− )) and let P be a property verified in a part of K. In the following Theorem we denote by Z1 (r, f ) the counting function of simple zeros of f and by N1 (r, f ) the counting function of simple poles of f in K resp. in d(0, R− ) . We shall also denote by Z(r, f : x satisfy P) resp. Z(r, f : x satisfy P) the counting function of zeros of f when the property P is satisfied, counting multiplicities resp. ignoring multiplicities . Theorem 4. (Milloux’s Inequality) Let f ∈ A(K) resp. f ∈ Au (d(0, R− )) and let a ∈ K − {0} be such that 0 is neither a zero of f , f − a and f . Then, for r > 0 resp. r ∈]0, R[ , T (r, f ) ≤ 2Z(r, f ) + Z(r, f − a) − Z r, f : f (x) = a, f (x), f (x) = (0, 0) −Z1 (r, f ) − log r + O(1). Proof. Let r > 0 resp. r ∈]0, R[ . We can easily check that (3)

Z(r, f ) − Z(r, f ) = Z(r, f ) − Z(r, f : f (x) = 0).

Moreover, since f is analytic, we deduce that (4)

Z(r, f ) = Z(r, f − a) + Z(r, f ) − Z(r, f : f (x) = a) + O(1)

So by (3) and (4), we deduce that Z(r, f ) = (5)

Z(r, f ) + Z(r, f − a) + Z(r, f ) − Z(r, f : f (x) = a) −Z(r, f : f (x) = 0) + O(1).

On the other hand, we note that the difference between Z(r, f ) and Z1 (r, f ) corresponds to the counting function of multiple zeros, ignoring their multiplicities. So (6)

Z(r, f ) − Z1 (r, f ) = Z(r, f : f (x) = 0) − Z(r, f : f (x) = f (x) = 0).

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Moreover Z(r, f ) = Z(r, f : f (x) = 0) + Z(r, f : f (x) = 0). But, by Lemma 4, Z(r, f ) ≤ Z(r, f ) − log r + O(1). Consequently Z(r, f ) ≤ Z(r, f : f (x) = 0) + Z(r, f : f (x) = 0) − log r + O(1). Therefore, by (6), we have Z(r, f ) ≤

Z(r, f ) − Z1 (r, f ) + Z(r, f : f (x) = f (x) = 0) + Z(r, f : f (x) = 0)

(7)

− log r + O(1).

Thus, by (5) and (7) Z(r, f )

2Z(r, f ) + Z(r, f − a) − Z1 (r, f ) + Z(r, f : f (x) = f (x) = 0) −Z(r, f : f (x) = a) − log r + O(1).

≤

(8)

Now, we see that

= ≥

Z(r, f : f (x) = a) − Z(r, f : f (x) = f (x) = 0) Z(r, f : f (x) = a, f (x) = 0) + Z(r, f : f (x) = a, f (x) = 0) Z(r, f : f (x) = a, f (x), f (x) = (0, 0)).

Therefore, by (8) and the latter remark, we can derive Z(r, f )

≤

2Z(r, f ) + Z(r, f − a) − Z1 (r, f ) −Z r, f : f (x) = a, f (x), f (x) = (0, 0) − log r + O(1)

and considering Lemma 1 we obtain the desired result: T (r, f ) ≤

2Z(r, f ) + Z(r, f − a) − Z1 (r, f ) −Z r, f : f (x) = a, f (x), f (x)) = (0, 0) − log r + O(1)

2. Uniqueness of Meromorphic Functions Following many studies on complex meromorphic functions, [5], [6], [8], [9] and [10], we shall examine properties of uniqueness on functions of the form f n f . As a Corollary of Theorem 3 [2], we may notice Theorem 5. Theorem 5. Let f, g ∈ M(K) resp. f, g ∈ Mu (d(0, R− )) and let n ≥ 2 (resp. n ≥ 3) be an integer. If f n f = g n g , then f = dg for some (n + 1) − th root of unity d. Proof. Indeed, if f n f = g n g there exists a constant c ∈ K such that f n+1 = + c. But if c = 0 then, by Theorem 3 [2], f and g are constant resp. f and g g lie in Mb (d(0, R− )) , a contradiction to our hypothesis. So, f n+1 = g n+1 . Thus, there exists d ∈ K with dn+1 = 1, such that f = dg. n+1


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2.1. Meromorphic Functions Sharing a Value C.M. As we know, a narrow relation exists between uniqueness of meromorphic functions and value sharing. So, the subject to consider is the following: given f, g ∈ M(K) resp. f, g ∈ M(d(0, R− )) such that f n and g n or f n f and g n g share one finite nonzero value C.M or I.M, for what positive integer n can we assert that f = g?. Let us remember the following definition. Definition. Let f, g ∈ M(K) resp. f, g ∈ M(d(0,R− )) be non-constant, I.M. , if f (z) − a a ∈ K∗ . We say that f and g share the value a C.M. resp. and g(z) − a have the same zeros with the same multiplicities resp. without taking multiplicities into account . Theorem 6. Let f, g ∈ M(K) be non-constant and let a, b ∈ K∗ satisfying ωα (f n − a) = ωα (g n − b) ∀α ∈ K, where n ≥ 3 is an integer. Then, there exists a d ∈ K such that dn = and f = d g. b Proof. Without loss of generality, we may assume that f and g have no zero and no pole at 0. Let a, b ∈ K∗ . Since ωα (f n − a) = ωα (g n − b) ∀α ∈ K, then f n − a and g n − b share in K the same zeros and poles with multiplicities. Therefore fn − a lies in M(K) but has no pole and no zero. So, it is a constant c ∈ K and gn − b hence f n − a = c (g n − b).

(9)

Clearly f n−1 f = cg n−1 g . Then, putting ln = c and g1 = lg, we have f n−1 f = Since n ≥ 3, by Theorem 5, there exists k ∈ K with kn = 1, such that f = kg1 , i.e. (kl)n = c and f = (kl)g. Thereby, by (9), we have cg n − a = cg n − cb a a that implies c = . Thus, there exists d = kl ∈ K with dn = , such that b b f = dg. g1n−1 g1 .

Theorem 7. Let f, g ∈ Mu (d(0, R− )) and let a, b ∈ K∗ satisfying ωα (f n − a) = ωα (g n − b) ∀α ∈ d(0, R− ) where n ≥ 4 is an integer. Then there a exists d ∈ K with dn = , such that f = dg. b Proof. Since ωα (f n − a) = ωα (g n − b) ∀α ∈ d(0, R− ), then f n − a and g n − b fn − a . share in d(0, R− ) the same zeros and poles counting multiplicities. Put ψ = n g −b Since ψ has no pole and no zero in d(0, R− ), it lies in Ab (d(0, R− )). a Suppose that ψ = . Putting u = a − ψb, we note that u ∈ Ab (d(0, R− )) and b u = 0. Assume, without loss of generality, that f , g and ψ have no zero and no pole at 0. Let r ∈]0, R[. Applying to f n Corollary 1, we have (10)

nT (r, f ) ≤ Z(r, f n ) + Z(r, f n − u) + N (r, f n ) + S(r)

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But Z(r, f n −u) ≤ T (r, u)+T (r, g), Z(r, f n ) ≤ T (r, f n ), N (r, f n ) ≤ T (r, f ) and T (r, f n ) = nT (r, f ) + O(1). Moreover, since f n = ψ g n + (a − ψb), we deduce that T (r, g) ≤ T (r, f ) + O(1). Thus, from (10), we have nT (r, f ) ≤ 3T (r, f ) + 5T (r, u). But u ∈ Ab (d(0, R− )) then, by Lemma 3, T (r, u) is bounded. Consequently nT (r, f ) ≤ 3T (r, f ) + O(1), a a contradiction to our hypothesis when n ≥ 4. Thus ψ = and hence there exists b a d ∈ K with dn = , such that f = dg. b With an analogous process as in Theorems 6 and 7, and considering, in this last, N (r, f ) = 0 we obtain the following result. Theorem 8. Let f, g ∈ A(K) resp. f, g ∈ Au (d(0, R− )) be non-constant and a, b ∈ K∗ satisfying ωα (f n − a) = ωα (g n − b) ∀α ∈ K where n ≥ 2 resp. n ≥ 3 a is an integer. Then there exists d ∈ K with dn = , such that f = dg. b In the proof of Corollary 4 we just have to assume a = b in Theorem 8. Corollary 4. Let f, g ∈ A(K) (resp. f, g ∈ Au (d(0, R− ))) be non-constant such that f n and g n share the value a ∈ K∗ C.M. where n ≥ 2 resp. n ≥ 3 is an integer. Then there exists d ∈ K with dn = 1, such that f = dg. Theorem 9. Let f, g ∈ A(K) be non-constant such that f n f and g n g share the value a ∈ K∗ C.M. where n ≥ 3 is an integer. Then f = dg for some (n + 1) th root of unity d. f nf − a ∈ M(K) gn g − a and it has no pole and no zero. So, there exists a constant l ∈ K such that f n f − a = l (g n g − a). Proof. Since f n f and g n g share the value a C.M.,

Suppose that l = 1. Now f n f = l (g n g ) + a(1 − l) and consequently, there exists l0 ∈ K such that f n+1 − lg n+1 = (n + 1)a(1 − l)x + l0 . But f n+1 − lg n+1 can n+1 (f − bi g) where each bi (i = 1, ..., n + 1) is a (n + 1) - th root of be written as i=1

l. Thus, (n + 1)a(1 − l)x + l0 = (f − b1 g)(f − b2 g)...(f − bn+1 g). Consequently at most one of these functions has a zero. But n ≥ 2, hence there exists at least two functions f − bi g that are constants. Thus, we deduce that f and g are constants, a contradiction to the hypothesis. Then l = 1 and (11)

f n+1 = g n+1 + l0 Suppose now that l0 = 0. Applying Theorem 1 to f n+1 , we have

T (r, f n+1 ) ≤ Z(r, f n+1 )+Z(r, f n+1 −l0 )−log r+O(1) ≤ T (r, f )+T (r, g)−log r+O(1). But, from (11), T (r, f ) = T (r, g) + O(1). Thus (n + 1)T (r, f ) ≤ 2T (r, f ) − log r + O(1),


169 9

a contradiction when n ≥ 1. Consequently l0 = 0 and f n+1 = g n+1 , this implies f = dg with d ∈ K such that dn+1 = 1. Thanks to Theorem 2, we can show a similar theorem in Au (d(0, R− )). Theorem 10. Let f, g ∈ Au (d(0, R− )) such that f n f and g n g share the value a ∈ K∗ C.M. where n ≥ 4 is an integer. Then f = dg for some (n + 1) - th root of unity d. f nf − a . Since f n f and g n g share the value a C.M., we gn g − a deduce that φ has no zeros and no poles in d(0, R− ) and then φ ∈ Ab (d(0, R− )) and T (r, φ) is bounded. Proof. Put φ =

Suppose φ = 1 and consider F = f n f and φ1 = a(1 − φ). Note that F ∈ Au (d(0, R− )) and φ1 ∈ Ab (d(0, R− )). This last implies that T (r, φ1 ) is bounded. Thus, applying Corollary 1 to F , we have (12)

T (r, F ) ≤ Z(r, F ) + Z(r, F − φ1 ) + O(1).

But Z(r, F ) ≤ T (r, f ) + T (r, f ), Z(r, F − φ1 ) ≤ 2T (r, g) + O(1) and T (r, F ) = nT (r, f ) + T (r, f ). Thus, from (12), we have (n − 1)T (r, f ) ≤ 2T (r, g) + O(1). n

Similarly, for G = g g , we have (n − 1)T (r, g) ≤ 2T (r, f ) + O(1). Consequently, adding this two inequalities, we obtain

(n − 1) T (r, f ) + T (r, g) ≤ 2 T (r, f ) + T (r, g) + O(1), a contradiction when n ≥ 4. Thus φ = 1. The end of this proof is similar to the proof of Theorem 9, considering that T (r, g) ≤ T (r, f ) + O(1). Theorem 11 is designed to establish a result comparable to Theorem 10, considering meromorphic functions. The proof of this will require Lemmas 6, 7, 8 and 9. Lemma 6. Let f, g ∈ A(d(0, R− )) be such that γ is a simple zero of f − 1 and g − 1. Then γ is also a zero of g f f g − +2 −2 f f −1 g g−1 Proof. Put f (x) =

+∞

an xn . Without loss of generality we assume γ = 0.

n=0

Thanks to Taylor’s development of f at 0, we have f1 ∈ A(d(0, R− )) and a1 = 0. This implies (13)

1 1 = + f1 (x) with f (x) − 1 a1 x

f (x) 1 =− + f1 (x). (f (x) − 1)2 a 1 x2

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Similarly

1 1 = + g1 (x) with g1 ∈ A(d(0, R− )) and b1 = 0. Then g(x) − 1 b1 x g (x) 1 =− + g1 (x). (g(x) − 1)2 b1 x2

(14)

Consequently, by (13) and (14), we deduce that (g(x) − 1)2 f (x) = · 2 (f (x) − 1) g (x)

1 a1 1 b1

− x2 f1 (x) − x2 g1 (x)

= u + vx2 + wx3 + ...

with u, v = 0. Then v w 1 f (x) (g(x) − 1)2 2 · = 1 + x x3 + ... · + u (f (x) − 1)2 g (x) u u and so, for x close to 0, we apply the ultrametric logarithm function log(1 + t) defined for t ∈ d(0, 1− ). Thus v2 w2 1 (g(x) − 1)2 v 2 w 3 f (x) 4 · + +...− − · = x x log x x6 −... u (f (x) − 1)2 g (x) u u 2u2 2u2 Consequently, applying the logarithmic derivative, we have 2v 3w 2v 2 3w2 f (x) 2f (x) g (x) 2g (x) 2 3 − − + = x+ x x x5 −... +...− − f (x) f (x) − 1 g (x) g(x) − 1 u u u2 u2 Thus, the function

f g g f −2 − −2 has a zero at x = γ. f f −1 g g−1

With an analogous process to Lemma 2.12 given by A Boutabaa and A. Escassut in [3], we can deduce the following result. Lemma 7. Let f ∈ M(K) resp. f ∈ Mu (d(0, R− )) be such that 0 is neither f a zero nor a pole of f . Then F = satisfies Z(r, F ) ≤ N (r, F ) − log r + O(1). f Henceforth, we put Z 2) (r, f ) := Z(r, f ) − Z1 (r, f ) and Z2) (r, f ) := Z(r, f ) − Z1 (r, f ). In the same way, we put N 2) (r, f ) := N (r, f ) − N1 (r, f ) and N2) (r, f ) := (r, f ) the counting N (r, f ) − N1 (r, f ). Moreover, we denote by Z0 (r, f ) resp. Z 0 function of the multiple zeros of f counting multiplicities resp. ignoring multiplicities which are not zeros of f and f − 1. Lemma 8. Let f, g ∈ M(K) resp. f, g ∈ Mu (d(0, R− )) be non-constant and such that 0 is neither a zero nor a pole of f , f , g and g . If f and g share the value 1 C.M., then one of the following three cases holds: (i) T (r, f ) ≤ N (r, f )+N 2) (r, f )+N (r, g)+N 2) (r, g)+Z(r, f )+Z 2) (r, f )+Z(r, g)+ Z 2) (r, g) − 2 log r + O(1) the same inequality holding for T (r, g). (ii) f ≡ g. (iii) f g ≡ 1.

171 11


Proof. Our proof takes reference from Lemma 5 given by C. C. Yang and X. Hua in [9], but we have to thoroughly check each inequality in order to deduce g ψ f f g − +2 . Clearly Ψ = conclusion involving − log r. Put Ψ = − 2 f f −1 g g−1 ψ (g − 1)2 f . with ψ = (f − 1)2 g For the proof, we consider two cases: Ψ = 0 and Ψ = 0. • Case 1: Ψ(x) = 0

∀x ∈ K

resp. ∀x ∈ d(0, R− ) .

Let r > 0 resp. r ∈]0, R[ . Applying Theorem 1 to f , we have (15) T (r, f ) ≤ Z(r, f ) + Z(r, f − 1) + N (r, f ) − Z(r, f : f (x) = 0, 1) − log r + O(1) ψ , by Lemma 7, we have Z(r, Ψ) ≤ N (r, Ψ) + O(1). But, if γ is a ψ simple zero of f − 1, since f and g sharing the value 1 C.M., by Lemma 6, γ is a zero of Ψ, i.e. Z1 (r, f − 1) ≤ Z(r, Ψ). Consequently Since Ψ =

Z1 (r, f − 1) ≤ N (r, Ψ) + O(1).

(16)

On the other hand, since f and g share the value 1 C.M., we deduce that

f f −1

g have a simple pole with the same residue at each point β such that g−1 f (β) = 1. Consequently Ψ has no pole at points β where f (β) = 1. Moreover, f at γ is −1 whereas the residue of when f has a simple pole γ, the residue of f −1 f at γ is −2. So, Ψ has no pole at γ. Thus, if Ψ has a pole or if f or g has a f multiple pole or if f or g has a zero and so, from (16), and

(17) Z1 (r, f − 1) ≤

N 2) (r, f ) − N 2) (r, g) + Z(r, f : f (x) = 0, 1) +Z(r, g : g(x) = 0, 1) + Z 2) (r, f ) + Z 2) (r, g) + O(1).

But Z(r, g : g(x) = 0, 1) + Z 2) (r, g − 1) + Z2) (r, g) − Z 2) (r, g) ≤ Z(r, g ) and so, from Lemma 4, (18)

Z(r, g : g(x) = 0, 1) + Z 2) (r, g − 1) ≤ Z(r, g) + N (r, g) − log r + O(1).

Moreover, since f and g share the value 1 C.M., we have Z(r, f − 1) = Z(r, g − 1). Thus Z(r, f − 1) ≤ Z1 (r, f − 1) + Z 2) (r, g − 1)

(19)

Therefore, from (15), (16), (17), (18) and (19), we have (i): T (r, f ) ≤

N (r, f ) + N 2) (r, f ) + N (r, g) + N 2) (r, g) + Z(r, f ) + Z 2) (r, f ) +Z(r, g) + Z 2) (r, g) − 2 log r + O(1).

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• Case 2: Ψ(x) = 0 ∀x ∈ K

resp. ∀x ∈ d(0, R− ) .

(g − 1)2 f ψ , there exists c ∈ K such that with ψ = ψ (f − 1)2 g f g 1 c =c and so, there exists d ∈ K such that = − d when 2 2 (f − 1) (g − 1) f −1 g−1 Ag + B with A, B, C, D ∈ K. Ψ = 0. Consequently f = Cg + D B − AD A C = and applying Theorem 1 to Suppose first that AC = 0. So f − C Cg + D f , we have (i): A T (r, f ) ≤ Z(r, f ) + Z r, f − + N (r, f ) − log r + O(1) C = Z(r, f ) + N (r, g) + N (r, f ) − log r + O(1). Since Ψ =

A B = g. If B = 0, we have (i): D D B + N (r, f ) − log r + O(1) Z(r, f ) + Z r, f − D Z(r, f ) + Z(r, g) + N (r, f ) − log r + O(1).

Suppose now A = 0 and C = 0. So f − T (r, f )

≤ =

A A If B = 0, f = g. Since f and g share 1 C.M., if = 1 we have f (x) = 1 ∀x ∈ K D D − resp. ∀x ∈ d(0, R ) and so g(x) = 1 ∀x ∈ K resp. ∀x ∈ d(0, R− ) . Thereby A f (x) = ∀x ∈ K resp. ∀x ∈ d(0, R− ) and so f is a constant, a contradiction. D A = 1 and thus f = g, i.e. we have (ii). Consequently D Finally, if A = 0 and C = 0, we have g +

B D = . If D = 0, applying C Cf

Theorem 1 to g, we have (i): T (r, g)

D ≤ Z(r, g) + Z r, g + + N (r, g) − log r + O(1) C = Z(r, g) + N (r, f ) + N (r, g) − log r + O(1).

B B . Suppose = 1. Since f and g share 1 C.M., necessarily Cf C B ∀x ∈ K resp. x ∈ d(0, R− ) . g(x) = 1 ∀x ∈ K resp. x ∈ d(0, R− ) and g(x) = C B = 1 and so f g = 1, i.e. we Thus g is a constant, a contradiction. Consequently C have (iii). Lemma 9. There are no f, g ∈ M(K) non-constant resp. f, g ∈ Mu (d(0, R− )) such that f f n g g n ≡ 1 when n ≥ 2 is an integer. Proof. Suppose that f has a zero γ of order s in K resp. in d(0, R− ) . Since f f n g g n = 1, we deduce that γ is a pole of g of order, for example t. So ns + s − 1 = nt + t + 1 that implies (s − t)(n + 1) = 2, a contradiction when n ≥ 2 and s, t ∈ Z. Similarly for g. Consequently f and g have no zeros in K resp. If D = 0, g =


173 13

in d(0, R− ) . Since f, g ∈ M(K) are non-constant resp. f, g ∈ Mu (d(0, R− )) without zeros in K resp. in d(0, R− ) , there exists non-constant functions φ, ψ ∈ 1 1 and g = . Consequently A(K) resp. φ, ψ ∈ Au (d(0, R− )) such that f = φ ψ 1 φ 1 ψ φ ψ 1 = f n f gn g = n 2 n 2 = and so φ ψ = (φψ)n+2 . Since φ, ψ ∈ φ φ ψ ψ (φψ)n+2 A(K) are non-constant resp. φ, ψ ∈ Au (d(0, R− )) , by Theorem 13.5 [4], we have 1 1 |φ |(r) ≤ |φ|(r) ∀r > 0 resp. ∀r ∈]0, R[ and |ψ |(r) ≤ |ψ|(r) ∀r > 0 resp. r r |φψ|(r) ∀r ∈]0, R[ . Therefore (|φψ|(r))n+2 = |φ ψ |(r) ≤ , a contradiction because r2 |φ|(r) and |ψ|(r) are unbounded when r tends to ∞ resp. to R− . In the complex case, C.C. Yang and X. Hua showed in Theorem 1 [9] that: “If two non-constant complex meromorphic functions f and g are such that f n f and g n g share the finite value a ∈ C when n ≥ 11, then either f = dg with dn+1 = 1 or f = c2 e−cz and g = c1 ecz where c, c1 and c2 are constants and (c1 c2 )n+1 c2 = −a”. Here, under similar hypothesis, we may get to the first conclusion, while the second is excluded. Theorem 11. Let f, g ∈ M(K) be non-constant resp. f, g ∈ Mu (d(0, R− )) such that f n f and g n g share the value a ∈ K∗ C.M. when n ≥ 11 is an integer. Then f = dg for some (n + 1) − th root of unity d. f n+1 g n+1 and G = . Since f n f and g n g share the a(n + 1) a(n + 1) value a C.M., F and G share the value 1 C.M. Let r > 0 resp. r ∈]0, R[ . Since N (r, F ) − N (r, f ) − N (r, f ) ≤ T (r, F ) + T (r, f ) − N (r, f ) − Z(r, f ) + O(1) and N (r, F ) = (n + 1)N (r, f ) + N (r, f ), we have Proof. Put F =

nN (r, f ) ≤ T (r, F ) + T (r, f ) − N (r, f ) − Z(r, f ) + O(1). On the other hand, since Z(r, F ) − Z(r, f ) ≤ T (r, F ) + T (r, f ) − N (r, f ) − Z(r, f ) + O(1) and Z(r, F ) = nZ(r, f ) + Z(r, f ), we have nZ(r, f ) ≤ T (r, F ) + T (r, f ) − N (r, f ) − Z(r, f ) + O(1). Consequently, nT (r, f ) ≤ T (r, F ) + T (r, f ) − N (r, f ) − Z(r, f ) + O(1). and so,

(20)

2T (r, f ) + Z(r, f ) n−3 2 (n − 1)T (r, f ) + Z(r, f ) + Z(r, f ) = n−1 n−1 n − 3 2 T (r, F ) − N (r, f ) + T (r, f ) + N (r, f ) + O(1) ≤ n−1 n−1 n − 3 1 2 T (r, F ) − N (r, f ) + T (r, F ) + N (r, f ) + O(1) ≤ n−1 n−1 n−1 3n − 5 n − 5 T (r, F ) + O(1). ≤ T (r, F ) + (n − 1)2 (n − 1)(n + 2)

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But N (r, f ) = N (r, F ) = N 2) (r, F ) ≤

(21)

1 T (r, F ) + O(1) n+2

and Z(r, F ) + Z 2) (r, F ) ≤ 2T (r, f ) + Z(r, f ) + O(1). Consequently, from (20), we deduce that Z(r, F ) + Z 2) (r, F ) ≤

(22)

4n2 − 5n − 5 T (r, F ) + O(1). (n + 2)(n − 1)2

Similarly for G , we have N (r, F ) = N 2) (r, F ) ≤

(23)

1 T (r, F ) + O(1) n+2

and Z(r, G ) + Z 2) (r, G ) ≤

(24)

4n2 − 5n − 5 T (r, G ) + O(1). (n + 2)(n − 1)2

Now, applying Lemma 8 to F and G , it follows that the following three cases are to be considered. • Case 1: From Lemma 8 (i), we have T (r, F )

≤ N (r, F ) + N 2) (r, F ) + N (r, G ) + N 2) (r, G ) + Z(r, F ) + Z 2) (r, F ) +Z(r, G ) + Z 2) (r, G ) − 2 log r + O(1).

Suppose that there exists an unbounded set I such that T (r, G ) ≤ T (r, F ) ∀r ∈ I. Let r ∈ I. Applying (21), (22), (23) and (24) into the above inequality we get n3 − 12n2 + 15n + 8 T (r, F ) ≤ −2 log r + O(1). (n + 2)(n − 1)2

(25)

Put P (n) = n3 − 12n2 + 15n + 8. Then P (11) = 52 and P (n) = 3n2 − 24n + 15 is positive when n ≥ 8. So P (n) is positive when n ≥ 11, a contradiction to (25). • Case 2:

f

Suppose Lemma 8 (ii), i.e. F = G . Then, there exists c ∈ K such that = g n+1 + c. If c = 0, applying Theorem 1 to f n+1 , we have

n+1

T (r, f n+1 )

≤ Z(r, f n+1 ) + Z(r, f n+1 − c) + N (r, f n+1 ) − log r + O(1) ≤ 2T (r, f ) + T (r, g) − log r + O(1).

But T (r, f n+1 ) = (n + 1)T (r, f ) + O(1) and, since f n+1 = g n+1 + c, T (r, f ) = T (r, g) + O(1). Consequently, from above inequality, we have (n + 1)T (r, f ) ≤ 3T (r, f ) − log r + O(1), a contradiction when n ≥ 11 and r tends to +∞ resp. to R− . So c = 0 and f n+1 = g n+1 . Thus, there exists d ∈ K with dn+1 = 1 and f = dg.


175 15

• Case 3: Suppose Lemma 8 (iii), i.e. F G = 1. So a−2 f n f g n g = 1. Let b be a (n + 1) − th root of a−1 . Put f˙ = bf and g˙ = bg. So f˙n f˙ g˙ n g˙ = 1, a contradiction to Lemma 9 because n ≥ 11.

