Uniform Frechet Algebras (North-Holland Mathematics Studies)

UNIFORM FRECHET ALGEBRAS

NORTH-HOLLAND MATHEMATICS STUDIES 162 (Continuation of the Notas de Matematica)

Editor: Leopoldo NACHBIN Centro Brasileiro de Pesquisas Fisicas Rio de Janeiro, Brazil and University of Rochester New York, U.S.A.

NORTH-HOLLAND -AMSTERDAM

NEW YORK

OXFORD TOKYO

UNIFORM FRECHET ALGEBRAS

Helmut GOLDMANN Marhemarisches lnstitut der Universiriir Bayreurh 68yreUth, ER.G.

1990

NORTH-HOLLAND -AMSTERDAM

NEW YORK

OXFORD TOKYO

ELSEVIER SCIENCE PUBLISHERS B.V. Sara Burgerhartstraat25 P.O. Box 211,1000 AE Amsterdam, The Netherlands Distributors for the United States and Canada: ELSEVIER SCIENCE PUBLISHING COMPANY, INC. 065 Avenue of the Americas New York, N.Y. 10010, U.S.A.

Ltbrary o f Congrrrr Cataloglng-in-Publication Data

Ooldaann, Helmut. Uniforn Frichet algabras / Helaut Goldaann. p. cn. (North-Holland nathrnetics studies ; 162) Includrs bibllographlcal referencrr. ISBN 0-444-88488-2 (U.S.) 1. Uniforn algabras. I. Tltlr. 11. Title: Frachrt algebras. 111. Sarles. OA328.084 1990 90-6870 612'.66--dc20

--

CIP

ISBN: 0 44488488 2 @

ELSEVIER SCIENCE PUBLISHERS B.V., 1990

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To my parents, and to a l l my friends for just being there.

This Page Intentionally Left Blank

PREFACE

By a uniform (Banach) algebra we mean a uniformly closed algebra of complex-val ued continuous functions on a compact Hausdorff space. These algebras have been studied intensively and there exists a vast literature on this subject. But there are important examples of algebras of continuous functions on noncompact spaces which are not Banach algebras, we mention for example Hol(X). the algebra of a l l holomorphic functions on a domain X in C". Hence it makes sense t o generalize the t e r m uniform algebra in an appropriate way. This leads to the definition o f uniform Frechet algebras. The t e r m refers to an algebra of complex-valued continuous functions on a hemicompact space which i s complete with respect to the compact open topology. (By a hemicompact space we mean a Hausdorff space X such that there exists a countable compact exhaustion (KnIn of X such that each compact subset K of X is contained in some K.,

e.g. l e t X be a locally compact and o-compact

Hausdorff space.) Examples are the algebra o f a l l continuous functions on a hemicompact k-space and Hol(X1. the algebra of a l l holomorphic functions on a hemicompact reduced complex space X. An important problem in the theory of uniform (Banach) algebras i s the question of the existence of analytic structure in the spectrum of a uniform algebra A. 1.e. the question as whether parts o f the spectrum can be endowed with the structure of a complex space so that a l l elements of A elements of A

-

-

more precisely a l l Gelfand transforms of

become analytic functions with respect t o this struc-

ture. We adopt this question for uniform Frechet algebras. So the main part of this book is devoted t o the problem when a given uniform Frechet algebra is topologically and algebraically isomorphic

ii

to Hol(X1. X a suitable complex space. We obtain a function algebraic characterization of certain classes o f algebras of holomorphic functions. In particular we give various characterizations of Hol ( X I in the case that X is

-

an n-dimensional Stein space, an n-dimensional Stein manifold,

- a Rlemann domain over -a

Cn,

domain of holomorphy in Cn,

- a polynomially convex domain in Cn, - a logarithmical l y convex complete Reinhardt domain, - a domain in C , etc.. These results have been mainly obtained by Arens. Brooks, Carpenter and Kramm. The material is presented in three sections. I n the f i r s t chapter we c o l l e c t definitions and results from the theory o f Banach algebras which are constantly used in the book. Since this material has been w e l l reported in many books, cf. for example [GAM], [STO], we omit the proofs of the results in most cases but give references. I n the second chapter we consider the algebra Hol(X) in greater detail. On the one hand it serves us as the f i r s t example of a uniform Fr6chet algebra, on the other hand, since we want t o characterize Hol(X) within the class of uniform Fr6chet algebras, we coll e c t properties of it which w i l l be discussed l a t e r in the general settlng of uniform Fr6chet algebras. The necessary background in the theory of several complex variables i s w e l l presented in t e x t books, e.g. in [G/R],