2.2. Analytic Functions Sharing a Value I.M.. In order to continue our study we analyze the uniqueness of analytic functions of the form f n f and g n g that share a value I.M. For this, we take as reference the definition given in the previous subsection and the study realized by Y. Xu and H. Qu in [8]. Let f, g ∈ M(K) resp. f, g ∈ M(d(0, R− )) be non-constant such that f and g share the value 1 I.M. Assuming that f − 1 and g − 1 have no zero and no pole at 0, we denote by Z c (r, f − 1, g − 1) the counting function of common zeros γ of both f − 1 and g − 1 when ωγ (f − 1) > ωγ (g − 1), each zero being counted without multiplicity, and we denote by Z11 (r, f − 1, g − 1) the counting function of common simple zeros of both f − 1 and g − 1. Before continuing, we must recall the following definition given by P. C. Hu and C. C. Yang in [7]. Let c ∈ K be a constant. If f ∈ M(d(0, R− )) and if 0 is neither a zero nor a pole of f and f − c we define Θ(c, f ) = 1 − lim sup r→R−

Z(r, f − c) T (r, f )

− Lemma 10. Let f, g ∈ A(K) be non-affine resp. f, g ∈ A u (d(0, R )) such that 0 is neither a zero nor a pole of f and g and satisfy lim inf T (r, f )−9Z(r, f ) > r→+∞ 8 8 . −∞ and lim inf T (r, g)−9Z(r, g) > −∞ resp. Θ(0, f ) > and Θ(0, g) > r→+∞ 9 9 If f and g share the value 1 I.M., then f = g. f g g φ f − + 2 . Clearly Φ = with Proof. Put Φ = − 2 f f −1 g g −1 φ f (g − 1)2 φ= . 2 (f − 1) g φ Suppose Φ = 0. Since Φ = , by Lemma 7, we have Z(r, Φ) ≤ N (r, Φ) + O(1). φ But, if γ is a common simple zero of both f − 1 and g − 1, from Lemma 6, γ is a zero of Φ and so Z11 (r, f −1, g −1) ≤ Z(r, Φ). Consequently Z11 (r, f −1, g −1) ≤ N (r, Φ) + O(1). But the poles of Φ occur at the zeros of f and g and at the zeros of f − 1 and g − 1 with not the same multiplicity. So, we deduce that N (r, Φ) ≤ Z(r, f ) + Z(r, g) + Z c (r, f − 1, g − 1) + Z c (r, g − 1, f − 1) + + Z r, f : f (x) = 1, (f (x), f (x)) = (0, 0) + + Z r, g : g (x) = 1, (g(x), g (x)) = (0, 0) .

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Consequently, from two above inequalities, we have (26)

Z 11 (r, f − 1, g − 1)

≤ Z(r, f ) + Z c (r, f − 1, g − 1) +Z r, f : f (x) = 1, (f (x), f (x)) = (0, 0) + Z(r, g) + Z c (r, g − 1, f − 1) +Z r, g : g (x) = 1, (g(x), g (x)) = (0, 0) .

On the other hand, by Theorem 4, T (r, f ) + T (r, g) ≤ (27)

2Z(r, f ) + 2Z(r, g) + Z(r, f − 1) −Z r, f : f (x) = 1, (f (x), f (x)) = (0, 0) + Z(r, g − 1) − Z r, g : g (x) = 1, (g(x), g (x)) = (0, 0) −2 log r + O(1).

Moreover Z(r, f − 1) + Z(r, g − 1) ≤ Z11 (r, f− −1, g − 1) + Z(r, f − 1) + Z(r, g − 1). But g − 1 ∈ A(K) resp. g − 1 ∈ A(d(0, R )) then, by Lemma 4, Z(r, g − 1) ≤ T (r, g) + O(1). Thus

Z(r, f − 1) + Z(r, g − 1) ≤ Z11 (r, f − 1) + Z(r, f − 1) + Z(r, g − 1). and so, from (27), we have T (r, f ) ≤ 2Z(r, f ) + 2Z(r, g) + Z11 (r, f − 1) + Z c (r, f − 1, g − 1) − − Z r, f : f (x) = 1, (f (x), f (x)) = (0, 0) − − Z(r, g : g (x) = 1, (g(x), g (x)) = (0, 0) − 3 log r + O(1). Consequently, considering (26) in the above inequality, we have T (r, f ) ≤ 3Z(r, f )+3Z(r, g)+2Z c (r, f −1, g −1)+Z c (r, g −1, f −1)−2 log r+0(1). But, by Lemma 4, Z c (r, f − 1, g − 1) ≤ Z(r, f ) − log r + O(1) and so T (r, f ) ≤ 5Z(r, f ) + 4Z(r, g) − 3 log r + O(1).

(28)

Similarly for g, we have T (r, g) ≤ 5Z(r, g) + 4Z(r, f ) − 4 log r + O(1).

(29)

Suppose first that f, g ∈ A(K). From (28) and (29), we have lim T (r, f ) − r→+∞ 9Z(r, f ) = −∞ and lim T (r, g) − 9Z(r, g) = −∞, a contradiction to our r→+∞

hypothesis. Now, suppose that f, g ∈ Au (d(0, R− )). From (28) and (29), we have T (r, f ) − 9Z(r, f ) + T (r, g) − 9Z(r, g) ≤ −6 log r + O(1). Consequently (30)

Z(r, f ) Z(r, g)

+ T (r, g) 1 − 9 < +∞. lim sup T (r, f ) 1 − 9 T (r, f ) T (r, g) r→R−

8 8 and Θ(0, g) > . So, we can check that there exists δ > 0 and 9 9 Z(r, f ) Z(r, g) ≥ δ ∀r ∈]ρ, R[ and 1 − 9 ≥ δ ∀r ∈]ρ, R[. ρ ∈]0, R[ such that, 1 − 9 T (r, f ) T (r, g) Z(r, f ) Z(r, g) = +∞ and lim T (r, g) 1 − 9 = +∞, Hence lim− T (r, f ) 1 − 9 T (r, f ) T (r, g) r→R r→R− But Θ(0, f ) >


177 17

a contradiction to (30). Consequently Φ = 0 whenever f, g ∈ A(K) resp. f, g ∈ A(d(0, R− )) . f (g − 1)2 φ with φ = and since Φ = 0, there exist φ (f − 1)2 g c, d ∈ K such that Since Φ =

dg + c − d 1 = f − 1 g − 1 with d g (x) = d − c ∀x ∈ K resp. ∀x ∈ d(0, R− ) . (31)

Consider the 3 following cases: • Case 1: d = 0 and c = d. c−d = 0. Applying Theorem 4 to g, we have d c − d − log r + O(1) T (r, g) ≤ 2Z(r, g) + Z r, g + d = 2Z(r, g) − log r + O(1).

From (31), we have g +

If g ∈ A(K), this contradicts the hypothesis because g is non-affine and lim inf T (r, g)− r→+∞ 2Z(r, g) ≤ −∞. Similarly, if g ∈ Au (d(0, R− )), this also contradicts the hypothesis quand r tends to R− because

Z(r, g) 1 + O(1) ≤ . 2 T (r, g)

• Case 2: d = 0 and c = d. 1 1 From (31), we have f = 1 + − with g (x) = 0 ∀x ∈ K resp. ∀x ∈ d dg d(0, R− ) . Suppose d = −1. So f g = 1 and thereby f and g are constants. Thus f (x) = ax + b and g(x) = cx + d which is excluded by hypothesis. Now, suppose 1 1 d = −1. So f − 1 + = − = 0. Applying Theorem 4 to f , we have d dg 1 − log r + O(1) T (r, f ) ≤ 2Z(r, f ) + Z r, f − 1 + d = 2Z(r, f ) − log r + O(1). Consequently, as in Case 1, we have a contradiction. • Case 3: d = 0 and c = d. From (31), there exists co such that f =

1 g + L(x) with L(x) = 1 − x + co . c c

Suppose first g = cf and so g = cf . Since f, g ∈ A(K) that L(x) = 0. We have − are non-affine resp. f, g ∈ Au (d(0, R )) , f and g are non-constant analytic functions and so f and g have no exceptional values. Consequently, there exists

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JACQUELINE OJEDA

at least one value γ ∈ K such that f (γ) = 1. But, since f and g share 1 I.M., we have g (γ) = 1. Thus c = 1 and f = g. Suppose now L(x) = 0. Clearly L(x) is a small function with respect to f . Thus, applying Corollary 1 to f , we have T (r, f ) ≤ Z(r, f ) + Z(r, g) + 3 log r + O(1).

(32)

If f, g ∈ Au (d(0, R− )), from the above inequality, we have Z(r, f )

Z(r, g)

+ T (r, g) 1 − 1 − + 3 log r + O(1). T (r, f ) ≤ T (r, f ) 1 − 1 − T (r, f ) T (r, g) g But, since f = + L(x) and since f and g are unbounded, we deduce that c Z(r, f ) Z(r, g) 3 log r + O(1) + 1− −1 ≤ . T (r, g) ≤ T (r, f ) + O(1). Thus 1 − T (r, f ) T (r, g) T (r, f ) − Consequently Θ(0, f ) + Θ(0, g) ≤ 1 when r tends to R , a contradiction because 8 8 Θ(0, f ) > and Θ(0, g) > . Suppose now f, g ∈ A(K). By hypothesis, we deduce 9 9 that there exists η ∈ R+ such that T (r, f ) ≥ 9Z(r, f ) − η and T (r, g) ≥ 9Z(r, g) − η. Moreover, since T (r, g) ≤ T (r, f ) + O(1), we have 2η 1 (33) T (r, f ) + T (r, g) > 3 Z(r, f ) + Z(r, g) − . T (r, f ) > 3 3 2η ≤ Z(r, f ) + Consequently, from (32) and (33), we have 3 Z(r, f ) + Z(r, g) − 3 3 Z(r, g) + 3 log r + O(1), this is Z(r, f ) + Z(r, g) < log r + O(1), a contradiction 2 because f and g are non-affine. n Theorem 12. Let f, g ∈ A(K) resp. f, g ∈ Au (d(0, R− )) such that f f and n ∗ g g share the value a ∈ K I.M. when n ≥ 8 resp. n ≥ 9 is an integer. Then f = dg for some (n + 1) − th root of unity d. f n+1 g n+1 Proof. Put F = and G = . Clearly F, G ∈ A(K) are nona(n + 1) a(n + 1) affine resp. F, G, F , G ∈ Au (d(0, R− )) and F and G share 1 I.M. Moreover the zeros of F and G have order ≥ n + 1. So, for r > 0 resp. r ∈]0, R[ , we have (34)

(n + 1)Z(r, F ) ≤ T (r, F )

and

(n + 1)Z(r, G) ≤ T (r, G)

respectively. Suppose first f, g ∈ A(K). From (34), since n ≥ 8, we have lim inf T (r, F )−(n+1)Z(r, F ) ≥ 0 and lim inf T (r, G)−(n+1)Z(r, G) ≥ 0.

r→+∞

r→+∞

Thus F and G satisfy the hypotheses of Lemma 10 and so F = G. Suppose now Z(r, F ) 1 ≤ and so Θ(0, F ) = f, g ∈ Au (d(0, R− )). From (34), we have T (r, F ) n+1 Z(r, F ) n n 1 − lim sup ≥ . Similarly for G, we have Θ(0, G) ≥ . Since T (r, F ) n+1 n+1 r→R− 8 8 n 9 n ≥ 9, we have n+1 ≥ 10 > 89 , hence Θ(0, F ) > and Θ(0, G) > . Consequently, 9 9 by Lemma 10, F = G again. As we can see in both cases F = G. Thus, f n+1 = g n+1 and f = dg for some (n + 1) − th root of unity d.


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Acknowledgements: This work was supervised by Alain Escassut, Université Blaise Pascal, Laboratoire de Mathématique, Clermont-Ferrand. My sincere gratitude. I am also grateful to Marie-Claude Sarmant and to the anonymous referee for many remarks. References [1] A. Boutabaa and A. Escassut, On uniqueness of p-adic meromorphic functions, Proceedings of the AMS 126(9), 2557 - 2568 (1998). [2] A. Boutabaa and A. Escassut, Applications of the p-adic Nevanlinna Theory to functional equations, Ann. Inst. Fourier, Grenoble 50, 751 - 766 (2000). [3] A. Boutabaa and A. Escassut, URS and URSIMS for p-adic meromorphic functions inside a disc, Proc. of the Edinburgh Mathematical Society 44, 485 - 504 (2001). [4] A. Escassut, Analytic Elements in p-adic Anaysis, World Scientific Publishing (1995). [5] W. K. Hayman, Meromorphic Functions, Clarendon Press, Oxford. (1964). [6] M. L. Fang and X. H. Hua, Entire functions that share one value. J. Nanjing Univ. Math. Biq. 13(1), 44 - 48 (1996). [7] P. C. Hu and C. C. Yang, Meromorphic functions over non archimedean fields, Kluwer Academy Publishers (2000). [8] Y. Xu and H. Qu, Entire functions sharing one value IM. Indian J. Pure Appl. Math. 31(7), 849 - 855 (2000). [9] C. C Yang and X. Hua, Uniqueness and value-sharing of meromorphic functions. Ann. Acad. Sci. Fenn. Math. 22, 395 - 406 (1997). [10] H. X. Yi and C. C. Yang, Uniqueness theorems of meromorphic functions. Science Press, China (1995). ´matique (UMR 6620), Universite ´ Blaise Pascal, Campus UniLaboratoire de Mathe versitaire des C´ ezeaux, 63177 Aubiere Cedex, France E-mail address: [email protected]



Tensor products of p-adic locally convex spaces having the strongest locally convex topology Cristina Pérez-Garc´ıa and Wim H. Schikhof Abstract. In this note we study the question whether the tensor product of two p-adic locally convex spaces having the strongest locally convex topology, again has the strongest locally convex topology. We show that, surprisingly, the answer depends on the algebraic dimension of the two spaces involved.

Introduction Tensor products form an important construction within the family of locally convex spaces in p-adic Functional Analysis. We point out their applications to the study of spaces of vector valued continuous functions, many of which can be described as tensor products (see [1], [3], [8], [9], [11], [12], [15]). Also, the strongest locally convex topology (s.l.c.t. in short) appears in the characterizations of certain classes of non-archimedean locally convex spaces (e.g. those whose topology coincides with the weak topology, [10], and inductive limits satisfying some Baire-like conditions, [2]). Several hereditary properties of tensor products were treated in [3], [6], [7], [11], [12] and [15]. Also, it is easily seen that the s.l.c.t. is preserved under taking subspaces, quotients, locally convex direct sums, inductive limits and finite products, but not under taking infinite products (an obvious example is K N ). However, as far as we know, the question whether the tensor product of two p-adic locally convex spaces having the s.l.c.t., again has the s.l.c.t., remained unsolved. In this note we show that, surprisingly, the answer depends on the algebraic dimension of the two spaces involved. Our study leads to the solution in “all” the possible cases, see the conclusion at the end of the paper. 1. Preliminaries Throughout this paper K = (K, | . |) is a non-archimedean non-trivially valued field that is complete with respect to the metric induced by the valuation | . |. 2000 Mathematics Subject Classification. 46S10. Key words and phrases. p-adic locally convex space, tensor product, strongest locally convex topology. Research partially supported by Ministerio de Educaci´ on y Ciencia, MTM2006-14786. c Mathematical 0000 (copyright Society holder) c 2010 American

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´ ÍA AND WIM H. SCHIKHOF CRISTINA PEREZ-GARC

Let E be a K-vector space. By dim E we mean the algebraic dimension of E; this cardinal will play a key role in this paper. The cardinal of a set J is denoted by J. As usually, ℵ0 := N. Also, recall that 2ℵ0 = NN . For basics on cardinals see e.g. [5]. For fundamentals on locally convex spaces over K we refer to [14]. Clearly the strongest locally convex topology (s.l.c.t. in short) on a K-vector space E is the locally convex topology generated by the collection of all (non-archimedean) seminorms on E. Let E, F be locally convex spaces over K. We endow the tensor product E ⊗ F with the so-called projective tensor product topology, which is the locally convex topology induced by the seminorms p ⊗ q, where p, q are continuous seminorms on E, F respectively (then E ⊗ F is called the projective tensor product of E and F ). Here p ⊗ q is defined as follows. For z ∈ E ⊗ F , let (p ⊗ q)(z) := inf{max1≤i≤n p(xi ) q(yi ) : n ∈ N, xi ∈ E, yi ∈ F, z =

n

xi ⊗ yi }.

i=1

Observe that a seminorm r on E ⊗F is continuous for the projective tensor product topology if and only if there exist continuous seminorms p on E, q on F such that r ≤ p ⊗ q. In fact, the “if” is clear. To prove the “only if”, suppose r is continuous i.e. there exist a constant C > 0 and continuous seminorms p1 , . . . , pn on E, q1 , . . . , qn on F (n ∈ N) such that r ≤ C max1≤i≤n pi ⊗ qi . Then r ≤ p ⊗ q with p := C max1≤i≤n pi , q := max1≤i≤n qi , which are continuous seminorms on E, F respectively. It is well known ([13], Proposition 1) that: (i) p ⊗ q is a cross seminorm i.e. (p⊗q)(x⊗y) = p(x) q(y) for each x ∈ E, y ∈ F . (ii) If E1 ⊂ E, F1 ⊂ F are subspaces equipped with the restricted topologies, then the natural inclusion E1 ⊗ F1 →E ⊗ F is a homeomorphic embedding. (iii) If E and F are Hausdorff then so is E ⊗ F ; the converse holds when E, F = {0}. Remark 1.1. The reader could think of considering also the natural nonarchimedean counterpart of the classical injective tensor product of E and F (see [4], Section 16.1). It is the K-vector space E ⊗F , but now endowed with the locally convex topology induced by the seminorms z ∈ E ⊗ F −→ sup{|(f ⊗ g)(z)| : f ∈ E ∗ , g ∈ F ∗ , |f | ≤ p, |g| ≤ q}, where E ∗ , F ∗ are the algebraic duals of E, F and p, q are continuous seminorms on E, F , respectively. However, it was proved in [3] that, contrary to the classical case, the projective and the injective tensor products coincide for a very large class of locally convex spaces E, F , the so-called polar spaces (introduced in [14]), among which we have those equipped with the s.l.c.t. Therefore, we only treat projective tensor products in this paper. The fact that the tensor product of two non-zero locally convex spaces has the s.l.c.t., implies that the spaces themselves also have the s.l.c.t., as we show in the next proposition. The converse will be discussed in Section 2. Proposition 1.2. Let E, F be non-zero locally convex spaces over K. Let E ⊗ F have the s.l.c.t. Then so have E and F .

TENSOR PRODUCTS OF p-ADIC LOCALLY CONVEX SPACES

183 3

Proof. As E ⊗ F is Hausdorff, so are E and F . Hence, E and F are linearly homeomorphic to subspaces of E ⊗ F . In fact, choose a ∈ E \ {0}, b ∈ F \ {0}. Then the maps E→E ⊗ F , x → x ⊗ b and F →E ⊗ F , y → a ⊗ y are easily seen to be linear homeomorphic embeddings. Thus, E and F have the s.l.c.t. Remark 1.3. If E = {0} or F = {0} then E ⊗ F = {0} and clearly the conclusion of Proposition 1.2 is false. 2. The main results In this section we get the purpose of this paper. That is, we answer the following QUESTION: Let E, F have the s.l.c.t. Does it follow that E ⊗ F has the s.l.c.t.? Hence, FROM NOW ON E, F ARE K-VECTOR SPACES EQUIPPED WITH THE S.L.C.T. Recall that, on a finite-dimensional space, the s.l.c.t. is the only Hausdorff locally convex topology. From this it is easily seen that if E and F are finitedimensional then E ⊗ F has the s.l.c.t. (indeed, E ⊗ F is finite-dimensional and Hausdorff). Some extensions of this simple fact are given in the next two results. Theorem 2.1. Let either E or F be finite-dimensional. Then E ⊗ F has the s.l.c.t. Proof. We have to prove only one of the cases e.g. when E is finite-dimensional (because E ⊗ F and F ⊗ E are linearly homeomorphic via the twist map given by x ⊗ y → y ⊗ x, x ∈ E, y ∈ F ). Let n := dim E (n ∈ N), {e1 , . . . , en } an algebraic base of E, and {fj : j ∈ J} an algebraic base of F (here J is a set with J = dim F ). Let r be a seminorm on E ⊗F . It suffices to find seminorms p on E, q on F such that r(x ⊗ y) ≤ p(x) q(y) for all x ∈ E, y ∈ F (from which one gets immediately that r ≤ p ⊗ q, and we are done). For that, take x=

n

λi ei ∈ E, y =

i=1

µj fj ∈ F

(λi , µj ∈ K, Jy ⊂ J finite)

j∈Jy

and define p(x) := max |λi | , 1≤i≤n

q(y) := max |µj | ρj , j∈Jy

where, for each j ∈ J, ρj := max r(ei ⊗ fj ). 1≤i≤n

Then the seminorms p, q satisfy the required conditions. In fact, for each x ∈ E, y ∈ F written as finite sums as above, we have λi µj (ei ⊗ fj )) ≤ max |λi | |µj | r(ei ⊗ fj ) ≤ p(x) q(y). r(x ⊗ y) = r( i,j

i,j

Theorem 2.2. Let E and F be countable-dimensional. Then E ⊗ F has the s.l.c.t.

´ ÍA AND WIM H. SCHIKHOF CRISTINA PEREZ-GARC

184 4

Proof. This proof runs similarly to the previous one. By Theorem 2.1 we may assume that E and F are infinite-dimensional i.e. dim E = dim F = ℵ0 . Let {ei : i ∈ N} and {fj : j ∈ N} be algebraic bases of E and F respectively. Let r be a seminorm on E ⊗ F . We find seminorms p on E, q on F such that r(x ⊗ y) ≤ p(x) q(y) for all x = i∈Nx λi ei ∈ E, y = j∈Ny µj fj ∈ F (λi , µj ∈ K; Nx , Ny ⊂ N finite). For that, we define for such x, y, p(x) := max |λi | ρi , i∈Nx

q(y) := max |µj | ρj , j∈Ny

where, for each m ∈ N, ρm := max{1, max r(ei ⊗ fj )}. 1≤i,j≤m

Then, since r(ei ⊗ fj ) ≤ max(ρi , ρj ) ≤ ρi ρj for all i, j, we can proceed as in the proof of Theorem 2.1 to obtain that the seminorms p, q meet the requirements. However, in many cases one has the opposite conclusion, as we prove below. Theorem 2.3. Let E and F be infinite-dimensional. Suppose either dim E ≥ 2ℵ0 or dim F ≥ 2ℵ0 . Then E ⊗ F does not have the s.l.c.t. Proof. As in Theorem 2.1, we have to prove only one of the cases e.g. when dim E ≥ 2ℵ0 . We follow two steps. 1. Assume dim E = 2ℵ0 , dim F = ℵ0 . To see that E ⊗ F does not have the s.l.c.t. we construct a seminorm r on E ⊗ F such that r ≤ p ⊗ q for no seminorms p on E, q on F . Let {ei : i ∈ NN } and {fj : j ∈ N} be algebraic bases of E and F respectively, let s ∈ (1, ∞). Then define r by the formula r( λij ei ⊗ fj ) = max |λij | si(j) i,j

i,j

(λij ∈ K, i ∈ NN , j ∈ N, and the above sum is finite). Now suppose there exist seminorms p on E, q on F such that r ≤ p ⊗ q. Then, (2.1)

si(j) = r(ei ⊗ fj ) ≤ (p ⊗ q)(ei ⊗ fj ) = p(ei ) q(fj ) for all i ∈ NN , j ∈ N.

Let i0 ∈ NN be given by i0 (j) := j [q(fj ) + 1] (j ∈ N), where [ ] indicates the entire part. Applying (2.1) for i := i0 we obtain si0 (j) ≤ p(ei0 ) q(fj ) for all j, that is, sj [q(fj )+1] ≤ p(ei0 ) for all j q(fj ) (note that from (2.1) we have q(fj ) > 0 for all j). But it is easily seen that the j [q(fj )+1] sequence j → s q(fj ) is unbounded, a contradiction. 2. Assume E and F are infinite-dimensional, dim E ≥ 2ℵ0 . Let E1 , F1 be subspaces of E, F respectively with dim E1 = 2ℵ0 , dim F1 = ℵ0 . Let us equip E1 and F1 with the restricted topologies i.e. with the s.l.c.t. By the first step, E1 ⊗ F1 does not have the s.l.c.t. Then neither has E ⊗ F , because it contains a subspace linearly homeomorphic to E1 ⊗ F1 . Conclusion We have proved that the answer to the question posed at the beginning of the section is:

TENSOR PRODUCTS OF p-ADIC LOCALLY CONVEX SPACES

185 5

YES When either (Y1) dim E < ∞, or (Y2) dim F < ∞, or (Y3) dim E = dim F = ℵ0 . NO When either (N1) dim E ≥ 2ℵ0 , dim F ≥ ℵ0 , or (N2) dim E ≥ ℵ0 , dim F ≥ 2ℵ0 . We do not know the answer for the case in which dim E and dim F are infinite cardinals such that ℵ0 < max(dim E, dim F ) < 2ℵ0 . Observe that if we assume the Continuum Hypothesis this unsolved case disappears! References [1] Aguayo, J. Strict topologies on spaces of continuous functions and u-additive measure spaces. J. Math. Anal. Appl. 220 (1998), 77-89. [2] De Grande-De Kimpe, N.; K¸ akol, J.; Perez-Garcia, C.; Schikhof, W.H. p-adic locally convex inductive limits. Lecture Notes in Pure and Appl. Math. 192, 159-222, Dekker, New York, 1997. [3] De Grande-De Kimpe, N.; Navarro, S. Non-Archimedean nuclearity and spaces of continuous functions. Indag. Math. (N.S.) 2 (1991), 201-206. [4] Jarchow, H. Locally Convex Spaces. B. G. Teubner, Stuttgart, 1981. [5] Jech, T. Set Theory. Academic Press, New York, 1978. [6] Katsaras, A.K. Tensor products and Λ0 -nuclear spaces in p-adic analysis. Ann. Math. Blaise Pascal 2 (1995), 155-168. [7] Katsaras, A.K. On p-adic locally convex spaces. Lecture Notes in Pure and Appl. Math. 222, 139-159, Dekker, New York, 2001. [8] Katsaras, A.K.; Beloyiannis, A. On non-Archimedean weighted spaces of continuous functions. Lecture Notes in Pure and Appl. Math. 192, 237-252, Dekker, New York, 1997. [9] Katsaras, A.K.; Beloyiannis, A. Tensor products of non-Archimedean weighted spaces of continuous functions. Georgian Math. J. 6 (1999), 33-44. [10] Perez-Garcia, C. Non-Archimedean polar spaces and the strongest locally convex topology. Bull. Soc. Math. Belg. S´ er. B 41 (1989), 183-195. [11] Perez-Garcia, C.; Schikhof, W.H. The Orlicz-Pettis property in p-adic analysis. Collect. Math. 43 (1992), 225-233. [12] Perez-Garcia, C.; Schikhof, W.H. Tensor product and p-adic vector valued continuous functions. p-adic Functional Analysis, 111-120, Universidad de Santiago, Chile, 1994. [13] van der Put, M.; van Tiel, J. Espaces nucl´ eaires non archim´ ediens. Indag. Math. 29 (1967), 556-561. [14] Schikhof, W.H. Locally convex spaces over nonspherically complete valued fields I-II. Bull. Soc. Math. Belg. S´ er. B 38 (1986), 187-224. [15] Schneider, P. Nonarchimedean Functional Analysis. Springer, Berlin, 2002. Department of Mathematics, Facultad de Ciencias, Universidad de Cantabria, Avda. de los Castros s/n, 39071 Santander, Spain E-mail address: [email protected] Department of Mathematics, Rabhoud University, 6535 ED Nijmegen, The Netherlands E-mail address: w [email protected]



Tensor products of p-adic measures Chrysostomos G. Petalas and Athanasios K. Katsaras Abstract. Tensor products of p-adic vector measures are introduced and some of their properties are investigated. It is shown that a Fubini’s Theorem holds for tensor products of τ -additive vector measures.

1. Preliminaries Throughout this paper, K will be a complete non-Archimedean valued field, whose valuation is non-trivial. By a seminorm, on a vector space over K, we will mean a non-Archimedean seminorm. Similarly, by a locally convex space we will mean a non-Archimedean locally convex space over K (see [15] or [16]). For E a locally convex space, we will denote by cs(E) the collection of all continuous ˆ denotes the completion of E. If F is another seminorms on E. If E is Hausdorff, E locally convex space, then, for p ∈ cs(E) and q ∈ cs(F ), p ⊗ q denotes the tensor product of the seminorms p, q. Recall that, for u ∈ E ⊗ F , p ⊗ q(u) = inf max p(xk )q(yk ), 1≤k≤n

where the infimum is taken over all possible representations u = nk=1 xk ⊗ yk of u. On the tensor product E ⊗ F we will consider the projective topology, which coincides with the topology generated by the seminorms p ⊗ q, p ∈ cs(E) and q ∈ cs(F ). For a zero-dimensional Hausdorff space X, βo X is the Banachewski compactification of X (see [4]), C(X, E) is the space of all continuous E-valued functions on X, while Cb (X, E) and Crc (X, E) are the subspaces of all f ∈ C(X, E) whose range is bounded or relatively compact in E, respectively. In case E = K, we write simply C(X), Cb (X) and Crc (X), respectively. For f ∈ KX and A ⊂ X, we define f A = sup{|f (x)| : x ∈ A} and f = f X . c For A ⊂ X, A will be its complement in X and χA the K-valued characteristic function of A. Next we will recall the definition of the topologies β and βo on Cb (X) (see [5] and [6]). Let Ω = Ω(X) be the family of all compact subsets of βo X which are disjoint from X. For Z ∈ Ω, let CZ be the set of all h ∈ Crc (X) for which the continuous extension hβo to all of βo X vanishes on Z. We denote by 2000 Mathematics Subject Classification. 46G10, 46S10. Key words and phrases. non-Archimedean fields, zero-dimensional spaces, p-adic measures, locally convex spaces, tensor products. c 2010 American c Mathematical 0000 (copyright Society holder)

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CHRYSOSTOMOS G. PETALAS AND ATHANASIOS K. KATSARAS

βZ the locally convex topology on Cb (X) generated by the seminorms ph , h ∈ CZ , ph (f ) = hf .The inductive limit of the topologies βZ , Z ∈ Ω, is the topology β (see [5]). As it is shown in [8], Theorem 2.2, an absolutely convex subset W of Cb (X) is a βZ neighborhood of zero iff, for each r > 0, there exist a clopen (i.e. both closed and open) subset A of X, whose closure in βo (X) is disjoint from Z, and > 0 such that {f ∈ Cb (X) : f A ≤ , f ≤ r} ⊂ W. The strict topology βo (see [5]) is defined by the seminorms f → φf , where φ ranges over the family Bou (X) of all φ ∈ KX which are bounded and vanish at infinity. As it is shown in [11], Theorem 4.18, the topologies β and βo on Cb (X) coincide. Assume next that X is a non-empty set and R a separating algebra of subsets of X, i.e. R is a family of subsets of X such that (1) X ∈ R , and, if A, B ∈ R , then A ∪ B, A ∩ B, Ac are also in R. (2) If x, y are distinct elements of X, then there exists a member of R which contains x but not y. Then R is a base for a Hausdorff zero-dimensional topology τR on X. For E a locally convex space, we denote by M (R, E) the space of all finitely-additive measures m : R → E such that m(R) is a bounded subset of E (see [11]). For a net (Vδ ) of subsets of X, we write Vδ ↓ ∅ if (Vδ ) is decreasing and ∩Vδ = ∅. An element m of M (R, E) is said to be τ -additive if m(Vδ ) → 0 for each net (Vδ ) in R with Vδ ↓ ∅. We will denote by Mτ (R, E) the space of all τ -additive members of M (R, E). For m ∈ M (R, E) and p ∈ cs(E), we define mp : R → R,

mp (A) = sup{p(m(V )) : V ∈ R, V ⊂ A}

and

mp = mp (X).

We also define Nm,p : X → R,

Nm,p (x) = inf{mp (V ) : x ∈ V ∈ R}.

Next we will recall the definition of the integral of an f ∈ KX with respect to some m ∈ M (R, E). Assume that E is a complete Hausdorff locally convex space. For A ⊂ X, let DA be the family of all α = {A1 , A2 , . . . , An ; x1 , x2 , . . . , xn }, where {A1 , A2 , . . . , An } is an R-partition of A and xk ∈ Ak . We make DA into a directed set by defining α1 ≥ α2 if the partition of A in α1 is a refinement nof the one in α2 . For α = {A1 , A2 , . . . , An ; x1 , x2 , . . . , xn }, we define ωα (f, m) = k=1 f (xk )m(Ak ). If the limit lim ωα (f, m) A and exists in E, we will say that f is m-integrable over denote this limit by A f dm (see [11]). For A = X, we write simply f dm. It is easy to see that if f is m-integrable over X, then it is m-integrable over every A ∈ R and A f dm = χA f dm. If f is bounded on A, then p f dm ≤ f A · mp (A). A

Assume next hat m is τ -additive and let S(R) be the space of all K-valued R-simple functions. Let Gm be the space of all f ∈ KX for which f Nm,p = sup |f (x)| · Nm,p (x) < ∞ x∈X

TENSOR PRODUCTS OF p-ADIC MEASURES

189 3

for all p ∈ cs(E). We consider on Gm the locally convex topology generated by the seminorms .Nm,p , p ∈ cs(E). The map ω : S(R) → E, ω(g) = g dm is continuous and so it has a continuous extension ω ¯ : S(R) → E. We will say that S(R = Lm . In this case we will f is (V R)-integrable with respect to m iff f ∈ denote ω ¯ (f ) by (V R) f dm (see [11]). As it is shown in [11], the space Lm , with the induced topology, is complete. In the same paper it is proved that, if f ∈ KX is m-integrable, then it is also (V R)-integrable and (V R) f dm = f dm. 2. Tensor Products of Measures Let R1 , R2 be separating algebras of subsets of the sets X, Y , respectively, and let R be the algebra of subsets of X × Y which is generated by the family R1 × R2 = {A × B : A ∈ R1 , B ∈ R2 }. Lemma 2.1. If R, R1 , and R2 are as above, then : (1) A subset G of X ×Y is in R iff it is a finite union of members of R1 ×R2 . (2) Every member of R is a finite union of pairwise disjoint members of R1 × R2 . Proof : (1). Let Φ be the family of all finite unions of members of R1 × R2 . It is clear that Φ contains X × Y and that it is closed under finite intersections and finite unions. Also, since (A × B)c = [Ac × Y ] [A × B c ], it follows easily that Φ is closed under ncomplimentation. It is clear now that Φ = R. (2). Let G = k=1 Ak × Bk , where Ak ∈ R1 , Bk ∈ R1 . We will show by induction on n that G is a finite union of pairwise disjoint members of R1 × R2 . Suppose that it is true when n = k and let n = k + 1. By our induction hypothesis, there are pairwise disjoint members Di × Fi of R1 × R2 . i = 1, . . . , N , such that N

Di × Fi =

i=1

k

A i × Bi .

i=1

Let Φi = {Dic × Y, Di × Fic }

for i = 1, . . . , N, N and let F be the family of all subsets of X × Y of the form i=1 Vi , where Vi ∈ Φi .

c k Clearly F = and the members of F are pairwise disjoint. Now 1 A i × Bi k+1 i=1

A i × B1 =

N

Di × Fi

{D ∩ (Ak+1 × Bk+1 ) : D ∈ F} .

1

k+1 This clearly shows that 1 Ai × Bi can be written as a finite union of pairwise disjoint members of R1 × R2 and the Lemma follows. Lemma 2.2. τR = τR1 × τR2 .

190 4


Proof : Since τo = τR1 × τR2 is zero dimensional and R1 × R2 ⊂ τo , it follows that τR ⊂ τo . On the other hand, let (x, y) ∈ G ∈ τo . There are A ∈ R1 , B ∈ R2 with (x, y) ∈ A × B ⊂ G. Since A × B ∈ R, it follows that R is a base for τo and so τo = τR . Lemma 2.3. Let E, F be Hausdorff locally convex spaces and m1 ∈ M (R1 , E), m2 ∈ M (R2 , F ). If {A, A1 , . . . , An } ⊂ R1 and {B, B1 , . . . , Bn } ⊂ R2 are such that the sets Ak × Bk , k = 1, . . . , n, are pairwise disjoint and their union is A × B, then m1 (A) ⊗ m2 (B) =

n

m1 (Ak ) ⊗ m2 (Bk ).

k=1

Proof : We will prove it by induction on n. Suppose that it holds for n = k and let n = k + 1. If one of the Ai × Bi is empty, we are done. Assume that none of them is empty. Then A ∩ Ai = Ai and B ∩ Bi = Bi . Now k

Ai × Bi = (A × B)

i=1

= (A × B)

c

[(Ak+1 × Bk+1 ) ]

c ) . (Ack+1 × Y ) (Ak+1 × Bk+1

Also (A ∩

Ack+1 )

× B = (A ∩

Ack+1 )

×B

k

A i × Bi

i=1

=

k

Ai ∩ Ack+1 × Bi

i=1

and

Ak+1 × (B ∩

c Bk+1 )

= Ak+1 × (B ∩

c Bk+1 )

k

A i × Bi

i=1

=

k

c (Ai ∩ Ak+1 ) × (Bi ∩ Bk+1 ).

i=1

By our induction hypothesis, we have m1 (A ∩ Ack+1 ) ⊗ m2 (B) =

k

m1 (Ai ∩ Ack+1 ) ⊗ m2 (Bi )

i=1

and c )= m1 (Ak+1 ) ⊗ m2 (B ∩ Bk+1

k

c m1 (Ai ∩ Ak+1 ) ⊗ m2 (Bi ∩ Bk+1 ).

i=1

Moreover, for i ≤ k,

m1 (Ai ) ⊗ m2 (Bi ) = m1 (Ai ∩ Ak+1 ) ⊗ m2 (Bi ) + m1 Ai ∩ Ack+1 ⊗ m2 (Bi ) = m1 (Ai ∩ Ak+1 ) ⊗ m2 (Bi ∩ Bk+1 ) c + m1 (Ai ∩ Ak+1 ) ⊗ m2 (Bi ∩ Bk+1 ) + m1 (Ai ∩ Ack+1 ) ⊗ m2 (Bi ).


191 5

Since one of the two sets Ai ∩ Ak+1 , Bi ∩ Bk+1 must be empty, we have that c ) + m1 (Ai ∩ Ack+1 ) ⊗ m2 (Bi ). m1 (Ai ) ⊗ m2 Bi ) = m1 (Ai ∩ Ak+1 ) ⊗ m2 (Bi ∩ Bk+1

Thus m1 (A) ⊗ m2 (B) = m1 (A ∩ Ack+1 ) ⊗ m2 (B) + m1 (Ak+1 ) ⊗ m2 (B) c ) = m1 (A ∩ Ack+1 ) ⊗ m2 (B) + m1 (Ak+1 ) ⊗ m2 (B ∩ Bk+1

+ m1 (Ak+1 ) ⊗ m2 (Bk+1 ) =

k

m1 (Ai ) ⊗ m2 (Bi ) + m1 (Ak+1 ) ⊗ m2 (Bk+1 ).

i=1

This clearly completes the proof. Lemma 2.4. Let R1 , R2 , R, m1 , m2 be as in the preceding Lemma. If Ak , Fi ∈ n R1 , Bk , Gi ∈ R2 , k = 1, . . . , n, i = 1, . . . , N , are such that k=1 Ak ⊗ Bk = N i=1 Fi ⊗ Gi and each of the families {Ak × Bk : k = 1, . . . , n} and {Fi × Gi : i = 1, . . . , N } consists of sets which are pairwise disjoint, then n

m1 (Ak ) ⊗ m2 (Bk ) =

N

m1 (Fi ) ⊗ m2 (Gi ).

i=1

k=1

Proof : A k × Bk =

N

(Fi ∩ Ak ) × (Gi ∩ Bk )

i=1

and

Fi × G i =

n

(Fi ∩ Ak ) × (Gi ∩ Bk ).

k=1

Now the result follows by applying the preceding Lemma. Theorem 2.5. If R1 , R2 , R, m1 , m2 are as in the preceding Lemma, then there exists a unique m ∈ M (R, E ⊗ F ) with m(A × B) = m1 (A) ⊗ m2 (B), when A ∈ R1 and B ∈ R2 . Proof n : For G ∈ R, there are Ai ∈ R1 , Bi ∈ R2 , i = 1, . . . , n, such that G = i=1 Ai × Bi and the sets Ai × Bi are pairwise disjoint. Define m(G) = n m (A 1 i ) ⊗ m2 (Bi ). In view of the preceding Lemma, m is well defined and 1 finitely additive. Also m(R) is a bounded subset of E ⊗F and so m ∈ M (R, E ⊗F ). Clearly m is the unique µ ∈ M (R, E ⊗ F ) such that µ(A × B) = m1 (A) ⊗ m2 (B) when A ∈ R1 and B ∈ R2 . We will call m1 ⊗ m2 the tensor product of m1 , m2 . Theorem 2.6. Let m1 ∈ M (R1 , E), m2 ∈ M (R2 ), F ), m = m1 ⊗ m2 , p ∈ cs(E) and q ∈ cs(F ). Then, for V1 ∈ R1 , V2 ∈ R2 , we have mp⊗q (V1 × V2 ) = (m1 )p (V1 ) · (m2 )q (V2 ). Moreover, for x ∈ X, y ∈ Y , we have Nm,p⊗q (x, y) = Nm1 ,p (x) · Nm2 ,q (y).

192 6


Proof : Let d = (m1 )p (V1 ) · (m2 )q (V2 ). It is clear that d ≤ mp⊗q (V1 × V2 ). On the other hand, let G ∈ R, G ⊂ V1 × V2 . There are pairwise disjoint Ai × Bi in n R1 × R2 such that G = 1 Ai × Bi , Ai ⊂ V1 , Bi ⊂ V2 . Then n

m1 (Ai ) ⊗ m2 (Bi ) p ⊗ q(m(G)) = p ⊗ q 1

≤ (m1 )p (V1 ) · (m2 )q (V2 ). Thus mp⊗q (V1 × V2 ) ≤ d and so d = mp⊗q (V1 × V2 ). Given > 0, there exist A ∈ R1 containing x and B ∈ R2 containing y such that (m1 )p (A) < Nm1 ,p (x) + ,

(m2 )q (B) < Nm2 ,q (y) + .

Thus Nm,p⊗q (x, y) ≤ mp⊗q (A × B) = (m1 )p (A) · (m2 )q (B) < [Nm1 ,p (x) + ] · [Nm2 ,q (y) + ] . Taking → 0, we get that Nm,p⊗q (x, y) ≤ Nm1 ,p (x) · Nm2 ,q (y). On the other hand, let Nm,p⊗q (x, y) < θ. There exists G ∈ R containing (x, y) such that mp⊗q (G) < θ. By Lemma 2.1, there exist A ∈ R1 containing x and B ∈ R2 containing y such that A × B ⊂ G. Now Nm1 ,p (x) · Nm2 ,q (y) ≤ (m1 )p (A) · (m2 )q (B) = mp⊗q (A × B) ≤ mp⊗q (G) < θ. It is clear now that Nm,p⊗q (x, y) ≥ Nm1 ,p (x) · Nm2 ,q (y) and the result follows. Throughout the rest of the paper, E, F will be complete Hausdorff locally convex spaces, X, Y non-empty sets, R1 , R2 separating algebras of subsets of X, Y , respectively, and R the algebra of subsets of X × Y generated by R1 × R2 . For f ∈ KX , g ∈ KY , we will denote by f g the function which is defined on X × Y by f g(x, y) = f (x)g(y). Theorem 2.7. Let m1 ∈ M (R1 , E), m2 ∈ M (R2 , F ) and m = m1 ⊗ m2 . If f ∈ KX is m1 -integrable and g ∈ KY is m2 -integrable, then f g is m-integrable and f g dm =

f dm1 ⊗

g dm2 .

Proof : Let p ∈ cs(E), q ∈ cs(F ) and > 0. By [11, Theorem 4.2], there are A ∈ R1 , B ∈ R2 such that (m1 )p (Ac ) = (m2 )q (B c ) = 0 and f , g are bounded on A, B, respectively. Let d > max{f A , gB } and choose 0 < 1 < min{1, } such that d1 · max{m1 p , m2 q } ≤ . By [11, Theorem 4.1], there exist an R1 partition {A1 , . . . , An } of X, which is a refinement of {A, Ac }, and an R2 -partition {B1 , . . . , BN } of Y , which is a refinement of {B, B c }, such that |f (x1 ) − f (x2 )| ·

193 7


(m1 )p (Ai ) < 1 , if x1 , x2 are in Ai , and |g(y1 ) − g(y2 )| · (m2 )q (Bj ) < 1 , if y1 , y2 are in Bj . Choose xi ∈ Ai , yj ∈ Bj . Then , by [11, Theorem 4,1], we have n

f (xi )m1 (Ai ) ≤ 1 p f dm1 − i=1

⎛ ⎞ N

q ⎝ g dm2 − g(yj )m2 (Bj )⎠ ≤ 1

and

j=1

k

r We may assume that i=1 Ai = A and j=1 Bj = B. Let 1 ≤ i ≤ n, 1 ≤ j ≤ N and let (z1 , z2 ) ∈ Ai × Bj . If either i > k or j > r, then mp⊗q (Ai × Bj ) = 0. Suppose that i ≤ k and j ≤ r. Then |f (xi )g(yj ) − f (z1 )g(z2 )| · mp⊗q (Ai × Bj ) ≤ d·max {|f (xi )−f (z1 )|·(m1 )p (Ai )·(m2 )q (Y ), |g(yj )−g(z2 )|·(m2 )q (Bj )·(m1 )p (X)} ≤ d1 · max{m1 p , (m2 q } = 2 ≤ . This, in view of [11, Theorem 4.1], implies that, if we consider m as a member of ˆ ), then h = f g is m-integrable and M (R, E ⊗F ⎞ ⎛

f (xi )g(yj )m1 (Ai )m2 (Bj )⎠ ≤ 2 . p ⊗ q ⎝ h dm − i,j

Let

u1 =

f dm1 −

k

f (xi )m1 (Ai ),

u2 =

g dm2 −

i=1

and

g(yj )m2 (Bj .

j=1

u=

r

h dm −

r k

f (xi )g(yj ) · m1 (Ai )m2 (Bj ).

i=1 j=1

Then − h dm + f dm1 ⊗ g dm2 =

r k = −u + u1 ⊗ u2 + u1 ⊗ j=1 g(yj )m2 (Bj ) + i=1 f (xi )m1 (Ai ) ⊗ u2 . But p ⊗ q(u1 ⊗ u2 ) ≤ 21 ≤ and p ⊗ q(u) ≤ . Also ⎛ ⎛ ⎞⎞ r

p ⊗ q ⎝u1 ⊗ ⎝ g(yj )m2 (Bj )⎠⎠ ≤ 1 d · m2 q ≤ j=1

and

p⊗q

k

f (xi )m1 (Ai ) ⊗ u2

≤ .

i=1

Thus p⊗q

h dm −

g dm2 ≤ . f dm1 ⊗

Since > 0 was arbitrary and E, F are Hausdorff, it follows that g dm2 h dm = f dm1 ⊗ which completes the proof.

194 8


3. The Case of τ -Additive Measures Theorem 3.1. If m1 ∈ Mτ (R1 , E) and m2 ∈ Mτ (R2 , F ), then ˆ ). m = m1 ⊗ m2 ∈ Mτ (R, E ⊗F Proof : Consider on X, Y the zero-dimensional topologies τR1 , τR2 , respectively and on X ×Y the product topology which coincides with τR . By [12, Theorem 2.4], there exists a linear homeomorphism ω : (Cb (X), βo ) ⊗ (Cb (Y ), βo ) −→ (Cb (X × Y, βo ) onto a dense subspace M of Cb (X × Y ), where ω(f ⊗ g) = f g, for f ∈ Cb (X), g ∈ Cb (Y ). The map f → f dm1 , from Cb (X) to E, is βo -continuous by [11, Theorem 4.13]. The same is true for the map g → g dm2 , from Cb (Y ) to F . Given p ∈ cs(E) and q ∈ cs(F ), there are φ1 ∈ Bou (X), φ2 ∈ Bou (Y ) such that g dm2 ≤ φ2 g p f dm1 ≤ φ1 f , q for all f ∈ Cb (X), g ∈ Cb (Y ). The bilinear map ˆ T : (Cb (X), βo ) × (Cb (Y ), βo ) → E ⊗F,

T (f, g) =

f dm1

⊗

g dm2

is continuous and so the induced linear map ˆ ψ : G = (Cb (X), βo ) ⊗ (Cb (Y ), βo ) −→ E ⊗F is continuous. Let

ˆ υ = ψ ◦ ω −1 : M −→ E ⊗F. Since M is βo -dense in Cb (X × Y ), there exists a continuous linear extension ˆ υ¯ : (Cb (X × Y ), βo ) −→ E ⊗F. ˆ ) such that In view of [11, Theorem 4.13], there exists a unique µ ∈ Mτ (R, E ⊗F υ¯(h) = h dµ for all h ∈ Cb (X × Y ). For A ∈ R1 , B ∈ R2 , taking as h the characteristic function of A × B, we get that χB dm2 = m1 (A) ⊗ m2 (B). µ(A × B) = χA dm1 ⊗ Thus µ(R) ⊂ E ⊗ F ) and µ = m by Theorem 2.5. For Ro a separating algebra of subsets of a set Z , G a complete Hausdorff locally convex space and u ∈ Mτ (Ro , G), we denote by Lµ the space of all f ∈ KZ which are (VR)-integrable with respect to µ. On Lµ we consider the locally convex topology generated by the seminorms Nµ,p , p ∈ cs(G). Theorem 3.2. Let m1 ∈ Mτ (R1 , E), m2 ∈ Mτ (R2 , F ) and m = m1 ⊗ m2 . Then the projective tensor product Lm1 ⊗Lm2 is topologically isomorphic to a dense subspace of Lm . Proof : Consider the bilinear map T : Lm1 × Lm2 −→ Lm ,

T (f, g) = f g.

Let p ∈ cs(E), q ∈ cs(F ), f ∈ Lm1 , g ∈ Lm2 and h = f g. Applying Theorem 2.6, we get that hNm,p⊗q = f Nm1 ,p · gNm2 ,q

195 9


and so T is continuous. Consequently the induced linear mat ψ : Lm1 ⊗ Lm2 −→ Lm

n is continuous. The map ψ is one-to-one. In fact, assume that k=1 fk gk = 0, n where fk ∈ Lm1 , gk ∈ Lm2 . We will show, by induction on n, that k=1 fk ⊗gk = 0. This is clearly true if n = 1 or if each fk is zero. Suppose that it is true for n − 1 and that, say, fn = 0. Then gn is a linear combination of g1 , . . . , gn−1 , i.e. n−1 gn = k=1 λk gk , λk ∈ K. Thus 0=

n

fk gk =

k=1

n−1

fk gk +

n−1

k=1

λk (fn gk ) =

k=1

0=

n−1

k=1

k=1

(fk + λk fn ) ⊗ gk =

(fk + λk fn ) gk .

k=1

By our induction hypothesis, we have n−1

n−1

fk ⊗ gk + fn ⊗

n−1

λk gk

k=1

=

n

fk ⊗ gk ,

k=1

which shows that ψ is one-to-one. Claim For qm = · Nm,p⊗q , qm1 = · Nm1 ,p , qm2 = · Nm2 ,q , we have

Indeed, if h =

qm (ψ(h)) = (qm1 ⊗ qm2 )(h).

n

k=1 fk

qm (ψ(h)) = qm

n

⊗ gk , then ψ(fk ⊗ gk )

k=1

≤ max qm (fk gk ) = max qm1 (fk ) · qm2 (gk ), k

k

which shows that qm (ψ(h)) ≤ (qm1 ⊗ qm2 )(h). On the other hand, for an arbitrary N 0 < t < 1, there exists a representation h = k=1 fk ⊗ gk of h such that the set {g1 , . . . , gN } is t-orthogonal with respect to the seminorm qm2 . Let u = ψ(h). For x ∈ X, let N

x x u : Y → K, u (y) = u(x, y) = fk (x)gk (y). k=1

Then sup |u(x, y)| · Nm,p⊗q (x, y) = Nm1 ,p (x) · sup |ux (y)| · Nm2 ,q (y) = Nm1 ,p (x) · qm2 (ux )

y∈Y

y∈Y

≥ t · Nm1 ,p (x) · max |fk (x)| · qm2 (gk ). k

Thus qm (u) ≥ t · max [qm1 (fk ) · qm2 (gk )] ≥ t · (qm1 ⊗ qm2 )(h). k

Since 0 < t < 1 was arbitrary, we get that qm (u) ≥ (qm1 ⊗ qm2 )(h) and the claim follows. Thus ψ : Lm1 ⊗ Lm2 −→ G = ψ(Lm1 ⊗ Lm2 ) is a topological isomorphism. Finally, G is dense in Lm . Indeed, for A ∈ R1 , B ∈ R2 , we have that χA×B = ψ(χA ⊗ χB ). Since each member of R is a finite union of sets of the form A × B, with A ∈ R1 , B ∈ R2 , it follows that S(R) ⊂ G and hence G is dense in Lm since this is true for S(R). This completes the proof.

196 10


4. A Fubini’s Theorem We will first define the integral of a vector-valued function with respect to a vector measure. Let Ro be a separating algebra of subsets of X, and µ ∈ M (Ro , F ). For A ∈ Ro , let DA be the family of all α = {A1 , . . . , An ; x1 , . . . , xn }, where X {A1 , . . . , An } is an Ro -partition of A and x and α = k ∈ Ak . For f ∈ E n {A1 , . . . , An ; x1 , . . . , xn }, we define ψα (f, µ) = k=1 f (xk ) ⊗ µ(Ak ) ∈ E ⊗ F . If the ˆ , then we will say that f is µ-integrable over A and we limα ψα (f, µ) exists in E⊗F will denote this limit by A f dµ. For A = X, we will write simply f dµ. It is easy to see that if f is µ-integrable over X, then f is µ-integrable over every A ∈ Ro and f dµ = χA f dµ. A

Using an argument analogous to the one used for scalar measures ([9], Theorem 2.1), we get the following Theorem 4.1. An f ∈ E X is µ-integrable with respect to some µ ∈ M (Ro , F ) iff, for each p ∈ cs(E) and each q ∈ cs(F ), there exists an Ro -partition {A1 , . . . , An } of X such that p(f (x) − f (y)) · mq (Ak ) ≤ , for all k, if x, y ∈ Ak . Moreover, in this case we have that n

p⊗q f dµ − f (xk ) ⊗ µ(Ak ) ≤ , k=1

ˆ . where we denote also by p ⊗ q the unique continuous extension to all of E ⊗F Assume next that µ ∈ Mτ (Ro , F ). For f ∈ E X , q ∈ cs(F ), p ∈ cs(E), let f Nµ,p,q = sup p(f (x)) · Nµ,q (x). x∈X

Let Zµ be the space of all f ∈ E with f Nµ,p,q < ∞ for all p ∈ cs(E) and q ∈ cs(F ). Each · Nµ,p,q is a seminorm on Zµ . Let S(Ro , E) be the space of all n E-valued Ro -simple functions. It is easy to see that, for f = k=1 χAk sk , we have that n

f dµ = sk ⊗ µ(Ak ) and p ⊗ q f dµ ≤ f Nµ,p,q X

k=1

for all p ∈ cs(E), q ∈ cs(F ). Let ˆ π : S(Ro , E) −→ E ⊗F,

π(f ) =

f dµ.