[GR/R],

[HUR]. I have included these prelim-

inary chapters t o make this book more accessible f o r a reader who i s not so familiar with these theories. The second part is devoted t o the study o f uniform Fr6chet algebras resp. more generally to the study o f Fr6chet algebras. One starting point are surely the papers of Arens [ARE 6 1 and Michael [MIC] in

iii

the early fifties. A basic r e s u l t is the projective limit representation of each Frechet algebra. It enables us t o develope this theory along the lines of the theory of Banach algebras. Many results which are known for Banach algebras remain true f o r Frgchet algebras with almost the same proofs. In particular we mention the holomorphic functional calculus and consequences o f it like Shilov's idempotent theorem etc.. Problems arise in connection with the question as whether each mu1tip1 icative I inear functional on a Frechet algebra i s automatically continuous. This question, known as Michael's problem, has been intensively studied (cf. the paper of Dixon and Esterle [DIE]), but only partial answers have been obtained. So t o carry over some results f o r Frechet algebras, like the uniqueness of norm theorem f o r semisimple Banach algebras, one has to develope completely new proofs. Differences between both theories appear in connecion with peak points. There exists no analogue f o r Frechet algebras to Rossi's maximum modulus principle. The third part is devoted t o questions of analyticity in one form or another. As mentioned above we characterize various classes of a l gebras of holomorphic functions. Moreover we study properties of holomorphic functions, like the maximum principle, Liouvil le's theorem, Montel's theorem etc., in the general setting o f uniform Frechet a l gebras and ask to what extend they are independent of the existence of analytic structure in the spectrum. We mention an example to Wermer

-

due

- of a uniform Frechet algebra which satisfies the maxi-

mum principle and the identity theorem but whose spectrum does not even contain an analytic disc. There exist several other books which deal with the theory o f more general topological algebras, like locally mu1tiplicatively convex algebras or natural systems, see [ENS]. [GUI], [MALI, [MIC],

[RIC 21, [ZEL]. Some of them contain also results on Frechet alge-

iv

bras. Most results of part 3 and some results of part 2 are presented f o r the f i r s t time in a book. I n [KRA 81 my academic teacher Bruno Kramm announced his book "Nuclear function algebras and Stein algebras

-

holomorphic struc-

ture in spectra", which should also appear in this series, but which has not been finished due to his tragic death in 1983. The present book is not the book Bruno had in mind. However parts of the book, in particular the part on the elementary theory of uniform FrQchet algebras, owe much t o a lecture o f Bruno, held at the university of Munich in 1980. I hope this book r e f l e c t s also some of the spirit of his work on this subject. Finally, it is a pleasure for me to thank Peter Pflug from Vechta f o r his constant encouragement during the preparation of this book. Many valuable suggestions and improvements are due t o him. I am also indebted t o Sandra Hayes from Munich for stimulating discussions. Robert Braun, Paul Fischer. Stefan MU1ler-Stach and Roland Weinfurtner did much of the proof reading, Elisabeth Guth typed a f i r s t version of the manuscript.

CONTENTS Preface

.

PART 1

..............................................................................

i

BANACH ALGEBRAS. ALGEBRAS OF HOLOMORPHIC FUNCTIONS. AN INTRODUCTION

.

CHAPTER 1 An excurs on Banach algebras

....................................................... The spectrum of a B-algebra .................................. The Shilov boundary ............................................... The holomorphic functional calculus ......................... Analytic structure in spectra ..................................

(1.1)

General theory

(1.2) (1.3) (1.4) (1.5)

3 9 18 25 30

.

CHAPTER 2 The algebra of holomorphic functions

....................................................... Analytic continuation ............................................... Stein spaces ..........................................................

(2.1)

General theory

(2.2) (2.3)

34 41

50

.

PART I1 GENERAL THEORY OF FRECHET ALGEBRAS

.

CHAPTER 3 Theory of Frechet algebras. basic results (3.1) (3.2) (3.3)

.................................................... The spectrum o f a FrQchet algebra .......................... Projective limits ...................................................... FrQchet algebras

59 71

84

.