Then π is continuous if we consider on S(Ro , E) the topology induced by the topology of Zµ . Thus there exists a continuous extension ˆ π ¯ : S(Ro , E) −→ E ⊗F. Definition 4.2. A function f ∈ E X is said to be (VR)-integrable with respect to some µ ∈ Mτ (Ro , F ) if it belongs to Dµ = S(Ro , E). In this case, π ¯ (f ) is called the (VR)-integrable of f and will be denoted by (V R) f dµ. Theorem 4.3. If f ∈ Dµ , then for all p ∈ cs(E), q ∈ cs(F ), we have p ⊗ q( (V R) f dµ ≤ f Nµ,p,q .


197 11

Proof : There exists a net (gδ ) in S(Ro , E) converging to f . Then (V R) f dµ = lim gδ dµ, and gδ Nµ,p,q → f Nµ,p,q . δ

Since

p⊗q

gδ dµ

≤ gδ Nµ,p,q ,

the Theorem follows. In view of [11, Theorem 2.8], the closure of the set {x : Nµ,q (x) > 0} q∈cs(F )

is the smallest closed support set for µ. Theorem 4.4. Let µ ∈ Mτ (Ro , F ) and f ∈ E X . If f is µ-integrable, then f is also (VR)-integrable and f dµ = (V R) f dµ. Proof : Assume that f is µ-integrable and let D be the directed set of all α = {A1 . . . , A n ; x1 , . . . , xn }, where {A1 , . . . , An } is an Ro -partition of X and xk ∈ Ak . n Let hα = k=1 χAk f (xk ). Let p ∈ cs(E), q ∈ cs(F ) and > 0. In view of Theorem 4.1, there exist an Ro -partition {B1 , . . . , BN } of X such that max sup p(f (x) − f (y)) · mq (Bk ) ≤ . k

x,y∈Bk

Let xk ∈ Bk and αo = {B1 , . . . , BN ; x1 , . . . , xN }. Then , for α = {A1 , . . . , An ; y1 , . . . yn } ∈ D, we have that

p⊗q

f dµ −

α ≥ αo ,

hα dµ

≤ .

It follows that f Nµ,p,q < ∞ and that f dµ = lim hα dµ = (V R) f dµ. α

Hence the Theorem holds. Lemma 4.5. Let m1 ∈ Mτ (R1 , E), m2 ∈ Mτ (R2 , F ) and m = m1 ⊗ m2 . Let also f ∈ KX×Y be (VR)-integrable with respect to m. For y ∈ Y , let f y = f (·, y). If there exists a q ∈ cs(F ) such that Nm2 ,q (y) > 0, then f y is (VR)-integrable with respect to m1 . Proof : Every h ∈ S(R) is of the form h = nk=1 λk χAk ×Bk , where λk ∈ K, Ak ∈ R1 , Bk ∈ R2 . It is clear that hy ∈ S(R1 ). Suppose that Nm2 ,q (y) = d > 0 and let > 0. Given p ∈ cs(E), there exists an h ∈ S(R) such that f − hNm,p⊗q ≤ d. Now, for x ∈ X, we have |f y (x) − hy (x)| · Nm1 ,p (x) = |f (x, y) − h(x, y)| · Nm,p⊗q (x, y)/d ≤ and so f y − hy Nm1 ,p ≤ . This clearly proves that f y is (VR)-integrable with respect to m1 .

198 12


Theorem 4.6. (Fubini’s Theorem). Let m1 , m2 , m be as in the preceding Lemma and let f ∈ KX×Y be (VR)-integrable with respect to m. Let g : Y → E be defined by g(y) = (V R) f y dm1 if y ∈ G = {z ∈ Y : ∃q ∈ cs(F )

with

Nm2 ,q (y) > 0}

and arbitrarily if y ∈ / G. Then g is (VR)-integrable with respect to m2 and (V R) g dm2 = (V R) f dm. Proof : There exists a net (hδ ) in S(R) such that hδ → f in Lm and (V R) f dm = lim hδ dm. δ

Define

gδ : Y → E,

hyδ dm1 .

gδ (y) =

Then, for p ∈ cs(E), q ∈ cs(F ) and y ∈ Y , we have p(gδ (y) − g(y)) · Nm2 ,q (y) ≤ f − hδ Nm,p⊗q . Indeed, if Nm2 ,q (y) = 0, then (f y − hyδ ) dm1

g(y) − gδ (y) = (V R) and so

p(gδ (y) − g(y)) ≤ f y − hyδ Nm1 ,p , which implies that p(gδ (y) − g(y) · Nm2 ,q (y) ≤ sup |f (x, y) − hδ (x, y)| · Nm1 ,p (x) · Nm2 ,q (y) x

≤ f − hδ Nm,p⊗q . This proves that g is (VR)-integrable with respect to m2 . Since, for A ∈ R1 , B ∈ R2 , u = χA×B and v(y) = uy dm1 , we have that u dm = v dm2 , it follows that

hδ dm =

and so

(V R)

gδ dm2 ,

f dm = lim

which completes the proof.

hδ dm = lim

gδ dm2 = (V R)

g dm2 ,


199 13

References [1] J. Aguayo, Vector measures and integral operators, in : Ultrametric Functional Analysis, Cont. Math., vol. 384(2005), 1-13. [2] J. Aguayo and T. E. Gilsdorf, Non-Archimedean vector measures and integral operators, in : p-adic Functional Analysis, Lecture Notes in Pure and Applied Mathematics, vol 222, Marcel Dekker, New York (2001), 1-11. [3] J. Aguayo and M. Nova, Non-Archimedean integral operators on the space of continuous functions, in : Ultrametric Functional analysis, Cont. Math., vol. 319(2002), 1-15. [4] G. Bachman, E. Beckenstein, L. Narici and S. Warner, Rings of continuous functions with values in a topological field, Trans. Amer. Math. Soc. 204 (1975), 91-112. [5] A. K. Katsaras, The strict topology in non-Archimedean vector-valued function spaces, Proc. Kon. Ned. Akad. Wet. A 87 (2) (1984), 189-201. [6] A. K. Katsaras, Strict topologies in non-Archimedean function spaces, Intern. J. Math. and Math. Sci. 7 (1), (1984), 23-33. [7] A. K. Katsaras, Separable measures and strict topologies on spaces of non-Archimedean valued functions, in : P-adic Numbers in Number Theory, Analytic Geometry and Functional Analysis, edided by S. Caenepeel, Bull. Belgian Math., (2002), 117-139. [8] A. K. Katsaras, Strict topologies and vector measures on non-Archimedean spaces, Cont. Math. vol. 319 (2003), 109-129. [9] A. K. Katsaras, Non-Archimedean integration and strict topologies, Cont. Math. vol. 384 (2005), 111-144. [10] A. K. Katsaras, P-adic spaces of continuous functions I, Ann. Math. Blaise Pascal 15 (2008), 109-133. [11] A. K. Katsaras, Vector valued p-adic measures (preprint). [12] A. K. Katsaras, P-adic spaces of continuous functions II, Ann. Math. Blaise Pascal 15 (2008), 169-188. [13] A. F. Monna and T. A. Springer, Integration non-Archimedienne, Indag. Math. 25, no 4(1963), 634-653. [14] J. B. Prolla, Approximartion of vector-valued functions, North Holland Publ. Co., Amsterdam, New York, Oxforfd, 1977. [15] W. H. Schikhof, Locally convex spaces over non-spherically complete fields I, II, Bull. Soc. Math. Belg., Ser. B, 38 (1986), 187-224. [16] A. C. M. van Rooij, Non-Archimedean Functional Analysis, New York and Bassel, Marcel Dekker, 1978. [17] A. C. M. van Rooij and W. H. Schikhof, Non-Archimedean Integration Theory, Indag. Math., 31(1969), 190-199. Department of Mathematics, University of Ioannina, 45110 Ioannina, Greece E-mail address: [email protected] Department of Mathematics, University of Ioannina, 45110 Ioannina, Greece E-mail address: [email protected]



p-adic arithmetic coding Anatoly Rodionov and Sergey Volkov Abstract. A new incremental algorithm for data compression is presented. For a sequence of input symbols the algorithm incrementally constructs a padic integer number as an output. The decoding process starts with a less significant part of a p-adic integer and incrementally reconstructs a sequence of input symbols. The algorithm is based on certain features of p-adic numbers and the p-adic norm. The p-adic coding algorithm may be considered as a generalization of a popular compression technique – arithmetic coding algorithms. It is shown that for p = 2 the algorithm works as an integer variant of arithmetic coding; for a special class of models it gives exactly the same codes as Huffman’s algorithm. For another special model and a specific alphabet it gives GolombRice codes.

1. Introduction The arithmetic coding algorithm in its modern version was published in Communications of ACM in June 1987 [9], but the authors, Ian Witten, Radford Neal and John Cleary, referred to [1] as to ”the first reference to what was to become the method of arithmetic coding“. The algorithm is now common knowledge; it was published in numerous textbooks (see for example [7], [8]), some reviews were published ([2], [6]), Dr. Dobbs Journal popularized it [3], wiki contains an article about it, and a lot of sources can be found on the web. So why one more paper on this subject and what is this ”p-adic arithmetic coding“? Let’s go back to the original idea of arithmetic coding. In arithmetic coding a message is represented as a subinterval [b, e) of the semi interval [0, 1). (We will give all definitions later). When a new symbol s arrives, a new subinterval [b(s), e(s)) of [b, e) is constructed. The common method of calculating a new subinterval is to divide a current interval into |A| subintervals (A is an alphabet, |A| – number of symbols in it), each subinterval represents a symbol from A and has length equal to the probability of this symbol. For a new symbol s a corresponding subinterval [b(s), e(s)) is constructed by the encoder. Thus encoding is a process of narrowing intervals starting from the union interval: [b0 , e0 ) ⊃ [b(s1 ), e(s1 )) ⊃ · · · ⊃ [b(st ), e(st )), where b(si ) and e(si ) are real numbers and 0 = b0 ≤ b(s1 ) ≤ · · · ≤ b(st ) < e(st ) ≤ · · · ≤ e(s1 ) ≤ e0 = 1. 2000 Mathematics Subject Classification. Primary 68P30; Secondary 11F85, 68W01. c c2010 American Mathematical 2009 Anatoly Rodionov, Sergey Society Volkov

1 201

202 2

ANATOLY RODIONOV AND SERGEY VOLKOV

The last subinterval of a series of constructed subintervals is used as a final output; another way of presenting the result is a pair – any point x from the last subinterval and message length. Usually, a special symbol EOM (End Of Message), which does not belong to the message’s set of symbols, is used as the termination symbol for a message. In this case, only a point x can be used as a coding result. Decoding is also a process of narrowing intervals. It starts with the union interval and a point x inside it. The decoder finds a symbol s1 by dividing current intervals into |A| subintervals and finds the subinterval that contains point x, say [b(s1 ), e(s1 )). Corresponding to this interval symbol s1 is pushed into an output buffer; [b(s1 ), e(s1 )) is used as a new current interval, and so on until the EOM symbol is received. But here is a problem – one has to use infinite precision real numbers to implement this algorithm and there is no such thing as effective infinite precision in real arithmetic. This problem was always considered as a technical one. The solution looks simple – just use integers instead. A first, and now canonical implementation, written in C language was published [9]. It was later reproduced in other languages, but no analysis of what happens to the algorithm after moving it from real numbers to integer numbers was published. In this paper we introduce p-adic arithmetic coding which is based on mapping a message to a p-adic interval. For defining a p-adic interval we need a lexicographical order ≤P on the p-adic integer numbers. Using the p-adic interval we formulate a p-adic arithmetic coding procedure, which is very close to the original arithmetic coding, but instead of real semi intervals, p-adic intervals are used. Again, the encoding is a process of narrowing intervals starting from the union interval: [B0 , E0 ) ⊃P [B(s1 ), E(s1 )] ⊃P · · · ⊃P [B(st ), E(st )], where B(si ), E(si ), 0 and −1P are p-adic integer numbers and 0 = B0 ≤P B(s1 ) ≤P · · · ≤P B(st )
K, where K is some natural number. We will use lexicographical ordering on p-adic numbers. ∞ ∞ Definition 2.1. Let X = i=0 xi P i , Y = i=0 yi P i . X
0. But f6 has a relative maximum at 0. The difficulties embodied in the examples above are not specific to R, but are common to all non-Archimedean ordered fields; and they result from the fact that R is disconnected in the topology induced by the order. This makes developing Analysis on the field more difficult than in the real case; for example, the existence of nonconstant functions whose derivatives vanish everywhere on an interval (as in example 3.4) makes integration much harder and renders the solutions of the simplest initial value problems (e.g. y = 0; y(0) = 0) not unique. To circumvent such difficulties, different approaches have been employed. For example, by imposing stronger conditions on the function than in the real case, the function satisfies an intermediate value theorem and an inverse function theorem [43]; by using a stronger concept of continuity and differentiability than in the real case, one-dimensional and multi-dimensional optimization results similar to those from Real Analysis have been obtained for R-valued functions [48, 49]; by carefully defining a measure on R in [44], we succeed in developing an integration theory with similar properties to those of the Lebesgue integral of Real Analysis; and by restricting solutions of initial value problems to analytic functions, the uniqueness of the solutions will be assured [10]. 4. Review of Power Series and R-Analytic Functions Power series on the Levi-Civita field R have been studied in details in [39, 41, 45, 46, 47]; work prior to that has been mostly restricted to power series with real coefficients. In [26, 27, 31, 25], they could be studied for infinitely small arguments only, while in [4], using the newly introduced weak topology (see Definition 4.4 below), also finite arguments were possible. Moreover, power series over complete valued fields in general have been studied by Schikhof [37], Alling [1] and others in valuation theory, but always in the valuation topology. In [41], we study the general case when the coefficients in the power series are Levi-Civita numbers (i.e. elements of R or C), using the weak convergence of [4]. We derive convergence criteria for power series which allow us to define a radius of convergence η such that the power series converges weakly for all points whose distance from the center is smaller than η by a finite amount and it converges in the order topology for all points whose distance from the center is infinitely smaller than η. In [45] it is shown that, within their radius of convergence, power series are infinitely often differentiable and the derivatives to any order are obtained by differentiating the power series term by term. Also, power series can be re-expanded around any point in their domain of convergence and the radius of convergence of the new series is equal to the difference between the radius of convergence of the original series and the distance between the original and new centers of the series. In the following, we summarize some of the key results in [41, 45, 46, 47]. We start with a brief review of the convergence of sequences in two different topologies.

222 8

KHODR SHAMSEDDINE AND MARTIN BERZ

Definition 4.1. A sequence (sn ) in R or C is called regular if the union of the supports of all members of the sequence is a left-finite subset of Q. (Recall that A ⊂ Q is said to be left-finite if for every q ∈ Q there are only finitely many elements in A that are smaller than q.) Definition 4.2. We say that a sequence (sn ) converges strongly in R or C if it converges with respect to the topology induced by the absolute value. As we have already mentioned in Section 2, strong convergence is equivalent to convergence in the topology induced by the valuation λ. It is shown in [3] that the fields R and C are complete with respect to the strong topology; and a detailed study of strong convergence can be found in [39, 41]. Since power series with real (complex) coefficients do not converge strongly for any nonzero real (complex) argument, it is advantageous to study a new kind of convergence. We do that by defining a family of semi-norms on R or C, which induces a topology weaker than the topology induced by the absolute value and called weak topology. Definition 4.3. Given r ∈ R, we define a mapping · r : R or C → R as follows: x r = max{|x[q]| : q ∈ Q and q ≤ r}. The maximum in Definition 4.3 exists in R since, for any r ∈ R, only finitely many of the x[q]’s considered do not vanish. Definition 4.4. A sequence (sn ) in R (resp. C) is said to be weakly convergent if there exists s ∈ R (resp. C), called the weak limit of the sequence (sn ), such that for all > 0 in R, there exists N ∈ N such that sm − s 1/ < for all m ≥ N . It is shown [4] that R and C are not Cauchy complete with respect to the weak topology and that strong convergence implies weak convergence to the same limit. A detailed study of weak convergence is found in [4, 39, 41]. 4.1. Power Series. In the following, we review strong and weak convergence criteria for power series, Theorem 4.5 and Theorem 4.6, the proofs of which are given in [41]. We also note that, since strong convergence is equivalent to convergence with respect to the valuation topology, Theorem 4.5 is a special case of the result on page 59 of [37]. Theorem 4.5. (Strong Convergence Criterion for Power Series) Let (an ) be a sequence in R (resp. C), and let −λ(an ) λ0 = lim sup in R ∪ {−∞, ∞}. n n→∞ ∈ R (resp. C) be fixed and let x ∈ R (resp. C) be given. Then the power Let x0 n series ∞ n=0 an (x − x0 ) converges strongly if λ(x − x0 ) > λ0 and is strongly divergent if λ(x − x0 ) < λ0 or if λ(x − x0 ) = λ0 and −λ(an )/n > λ0 for infinitely many n. Theorem 4.6. (Weak Convergence Criterion for Power Series) Let (an ) be a sequence in R (resp. C), and let λ0 = lim supn→∞ (−λ(an )/n) ∈ Q. Let x0 ∈ R (resp. C) be fixed, and let x ∈ R (resp. C) be such that λ(x − x0 ) = λ0 . For each n ≥ 0, let bn = an dnλ0 . Suppose that the sequence (bn ) is regular and write

223 9

ANALYSIS ON THE LEVI-CIVITA FIELD, A BRIEF OVERVIEW

∞ supp(bn ) = {q1 , q2 , . . .}; with qj1 < qj2 if j1 < j2 . For each n, write bn = n=0 ∞ qj j=1 bnj d , where bnj = bn [qj ]. Let (4.1)

η=

1

sup lim supn→∞ |bnj |1/n : j ≥ 1

in R ∪ {∞},

∞ with the conventions 1/0 = ∞ and 1/∞ = 0. Then n=0 an (x − x0 )n converges absolutely weakly if |(x − x0 )[λ0 ]| < η and is weakly divergent if |(x − x0 )[λ0 ]| > η. Remark 4.7. The number η in Equation (4.1) is referred to as the radius of n weak convergence of the power series ∞ n=0 an (x − x0 ) . As an immediate consequence of Theorem 4.6, we obtain the following result which allows us to extend real and complex functions representable by power series to the Levi-Civita fields R and C. This result is of particular interest for the application discussed in Section 7. 4.8. (Power Series with Purely Real or Complex Coefficients) Let ∞Corollary n be a power series with purely real (resp. complex) coefficients and n=0 an X with classical n radius of convergence equal to η. Let x ∈ R (resp. C), and let An (x) = j=0 aj xj ∈ R (resp. C). Then, for |x| < η and |x| ≈ η, the sequence (An (x)) converges absolutely weakly. We define the limit to be the continuation of the power series to R (resp. C). 4.2. R-Analytic Functions. In this section, we review the algebraic and analytical properties of a class of functions that are given locally by power series and we refer the reader to [45] for a more detailed study. Definition 4.9. Let a, b ∈ R be such that 0 < b − a ∼ 1 and let f : [a, b] → R. Then we say that f is expandable or R-analytic on [a, b] if for all x ∈ [a, b] there exists a finite δ > 0 in R, and there exists a regular sequence (an (x)) in R such that, ∞ n under weak convergence, f (y) = n=0 an (x) (y − x) for all y ∈ (x − δ, x + δ) ∩ [a, b]. Definition 4.10. Let a < b in R be such that t = λ(b − a) = 0 and let f : [a, b] → R. Then we say that f is R-analytic on [a, b] if the function F : [d−t a, d−t b] → R, given by F (x) = f (dt x), is R-analytic on [d−t a, d−t b]. It is shown in [45] that if f is R-analytic on [a, b] then f is bounded on [a, b]; also, if g is R-analytic on [a, b] and α ∈ R then f + αg and f · g are R-analytic on [a, b]. Moreover, the composition of R-analytic functions is R-analytic. Finally, using the fact that power series on R are infinitely often differentiable within their domain of convergence and the derivatives to any order are obtained by differentiating the power series term by term [45], we obtain the following result. Theorem 4.11. Let a < b in R be given, and let f : [a, b] → R be R-analytic on [a, b]. Then f is infinitely often differentiable on [a, b], and for any positive integer m, we have that f (m) is R-analytic on [a, b]. Moreover, if f is given locally ∞ n around x0 ∈ [a, b] by f (x) = n=0 an (x0 ) (x − x0 ) , then f (m) is given by f (m) (x) =

∞

n−m

n (n − 1) · · · (n − m + 1) an (x0 ) (x − x0 )

.

n=m

In particular, we have that am (x0 ) = f (m) (x0 ) /m! for all m = 0, 1, 2, . . ..

224 10


In [47], we focus on the proof of the intermediate value theorem for R-analytic functions (functions that are given locally by power series). Given a function f that is R-analytic on an interval [a, b] and a value S between f (a) and f (b), we use iteration to construct a sequence of numbers in [a, b] that converges in the order topology to a point c ∈ [a, b] such that f (c) = S. The proof is quite involved, making use of many of the results proved in [41, 45] as well as some results from Real Analysis, including the intermediate value theorem for real-valued functions, continuous on closed and finite real intervals. Ongoing research aims at proving the Extreme Value Theorem for R-analytic functions, stated as a conjecture below. Once this conjecture has been proved, the Mean Value Theorem will follow readily. Conjecture 4.12. (Extreme Value Theorem) Let a < b in R be given, and let f : [a, b] → R be R-analytic on [a, b]. Then f assumes a maximum and a minimum on [a, b]. Finally, in [46] we generalize the results in [41, 45, 47] to power series with rational exponents over R. 5. Measure Theory and Integration Before we define a measure on R, we introduce the following notations which will be adopted throughout this section: I(a, b) will be used to denote any one of the intervals [a, b], (a, b], [a, b) or (a, b), unless we explicitly specify a particular choice of one of the four intervals. Also, to denote the length of a given interval I, we will use the notation l(I). Definition 5.1. Let A ⊂ R be given. Then we say that A is measurable if for every > 0 in R, there exist a sequence of mutually disjoint intervals (In ) and ∞ asequence of mutually disjoint intervals (Jn ) such In ⊂ A ⊂ ∪ ∞ n=1 Jn , ∞ ∞ that ∪n=1 ∞ ∞ l(I ) and l(J ) converge in R, and l(J ) − l(I ) ≤ . n n n n n=1 n=1 n=1 n=1 Given a measurable set A, then for every k ∈ N, we can select a sequence kof k and a sequence of mutually disjoint intervals Jn mutually disjoint intervals I n ∞ ∞ such that n=1 l Ink and n=1 l Jnk converge in R for all k, ∞ ∞ l Jnk − l Ink ≤ dk

k ∞ k+1 k+1 k ⊂ A ⊂ ∪∞ ⊂ ∪∞ ∪∞ n=1 In ⊂ ∪n=1 In n=1 Jn n=1 Jn and

n=1

n=1

for all k ∈ N. Since the order topology, it follows that R is Cauchy-complete ∞ in ∞ limk→∞ n=1 l Ink and limk→∞ n=1 l Jnk both exist and they are equal. We call the common value of the limits the measure of A and we denote it by m(A). Thus, ∞ ∞ m(A) = lim l Ink = lim l Jnk . k→∞

n=1

∞

k→∞

k

n=1

Moreover, since the sequence n=1 l In k∈N is nondecreasing and since the se∞ k quence n=1 l Jn k∈N is nonincreasing, we have that ∞ ∞ l Ink ≤ m(A) ≤ l Jnk for all k ∈ N. n=1

n=1


225 11

Contrary to the real case, sup { ∞ n=1 l(In ) : In ’s are mutually disjoint intervals ∞ and ∪∞ I ⊂ A} and inf { l(J ) : A ⊂ ∪∞ n n n=1 n=1 Jn } need not exist for a given set n=1 A ⊂ R. However, we show in [44] that if A is measurable then both the supremum and infimum exist and they are equal to m(A). This shows that the definition of measurable sets in Definition 5.1 is a natural generalization of the Lebesgue measure of real analysis that corrects for the lack of suprema and infima in non-Archimedean totally ordered fields. Proposition 5.2. Let A ⊂ R be measurable. Then ∞ m(A) = inf l(Jn ) : for all n Jn is an interval, A ⊂ ∪∞ n=1 Jn and n=1 ∞

= sup

l(Jn ) converges

n=1 ∞

l(In ) : In ’s are mutually disjoint intervals, ∪∞ n=1 In ⊂ A, and

n=1 ∞

l(In ) converges .

n=1

We prove that the measure defined above has similar properties to those of the Lebesgue measure of Real Analysis. Namely (see [44] for the details), we show that any subset of a measurable set of measure 0 is itself measurable and has measure 0. We also show that any countable unions of measurable sets whose measures form a null sequence is measurable and the measure of the union is less than or equal to the sum of the measures of the original sets; moreover, the measure of the union is equal to the sum of the measures of the original sets if the latter are mutually disjoint. Then we show that any finite intersection of measurable sets is also measurable and that the sum of the measures of two measurable sets is equal to the sum of the measures of their union and intersection. Like in R, we first introduce a family of simple functions on R from which we obtain a larger family of measurable functions. In the Lebesgue measure theory on R, the simple functions consist only of step functions (piece-wise constant functions); and all measurable functions including all monomials, polynomials and power series are obtained as uniform limits of simple functions. It can be easily shown that in R the order topology is too strong and none of the monomials can be obtained as a uniform limit of polynomials of lower degrees. So using the step functions as our simple functions would yield a too small class of functions that we can integrate. So we introduce a larger family of simple functions. Here we define such a family of simple functions in an abstract way, which we will use throughout the discussions in this section; and but we give two examples below. Definition 5.3. Let a < b in R be given and S(a, b) a family of functions from I(a, b) to R. Then we say that S(a, b) is a family of simple functions on I(a, b) if the following are true: (1) S(a, b) is an algebra that contains the identity function; (2) for all f ∈ S(a, b), f is Lipschitz on I(a, b) and there exists an antiderivative F of f in S(a, b);

226 12


(3) for all differentiable f ∈ S(a, b), if f = 0 on I(a, b) then f is constant on I(a, b); moreover, if f ≤ 0 on I(a, b) then f is nonincreasing on I(a, b). If f ∈ S(a, b), we say that f is simple on I(a, b). It follows from the first condition in Definition 5.3 that any constant function on I(a, b) is in S(a, b); moreover, if f, g ∈ S(a, b) and if α ∈ R, then f + αg ∈ S(a, b). Also, it follows from the third condition that the anti-derivative in the second condition is unique up to a constant. A close look at Definition 5.3 reveals that the polynomials algebra on I(a, b) is the smallest family of simple functions on I(a, b). Another example is the family of R-analytic functions on I(a, b), which was discussed in Section 4 above. While the third condition in Definition 5.3 is automatically satisfied in real analysis, this is not the case in R, as shown by Example 3.3 and Example 3.4. Definition 5.4. Let A ⊂ R be a measurable subset of R be bounded on A. Then we say that f is measurable on A if there ∞exists a sequence of mutually disjoint ∞ intervals (In ) such n, n=1 l (In ) converges in R, m(A) − n=1 l(In ) ≤ and f is n.