CHAPTER 4 General theory of uniform Frechet algebras

(4.3)

........................................ Extension of a uniform Frechet algebra .................... Convexity for uniform FrQchet algebras ....................

(4.4)

Uniform Fr6chet algebras with locally compact spec-

(4.1) (4.2)

Uniform Frechet algebras

trum

.....................................................................

91 97 103 109

vi

.

CHAPTER 5 Finitely generated Fr6chet algebras

......................... Fr6chet algebras ..........

(5.1)

Finitely generated Frechet algebras

113

(5.2)

Rationally finitely generated

122

.

CHAPTER 6 Applications of the projective limit representation

........................

129

..................................

141

(6.1)

The holomorphic functional calculus

(6.2)

The theorem o f Arens-Royden

.

CHAPTER 7 A Fr6chet algebra whose spectrum is not a k-space

.............................................. Surjectivity of the transpose map .............................. The weak Nullstellensatz .......................................

(7.1)

The example o f Dors

(7.2) (7.3)

147 150 155

.

CHAPTER 8 Semisimple Fr6chet algebras

........................................... derivations ..........................................

(8.1)

Uniqueness of topology

161

(8.2)

Continuity of

165

.

CHAPTER 9 Shilov boundary and peak points for Frechet algebras

................. 171 ............................. 180

(9.1)

The Shilov boundary of a Fr6chet algebra

(9.21

Peak points for Fr6chet algebras

.

CHAPTER 10 Michael's problem (10.1)

Results on the automatic continuity of characters

(10.2)

The approach of Dixon and Esterle

...... 185

...........................

190

.

PART 3 ANALYTIC STRUCTURE IN SPECTRA

.

CHAPTER 11 Stein algebras (11.1)

Analytic structure in spectra of uF-algebras

............. 197

.

CHAPTER 12 Characterizing some particular Stein algebras (12.1)

Polynomially convex analytic subsets

................,.......

205

vii

....................

208

Open subsets of the plane

213

(12.4)

Domains of holomorphy

...................................... ...........................................

215

(12.5)

Logarlthmical l y convex complete Reinhardt domains

... 219

(12.2)

Polynomially convex open subsets of Cn

(12.3)

.

CHAPTER 13 Liouville algebras (13.1)

Liouville algebras

...................................................

223

.

CHAPTER 14 Maximum modulus principle (14.1)

Maximum modulus algebras

(14.2)

Maximum modulus algebras

..................................... 231 and subharmonicity ......... 237

CHAPTER 15. Maximum modulus algebras and analytic structure

........................................ algebras and finite mappings ........

(15.1)

Riemann domains over Cn

243

(15.2)

Maximum modulus

248

.

CHAPTER 16 Higher Shilov boundaries (16.1)

Higher Shilov boundaries for uB-algebras

(16.2)

Higher Shilov boundaries for uF-algebras

................. 263 ................. 268

.

CHAPTER 17 Local analytic structure in the spectrum of a uniform Frechet algebra (17.1)

Local rings of functions of uniform Fr6chet algebras

273

.

CHAPTER 18 Reflexive uniform Frgchet algebras (18.1)

Reflexive uniform Frechet algebras

(18.2)

Gleason parts

..........................

285

........................................................

287

.

CHAPTER 19 Uniform Frechet Schwartz algebras

......................................

(19.1)

Fr6chet Schwartz algebras

(19.2)

Strongly uniform Frechet algebras

(19.3)

Chevalley dimension for uniform Frechet algebras

...........................

304 306

..... 310

viii (19.4)

A characterization of pure dimensional Stein algebras 315

(19.5)

Riemann algebras

...................................................

APPENDIX A

.

Subharmonic functions. Poisson integral

APPENDIX B

.

Functional

........... 331 analysis ......................................... 335

...................................................................

336

.........................................................................

330

.................................................................................

350

List of symbols References Index

325

1

PART 1 BANACH ALGEBRAS, ALGEBRAS OF HOLOMORPHIC FUNCTIONS AN INTRODUCTION

This part is devoted to classical definitions and properties of Banach algebras and algebras of holomorphic functions. W e put together main definitions and results which are of constant use in the book. Most o f the results were given without proof but with references.

This Page Intentionally Left Blank

3

CHAPTER 1 AN EXCURS ON BANACH ALGEBRAS

Banach algebras w i l l serve us as an important class o f examples o f Fr6chet algebras. Moreover it w i l l turn out that in some sense they are the local building stones of Fr6chet algebras. So, t o develope the theory of Fr6chet algebras we shall frequently have to use results from the theory of Banach algebras. In this chapter we put together results which w i l l be needed later. Of course this excurs is by no means complete and cannot replace the detailed study of Banach algebras, which i s presented in many textbooks (see the l i s t in the bibliography), but intends to make the book more readable for the reader, who Is not familiar with this theory

.

In the f i r s t part of this chapter we give an introduction t o the general theory of Banach algebras. I n the second part we state deeper results. Iike the holomorphic functional calculus, Rossi's maximum modulus theorem etc.

. In this

part we omit the proofs but give al-

ways a detailed reference. I n most cases we cite the books of Stout ISTOl and Gamelin IGAMI. (1.1) General theory (1.1.1) Throughout this book we only consider associative and commu-

tative algebras over the field of complex numbers. With the exception of chapter 3 we also assume that our algebras have an identity, i.e. there is an element 1 in the algebra A such that 1 - f = f for a l l

f

E

A.

Chapter 1

4

(1.1.2) Definition. I) A Banach algebra (B-algebra) is an algebra A, which Is a Banach space and in which multiplication and the norm are linked together by the inequality IIfgll

IIfII 11g11.

f,g

E

A.

Moreover lllll = 1. ill A Banach algebra is called uniform Banach algebra (uB-algebra) if

IIf211 = Ilfl12 for

ali f

0

A.

(1.1.3) Examples. 1) Denote by Ck([O,l])

the algebra of a l l k-times

continuously differentiable (complex-valued) functions on the unit k interval with pointwise defined operations. Then C ([0,1]) Is a B-algebra with respect t o the norm Pk+l(f) = 2 k SUP { I f (i)( t ) l : t

E

[0,1]. i = 0.1.

where f(') denotes the i-th derivative of f. C k C[O.l])

....k) is not a uniform

algebra. 11) Let K be a nonempty compact Hausdorff space. Let C(K) denote

the algebra of a l l (complex-valued) continuous functions on K with pointwise defined operations. Then C(K) becomes a uB-algebra when endowed with the sup-norm

IIfll, ill) Let K

= sup{lf(t)l

:

t e K).

Cn be a compact set. We define

c

P(K) = { f

E

C(K)

:

f can be approximated uniformly on K by

polynomials), and

R(K) = ( f

E

C(K)

4

f can be approximated uniformly on K by

rational functions which are analytic on K). P(K) and R(K) are uB-algebras with respect to the sup-norm and we have P(K)

c

R(K)

c

C(K).

For other examples cf. [STO] and [GAM].

General theory o f Banach algebras

5

(1.1.4) Let B be a B-algebra. We introduce the spectrum d b ) of an element b

P

Tb

B. a(b) is just the spectrum of the operator

B -a

:

B.

->

X

bx.

We shall show that a(b) is a nonempty compact set. As a consequence we get the theorem of Gelfand-Mazur which states that each B-algebra which is a field is isomorphic to C. Definition. a(b) = { X

P

4:

:

X-1-b is not invertible in B).

(1.1.5)Proposition.The set of a l l invertible elements of a B-algebra B

B - l = {b

P

B: There is b-'

E

B so that b-b-' = 1)

is an open subset of B. Proof. Let b

(*I

P

B-'. Let a

Ilb-all

c

B such that

< 1 1 llb-lll.

For n c N set

then f o r n>m

It follows that (snIn is a Cauchy sequence in B. since Ilb-lllllb-all

n(B), cp ->

ker c p .

is bijective. Proof. Let

(Q E

M(B), then

B/ker cp ->

C. b + ker

CQ

->

cp(b).

is a ring isomorphism, hence ker cp is a maximal Ideal. i.e. T is w e l l defined. Clearly T is injective. Now l e t m

Q

n ( B ) be an arbitrary element. The canonical map

Chapter 1

14

B ->

B/m induces a nontrivial homomorphism pm: B ->

considerations above. We have ker

vm =

C by the

m. So T is surjective, too. 0

(1.2.7) I n the next theorem we consider the connection between M(B) and the spectrum a(b) of an element b

E

B.