and let f : A → R for all > 0 in R, that In ⊂ A for all simple on In for all

In [44], we derive a simple characterization of measurable functions and we show that they form an algebra. Then we show that a measurable function is differentiable almost everywhere and that a function measurable on two measurable subsets of R is also measurable on their union and intersection. We define the integral of a simple function over an interval I(a, b) and we use that to define the integral of a measurable function f over a measurable set A. Definition 5.5. Let a < b in R, let f : I(a, b) → R be simple on I(a, b), and let F be a simple anti-derivative of f on I(a, b). Then the integral of f over I(a, b) is the R number I(a,b)

f = lim F (x) − lim F (x). x→a

x→b

The limits in Definition 5.5 account for the case when the interval I(a, b) does not include one or both of the end points; and these limits exist since F is Lipschitz on I(a, b). Now let A ⊂ R be measurable, let f : A → R be measurable and let M be a bound for |f | on of mutually A. Then for every k ∈ kN, thereexists a sequence disjoint intervals Ink n∈N such that ∪∞ In ⊂ A, ∞ l Ink converges, m(A) − n=1 n=1 ∞ k k k n=1 l In ≤ d , and f is simple on In for all n ∈ N. Without loss of generality, k we k k+1 ⊂ I for all n ∈ N and for all k ∈ N. Since lim l In = 0, may assume that I n→∞ n n k and since I k f ≤ M l In (proved in [44] for simple functions), it follows that n f = 0 for all k ∈ N. lim n→∞

∞

k In

f converges in R for all k ∈ N [41]. ∞ Next we show that the sequence f k n=1 I

Thus,

k n=1 In

n

K

be given in R; and let K ∈ N be such that M d

k∈N

converges in R. So let > 0

≤ . Let k > j ≥ K be given in


227 13

∞ k ∞ j N. Then be written as a union of mutually disjoint intervals, j,k ∪n=1 In \ ∪n=1 Incan j,k say In n∈N , such that ∞ converges, and l I n n=1 ∞ ∞ ∞ ∞ l Inj,k = l Ink − l Inj ≤ m(A) − l Inj ≤ dj ≤ dK . n=1

Thus,

n=1

n=1

n=1

∞ ∞ ∞ ∞ f− f = f ≤ f j j,k j,k k In In In In n=1

n=1

n=1

≤

∞

n=1

∞ M l Inj,k = M l Inj,k

n=1

n=1

≤ M dK ≤ , where we have used the fact that an infinite series converges if and only if it con ∞ is Cauchy; and hence verges absolutely [41]. Thus, the sequence n=1 I k f n

k∈N

it converges in R. We define the unique limit as the integral of f over A. Definition 5.6. Let A ⊂ R be measurable and let f : A → R be measurable. Then the integral of f over A, denoted by A f , is given by ∞ f= lim f. ∞ A

l(In ) → m(A) n=1∞ ∪n=1 In ⊂ A (In ) are mutually disjoint f is simple on In ∀ n

n=1

In

It turns out that the integral in Definition 5.6 satisfies similar properties to those of the Lebesgue integral on R; see [44] for the details. In particular, we prove the linearity property of the integral and that if |f | ≤ M on A then A f ≤ M m(A), where m(A) is the measure of A. We also show that the sum of the integrals of a measurable function over two measurable sets is equal to the sum of its integrals over the union and the intersection of the two sets. Moreover, we prove the following theorem. Theorem 5.7. Let A ⊂ R be measurable, let f : A → R, for each k ∈ N let fk : A → R be measurable on A, and let the sequence (fk ) converge uniformly to f on A. Then lim f exists. Moreover, if f is measurable on A, then k→∞ A k limk→∞ A fk = A f . 6. Optimization In [48], we consider unconstrained one-dimensional optimization on R. We study general optimization questions and derive first and second order necessary and sufficient conditions for the existence of local maxima and minima of a function on a convex subset of R. We show that for first order optimization, the results are similar to the corresponding real ones. However, for second and higher order optimization, we show that conventional differentiability is not strong enough to just extend the real-case results (see Example 3.5 and Example 3.6); and a stronger concept of differentiability, the so-called derivate differentiability (see Definition 6.3 below), is used to solve that difficulty. We also characterize convex functions on convex sets of R in terms of first and second order derivatives.

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In the following, we review the definitions of derivate continuity and differentiability in one dimension, as well as some related results and we refer the interested reader to [6, 39] for a more detailed study. Definition 6.1. Let D ⊂ R be open and let f : D → R. Then we say that f is derivate continuous on D if there exists M ∈ R, called a Lipschitz constant of f on D, such that f (y) − f (x) ≤ M for all x = y in D. y−x It follows immediately from Definition 6.1 that if f : D → R is derivate continuous on D then f is uniformly continuous (in the conventional sense) on D. Remark 6.2. It is clear that the concept of derivate continuity in Definition 6.1 coincides with that of uniform Lipschitz continuity when restricted to R. We chose to call it derivate continuity here so that, after having defined derivate differentiability in Definition 6.3 and higher order derivate differentiability in Definition 6.5, we can think of derivate continuity as derivate differentiability of “order zero”, just as is the case for continuity in R. Definition 6.3. Let D ⊂ R be open, let f : D → R be derivate continuous on D, and let ID denote the identity function on D. Then we say that f is derivate (x) : D \ {x} → R is derivate differentiable on D if for all x ∈ D, the function fI−f D −x (x) to D (see continuous on D \ {x}. In this case, the unique continuation of fI−f D −x [39]) will be called the first derivate function (or simply the derivate function) of f at x and will be denoted by F1,x ; moreover, the function value F1,x (x) will be called the derivative of f at x and will be denoted by f (x).

It follows immediately from Definition 6.3 that if f : D → R is derivate differentiable then f is differentiable in the conventional sense; moreover, the two derivatives at any given point of D agree. The following result provides a useful tool for checking the derivate differentiability of functions; we refer the interested reader to [39, 48] for its proof. Theorem 6.4. Let D ⊂ R be open and let f : D → R be derivate continuous on D. Suppose there exists M ∈ R and there exists a function g : D → R such that f (y) − f (x) ≤ M |y − x| for all y = x in D. − g (x) y−x Then f is derivate differentiable on D, with derivative f = g. Definition 6.5 (n-times Derivate Differentiability). Let D ⊂ R be open, and let f : D → R. Let n ≥ 2 be given in N. Then we define n-times derivate differentiability of f on D inductively as follows: Having defined (n − 1)-times derivate differentiability, we say that f is n-times derivate differentiable on D if f is (n − 1)-times derivate differentiable on D and for all x ∈ D, the (n − 1)st derivate function Fn−1,x is derivate differentiable on D. For all x ∈ D, the derivate function Fn,x of Fn−1,x at x will be called the nth derivate function of f at x, and (x) will be called the nth derivative of f at x and the number f (n) (x) = n!Fn−1,x (n) denoted by f (x). One of the most useful consequences of the derivate differentiability concept is that it gives rise to a Taylor formula with remainder while the conventional


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(topological) differentiability does not. We only state the result here and refer the reader to [6, 39] for its proof. We also note that, as an immediate result of Theorem 6.6, we obtain local expandability in Taylor series around x0 ∈ D of a given function that is infinitely often derivate differentiable on D [6, 39]. Theorem 6.6 (Taylor Formula with Remainder). Let D ⊂ R be open and let f : D → R be n-times derivate differentiable on D. Let x ∈ D be given, let Fn,x be the nth order derivate function of f at x, and let Mn,x be a Lipschitz constant of Fn,x on D. Then for all y ∈ D, we have that f (y) = f (x) +

n f (j) (x) j=1

j!

(y − x)j + rn+1 (x, y) (y − x)n+1 ,

with λ (rn+1 (x, y)) ≥ λ (Mn,x ). Using Theorem 6.6, we are able to generalize in [48] most of one-dimensional optimization results of Real Analysis. For example, we obtain the following two results which state necessary and sufficient conditions for the existence of local (relative) extrema. Theorem 6.7 (Necessary Conditions for Existence of Local Extrema). Let a < b be given in N , let m ≥ 2, and let f : I(a, b) → N be m-times derivate differentiable on I(a, b). Suppose that f has a local extremum at x0 ∈ (a, b) and l ≤ m is the order of the first nonvanishing derivative of f at x0 . Then l is even. Moreover, f (l) (x0 ) is positive if the extremum is a minimum and negative if the extremum is a maximum. Theorem 6.8 (Sufficient Conditions for Existence of Local Extrema). Let a < b be given in N , let k ∈ N, and let f : I(a, b) → N be 2k-times derivate differentiable on I(a, b). Let x0 ∈ (a, b) be such f (j) (x0 ) = 0 for all j ∈ {1, . . . , 2k − 1} and f (2k) (x0 ) = 0. Then f has a local minimum at x0 if f (2k) (x0 ) > 0 and a local maximum if f (2k) (x0 ) < 0. In [49], we generalize the concepts of derivate continuity and differentiability to higher dimensions; and this yields a Taylor Formula with a bounded remainder term for C m functions (in the derivate sense) from an open subset of Rn to R. Theorem 6.9 (Taylor Formula for Functions of Several Variables). Let D ⊂ Rn be open, let x0 ∈ D be given and let f : D → R be C q on D. Then there exist M, δ > 0 in R such that Bδ (x0 ) ⊂ D and, for all x ∈ Bδ (x0 ), we have that ⎛ ⎞ q n 1 j ⎝ ∂l1 · · · ∂lj f (x0 )πk=1 f (x) = f (x0 ) + (xlk − x0,lk ) ⎠ j! j=1 l1 ,...,lj =1

+Rq+1 (x0 , x), where |Rq+1 (x0 , x)| ≤ M |x − x0 |q+1 . Then we use that to derive necessary and sufficient conditions of second order for the existence of a minimum of an R-valued function on Rn subject to equality and inequality constraints. More specifically, we solve the problem of minimizing a

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function f : Rn → R, subject to the following set of constraints: ⎧ ⎧ ⎪ ⎪ ⎨ g1 (x) ≤ 0 ⎨ h1 (x) = 0 . .. .. and , (6.1) . ⎪ ⎪ ⎩ ⎩ hm (x) = 0 gp (x) ≤ 0 where all the functions in Equation (6.1) are from Rn to R. A point x0 ∈ Rn is said to be a feasible point if it satisfies the constraints in Equation (6.1). Definition 6.10. Let x0 be a feasible point for the constraints in Equation (6.1) and let I(x0 ) = {l ∈ {1, . . . , p} : gl (x0 ) = 0}. Then we say that x0 is regular for the constraints if {∇hj (x0 ) : j = 1, . . . , m; ∇gl (x0 ) : l ∈ I(x0 )} forms a linearly independent subset of vectors in Rn . The following theorem provides necessary conditions of second order for a local minimizer x0 of a function f subject to the constraints in Equation (6.1). The result is a generalization of the corresponding real result [29, 14] and the proof (see [49]) is similar to that of the latter; but one essential difference is the form of the remainder formula. In the real case, the remainder term is related to the second derivative at some intermediate point, while here that is not the case. However, the concept of derivate differentiability puts a bound on the remainder term; and this is instrumental in the proof of the theorem. p 2 Theorem 6.11. Suppose that f , {hj }m j=1 , {gl }l=1 are C on some open set n D ⊂ R containing the point x0 and that x0 is a regular point for the constraints in Equation (6.1). If x0 is a local minimizer for f under the given constraints, then there exist α1 , . . . , αm , β1 , . . . , βp ∈ R such that (i) βl ≥ 0 for all l ∈ {1, . . . , p}, (ii) βl gl (x0 ) = 0 for all l ∈ {1, . . . , p}, m p (iii) ∇f (x0 ) + j=1 αj ∇hj (x0 ) + l=1 βl ∇gl (x0 ) = 0, and m p (iv) y T ∇2 f (x0 ) + j=1 αj ∇2 hj (x0 ) + l=1 βl ∇2 gl (x0 ) y ≥ 0 for all y ∈ Rn satisfying ∇hj (x0 )y = 0 for all j ∈ {1, . . . , m}, ∇gl (x0 )y = 0 for all l ∈ L = {k ∈ I(x0 ) : βk > 0} and ∇gl (x0 )y ≤ 0 for all l ∈ I(x0 ) \ L.

In the following theorem, we present second order sufficient conditions for a feasible point x0 to be a local minimum of a function f subject to the constraints in Equation (6.1). It is a generalization of the real result [14] and reduces to it, when restricted to functions from Rn to R. In fact, since in condition (iv) below is allowed to be infinitely small, the condition |∇hj (x0 )y | < would reduce to ∇hj (x0 )y = 0, when restricted to R. Similarly, one can readily see that the other conditions are mere generalizations of the corresponding real ones. However, the proof (see [49]) is different than that of the real result since the supremum principle does not hold in R. p 2 Theorem 6.12. Suppose that f , {hj }m j=1 , {gl }l=1 are C on some open set n D ⊂ R containing the point x0 and that x0 is a feasible point for the constraints in Equation (6.1) such that, for some α1 , . . . , αm , β1 , . . . , βp ∈ R and for some , γ > 0 in R, we have that (i) βl ≥ 0 for all l ∈ {1, . . . , p}, (ii) βl gl (x0 ) = 0 for all l ∈ {1, . . . , p}, m p (iii) ∇f (x0 ) + j=1 αj ∇hj (x0 ) + l=1 βl ∇gl (x0 ) = 0, and


231 17

2 (iv) yT ∇2 f (x0 ) + m x0 ) + pl=1 βl ∇2 gl (x0 ) y ≥ γ for all y ∈ j=1 αj ∇ hj ( Rn satisfying |y | = 1, |∇hj (x0 )y | < for all j ∈ {1, . . . , m}, |∇gl (x0 )y | < for all l ∈ L = {k : βk > 0} and ∇gl (x0 )y < for all l ∈ I(x0 ) \ L, where I(x0 ) = {k : gk (x0 ) = 0}. Then x0 is a strict local minimum for f under the constraints of Equation (6.1).

7. Computation of Derivatives of Real Functions The general question of efficient differentiation is at the core of many parts of the work on perturbation and aberration theories relevant in Physics and Engineering; for an overview, see for example [8]. In this case, derivatives of highly complicated functions have to be computed to high orders. However, even when the derivative of the function is known to exist at the given point, numerical methods fail to give an accurate value of the derivative; the error increases with the order, and for orders greater than three, the errors often become too large for the results to be practically useful. On the other hand, while formula manipulators like Mathematica are successful in finding low-order derivatives of simple functions, they fail for high-order derivatives of very complicated functions. Moreover, they fail to find the derivatives of certain functions at given points even though the functions are differentiable at the respective points. This is generally connected to the occurrence of nondifferentiable parts that do not affect the differentiability of the end result as well as the occurrence of branch points in coding as in IF-ELSE structures. Using Calculus on R and the fact that the field has infinitely small numbers represents a new method for computational differentiation that avoids the wellknown accuracy problems of numerical differentiation tools. It also avoids the often rather stringent limitations of formula manipulators that restrict the complexity of the function that can be differentiated, and the orders to which differentiation can be performed. By a computer function, we denote any real-valued function that can be typed on a computer. The R numbers as well as the continuations to R of the intrinsic functions (and hence of all computer functions) have all been implemented for use on a computer, using the code COSY INFINITY [9, 30]. Using the calculus on R, we formulate a necessary and sufficient condition for the derivatives of a computer function to exist, and show how to find these derivatives whenever they exist [40, 42]. The new technique of computing the derivatives of computer functions, which we summarize below, achieves results that combine the accuracy of formula manipulators with the speed of classical numerical methods, that is the best of both worlds. Lemma 7.1. Let f be a computer function. Then f is defined at x0 if and only if f (x0 ) can be computed on a computer. This lemma hinges on a careful implementation of the intrinsic functions and operations, in particular in the sense that they should be executable for any floating point number in the domain of definition that produces a result within the range of allowed floating point numbers.

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Lemma 7.2. Let f be a computer function, and let x0 be such that f (x0 − d), f (x0 ), and f (x0 + d) are all defined. Then f is continuous at x0 if and only if f (x0 − d) =0 f (x0 ) =0 f (x0 + d). If f (x0 ) and f (x0 +d) are defined, but f (x0 −d) is not, then f is right-continuous at x0 if and only if f (x0 + d) =0 f (x0 ). Finally, if f (x0 ) and f (x0 − d) are defined, but f (x0 + d) is not, then f is left-continuous at x0 if and only if f (x0 − d) =0 f (x0 ). Theorem 7.3. Let f be a computer function that is continuous at x0 , and let f (x0 − d) and f (x0 + d) be both defined. Then f is differentiable at x0 if and only if f (x0 + d) − f (x0 ) f (x0 ) − f (x0 − d) and d d are both at most finite in absolute value, and their real parts agree. In this case, f (x0 + d) − f (x0 ) f (x0 ) − f (x0 − d) =0 f (x0 ) =0 . d d If f is differentiable at x0 , then f is twice differentiable at x0 if and only if f (x0 + 2d) − 2f (x0 + d) + f (x0 ) f (x0 ) − 2f (x0 − d) + f (x0 − 2d) and 2 d d2 are both at most finite in absolute value, and their real parts agree. In this case f (x0 + 2d) − 2f (x0 + d) + f (x0 ) f (x0 ) − 2f (x0 − d) + f (x0 − 2d) =0 f (2) (x0 ) =0 . d2 d2 In general, if f is (n − 1) times differentiable at x0 , then f is n times differentiable at x0 if and only if n n n n n−j j f (x f (x0 − jd) (−1) + jd) (−1) 0 j=0 j=0 j j and dn dn are both at most finite in absolute value, and their real parts agree. In this case, n n n n n−j j f (x f (x0 − jd) (−1) + jd) (−1) 0 j=0 j=0 j j (n) =0 f (x0 ) =0 . dn dn Since knowledge of f (x0 − d) and f (x0 + d) gives us all the information about a computer function f in a real positive radius σ around x0 , we have the following result which states that, from the mere knowledge of f (x0 − d) and f (x0 + d), we can find at once the order of differentiability of f at x0 and the accurate values of all existing derivatives. Theorem 7.4. Let f be a computer function that is continuous at x0 . Then f is n times differentiable at x0 if and only if f (x0 − d) and f (x0 + d) are both defined and can be written as n n (−1)j αj dj and f (x0 + d) =n f (x0 ) + αj dj , f (x0 − d) =n f (x0 ) + j=1

where the αj ’s are real numbers. Moreover, in this case f j ≤ n.

j=1 (j)

(x0 ) = j! αj for 1 ≤


233 19

Now consider, as an example, the function 3+cos(sin(ln|1+x|)) sin x3 + 2x + 1 + sin(cos(tan(exp(x)))) exp(tanh(sinh(cosh( cos(sin(exp(tan(x+2)))) )))) −1 (7.1) g(x) = . 2 + sin sinh cos tan (ln (exp(x) + x2 + 3)) Using the R calculus, we find g (n) (0) for 0 ≤ n ≤ 10. These numbers are listed in table 1; we note that, for 0 ≤ n ≤ 10, we list the CPU time needed to obtain all Table 1. g (n) (0), 0 ≤ n ≤ 10, computed with R calculus g (n) (0) 1.004845319007115 0.4601438089634254 −5.266097568233224 −52.82163351991485 −108.4682847837855 16451.44286410806 541334.9970224757 7948641.189364974 −144969388.2104904 −15395959663.01733 −618406836695.3634

Order n 0 1 2 3 4 5 6 7 8 9 10

CPU Time 1.820 msec 2.070 msec 3.180 msec 4.830 msec 7.700 msec 11.640 msec 18.050 msec 26.590 msec 37.860 msec 52.470 msec 72.330 msec

derivatives of g at 0 up to order n and not just g (n) (0). For comparison purposes, we give in table 2 the function value and the first six derivatives computed with Table 2. g (n) (0), 0 ≤ n ≤ 6, computed with Mathematica Order n 0 1 2 3 4 5 6

g (n) (0) 1.004845319007116 0.4601438089634254 −5.266097568233221 −52.82163351991483 −108.4682847837854 16451.44286410805 541334.9970224752

CPU Time 0.11 sec 0.17 sec 0.47 sec 2.57 sec 14.74 sec 77.50 sec 693.65 sec

Mathematica. Note that the respective values listed in tables 1 and 2 agree. However, Mathematica used much more CPU time to compute the first six derivatives, and it failed to find the seventh derivative as it ran out of memory. We also list in table 3 the first ten derivatives of g at 0 computed numerically using the numerical differentiation formulas ⎞ ⎛ n n g (n) (0) = (∆x)−n ⎝ (−1)n−j g (j∆x)⎠ , ∆x = 10−16/(n+1) , j j=0

for 1 ≤ n ≤ 10, together with the corresponding relative errors obtained by comparing the numerical values with the respective exact values computed using R calculus.

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Table 3. g (n) (0), 1 ≤ n ≤ 10, computed numerically Order n 1 2 3 4 5 6 7 8 9 10

g (n) (0) 0.4601437841866840 −5.266346392944456 −52.83767867680922 −87.27214664649106 19478.29555909866 633008.9156614641 −12378052.73279768 −1282816703.632099 83617811421.48561 91619495958355.24

Relative Error 54 × 10−9 47 × 10−6 30 × 10−5 0.20 0.18 0.17 2.6 7.8 6.4 149

On the other hand, formula manipulators fail to find the derivatives of certain functions at given points even though the functions are differentiable at the respective points. For example, the functions ⎧ 1−exp (−x2 ) ⎨ · g(x) if x = 0 x g1 (x) = |x|5/2 · g(x) and g2 (x) = , ⎩ 0 if x = 0 where g(x) is the function given in Equation (7.1), are both differentiable at 0; but the attempt to compute their derivatives using formula manipulators fails. This is not specific to g1 and g2 , and is generally connected to the occurrence of nondifferentiable parts that do not affect the differentiability of the end result, of which case g1 is an example, as well as the occurrence of branch points in coding as in IF-ELSE structures, of which case g2 is an example.

8. Existence and Uniqueness of Solutions of Ordinary Differential Equations In [10], we consider differential equations over R with a right hand side that is infinitely often derivate differentiable. We show that such an ODE admits solutions that are themselves infinitely often derivate differentiable. To this end, we develop a theory of multivariate infinitely often derivate differentiable functions and show that they can be locally represented as Taylor series. We then re-phrase the ODE problem as a fixed point problem of a Picard operator in the common way. After various transformations and utilizing well-known existence and uniqueness properties of ODEs over R, the problem is transformed to a fixed point problem with an infinitely small contraction factor. We show that the sequence of functions obtained by iteration converges uniformly in the order topology, that the resulting limit is itself infinitely often derivate differentiable, and that this limit indeed solves the ODE. It is then shown that while there are other solutions with lesser smoothness requirements, the solution so obtained is unique among all the infinitely often derivate differentiable functions.


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9. Current and Future Research Several research projects are currently under way. Building on the existing knowledge of the field, which has been summarized in the sections above, we are working on developing a multivariate calculus theory on Rn as well as on completing the study of power series and R-analytic functions by proving the Extreme Value Theorem and the Mean Value Theorem. We plan also to do more analysis on C and look into more potential applications of the implementation of the R numbers on a computer. Moreover, in collaboration with Jose Aguayo and Miguel Nova, we are looking into the possibility of developing a non-Archimedean Hilbert Space theory by considering the space c0 of null sequences of elements of R, equipped with the ∞ non-Archimedean inner product (xn ), (yn ) := n=1 xn yn ; this may lead to some useful applications in Physics, particularly in Quantum Mechanics. Acknowledgments The authors wish to thank the kind referee for the useful comments and suggestions which helped improve the quality of the paper. References [1] N. L. Alling. Foundations of Analysis over Surreal Number Fields. North Holland, 1987. [2] S. Basu, R. Pollack, and M. Roy. Algorithms in Real Algebraic Geometry. Springer, 2003. [3] M. Berz. Analysis on a nonarchimedean extension of the real numbers. Lecture Notes, 1992 and 1995 Mathematics Summer Graduate Schools of the German National Merit Foundation. MSUCL-933, Department of Physics, Michigan State University, 1994. [4] M. Berz. Calculus and numerics on Levi-Civita fields. In M. Berz, C. Bischof, G. Corliss, and A. Griewank, editors, Computational Differentiation: Techniques, Applications, and Tools, pages 19–35, Philadelphia, 1996. SIAM. [5] M. Berz. Analytical and computational methods for the Levi-Civita fields. In Proc. Sixth International Conference on Nonarchimedean Analysis, pages 21–34, New York, NY, 2000. Marcel Dekker. [6] M. Berz. Nonarchimedean analysis and rigorous computation. International Journal of Applied Mathematics, 2:889–930, 2000. [7] M. Berz. Cauchy theory on Levi-Civita fields. Contemporary Mathematics, 319:39–52, 2003. [8] M. Berz, C. Bischof, A. Griewank, G. Corliss, and Eds. Computational Differentiation: Techniques, Applications, and Tools. SIAM, Philadelphia, 1996. [9] M. Berz, G. Hoffst¨ atter, W. Wan, K. Shamseddine, and K. Makino. COSY INFINITY and its applications to nonlinear dynamics. In M. Berz, C. Bischof, G. Corliss, and A. Griewank, editors, Computational Differentiation: Techniques, Applications, and Tools, pages 363–367, Philadelphia, 1996. SIAM. [10] M. Berz and K. Shamseddine. Existence and uniqueness of solutions of differential equations on the Levi-Civita field. in preparation. [11] J. H. Conway. On Numbers and Games. North Holland, 1976. [12] M. Davis. Applied Nonstandard Analysis. John Wiley and Sons, 1977. [13] H.-D. Ebbinghaus et al. Zahlen. Springer, 1992. [14] Anthony V. Fiacco and Garth P. McCormick. Nonlinear Programming; Sequential Unconstrained Minimization Techniques. SIAM, Philadelphia, 1990. [15] L. Fuchs. Partially Ordered Algebraic Systems. Pergamon Press, Addison Wesley, 1963. [16] H. Gonshor. An Introduction to the Theory of Surreal Numbers. Cambrindge University Press, 1986. ¨ [17] H. Hahn. Uber die nichtarchimedischen Gr¨ oßensysteme. Sitzungsbericht der Wiener Akademie der Wissenschaften Abt. 2a, 117:601–655, 1907. [18] E. Hewitt and K. Stromberg. Real and Abstract Analysis. Springer, 1969. [19] D. E. Knuth. Surreal Numbers: How two ex-students turned on to pure mathematics and found total happiness. Addison-Wesley, 1974.

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[20] W. Krull. Allgemeine Bewertungstheorie. J. Reine Angew. Math., 167:160–196, 1932. [21] D. Laugwitz. Eine Einf¨ uhrung der Delta-Funktionen. Sitzungsberichte der Bayerischen Akademie der Wissenschaften, 4:41, 1959. [22] D. Laugwitz. Anwendungen unendlich kleiner Zahlen I. Journal f¨ ur die reine und angewandte Mathematik, 207:53–60, 1961. [23] D. Laugwitz. Anwendungen unendlich kleiner Zahlen II. Journal f¨ ur die reine und angewandte Mathematik, 208:22–34, 1961. [24] D. Laugwitz. Ein Weg zur Nonstandard-Analysis. Jahresberichte der Deutschen Mathematischen Vereinigung, 75:66–93, 1973. [25] D. Laugwitz. Tullio Levi-Civita’s work on nonarchimedean structures (with an Appendix: Properties of Levi-Civita fields). In Atti Dei Convegni Lincei 8: Convegno Internazionale Celebrativo Del Centenario Della Nascita De Tullio Levi-Civita, Academia Nazionale dei Lincei, Roma, 1975. [26] T. Levi-Civita. Sugli infiniti ed infinitesimi attuali quali elementi analitici. Atti Ist. Veneto di Sc., Lett. ed Art., 7a, 4:1765, 1892. [27] T. Levi-Civita. Sui numeri transfiniti. Rend. Acc. Lincei, 5a, 7:91,113, 1898. [28] A. H. Lightstone and A. Robinson. Nonarchimedean Fields and Asymptotic Expansions. North Holland, New York, 1975. [29] David G. Luenberger. Linear and Nonlinear Programming. Addison-Wesley, Reading, Massachusetts, 2nd edition, 1984. [30] K. Makino and M. Berz. COSY INFINITY version 9. Nuclear Instruments and Methods, A558:346–350, 2005. [31] L. Neder. Modell einer Leibnizschen Differentialrechnung mit aktual unendlich kleinen Gr¨ oßen. Mathematische Annalen, 118:718–732, 1941-1943. [32] S. Priess-Crampe. Angeordnete Strukturen: Gruppen, K¨ orper, projektive Ebenen. Springer, Berlin, 1983. [33] P. Ribenboim. Fields: Algebraically Closed and Others. Manuscripta Mathematica, 75:115– 150, 1992. [34] A. Robinson. Non-standard analysis. In Proceedings Royal Academy Amsterdam, Series A, volume 64, page 432, 1961. [35] A. Robinson. Non-Standard Analysis. North-Holland, 1974. [36] W. Rudin. Real and Complex Analysis. McGraw Hill, 1987. [37] W. H. Schikhof. Ultrametric Calculus: An Introduction to p-Adic Analysis. Cambridge University Press, 1985. [38] C. Schmieden and D. Laugwitz. Eine Erweiterung der Infinitesimalrechnung. Mathematische Zeitschrift, 69:1–39, 1958. [39] K. Shamseddine. New Elements of Analysis on the Levi-Civita Field. PhD thesis, Michigan State University, East Lansing, Michigan, USA, 1999. also Michigan State University report MSUCL-1147. [40] K. Shamseddine and M. Berz. Exception handling in derivative computation with nonArchimedean calculus. In M. Berz, C. Bischof, G. Corliss, and A. Griewank, editors, Computational Differentiation: Techniques, Applications, and Tools, pages 37–51, Philadelphia, 1996. SIAM. [41] K. Shamseddine and M. Berz. Convergence on the Levi-Civita field and study of power series. In Proc. Sixth International Conference on Nonarchimedean Analysis, pages 283–299, New York, NY, 2000. Marcel Dekker. [42] K. Shamseddine and M. Berz. The differential algebraic structure of the Levi-Civita field and applications. International Journal of Applied Mathematics, 3:449–465, 2000. [43] K. Shamseddine and M. Berz. Intermediate values and inverse functions on non-Archimedean fields. International Journal of Mathematics and Mathematical Sciences, 30:165–176, 2002. [44] K. Shamseddine and M. Berz. Measure theory and integration on the Levi-Civita field. Contemporary Mathematics, 319:369–387, 2003. [45] K. Shamseddine and M. Berz. Analytical properties of power series on Levi-Civita fields. Annales Math´ ematiques Blaise Pascal, 12(2):309–329, 2005. [46] K. Shamseddine and M. Berz. Generalized power series on a non-Archimedean field. Indagationes Mathematicae, 17(3):457–477, 2006. [47] K. Shamseddine and M. Berz. Intermediate value theorem for analytic functions on a LeviCivita field. Bulletin of the Belgian Mathematical Society- Simon Stevin, 14:1001–1015, 2007.


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[48] K. Shamseddine and V. Zeidan. One-dimensional optimization on non-Archimedean fields. Journal of Nonlinear and Convex Analysis, 2:351–361, 2001. [49] K. Shamseddine and V. Zeidan. Constrained second order optimization on non-Archimedean fields. Indagationes Mathematicae, 14:81–101, 2003. [50] K. Stromberg. An Introduction to Classical Real Analysis. Wadsworth, 1981. [51] K. D. Stroyan and W. A. J. Luxemburg. Introduction to the Theory of Infinitesimals. Academic Press, 1976. Department of Physics and Astronomy, University of Manitoba, Winnipeg, Manitoba R3T 2N2, Canada E-mail address: [email protected] Department of Physics and Astronomy, Michigan State University, East Lansing, MI 48824, USA E-mail address: [email protected]



Criteria for non-repelling fixed points Per-Anders Svensson Abstract. Let g(x) be a monic irreducible polynomial, defined over the ring of integers of a p-adic field K. For each a ∈ K, we put ha (x) = x + ag(x). We study the discrete dynamical system defined by the polynomial ha (x), on the extension K(α) of K, where α is a zero of g(x) and thus a fixed point of ha . The investigation is divided into four parts, each dealing with the particular value of a: a = 1, |a| = 1, |a| < 1, or |a| > 1.

1. Introduction In this paper we will study the nature of the fixed points of a certain class of discrete dynamical systems, defined over a non-archimedean field. The interest for such dynamical systems, and their applications, has increased rapidly during the last twenty years, see e.g. Anashin [1, 2, 3], Arrowsmith & Vivaldi [4], Benedetto [5], Khrennikov [9], Khrennikov & Nilsson [10], Lubin [12], Nyqvist [13], Svensson [15], and Verstegen [20]. Let K be a p-adic field. This means among other things, that char K = 0 and that K is complete with respect to a non-trivial, non-archimedean, and discrete valuation. Let OK denote the ring of integers in K, PK the corresponding maximal ideal, and finally Kp = OK /PK the residue class field, which by definition is finite. We suppose that Kp contains q elements, where q = pm for some positive integer m. Here p = char Kp . The multiplicative group Kp∗ of Kp is cyclic, and if a is an element in Kp∗ , we write o(a) for the order of the cyclic subgroup in Kp∗ , generated by a. The valuation of an element b ∈ K will be denoted |b|. We assume that the valuation is normalized in such a way, that |p| = p−1 (where p = char Kp ). The set of all valuations of the non-zero elements in K forms a multiplicative group, which we denote by VK . This is the so called valuation group of K. The ideal PK is by definition principal; we fix a generator π of PK and write ordπ (b) for the order of b at π. The connection between |b| and ordπ (b) is given by |b| = |π|− ordπ (b) , for all b = 0. Thus the valuation group VK is cyclic and generated by |π|. If b ∈ OK we let ¯b denote the canonical image of b in Kp , and in a similar way, we write f¯(x) for the canonical image in Kp [x] of a polynomial f (x) ∈ OK [x]. 1991 Mathematics Subject Classification. 37E15, 11S05, 46S10. Key words and phrases. dynamical systems, p-adic fields, fixed points. c Mathematical 0000 (copyright Society holder) c 2010 American

1 239

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PER-ANDERS SVENSSON

The most well-known example of a p-adic field is Qp , the field of p-adic numbers. Its ring of integers is denoted Zp . The residue class field Zp /pZp (here pZp denotes the maximal ideal of Zp ) is isomorphic to the finite field Fp containing p elements. Given an element α in a p-adic field K, and a real number r > 0, we will use the notations Br (α, K) = {β ∈ K : |β − α| < r}, B r (α, K) = {β ∈ K : |β − α| ≤ r},

and

Sr (α, K) = {β ∈ K : |β − α| = r} for the open ball, closed ball, and sphere, respectively, with radius r centered at α. Recall however, that each open ball is closed, and vice versa, in a p-adic field. Let g(x) ∈ OK [x] be a monic irreducible polynomial of degree n. We pick a zero α of g(x) (from a fixed algebraic closure of K), and put L = K(α). We recall that α is an attracting, indifferent, or repelling fixed point, depending on whether |h (α)| < 1, |h (α)| = 1, or |h (α)| > 1, respectively. If α is attracting, it is well-known that there is a neighborhood A(α, L) = {β ∈ L | lim hn (β) = α} n→∞

of α, containing all elements β ∈ L that is attracted by α.1 The set A(α, L) is called the basin of attraction of α in L. If α is indifferent, then it is the center of a Siegel disc, i.e. an open ball Br (α, L) such that hn [Sρ (α, L)] ⊆ Sρ (α, L) for all ρ < r. The union of all Siegel discs of α in L is called the maximal Siegel disc of α in L, and will be denoted Si(α, L). Finally, if α is repelling, then there is a neighborhood V ⊆ L such that |hn (β) − α| > |β − α| for each β = α in V . The dependence of g(x) on the dynamical system h(x) = x + g(x) on L has previously been studied in e.g. Svensson [15, 17], Svensson & Nyqvist [18], and Khrennikov & Svensson [11]. We will in this paper generalize our investigations, and study how an element a ∈ K affects the dynamics of the system ha (x) = x + ag(x). We will begin by restating earlier results for the special case when a = 1. Then we try to generalize those results to the general case, dividing our investigation into three parts, depending on whether |a| = 1, |a| < 1, or |a| > 1. 2. The case a = 1 Since this section just contains restatements of previous results, all proofs are omitted. 1By hn we mean the n-fold composition of h with itself.

CRITERIA FOR NON-REPELLING FIXED POINTS

241 3

Theorem 2.1. Suppose g(x) ∈ OK [x] is monic and irreducible over K. Then a fixed point of h(x) = x + g(x) is attracting, if and only if g¯(x) ∈ Kp [x] is irreducible and can be written as g¯(x) = ψ(xp ) − x, where ψ(x) ∈ Kp [x] is a non-constant polynomial. Proof. See Khrennikov & Svensson [11, Theorem 4.4].

Theorem 2.2. Let h(x) = x + g(x), where g(x) is a monic irreducible polynomial of degree n in OK [x]. Suppose g(x) defines an unramified extension of K, and put L = K(α), where α is a zero of g(x). Then the following statements are true. (1) If g¯(x) is irreducible and fulfills the condition of Theorem 2.1, then α is attracting, and B1 (α, L) ⊆ A(α, L) ⊂ B 1 (α, L). (2) If g¯(x) is not irreducible, then α is indifferent. Furthermore, if the zeros of g(x) are equidistant, then ⎧ qn − 1 ⎨B (α, L), if q ≡ ≡ 1 (mod 2) 1 (n − 1, q n − 1) Si(α, L) = ⎩ B1 (α, L) otherwise, where q is the number of elements in Kp . Proof. See Khrennikov & Svensson [11, Theorem 4.6].

Theorem 2.3. Suppose g(x) defines a totally ramified extension L/K of degree n, and let q be the number of elements in Lp = Kp . If α is a fixed point of the dynamical system h(x) = x + g(x), then α is indifferent, and ⎧ q−1 ⎨B (α, L), if q ≡ ≡ 1 (mod 2) 1 (n − 1, q − 1) Si(α, L) = ⎩ B1 (α, L) otherwise. Proof. This theorem was proved in Svensson [16], with the added assumption that deg g(x) is a prime (see Theorem 5.4 of that paper). However, this requirement on g(x) may be skipped without spoiling the proof. 3. The case |a| = 1 We will now make the first step in trying to generalize the results of the previous section. Still, we assume that g(x) ∈ OK [x] is monic and irreducible polynomial of degree n. We let a ∈ OK be a unit, and put (3.1)

ha (x) = x + ag(x).

The first theorem in this section generalizes Theorem 2.1. Theorem 3.1. Let α be a fixed point of the dynamical system (3.1). Then α is attracting, if and only if g¯(x) is irreducible, and in the form (3.2)

¯−1 x, g¯(x) = ψ(xp ) − a

for some non-constant ψ(x) ∈ Kp [x]. The proof is just a minor modification of the one that can be found in Khrennikov & Svensson [11]. For the sake of completeness, we present the full proof here.

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Proof of Theorem 3.1. It is readily checked that if g¯(x) has the form (3.2), then |ha (α)| < 1, whence α is attracting. Assume now that α is attracting. We first show that g¯(x) is irreducible. Now g¯(x) cannot be a product of two non-constant relatively prime polynomials in Kp [x], since one then could use Hensel’s lemma (as presented in e.g. Hasse [7, pp. 169]) to obtain a non-trivial factorization of g(x). Thus we must have g¯(x) = r(x)m for some non-constant polynomial r(x) ∈ Kp [x] and positive integer m. Here we may assume that r(x) is the minimal polynomial of α over Kp . Thus we want to show that m = 1. Having g¯(x) = r(x)m , we find that g¯ (x) = mr(x)m−1 r (x). This would be the zero polynomial, if p | m or if r(x) = s(xp ) for some non-constant s(x) ∈ Kp [x]. ¯ (x) = 1, and thereby |h (α)| = 1, contradicting that α is attracting. But then h a a Therefore we conclude that if m > 1, then r(x) is a non-trivial factor of g¯ (x). ¯ (x), since h ¯ (α) = 0 due to the assumption that α is We also have r(x) | h a a ¯ attracting. But since ha (x) = 1 + a ¯g¯ (x), this would mean that r(x) | 1. Thus assuming that m > 1 leads to a contradiction. The fact that we now know that g¯(x) is the minimal polynomial of α over Kp , ¯ (α) = 0, yields h ¯ (x) | g¯(x). Since deg h ¯ (x) < deg g¯(x), this is not and that h a a a p ¯ possible, unless ha (x) = φ(x ) for some φ(x) ∈ Kp [x]. But then ¯ a (x) − x = φ(xp ) − x, a ¯g¯(x) = h and we obtain (3.2), by putting ψ(x) = a ¯−1 φ(x).

Next comes the generalization of Theorem 2.2. Theorem 3.2. Let ha (x) = x + ag(x), where g(x) is a monic irreducible polynomial of degree n in OK [x]. Suppose g(x) defines an unramified extension of K, and put L = K(α), where α is a zero of g(x). Then the following statements are true. (1) If g¯(x) is irreducible and fulfills the condition of Theorem 3.1, then α is attracting, and B1 (α, L) ⊆ A(α, L) ⊂ B 1 (α, L). (2) If g¯(x) is not irreducible, then α is indifferent. Moreover, if |αi − αj | < 1 for any two different zeros αi and αj of g(x), then Si(α, L) = B1 (α, L) if and only if o(b) | (q n − 1)/(n − 1, q n − 1), where q is the number of elea−1 ∈ Kp . Otherwise Si(α, L) = B 1 (α, L). ments in Kp , and b = −¯ Proof. Since (1) is straightforward, we will only prove (2) here. We find that B1 (α, L) ⊆ Si(α, L) and that no element outside B 1 (α, L) can belong to Si(α, L), exactly in the same way as in the proof of Theorem 2.2, which can be found in Khrennikov & Svensson [11]. Suppose Si(α, L) = B1 (α, L). Then there is at least one β ∈ OL such that |h(β) − α| < |β − α| = 1. Thus in the residue class field Lp , we will have the ¯ a (β) = α. Since the distance between any two different zeros situation β = α but h ¯ a (x) = x + a of g(x) are strictly less than 1, we have h ¯(x − α)n ∈ Kp [x]. Hence β = α, β + a ¯(β − α)n = α ⇐⇒ (β − α)n−1 = b, whence the polynomial xn−1 −b has a zero in Lp . The field Lp contains q n elements, since L/K is an unramified extension of degree n, and |Kp | = q. We conclude that n the resultant res(xn−1 − b, xq −1 − 1) has to be zero. By using the formula for the


243 5

resultant of two binomials, see e.g. Swan [19, Lemma 3], we find that res(xn−1 − b, xq

n

−1

− 1) = 0 ⇐⇒ b(q

n

−1)/d

= 1,

where d = (n − 1, q − 1). Thus (q − 1)/d is divisible by o(b). Conversely, if o(b) | (q n − 1)/(n − 1, q n − 1), then the existence of an β ∈ S1 (α, L) such that ha (β) ∈ / S1 (α, L) is proved by following the above reasoning in the opposite direction. n

n

Remark 3.3. The demands on the zeros of g(x) is weaker in Theorem 3.2, than in Theorem 2.2, and we may make the same enervation of the conditions on g(x) in the latter theorem, without spoiling the proof. Remark 3.4. Putting a = 1 yields m = 2 in Theorem 3.2, and consequently Si(α, L) = B1 (α, L) if and only if (q n − 1)/(n − 1, q n − 1) is even. This is equivalent to the earlier result in Theorem 2.2(2). Now we assume that g(x) defines an totally ramified extension of K. Recall that any such extension can be generated by an Eisenstein polynomial in Kp [x], i.e. a polynomial xn + an−1 xn−1 + · · · + a1 x + a0 such that ordπ (ai ) ≥ ordπ (a0 ) = 1, for i = 1, 2, . . . , n − 1. The generalization of Theorem 2.3 goes like this. Theorem 3.5. Suppose g(x) is irreducible and defines a totally ramified extension L/K of degree n, and let q be the number of elements in Kp . Let α be a fixed point of the dynamical system h(x) = x + ag(x). Then α is indifferent, and Si(α, L) = B1 (α, L) if and only if o(b) | (q − 1)/(n − 1, q − 1), where b = −¯ a−1 ∈ Kp . Otherwise Si(α, L) = B 1 (α, L). The proof is similar to the proof of Theorem 2.3, which is presented in Svensson [16], but for completeness, we give the full proof below. Proof of Theorem 3.5. First we assume that |α| < 1 for each zero of g(x). Then |ha (α)| = 1, whence each fixed point of ha is indifferent. We conclude that B1 (α, L) ⊆ Si(α, L) ⊆ B 1 (α, L) in the same manner as in the previous theorem. Assuming that Si(α, L) = B1 (α, L), means that there is a ¯ a (β) = α ∈ Lp . But h ¯ a (x) = a ¯xn + x and α = 0, whence β ∈ S1 (α, L) such that h

β ∈ Lp is a zero of xn−1 − b. Hence we are in a situation, similar to the one in the proof of Theorem 3.2, and we can finish the proof (for the case when |α| < 1) in exactly the same way, by noting that for a totally ramified extension, we have |Lp | = |Kp | = q. Finally we turn to the case when |α| = 1. Let Π be a generator of the ideal PL . Then (3.3)

α = ζ(1 + γΠ),

for some γ ∈ OL , where ζ ∈ OK is a (q − 1)st root of unity (see e.g. Hasse [7]). If g(x) = xn + bn−1 xn−1 + · · · + b1 x + b0 , then ζ −1 α has the minimal polynomial m(x) = xn + ζ −1 bn−1 xn−1 + · · · + ζ −n+1 b1 x + ζ −n b0 . Let s(x) = xn + an−1 xn−1 + · · · + a1 x + a0

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PER-ANDERS SVENSSON

be the minimal polynomial of γΠ over K. Then m(x) = s(x − 1) = xn +

s(n−1) (−1) n−1 + · · · + s (−1)x + s(−1), x (n − 1)!

whence

ζ n−i s(i) (−1) i! for each i = 0, 1, . . . , n − 1. On the other hand, since |γΠ| ≤ |Π| < 1, we must have |ai | < 1 for all i, and consequently s¯(x) = xn ∈ Kp [x]. Thereby bi =

s¯(i) (−1) = which yields g¯(x) =

n! (−1)n−i , (n − i)!

n n ¯ n. ¯ n−i xi = (x − ζ) (−ζ) i i=0

Hence

¯ n−1 . ¯ (α) = 1 + n¯ a(α − ζ) h a ¯ and it follows that α is indifferent. By (3.3) we have α = ζ, We have at least B1 (α, L) ⊆ Si(α, L) ⊆ B 1 (α, L) for the same reason as before. ¯ a (β) = α, then β − α is a zero of xn−1 − b. Thus the If β ∈ OL fulfills β = α but h we can make conclusion about the maximal Siegel disk, as in the case |α| < 1. Remark. Note that if a ≡ −1 (mod π), then the case Si(α, K) = B 1 (α, K) can never occur, since we then will have b = 1 in Theorem 3.5. On the other hand, if b = −¯ a−1 is a primitive element of Kp , we must have (n − 1, q − 1) = 1 to obtain Si(α, K) = B1 (α, K). This cannot happen, for instance if n and q both are odd. 4. The case |a| < 1 In this section it is assumed that |a| < 1. Still, for any fixed point α of the dynamical system ha , we will have |α| ≤ 1 just like before, since α is a zero of a monic irreducible polynomial with integer coefficients. Lemma 4.1. Any fixed point of the dynamical system ha (x) = x + ag(x), where |a| < 1, is indifferent. Proof. Obvious.

When we in the two previous section determined the size of the basin of attraction or maximal Siegel disk to a certain fixed point, we were actually using the following result, that was proved in Khrennikov [8]. Theorem 4.2. Let h be a dynamical system on a p-adic field L and α ∈ L a fixed point of h. If r is a positive real number, such that (m) h (α) m−1 (4.1) max r < 1, m≥2 m! then B r (α, L) ⊆ A(α, L), whenever α is attracting, and B r (α, L) ⊆ Si(α, L), whenever α is indifferent. For the dynamical systems ha (x) = x + ag(x) we are investigating, one can show that (4.1) is equivalent to r < 1, if |a| = 1, see e.g. Svensson [16]. If |a| < 1, this condition on r will however depend on |a| in the following way.


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Lemma 4.3. Let K be a p-adic field and g(x) ∈ OK [x] a monic polynomial of degree n ≥ 2, and put ha (x) = x + ag(x), where a ∈ PK . Let α be an element in an extension of K, such that |α| ≤ 1. Then (4.1) holds, if and only if r < |a|−1/(n−1) .

(4.2)

Proof. (⇒) If (4.1) holds, then especially h(n) (α) n−1 a < 1. r n! (n)

But ha (α) = an!, whence |a|r n−1 < 1. (⇐) Assume that (4.2) is true. Put g(x) = xn + bn−1 xn−1 + · · · + b1 x + b0 . For 2 ≤ m ≤ n we then obtain n−m h(m) (α) a m−1 m + i i m−1 bm+i α r = a ≤ |a| · r m−1 r m! m i=0 < |a| · (|a|−1/(n−1) )m−1 = |a|(n−m)/(n−1) ≤ 1,

implying (4.1).

Studying the dynamics of ha on a sphere Sr (α, L), where α is a fixed point of ha and r = |a|−1/(n−1) can sometimes be simplified, if turns out that r ∈ / VL , the valuation group of L. Lemma 4.4. Let L/K be an extension of K of degree n ≥ 2. Suppose a ∈ PK = π. Then |a|−1/(n−1) ∈ VL if and only if n − 1 | ordπ (a). Proof. Let s = ordπ (a). Then s ≥ 1 and |a| = |π|s . Let Π be a generator of PL , the maximal ideal of L. Then |Π|e = |π|, where e is the ramification index of the extension L/K. If n − 1 | s, then (n − 1)t = s for some integer t. Thereby |a|−1/(n−1) = |a|−t/s = |π|−t = |Π|−et ∈ VL . Conversely, if |a|−1/(n−1) ∈ VL then |a|−1/(n−1) = |Π|u for some integer u. Since on the other hand |a|−1/(n−1) = |π|−s/(n−1) = |Π|−se/(n−1) , this means that (4.3)

se ≡ 0

(mod n − 1).

Let f be the residue class degree of the extension L/K. Then it is well-known that ef = n (see e.g. Cassels [6]). Using this, we obtain from (4.3) that 0 ≡ 0 · f ≡ sef ≡ sn ≡ s and thereby the lemma is established.

(mod n − 1),

If the above lemma does not apply, then it is clear that Si(α, L) = B r (α, L), where r ∈ VL is the largest member of the valuation group that does not exceed |a|−1/(n−1) . Theorem 4.5. Let s = ordπ (a) and suppose n − 1 | s. Let ζ ∈ OK be the uniquely determined (q − 1)st root of unity such that a ≡ ζ (mod π s ). Then Si(α, L) = B r (α, L), where r = |a|−1/(n−1) , unless xn−1 + ζ −1 has a zero in L, in which case Si(α, L) = B r (α, L).

246 8

PER-ANDERS SVENSSON

Proof. By Theorem 4.2 and Lemma 4.3 we have at least Br (α, L) ⊆ Si(α, L). Suppose β ∈ Sr (α, L). Since r > 1 and |α| ≤ 1 we have |β − α| = |β| = r, whence β = α + βu for some unit u ∈ OL . This implies that ha (β) − α = βu + a

n g (k) (α) k=0

k!

(βu)k .

Since |β| > 1 and g(x) is a monic polynomial with integer coefficients, n g (k) (α) (βu)k = |aβ n | = r. a k! k=0

Thus |ha (β) − α| < |β − α| if and only if |β + aβ n | < |β|, i.e. |1 + aβ n−1 | < 1. By the assumptions of the theorem, we may write a = π s ζη, where η ∈ 1 + PK . Put β = π −s/(n−1) ξ, where ξ ∈ L is a zero of xn−1 + ζ −1 (if such a zero exists). Then |1 + aβ n−1 | < 1. On the other hand, if no such zero exists, then ζξ n−1 = −1 for all (q f − 1)st roots of unit in L, where f = f (L/K) is the residue class degree of the extension L/K. Therefore |1 + aβ n−1 | = 1 for all β such that |β| = r. We give two examples in which each one of these occasions occurs. Example 4.6. Let ha (x) = 9x3 + x − 27 ∈ Q3 [x]. Then ha (x) = x + ag(x), where a = 9 and g(x) = x3 − 3. Of course g(x) is irreducible over Q3 and generates √ a totally ramified extension L of degree n = 3. A fixed point of h9 is α = 3 3. We pick an element β ∈ L such that |β − α| = |a|−1/(n−1) = 3. Then there is a unit u ∈ OL such that u β = + α. 3 This means that u 3 u h9 (β) − α = 9 +α + + α − 27 − α 3 3 u3 u 3 2 + − 27 = 9α + 9uα + 3u2 α + 3 3 = 32 uα2 + 3u2 α + 3−1 u(u2 + 1) Now |u2 + 1| = 1 for all units u. Thereby |h9 (β) − α| = |β − α| = 3, whence Si(α, L) = B 3 (α, L). Example 4.7. Let g(x) be as in the previous example, but replace a = 9 by a = 18. Then with the same reasoning h18 (β) − α = 2 · 32 uα2 + 2 · 3u2 α + 3−1 u(2u2 + 1). The polynomial 2x2 + 1 splits over Q3 , so by choosing u as one of its zeros, we obtain |h18 (β) − α| < |β − α| as a result. Thus Si(α, L) = B3 (α, L) in this case.

247 9


5. The case |a| > 1 It remains to check what happens if |a| > 1. In this case, all possible cases of the nature of a fixed point α may occur, as we can see in the following example. Example 5.1. Let g(x) = x3 + 3x + 3 ∈ Z3 [x] (which is irreducible over Q3 since it is Eisenstein) and study the dynamical system 2 1 h2/3 (x) = x + g(x) = (2x3 + 9x + 9). 3 3 If α is a zero of g(x), then |α| < 1, whence |h2/3 (α)| = |2α2 + 3| < 1, and we have an attracting fixed point. Above we have a = 2/3. If instead a = 1/3, we obtain indifferent fixed points, and if a = 1/9, the fixed points are repelling. Lemma 5.2. Let α be a fixed point of ha (x) = x + ag(x), where |a| > 1 and g(x) is defined by (5.1). Then α is not repelling, if and only if |ibi | ≤ |a−1 | for all i = 1, 2, . . . , n. Proof. Follows immediately from the fact that |ha (α)| = |a / VK if i ∈ {2, 3, . . . , n}. and |αi−1 | ∈

n i=1

ibi αi−1 + 1|

For simplicity, we will limit our investigation to the case when g(x) defines a totally ramified extension of degree n; we merely assume that g(x) is actually a monic Eisenstein polynomial (as in Example 5.1). Thus, we assign (5.1)

g(x) = bn xn + bn−1 xn−1 + · · · + b1 x + b0 ,

where bn = 1 and |bi | ≤ |b0 | = |π| for i = 1, 2, . . . , n − 1. Moreover, we will also assume that K = Qp , and thus π = p and Kp = Fp . Note that Lemma 5.2 implies especially that deg g(x) has to be a multiple of |a|, if we want the fixed points of ha to be non-repellers. We will actually assume that deg g(x) = |a| = pk for some positive integer k. In what follows, we will need to calculate the p-adic valuation of certain binomial coefficients. For this, we recall the following well-known result, see e.g. Schikhof [14, p. 70]. Lemma 5.3. For each positive integer n, let σ(n) denote the sum of its digits, when written in the base p, where p is a prime. Then ordp (n!) =

n − σ(n) . p−1

Lemma 5.4. Let p be a prime and k a positive integer. Let u be an integer such that 1 ≤ u ≤ p − 1. Then k up ordp + ordp (v) = k v for each integer v such that 1 ≤ v ≤ upk .

248 10

PER-ANDERS SVENSSON

Proof. Let = ordp (v). We then can write v=

k

ci p i ,

i=

where 0 ≤ ci ≤ p − 1 for all i and c = 0. (Actually, since v ≤ upk , we have 0 ≤ ck ≤ u.) Then upk − v has the p-nary expansion upk − v = (u − ck − 1)pk +

k−1

(p − ci − 1)pi + (p − c )p .

i=+1

By Lemma 5.3, k−1 upk − u pi , =u p−1 i=0

k 1 v− ordp (v!) = ci , p−1

ordp [(upk )!] =

i=

and ordp [(upk − v)!] =

k−1 upk − v − (u − ck − 1) + i=+1 (p − ci − 1) + (p − c )

1 = p−1 k−1

p−1

u(p − 1) − v − (k − )(p − 1) + k

1 p −k+ − =u p − 1 i=0 i

v−

k

k

ci

i=

ci

i=

= ordp [(upk )!] − ordp (v!) − k + . Putting all this together, we find that k up = ordp [(upk )!] − ordp (v!) − ordp [(upk − v)!] = k − , ordp v

as claimed. We now introduce a function f , defined on the set {2, 3, . . . , pk }, as follows: −i/pk m+i k bm+i p (5.2) f (m) = p . max m 0≤i≤pk −m

It follows from Theorem 4.2, that the closed ball of radius r, centered at a nonrepelling fixed point, is contained in the basin of attraction or maximal Siegel disk of this fixed point, whenever max f (m)r m−1 < 1. Lemma 5.5. Let f be defined by (5.2). Then f (m) > 1 ⇐⇒ ordp (m) ≥ 1. Proof. Suppose ordp (m) ≥ 1. Then m = upj for some j ≥ 1, where p | u. By choosing i = pk − upj , we conclude that k k −(pk −upj )/pk j k p k−1+upj−k p = pj−1+upj−k , f (up ) ≥ p k p = p b p upj upj

249 11


where the last equality follows from Lemma 5.4. Hence f (m) > 1. On the other hand, suppose f (m) > 1. Then m+i k bm+i p−i/p > p−k (5.3) m for at least one i ∈ {0, 1, . . . , pk −m}. If ordp (m+i) = 0 for this i, then |bm+i | ≤ p−k which contradicts (5.3). Thus we assume that ordp (m + i) = s for some integer s ≥ 1. Applying Lemma 5.4 now yields m+i k k −k bm+i p−i/p = |bm+i | · pordp (m)−s−i/p . p ip−k . Thus ordp (m) ≥ 1, and the proof is finished. We will now investigate the size of the basin of attraction or maximal Siegel disk of a non-repelling fixed point, in the case when |a| = pk for some small values of the positive integer k. Only the cases k = 1 and k = 2 will be considered. The case k = 1 is quite straightforward: Theorem 5.6. Suppose ha (x) = x + ag(x) ∈ Qp [x] has non-repelling fixed points, where g(x) has degree p, and |a| = p. Let α be such a fixed point and let M denote the maximal Siegel disk (in case α is indifferent) or the basin of attraction (in case α is attracting) that is contained in the extension L = Qp (α). Then B r (α, L) ⊆ M , if r < p−1/(p−1) . Proof. We only need to prove that

h(m) (α) m−1 a max < 1 ⇐⇒ r < p−1/(p−1) . r m≥2 m! Due to Lemma 5.5, the coefficients bi of g(x) can be assumed to be zero, whenever p | i (and i = 0). Thus g(x) = xp + b0 , and thereby (m) ha (α) p =a αp−m , m! m if m ≥ 2. Thereby h(m) (α) p −(p−m)/p m−1 a m−1 p < 1 ⇐⇒ p r 0 and (X − 1)(0) = 1. Let q ∈ K be not a root of unity such that |q − 1| < 1. The q-binomial number (q m − 1)(n) m = n , (m, n) ∈ N×N has a continuous extension in such a way that, n q (q − 1)(n) (q x − 1)(n) x x for x ∈ Zp , one has = n . It will be convenient to set C = . n,q n q n q (q − 1)(n) the sequence Let q ∈ K be a pN -th primitive root of unity. Let us consider (qα )α≥0 , where qα = q + pα . One has lim qα = q and Cn,qα1 − Cn,qα1 ≤ α→+∞

2000 Mathematics Subject Classification. 47B37, 47B38, 47B39, 47B48, 81R15, 81R50. Key words and phrases. q-deformation, Weyl algebra, orthonormal bases, quantum plane, pN -th primitive root of unity. c Mathematical c 0000 (copyright Society holder) 2010 American

1 253

254 2

FANA TANGARA

|qα1 − qα2 | < 1, ∀α1 ≥ 0, α2 ≥ 0. Then the limit lim Cn,qα (x) exists and one has α→+∞ x1 Cn0 ,q (x0 ), for lim Cn,qα (x) = Cn,q (x) = α→+∞ n1 x = x1 pN + x0 ∈ Zp , 0 ≤ x0 ≤ pN − 1 and n = n0 + n1 pN ∈ N, 0 ≤ n0 ≤ pN − 1. It is now well known (cf. [1] K. Conrad) that the sequence (Cn,q )n≥0 is an orthonormal basis of C(Zp , K) : this means that any element f ∈ C(Zp , K) can be written as a convergent sum f = an Cn,q , an ∈ K, lim |an | = 0 and f = sup |an |. n≥0

n→+∞

n≥0

Let L C(Zp , K) be the Banach algebra of the continuous linear endomorphisms R(f ) for every of C(Zp , K). The norm on L C(Zp , K) is given by R = sup f f =0 R ∈ L C(Zp , K) . Let Zq be the continuous linear operator defined by setting, for f ∈ C(Zp , K) and x ∈ Zp , Zq(f )(x) = q x f (x). We denote by Γq (Zp , K) the closed subalgebra of L C(Zp , K) generated by Zq . Assuming q to be a primitive pN -root of unity, one can prove that the elements of Γq (Zp , K) are the operators determined by multiplication with a pN Zp -invariant continuous function of Zp into K. Let K < X, Y >, be the free algebra generated by X and Y and let us denote by Jq the two-sided ideal of K < X, Y > generated by Y X − qXY − 1. The algebra Aq [X, Y ] = K < X, Y > /Jq is a q-deformation of the Weyl algebra. The quantum plane algebra or the q-deformation of the affine plane algebra is defined as being the quotient algebra Kq [X, Y ] = K < X, Y > /{Y X − qXY }. It is well known that Aq [X, Y ] (resp. Kq [X, Y ]) is isomorphic to the Ore extension K[X][Y, τq , δq ] (resp. K[X][Y, τq , 0]), where τq (h)(X) = h(qX) and δq is the Jackson q-derivation h(qX) − h(X) i j . Hence the sequence (X Y )(i,j)∈N×N is a defined by δq (h)(X) = qX − X linear basis of Aq [X, Y ] (resp. Kq [X, Y ]), where X is the class of X and Y the class of Y (see for instance [9]). The algebra Aq [X, Y ] is also called in the literature the quantum Weyl algebra (cf. [6],[7],[10]) or the q-deformed Heisenberg algebra (cf. [8]). In [8], for a general ground field K and a general element q of K, L. Hellström et al give a necessary and sufficient condition for the q-deformed Heisenberg algebra to be simple and show that this algebra is non-simple and non-Artinian for q = 1 and q = 0. The aim of this paper is not to study the structure of two-sided ideals of this algebra but to exhibit an orthogonal basis for it. Let τ1 be the operator of translation defined by setting τ1 (f )(x) = f (x + 1) and ∇q f (x + 1) − f (x) the analogue of the Jackson q-derivation defined by ∇q (f )(x) = , (q − 1)q x for f ∈ C(Zq , K). It is readily seen that τ1 ◦ Zq = qZq ◦ τ1 , ∇q ◦ τ1 = qτ1 ◦ ∇q and ∇q ◦ Zq − qZq ◦ ∇q = id. Let Aq be the algebra of continuous linear operators generated by Zq and ∇q . In a recent work, we have studied this algebra and shown that it is a concrete realization of the quantum Weyl algebra as an algebra of p-adic continuous linear operators, whenever q is not a root of unity. We have also shown that this algebra is isomorphic to Aq [X, Y ]. Indeed, considering the map ϕ which sends X onto Zq and Y the class of Y onto ∇q , one defines an algebra homomorphism of Aq [X, Y ] onto Aq

255 3

A q-DEFORMATION OF THE WEYL ALGEBRA i

j

which is bijective, because (X Y )i≥0,j≥0 (resp. (Zqi ∇jq )i≥0,j≥0 ) is a linear basis of Kq [X, Y ] (resp. Aq ). In the same condition we have shown that the algebras of continuous linear operators Pq1 generated by τ1 and Zq is isomorphic to the quantum plane algebra and the closure of Aq is equal to that of Pq1 , but Pq1 is not equal to Aq . Here we study these algebras, when q = 1 is a pN -th primitive root of unity. In the next section, we give some rules of commutation of operators. If for the case q non root of unity, we need the completion of the two algebras Pq1 and Aq to have the equality, here we have directly equality between the two algebras. We show that Aq = Pq1 is no longer isomorphic to the q-deformation of the Weyl algebra Aq [X, Y ], but it is isomorphic to the quotient algebra Aq [X, Y ]/{X

pN

− 1} which

pN

is also isomorphic to the algebra Kq [X, Y ]/{X − 1}. On the other hand, the algebra of continuous linear operators Pq2 generated by τ1 and ∇q is known to be an other model of the quantum plane algebra, whenever q is not a root of unity (see for instance [11]). Here we show that this algebra is not isomorphic to Kq [X, Y ], when q = 1 is a pN -th primitive root of unity. It is a subalgebra of Aq isomorphic to Kq [X, Y ]/{Y

pN

− (q − 1)−p q N

−pN (pN −1) 2

(X

pN

− 1)}.

In the third section, after some calculus of norm of operators, interesting or q the closure of Aq and P q2 the closure of Pq2 , when thonormal family are found for A N q = 1 is a p -th primitive root of unity. We show that the algebras Aq and Pq2 have the same center Z(Aq ) = K.id. Moreover, the closure of this center is characterized. In the sequel, except specific mention, q = 1 will be a pN -th primitive root of unity.

2. The q-deformation of the Weyl algebra

The algebra Aq generated by Zq and ∇q is a concrete realization of the qdeformation of the Weyl algebra generated by two variables, whenever q ∈ K is not a root of unity. In this section we study this algebra when q = 1 is a pN -th primitive root of unity.

2.1. The algebra Γq (Zp , K). We study here the closed subalgebra Γq (Zp , K) of L C(Zp , K) generated by Zq , when q is a pN -th primitive root of unity. The operators Zqs and Qs (Zq ) are respectively defined, for 1 ≤ s ≤ pN − 1 by (Zq − id)(s) and Zq0 = Q0 (Zq ) = id. Zqs = Zq ◦ Zqs−1 = Zqs , Qs (Zq ) = (q s − 1)(s) One will often omit the sign ◦ to note the law of composition in the algebras of

256 4

FANA TANGARA

linear operators. When q is not a root of unity, there is a bijective correspondence between the sequence of monomials (X n )n≥0 and the sequence (Zqn )n≥0 . One can prove that the algebra Γq (Zp , K) is isomorphic to C(Zp , K), the Banach algebra of continuous function from Zp into K. But when q is a pN -th primitive root of unity, one has N Zqkp +r = Zqr , ∀k ≥ 0, 0 ≤ r ≤ pN − 1, then the sequence (Zqn )n≥0 is the finite sequence (Zqn )0≤n≤pN −1 . Furthermore it is a linear basis on Γq (Zp , K). Hence the algebra Γq (Zp , K) is here a finite dimensional vector space of dimension pN . Let us consider C(Zp , K)p Zp the space of pN Zp -invariant continuous functions: N that is, f ∈ C(Zp , K)p Zp if and only if f (x + ypN ) = f (x), ∀x, y ∈ Zp . Let IpN = [0, pN − 1], it is readily seen that the finite sequence of functions (χa+pN Zp )a∈IpN is N

an orthonormal basis of C(Zp , K)p Zp , where the function χa+pN Zp is the characN teristic function of a + pN Zp . Then the space C(Zp , K)p Zp is of finite dimension N p , and it is a subalgebra of C(Zp , K). On the other hand, one sees that the functions Cs,q , s ∈ IpN are pN Zp -invariant continuous functions, when q is a pN -th primitive root of unity. Then every Cs,q N p −1 can be written on the forme Cs,q = ba (s)χa+pN Zp , with ba (s) = Cs,q (a). Since N

a=s

the family (Cn,q )n≥0 is an orthonormal basis of C(Zp , K), one sees that the finite sequence (Cs,q )s∈IpN is an orthonormal family in C(Zp , K). Thus it is an orthonorN

mal basis of C(Zp , K)p

Zp

, whenever q is a pN -th primitive root of unity.

Proposition 2.1. Let q = 1 be a pN -th primitive root of unity in K. The N algebras C(Zp , K)p Zp and Γq (Zp , K) are isometrically isomorphic. Moreover, the sequence (Qs (Zq ))s∈IpN is an orthonormal basis of Γq (Zp , K). Proof. Considering the map ϕ which sends the continuous function q sx onto the continuous linear operator Zqs , ∀s ∈ IpN , one defines an algebra homomorphism N of C(Zp , K)p Zp onto Γq (Zp , K). Moreover, one has ϕ(Cs,q ) = Qs (Zq ), ∀s ∈ IpN . Let us consider the function h =

N p −1

sx

as q , one has ϕ(h) =

s=0

N p −1

as Zqs . Since

s=0

the sequence (Zqs )s∈IpN is a linear basis of Γq (Zp , K), then ϕ(h) = 0 if and only if h = 0. Hence ϕ is injective. N p −1 as Zqs be an element of Γq (Zp , K), one has On the other hand, let T = s=0 pN −1

T = ϕ(

as q sx ). Then ϕ is surjective and one concludes that ϕ is a bijec-

s=0

tive homomorphism. Let us consider now the continuous function h =

N p −1

s=0

as Cs,q , one has h =

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A q-DEFORMATION OF THE WEYL ALGEBRA

max |as | and ϕ(h) =

s∈IpN

N p −1

as Qs (Zq ). For any continuous function f , one has

s=0

ϕ(h)(f ) =

N p −1

as Cs,q · f . Hence ϕ(h) ≤ h = max |as |. On the other s∈IpN

s=0

hand, applying ϕ(h) to the continuous function C0,q , one obtains ϕ(h)(C0,q ) = N p −1 as Cs,q = h. Then h = ϕ(h)(C0,q ) ≤ ϕ(h). It follows that h = ϕ(h) s=0 N

and the algebra C(Zp , K)p

Zp

is isometrically isomorphic to the algebras Γq (Zp , K).

Since the sequence (Cs,q )s∈IpN is an orthonormal basis of C(Zp , K)p Zp , one deduces that the sequence ϕ(Cs,q ) = Qs (Zq ) is an orthonormal basis of N

s∈IpN

Γq (Zp , K). In this case, the elements of the algebra Γq (Zp , K) are the operators determined by multiplication with the pN Zp -invariant continuous functions of Zp into K. 2.2. The algebra Aq generated by Zq and ∇q . Let Aq [X, Y ] = K < X, Y > /Jq , be the q-deformation of the Weyl algebra generated by two variables, where Jq is the two-side ideal of K < X, Y > generated by Y X − qXY − 1. In this section we study the algebra Aq generated by Zq and ∇q . When q is not a root of unity, this algebra is isomorphic to Aq [X, Y ]. We show here that Aq is not isomorphic to Aq [X, Y ], when q = 1 is a pN -th primitive root of unity. In pN

Proposition 2.4, we show that Aq is isomorphic to Aq [X, Y ]/{X − 1}. On the other hand, the algebra Pq1 , generated by Zq and τ1 , is well known to be a model of quantum plane algebra, whenever q is not a root of unity (cf. [9]). In a recent work, we have shown that Pq1 was different from Aq , when q is not a root of unity. But the closure of Pq1 is equal to the one of Aq . In contrast, here we show that the algebra Pq1 is equal to Aq , when q = 1 is a pN -th primitive root of unity. (n)

The operators τ1n , ∆q

and ∇nq are defined, for n ≥ 1 by τ1n = τ1 ◦ τ1n−1 = τn ,

(n)

(0)

, (τ10 = ∆q ∆q = (τ1 − id)(n) and ∇nq = ∇q ◦ ∇n−1 q following rules of commutation.

= ∇0q = id). We have the

Proposition 2.2. Let q = 1 be a pN -th primitive root of unity. (i) One has τ1 Zq = qZq τ1 and more generally, one has τ1i Zqj = q ij Zqj τ1i , ∀i ≥ 0, 0 ≤ j ≤ pN − 1. (ii) Let i and n be two integers such that 0 ≤ i ≤ pN − 1 and n ≥ 0, one has i (n) αi,n (s)∆q(n−s) , where αi,n (0) = q in and ∆q Zqi = Zqi s=0

αi,n (s) = q i(n−s) Cs,q (i)

s−1

(q n − q ), ∀1 ≤ s ≤ i.

=0

258 6

FANA TANGARA

(iii) One has also ∇nq Zqi =

i

βi,n (s)Zqi−s ∇n−s , ∀n ≥ 0, 0 ≤ i ≤ pN −1, where q

s=0

βi,n (s) = q (i−s)(n−s) Cs,q (n)

s−1

=0

q i− − 1 , ∀0 < s ≤ i and βi,n (0) = q in . q−1

Proof. (i) Let f ∈ C(Zp , K), one has τ1 Zq (f )(x) = q x+1 f (x + 1) and Zq τ1 (f )(x) = q x f (x + 1). Then τ1 Zq = qZq τ1 and the property follows. (n) (n) (n−1) (ii) One deduces from (i) that ∆q Zq = Zq (q n ∆q + q (n−1) (q n − 1)∆q ). i (n) Suppose by induction hypothesis that ∆q Zqi = Zqi αi,n (s)∆q(n−s) for s=0

0 ≤ i < pN − 1. One obtains (n) ∆q Zqi+1 i+1 n(i+1) (n) n−s n−s+1 (n−s) i+1 = Zq ∆q + q −1)αi,n (s−1) ∆q q αi,n (s)+(q = Zqi+1

i+1

s=1

αi+1,n (s)∆q(n−s) .

s=0

(iii) Let us remind that ∇nq = (q − 1)−n q

−n(n−1) 2

Zq−n ∆q , ∀n ≥ 0 (c.f [15]). (n)

−n(n−1)

Then, one deduces from (ii) that ∇nq Zqi = (q − 1)−n q 2 Zq−n ∆q Zqi = i βi,n (s)Zqi−s ∇n−s , ∀n ≥ 0, 0 ≤ i ≤ pN − 1, where βi,n (0) = q in and q (n)

s=0

βi,n (s) = (q − 1)−s q

s(s−2n+1) 2

αi,n (s), ∀0 ≤ s ≤ i.

Remark 2.3. Since N

N

s−1

=0 (kpN ) i Zq

τ1kp Zqi = Zqi τ1kp , ∆q

N

(q kp − q l ) = Cs,q (kpN ) = 0, ∀1 ≤ s ≤ i, one has (kpN )

= Zqi ∆q

N

N

and ∇kp Zqi = Zqi ∇kp , ∀k ≥ 0, i ∈ IpN . q q

N Proposition 2.4. Let q = 1 be a p -th primitive root of unity. The subalgebra Aq of L C(Zp , K) is a vector space which admits the sequence of operators (Zqi ∇jq )(i,j)∈IpN ×N as a linear basis. Moreover, Aq is isomorphic to the algebra

Aq [X, Y ]/{X

pN

− 1}. N

Proof. Since Zqp = id, one sees that the sequence (Zqi ∇jq )(i,j)∈IpN ×N is a system of generators of Pq1 . Let us prove that the sequence (Zqi ∇jq )(i,j)∈IpN ×N is free. We remind here that ∇jq (Ck,q )(x) = (q − 1)−j q

j(j+1) −jk 2

Ck−j,q (x), for j ≤ k

and ∇jq (Ck,q )(x) = 0 when j > k (see e.g. [11] or [12]). Let Rm =

N p −1 m

i=0 j=0

bi,j Zqi ∇jq

259 7


be an element of Aq . Assume Rm = 0, then Rm (C0,q )(x) = N

sees easily (cf. [12] Property 1) that (∇pq

−1

N p −1

bi,0 q ix = 0. One

i=0 N p −1

◦ Rm )(C0,q )(x) =

=1

Since

N p −1

=1

q − 1 b N . q − 1 p −1,0

q −1 = 0, one has bpN −1,0 = 0; and step by step, one obtains bi,0 = 0, q−1

∀0 ≤ i ≤ pN − 1. By the same way, applying Rm to C1,q , one obtains, bi,1 = 0, for every integer i, 0 ≤ i ≤ pN − 1. Hence, applying successively Rm to the Cj,q , 0 ≤ j ≤ m, one sees that, for any integers i and j, such that 0 ≤ i ≤ pN − 1 and 0 ≤ j ≤ m, one has bi,j = 0. One concludes that the family (Zqi ∇jq )(i,j)∈IpN ×N is free. Let Aq [X, Y ] = K < X, Y > /{Y X − qXY − 1}, be the q-deformation of the Weyl algebra generated by two variables. It is well known that Aq [X, Y ] is isomorphic to the Ore extension K[X][Y, τq , δq ], where τq and δq are the linear operators defined f (qX) − f (X) (cf. [9]). respectively by setting τq (f )(X) = f (qX) and δq (f )(X) = qX − X i j Then the sequence (X Y )(i,j)∈N×N is a linear basis of Aq [X, Y ]. Considering the map ϕ which sends X onto Zq and Y onto ∇q , one defines an algei j bra homomorphism of Aq [X, Y ] onto Aq which is surjective, because (X Y )(i,j)∈N×N is well known to be a linear basis of Aq [X, Y ]. One sees easily that {X two side ideal generated by X

pN

pN

pN

− 1} the

− 1 is a subset of ker ϕ. Let us prove that ker ϕ is

a subset of {X − 1}. N mp n +r i j ai,j X Y be an element of ker ϕ, with 0 ≤ r ≤ pN − 1, n ≥ 0, Let T = i=0

j=0 −1 n p m

akpN +i,j Zqi ∇jq = 0, with ampN +i,j = 0,

N

m ≥ 0. One has ϕ(T ) =

j=0 i=0

k=0

∀r < i ≤ pN − 1. Since the sequence (Zqi ∇jq )0≤i≤pN −1,j≥0 is a linear basis of Aq , m−1 akpN +i,j , ∀0 ≤ i ≤ pN − 1, 0 ≤ j ≤ n. Hence one obtains ampN +i,j = − k=0

T =

N p −1 m−1 n

akpN +i,j X

kpN +i

(1 − X

(m−1−k)pN

j

)Y , for 0 ≤ r ≤ pN − 1. Then

i=0 k=0 j=0

T is an element of {X generated by X

pN

pN

− 1}. One concludes that ker ϕ is the two side ideal

− 1 and Aq [X, Y ]/{X

pN

− 1} is isomorphic to Aq .

Remark 2.5. Let n and m be two positive integers. On deduces from Proposition 2.2 that for every operator R ∈ Aq , there exists R ∈ Aq such that R(Zqn ∇m q ) = (Zqn ∇m q )R . Proposition 2.6. The algebra Aq is non-simple and non-Artinian.

260 8

FANA TANGARA

Proof. One deduces from Remark 2.5 that the sets Jn = {R ◦ ∇nq , R ∈ Aq }, n ∈ N∗ are proper two-sided ideals. Then Aq is non-simple. Moreover, one has the infinite descending chain of ideals Aq = J0 ⊃ J1 ⊃ · · · ⊃ Jn ⊃ Jn+1 ⊃ . . . Hence Aq is non-Artinian. Theorem 2.7. Let q = 1 be a pN -th primitive root of unity. The algebra Aq is equal to the subalgebra Pq1 of L C(Zp , K) generated by Zq and τ1 . Proof. It is readily seen that ∇q = (q − 1)−1 Zqp −1 (τ1 − id) is an element of Pq1 . Then Aq is a subalgebra of Pq1 . In the other hand, one can prove that τ1 = ∇q Zq − Zq ∇q belongs to Aq . Hence the algebra Pq1 generated by Zq and τ1 is a subalgebra of Aq . One concludes that Aq = Pq1 . N

This equality gives us the possibility to exhibit, in the next section, two different orthonormal bases for the closure of Aq . By definition, one sees that the sequence (Zqi τ1j )(i,j)∈IpN ×N (resp. (Zqi ∇jq )(i,j)∈IpN ×N ) is a linear basis of Aq = Pq1 . Let us notice by Mf the operator of multiplication by the continuous function N f , one sees that the operator id + Zq + · · · + Zqp −1 = MpN χpN Z , where χpN Zp is p

N

the characteristic function of pN Zp . Indeed, since q p x − 1 = (q x − 1)(1 + q x + N N · · · + q (p −1)x ) = 0, ∀x ∈ Zp , one has 1 + q x + · · · + q (p −1)x = 0, ∀x ∈ / pN Zp and x (pN −1)x N N 1 + q + ···+ q = p , ∀x ∈ p Zp .

Proposition 2.8. Let W1 = {R ∈ Aq / (Zq − id)R = 0}, (resp. W2 = {R ∈ Aq / R(Zq − id) = 0}) be the set of right ( resp. left) divisors of zero of Zq − id in Aq . Then W1 (resp. W2 ) is equal to the right ideal (resp. the left ideal) of Aq generated by MpN χpN Z . p

Proof. Since (Zq − id)(id + Zq + · · · + Zqp −1 ) = (id + Zq + · · · + Zqp −1 )(Zq − N id) = Zqp −id = 0, then one sees that the right ideal (resp. the left ideal) generated N by id + Zq + · · · + Zqp −1 is a subset of W1 (resp W2 ). N

−1 m p

N

N

Furthermore, if R =

bi,j Zqi τ1j is such that (Zq − id)R = 0, then one has

j=0 i=0

bi−1,j = bi,j , ∀1 ≤ i ≤ pN − 1, 0 ≤ j ≤ m and bpN −1,j = b0,j , ∀0 ≤ j ≤ m. One m N b0,j τ1j . It follows that W1 is equal to the obtains R = (id + Zq + · · · + Zqp −1 ) j=0 N

right ideal generated by id + Zq + · · · + Zqp −1 m p

−1

.

N

In the same way, if R =

bi,j Zqi τ1j ∈ W2 , then q j bi−1,j = bi,j , ∀1 ≤ i ≤

j=0 i=0

pN − 1, 0 ≤ j ≤ m and q j bpN −1,j = b0,j , ∀0 ≤ j ≤ m. One obtains bi,j = q ij b0,j , ∀1 ≤ i ≤ pN − 1, 0 ≤ j ≤ m.

261 9

A q-DEFORMATION OF THE WEYL ALGEBRA m

It follows that R =

b0,j

j=0 −1 m p N

m

bi,j Zqi τ1j =

j=0 i=0

N p −1

q ij Zqi τ1j . One deduces from Proposition 2.2 (i) that

i=0 N

b0,j τ1j (id + Zq + · · · + Zqp

−1

).

j=0

Remark 2.9. The product of operators

N p −1

N

(Zq − q j id) = Zqp

− id = 0.

j=0

Let q = 1 be a pN -th primitive root of unity. Let us consider now Pq2 the subal gebra of L C(Zp , K) , generated by τ1 and ∇q . Since ∇q = (q − 1)−1 Zq−1 (τ1 − id) = N (q − 1)−1 Zqp −1 (τ1 − id), then Pq2 is a subalgebra of the algebra Pq1 = Aq . −n(n−1)

One deduces from [15] Lemme 1 (iii) that ∇nq = (q − 1)−n q 2 Zq−n ∆q , N ∀n ≥ 0. Since q is a pN -th primitive root of unity, N = 0, one has ∇pq = (q − −pN (pN −1)

(pN )

(pN )

2 ∆q = (q − 1)−p ∆q 1)−p q On the other hand, one has N

N

and X n =

k

k=0

(2N )

for p = 2 and ∇2q = −(q − 1)−2 ∆q N

(X − 1)(n) = (X − 1) . . . (X − q n−1 ) =

n

(−1)n−s q

N

(n−s)(n−s−1) 2

s=0

n n

(n)

.

n Xs s q

(X − 1)(k) , ∀n ≥ 0. Hence

q

= (τ1 − id)

∆q(n)

(n)

=

n

(−1)

n−s

q

(n−s)(n−s−1) 2

s=0

n τs s q 1

(n−s)(n−s−1) n 2 n (Zq − id)(n) n−s q Qn (Zq ) = = (−1) Zqs , n − 1)(n) s (q n − 1)(n) (q q s=0 n n n (pN ) = ∆(k) , and Zqn = (q n − 1)(k) Qk (Zq ). In particular ∆q = k q q

τ1n

k=0

k=0

N

(τ1 − id)(p

)

N

= τ1p

− id. It follows that the sequences (τ1i ∇jq )(i,j)∈N×IpN and

(i)

(∆q ∇jq )(i,j)∈N×IpN are linear bases of Pq2 . While Pq2 is isomorphic to the quantum plane algebra Kq [X, Y ] = K < X, Y > /{Y X − qXY }, when q is not a root of unity, here it is isomorphic to the algebra Kq [X, Y ]/{Y

pN

− (q − 1)−p q N

−pN (pN −1) 2

(X

pN

− 1 )}.

Proposition 2.10. The sequence Qi (Zq )∇i+j q

(i,j)∈IpN ×N

Pq2 . (n)

Proof. Since ∆q one obtains

= (q − 1)n q

n(n−1) 2

is a linear basis of

Zqn ∇nq , ∀n ≥ 0 (cf. [15] Lemme 1 (iii)),

262 10

FANA TANGARA

(i)

∆q ∇jq = (q − 1)i q

i(i−1) 2

Zqi ∇i+j = (q − 1)i q q

i(i−1) 2

i

(q i − 1)(k) Qk (Zq )∇q(i+j) and

k=0

(i−1)(i−2s) 2 i−s q −s i = (−1) (q − 1) ∆sq ∇i+j−s . It follows that the Qi (Zq )∇i+j q q i − 1)(i) s (q q s=0 sequence Qi (Zq )∇i+j is a linear basis of Pq2 . q i

(i,j)∈IpN ×N

3. The completion of the algebra Aq = Pq1

In this section, we study the completion with respect to the operator norm of the algebra Aq . In fact, this completion can be identified with a closed subalgebra q . We do some calculus of norms of operators of L C(Zp , K) . We denote it by A and exhibit two orthonormal bases for this algebra. We exhibit also an orthonormal basis for the closure of the center of Aq . Proposition 3.1. For 0 ≤ i ≤ pN − 1, j ≥ 1, one has j

∆q(j) Qi (Zq ) =

βi,j (k, s)Qi−k (Zq )∆(s) q ,

k=0 s=0

where , = min(i, j) and βi,j (k, s) =

j

(−1)j−t q

(j−t)(j−t+1) 2

q (t−k)(i−k) Ct,q (j)Ck,q (t)Cs,q (t).

t=s

(Zq − id) . . . (Zq − q i−1 id) , one sees that τ1 Qi (Zq ) = (q i − 1) . . . (q i − q i−1 ) (q i Qi (Zq ) + Qi−1 (Zq ))τ1 . By induction on t, one obtains Proof. Since Qi (Zq ) =

τ1t Qi (Zq ) =

η

q (t−k)(i−k) Ck,q (t)Qi−k (Zq )τ1t ,

k=0

where η = min(i, t). Let us prove now that

(j) ∆q Qi (Zq )

=

j

βi,j (k, s)Qi−k (Zq )∆sq , with = min(i, j)

k=0 s=0

and βi,j (k, s) =

j

(−1)j−t q

(j−t)(j−t+1) 2

q (t−k)(i−k) Ct,q (j)Ck,q (t)Cs,q (t).

t=s j

(j)

Let us remind that ∆q =

(−1)j−t q

(j−t)(j−t−1) 2

Ct,q (j)τ1t and τ1t =

t=0

t

Cs,q (t)∆(s) q .

s=0

It follows that ∆q(j) Qi (Zq )

=

j

(−1)j−t q

(j−t)(j−t−1) 2

Ct,q (j)τ1t Qi (Zq )

t=0

=

j η

(−1)j−t q

t=0 k=0

(j−t)(j−t−1) 2

q (t−k)(i−k) Ct,q (j)Ck,q (t)Qi−k (Zq )τ1t .

263 11


One obtains ∆q(j) Qi (Zq ) =

j (j−t)(j−t−1) 2 (−1)j−t q q (t−k)(i−k) Ct,q (j)Ck,q (t)Qi−k (Zq )τ1t . k=0 t=k

It follows that ∆q(j) Qi (Zq ) j j

=

(−1)j−t q

(j−t)(j−t−1) 2

q (t−k)(i−k) Ct,q (j)Ck,q (t)Cs,q (t)Qi−k (Zq )∆(s) q .

k=0 s=0 t=s

Lemma 3.2. Let R be an element of Γq (Zp , K), one has and

lim

m→+∞

∇m q

◦ R(f ) = 0.

lim

m→+∞

∆q(m) ◦R(f )

=0

Proof. Let f be a continuous function from Zp into K. Then R(f ) is also a continuous function from Zp into K. However, continu it is well known that for any m (m) ous function g of Zp into K, one has lim ∆q (g) = 0 and lim ∇q (g) = m→+∞ m→+∞ 0 (cf. for instance [1], [11], [12] or [13]), hence lim ∆q(m) (R(f )) = 0 = m→+∞ m lim ∇q (R(f )). m→+∞

There is a statement better than the Proposition 2.4. Theorem 3.3. The family of continuous linear operators (q − 1)j Qi (Zq )∇jq (i,j)∈IpN ×N

is an orthonormal family in L C(Zp , K) . Moreover, it is an orthonormal basis q can be written as q , the closure of Aq in L C(Zp , K) ; that is, any R ∈ A of A −1 p a convergent sum R = (q − 1)j bi,j Qi (Zq )∇jq , with bi,j ∈ K and R = N

sup (i,j)∈IpN ×N

|bi,j |.

j∈N i=0

j(j+1)

−jn 2 Proof. One has (q − 1)j Qi (Zq )∇jq (Cn,q Ci,qCn−j,q , ∀n ≥ j and ) = q j j j (q − 1) Qi (Zq )∇q (Cn,q ) = 0, ∀n < j. Hence (q − 1) Qi (Zq )∇jq ≤ 1. On the other j(j−1)

j(j−1)

hand (q − 1)j Qi (Zq )∇jq (Cj,q )(i) = q − 2 Ci,q (i) = q − 2 . Then 1 = (q − 1)j Qi (Zq )∇jq (Cj,q )(i) ≤ (q − 1)j Qi (Zq )∇jq . Thus one obtains (q − 1)j Qi (Zq )∇jq = 1. Let us show now that the sequence (q −1)j (Qi (Zq )∇jq )(i,j)∈IpN ×N is an orthonormal family in L C(Zp , K) . Let Rm =

pN −1 m

bi,j (q − 1)j bi,j Qi (Zq )∇jq be an element of Aq . One has Rm ≤

i=0 j=0

max

(i,j)∈IpN ×[0,m]

|bi,j |. Applying Rm to C0,q , one obtains Rm (C0,q ) =

N p −1

i=0

bi,0 Ci,q .

264 12

FANA TANGARA

Since the sequence (Ci,q )i≥0 is an orthonormal basis of C(Zp , K), then ≤ Rm . By the same way, applying Rm −

N p −1

max

0≤i≤pN −1

|bi,0 |

bi,0 Qi (Zq ) to C1,q one has

i=0

max

0≤i≤pN −1

N p −1 j−1

|bi,1 | ≤ Rm . Applying, recursively Rm −

bi,k (q−1)k bi,j Qi (Zq )∇kq

i=0 k=0

to Cj,q , one obtains max |bi,j | ≤ Rm , ∀0 ≤ j ≤ m. Then one sees that the 0≤i≤pN −1 j sequence (q − 1) Qi (Zq )∇jq is an orthonormal family in L C(Zp , K) . (i,j)∈IpN ×N Let Tq be the set of the elements R of L C(Zp , K) that can be written as convergent sum R = bi,j (q − 1)j Qi (Zq )∇q(j) .One sees that the sequence (i,j)∈I

×N

pN j j is an orthonormal basis of Tq and Tq is a closed (q − 1) Qi (Zq )∇q (i,j)∈IpN ×N subspace of L C(Zp , K) , then it is complete. Since the sequence (Zqi ∇jq )(i,j)∈IpN ×N is a linear basis of Aq (see Proposition 2.4) and i−1 i

1 i i−s (i−s)(i−s−1) 2 (−1) q Z s, Qi (Zq ) = q i − q s=0 s q q =0 is also a linear basis of Aq . Hence then the sequence (q − 1)j Qi (Zq )∇jq (i,j)∈IpN ×N q , the closure of Aq , is equal to Tq and the sequence (q−1)j Qi (Zq )∇j A q

(i,j)∈IpN ×N

q . is an orthonormal basis of A

q , closure of Aq in L C(Zp , K) can be identified Remark 3.4. The algebra A with the completion of Aq , . q with Tq . Proof. This follows from the identification of A is an orthonormal basis Corollary. The family (q − 1)i+j Qi (Zq )∇i+j q (i,j)∈IpN ×N

q2 the closure of Pq2 in the algebra L C(Zp , K) . Furthermore, this closure is of P the completion of the normed algebra Pq2 . is an orthonormal (q − 1)j Qi (Zq )∇jq (i,j)∈IpN ×N q , then its subfamily (q−1)i+j Qi (Zq )∇i+j basis of A is an orthonormal q Proof. Since the family

(i,j)∈IpN ×N

q . One the other hand, one deduces from Proposition 2.10 that the family family in A

is a linear basis of Pq2 . It follows that the closure (q −1)i+j Qi (Zq )∇i+j q (i,j)∈IpN ×N

q2 of Pq2 in L C(Zp , K) is equal to the set of the elements R of L C(Zp , K) that P bi,j (q − 1)i+j Qi (Zq )∇i+j can be written as convergent sum R = q . Hence (i,j)∈IpN ×N

265 13


q2 . It follows that is an orthonormal basis of P (q − 1)i+j Qi (Zq )∇i+j q (i,j)∈IpN ×N

q2 is a completion of Pq2 , . P q . We give here an other orthonormal basis of A (j)

Theorem 3.5. The family of operators (Qi (Zq )∆q )(i,j)∈IpN ×N is an orthonor q , the completion of the mal family in L C(Zp , K) . It is an orthonormal basis of A normed algebra Aq . (j) Proof. It is readily seen that Qi (Zq )∆q ≤ 1.

(j) On the other hand, one deduces from [13] Property that 1 = Qi (Zq )∆q (Cj,q )(i) ≤ (j) Qi (Zq )∆q . (j)

Let us show now that (Qi (Zq )∆q )(i,j)∈IpN ×N is an orthonormal family. Let us consider the operator Rm =

N p −1 m

bi,j Qi (Zq )∆q(j) . One has Rm ≤

i=0 j=0

max |bi,j |. Let us prove that

max

0≤i≤pN −1 0≤j≤m

p −1

max

max |bi,j | ≤ Rm .

0≤i≤pN −1 0≤j≤m

N

One has Rm (C0,q ) =

bi,0 Ci,q . Since (Cn,q )n≥0 is an orthonormal basis of

i=0

C(Zp , K), one has Rm (C0,q ) =

In the same way, one shows that obtains

max

0≤i≤pN −1

max

|bi,0 | and

0≤i≤pN −1

max

0≤i≤pN −1

|bi,0 | ≤ Rm .

|bi,1 | ≤ Rm and step by step, one (j)

max |bi,j | ≤ Rm . Since (Qi (Zq )∆q )(i,j)∈IpN ×N is a linear

max

0≤i≤pN −1 0≤j≤m Aq = Pq1 and is

q is also equal to an orthonormal family, one sees that A basis of the set of the elements R of L C(Zp , K) that can be written as convergent sum R =

N p −1

(j)

bi,j Qi (Zq )∆q(j) . Hence the sequence (Qi (Zq )∆q )(i,j)∈IpN ×N is an

i=0 j∈N

q . orthonormal basis of A

Let us remind that, when q = 1 is a pN -th primitive root of unity, any N p −1 ai Qi (Zq ), with element T ∈ Γq (Zp , K) can be written on the form T = T =

sup 0≤i≤pN −1

i=0

|ai |.

Corollary 1. Let us consider, in L C(Zp , K) , the subspace Γq (Zp , K){∇q } = Rj (Zq )∇jq , Rj (Zq ) ∈ Γq (Zp , K), lim Rj (Zq ) |q − 1|−j = 0}. Then {S = j≥0

q . Γq (Zp , K){∇q } = A

j→+∞

266 14

FANA TANGARA

Again let q = 1 be a pN -th primitive root of unity, we have the following corollary. () Corollary 2. Let K{∆q } = {Q = an ∆q(n) / lim an = 0} and K{∇q } = {R =

n→+∞

n≥0

an ∇jq /

−j

lim |q − 1|

n→+∞

()

|an | = 0}. Then K{∆q } (resp. K{∇q }) is a

(n) closed subalgebra of L C(Zp , K) and (∆q )n≥0 (resp. (q − 1)n ∇nq n≥0

) is an

n≥0

() q is orthonormal basis of K{∆q } (resp. K{∇q }). Moreover the Banach space A ()

isometrically isomorphic to the topological tensor products Γ(Zp , K)⊗K{∆ q } = ()

Γ(Zp , K) ⊗ K{∆q } and Γ(Zp , K)⊗K{∇ q } = Γ(Zp , K) ⊗ K{∇q }.

Proof. Since the vector space Γq (Zp , K) is a finite dimensional, one sees ()

that the topological tensor product of Banach spaces Γq (Zp , K)⊗K{∆ q } (resp. ()

Γq (Zp , K)⊗K{∇ q }) is equal to the algebraic tensor product Γq (Zp , K) ⊗ K{∆q } (j) (resp. Γq (Zp , K) ⊗ K{∇q }). On the other hand, since Qi (Zq )∆q (i,j)∈I ×N N p (j) q and Qi (Zq )⊗ (resp. (q−1)j Qi (Zq )∆q (i,j)∈I ×N ) is an orthonormal basis of A pN (j) ∆q (i,j)∈I N ×N (resp. Qi (Zq ) ⊗ (q − 1)j ∇jq (i,j)∈I N ×N ) is an orthonormal bap

p

()

sis basis of Γq (Zp , K) ⊗ K{∆q } (resp. Γq (Zp , K) ⊗ K{∇q }), one sees that the () q are isometrically Banach spaces Γq (Zp , K) ⊗ K{∆q }, Γq (Zp , K) ⊗ K{∇q } and A isomorphic.

N.B. Let q = 1 be a pN -th primitive root of unity. Let n ≥ 0 and 0 ≤ r ≤ pN − 1, (pN ) n (r) (npN +r) = ∆q ∆q (c.f [11] Proposition 1.0.1). one has ∆q Proposition 3.6. The center Z(Aq ) of the algebra Aq is equal to the algebra (pN )

of continuous linear operators generated by ∆q

. (pN )

Proof. Let F be the subalgebra of Aq generated by ∆q . It is readily seen that F ⊂ Z(Aq ). Let us prove now that the algebra Z(Aq ) ⊂ F. pN −1 m Suppose that Rm = bi,j Qi (Zq )∆q(j) is an element of Z(Aq ). Then Rm ◦ i=0 j=0

τ1 = τ1 ◦ Rm . Since τ1 Qi (Zq ) = (q i Qi (Zq ) + Qi−1 (Zq ))τ1 , ∀1 ≤ i ≤ pN − 1, N N −1 p −2 m m p i (j) one obtains (q − 1)bi,j Qi (Zq )∆q + bi+1,j Qi (Zq )∆q(j) τ1 = 0. i=0 j=0

Since the sequence

i=0 j=0

(j) (Qi (Zq )∆q )(i,j)∈IpN ×N

q , and the is an orthonormal basis of A

N

operator τ1 is invertible, it follows that (q p

−1

− 1)bpN −1,j = 0, ∀0 ≤ j ≤ mpN + t

267 15


and (q i − 1)bi,j + bi+1,j = 0, ∀0 ≤ i ≤ pN − 2, 0 ≤ j ≤ m. Since q i − 1 = 0, m ∀1 ≤ i ≤ pN − 1, one obtains bi,j = 0, ∀i = 0, then Rm = b0,j ∆q(j) . j=0 (j)

(j)

(j−1)

). On the other hand Rm ◦Zq = Zq ◦Rm and ∆q Zq = Zq (q j ∆q +q j−1 (q j −1)∆q m j (q − 1)b0,j + q j (q j+1 − 1)b0,j+1 ∆q(j) = 0. It follows that b0,j = 0, Hence Zq j=0

for any integer j prime to pN . One concludes that Rm can be written in the form n N Rm = bk ∆q(kp ) , with npN ≤ m, that is Rm ∈ F. k=0

Corollary. Let Z(Pq2 ) be the center of Pq2 . Then Z(Pq2 ) = Z(Aq ), where Z(Aq ) is the center of Aq . Proof. Since Pq2 is a subalgebra of Aq , then Z(Pq2 ) ⊂ Z(Aq ). On the other (pN )

(pN )

∈ Pq2 . Hence Z(Aq ) the algebra generated by ∆q is a hand, one has ∆q subalgebra of the algebra Pq2 . It follows that Z(Aq ) is a subset of the center of Pq2 . (kpN )

Remark 3.7. Let us notice that ∆q

(Cn,q ) = Cn−kpN ,q , ∀n ≥ kpN and

N

(kp )

∆q

(Cn,q ) = 0 otherwise.

Proof. Indeed, N

∆q(kp

)

N

(Cn,q )(x) = q kp

(kpN −n) kpN x

q

Cn−kpN ,q (x) = Cn−kpN ,q (x),

for n ≥ kp . N

We have the following theorem. Theorem 3.8. Let Z(A q ) be the closure of Z(Aq ). Then the sequence N n ∆q(p ) n≥0

is an orthonormal basis of Z(A q ). (j) Proof. Since the sequence (∆q )j≥0 is an orthonormal family in L C(Zp , K) , (npN ) is also an orthonor(see for instance [13] Theorem 1), its subfamily ∆q n≥0 N (pN ) n (np ) = ∆q and mal family inL C(Zp , K) . On the other hand one has ∆q (pN ) n the sequence ∆q is a linear basis of Z(Aq ). Hence one sees that the n≥0 N n (p ) is an orthonormal basis of the closure Z(A sequence ∆q q ) of Z(Aq ) in n≥0 L C(Zp , K) .

268 16

FANA TANGARA

Proposition 3.9. The closure Z(A q ) of Z(Aq ) is equal to the center Z(Aq ) of Aq . It is the completion of the algebra Z(Aq ) endowed with the operator norm. Proof. It is readily seen that Z(A q ) is a subalgebra of Z(Aq ). Then it remains q ) ⊆ Z(Aq ). to show that Z(A N p −1 q ). On the one hand, one bi,j Qi (Zq )∆q(j) be an element of Z(A Let R = i=0 j∈N

has R ◦ τ1 = τ1 ◦ R. One deduces from τ1 Qi (Zq ) = (q i Qi (Zq ) + Qi−1 (Zq ))τ1 , pN −2 N N (q i − 1)bi,j + bi+1,j Qi (Zq )∆q(j) + ∀1 ≤ i ≤ p − 1 that (q p −1 − i=0

j∈N

j∈N

1)bpN −1,j QpN −1 (Zq )∆q(j) τ1 = 0. However the operator τ1 is invertible and the (j) q , hence bi,j = 0, is an orthonormal basis of A sequence Qi (Zq )∆q (i,j)∈IpN ×N ∀1 ≤ i ≤ pN − 1, j ≥ 0 and R = b0,j ∆q(j) , with lim b0,j = 0. j→+∞

j∈N (j) (j−1) and Zq ◦ R = R ◦ Zq , Furthermore, since = Zq q j ∆q + q j−1 (q j − 1)∆q j j j+1 (j) − 1)b0,j+1 ∆q = 0. Noticing that the (q − 1)b0,j + q (q ∀j ≥ 1, one has Zq (j) ∆q Zq

(j) operator Zq is invertible and that the sequence ∆q j≥0

is an orthonormal family

j∈N

q , one sees that (q j − 1)b0,j + q j (q j+1 − 1)b0,j+1 = 0, ∀j ≥ 0. From these idenin A tities, one deduces first that if j = kpN , then b0,kpN +1 = 0 and second, step by step k N b0,kpN ∆q(p ) . Since that b0,kpN +s = 0, ∀1 ≤ s ≤ pN − 1. It follows that R = R=

m

lim

m→+∞

k≥0

m k k N (pN ) b0,kpN ∆q , with b0,kpN ∆q(p ) an element of Z(Aq ), one

k=0

k=0

obtains that R is in the closure Z(A q ) of Z(Aq ). T hat finishes the proof of the proposition. q ). q of Aq is a module of finite type over Z(A Corollary. The closure A Proof. Let us remind that if q = 1 is a pN -th primitive root of unity, then k (kpN +j) (j) (kpN ) (kpN ) (pN ) ∆q = ∆q ∆q , with ∆q = ∆q , ∀k ≥ 0, 0 ≤ j ≤ pN − 1. pN −1 pN −1

Let R =

i=0

N

bi,j (k)Qi (Zq )∆(kp q

+j)

q . be an element of A

j=0 k∈N

One has R=

N N p −1 p −1

i=0

j=0 k∈N

N ) bi,j (k)∆(kp Qi (Zq )∆q(j) q

=

N N p −1 p −1

i=0

j=0

N

Ri,j (∆q(p

)

)Qi (Zq )∆q(j) ,


where N

Ri,j (∆q(p

)

)=

269 17

N k N ) q ), ∀(i, j) ∈ IpN ×IpN . bi,j (k) ∆q(p ) = bi,j (k)∆(kp ∈ Z(A q

k∈N

k∈N

Acknowledgements. Bertin Diarra has taken a very important part in the preparation of this paper. I thank him for his advice on improvements of many results stated here. References [1] K. Conrad, A q-analogue of Mahler Expansion, Advances in Mathematics 153, 185-230 (2000). [2] R. Coquereaux, A. O. Garcia and R. Trinchero, Differential calculus and connections on a quantum plane at a cubic root of unity, Rev. Math. Phys. 12 (2000), 227-285. [3] B. Diarra, Sur quelques repr´ esentations lin´ eaires p-adiques de Zp . Proceedings Kon. Ned. Akad. van Wetensch (Amsterdam), Series A. 82 (4), (1979), 481-493. [4] B. Diarra, The continuous coalgebra endomorphisms of C(Zp , K), Bull. Belg. Math. Soc., Simon Stevin - Supplement, December (2002), 63-79. [5] B. Diarra and F. Tangara, The p-adic quantum plane algebras and quantum Weyl algebra . preprint [6] R. D´ıaz and E. Pariguan, Symetric quantum Weyl algebras, Annales Math´ ematiques Blaise Pascal 12, n◦ 2(2004), 187-203. [7] A. Giaquinto and J. J. Zhang, Quantum Weyl Algebras, J. Algebra 174, 3 (1995), 861-881. [8] L. Hellstr¨ om and S. Silvestrov, Two-sided ideals in q-deformed Heisenberg algebras, Expositiones Mathematicae, 23. 2(2005) 99-125 [9] C. Kassel, Quantum groups, GTM 155, Springer Verlag (1995). [10] H.-y. Pan and Z. S. Zhao, Operator realizations of the q-deformed Heisenberg algebra Physics Letters A 282 (2001) 251-256. [11] F. Tangara, Bases orthonormales et calcul ombral en analyse p-adique, Th` ese de l’Université Blaise Pascal, Clermont Ferrand (2006). [12] F. Tangara, Some Continuous Linear Operators and Orthogonal q-Bases on C(Zp , K), Contemp. Math., AMS, 384 (2005), 335-351 [13] F. Tangara, Orthonormal q-Bases and Linear Continuous Operators on C(Zp , K), to appear in Journal of Analysis [14] F. Tangara, The p-adic quantum Weyl algebra, preprint [15] A. Verdoodt, The use of operators for the construction of normal bases for the space of continuous functions on Vq , Bull. Belg. Soc. 1 (1994), 685-699.

´matiques et d’Informatique, D´ epartement de Mathe Facult´ e des Sciences et Techniques, Universit´ e de Bamako, B.P.E. 3206 Bamako Mali E-mail address: [email protected]

This volume contains the proceedings of the Tenth International Conference on p-adic and Non-Archimedean Analysis, held at Michigan State University in East Lansing, Michigan, on June 30–July 3, 2008. This volume contains a kaleidoscope of papers based on several of the more important talks presented at the meeting. It provides a cutting-edge connection to some of the most important recent developments in the field. Through a combination of survey papers, research articles, and extensive references to earlier work, this volume allows the reader to quickly gain an overview of current activity in the field and become acquainted with many of the recent sub-branches of its development.

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