Theorem. Let B be a B-algebra, and l e t b

E

B be an element. Then

6 M B ) ) = db). Proof. We have $(M(B)) c a(b) by the considerations in (1.2.1).

If z

E

a(b), then 2.1-b is not invertible, i.e. z - l - b is contained in a

maximal ideal m. By theorem (1.2.6) there is p

E

M(B) such that

6(cp) = z.

rn = ker p, hencecp(z.1-b) = 0,resp.

0

(1.2.8)Theorem. Let B be a B-algebra. Then M(B) i s a nonempty compact space. Proof. M(B)*0 by theorem (1.2.7) and theorem (l.l.S).By

(1.2.1) M(B)

is contained in E. the closed unit disc of the dual space of B. Since

E isweak* compact it suffices to show that M(B) i s a weak* closed subset of E. Let p

E

E\M(B). Hence there are f.g p(fg)

Then

- for

E

B such that

* p(f)p(g).

sufficiently small e>O

-

we have &(fg)*&(f)S(g) for a l l 6

of the weak* open set

u = { 6: 16(g)-p(g)i<E. Hence U n M(B) = 0 .

i6(f)-p(f)i<E8 16(fg)-p(fg)l

C(M(B)). b ->

6,

is an isomorphism onto a closed pointseparating subalgebra o f C(M(B)). and we have llbll = Proof. Clearly

I I ~ I Ifor ~ each ~ ~ b) E B.

separates the points of M(B) and contains the con-

stants. We have A

b(M(B)) = a(b)

for each b

E

B by theorem (1.2.71, so

for each b

E

B by remark (1.1.8).

(1.2.11)

0

We shall show next that for certain 8-algebras any two

norms under which the algebra is a B-algebra are equivalent. I t is clear that one cannot expect that such a r e s u l t holds f o r a l l B-algebras, since one can make any Banach space into a B-algebra by defining a l l products t o be zero and adjoining an identity. Deflnition. Let A be a B-algebra. The intersection of a l l maximal ideals of A i s called the radical of A and is denoted by rad A. A i s

16

Chapter 1

called semisimple if rad A = (01. By theorem (1.2.6) rad A = cp

fl E

Q(A)

ker cp.

It follows that A is semisimple iff the Gelfand map

r

1

P.

c(M(A)), f->

A ->

i s injective. Note that in particular each uB-algebra is semisimple. Theorem. Let A and B be B-algebras. If B is semisimple then each homomorphism cp

:

A ->

B is automatically continuous.

Proof. We show that the graph of cp is closed and obtain our r e s u l t from the closed graph theorem (8.2).

-

Let (an,cp(an)In be a sequence w i t h an -> cp(an)

b as

I f J,

n->ao.

J,ocp(an)

-+

E

M(B) then Q ~ c pc M(A). hence

$*cp(a) as n->-

by theorem (1.2.1). Since J,ecp(an) -> J,(b) = J,(cp(a)) for a l l J, Since

a and

o

J,(b) we have

M(B).

B is semisimple we get b = cp(a), as desired. 0

As a corollary we get the uniqueness of the norm topology f o r semisimple B-algebras. This r e s u l t is due t o Gelfand. (1.2.12)Corollary. Let A be a semisimple B-algebra with respect t o a norm II.11. If A is a B-algebra with respect t o a second norm then

11-11

and

II-II, are

Proof. The map id

:

II-II,,

equivalent.

(A,

11.11,)

->

(A,

II-II),a

->

a i s bijective. It is

continuous by the theorem (1.2.11 1 and hence a homeomorphism by the open mapping theorem (B.1).

0

17

Spectra of Banach algebras (1.2.13) We return t o polynomial convexity and consider finitely generated B-algebras. Definition. i) Let B be a B-algebra. and l e t fl, a(fl ,...,fn) = {(cp(f,)

,...,cp(fn))

E

Cn:

....fn

(Q E

E

8. The set

M(A))

is called the joint spectrum of flm...,fn. ii) 8 is called n-generated by fls...,fn

if the polynomials in fl.

...,fn

lie dense in B. Theorem. Let B be an n-generated B-algebra with generating elements fl

,...,fn. Then o(fl ,...,fn) c

Cn is a polynomially convex com-

pact set and the map

?

:

a(fl ,...,fn),

M(A) ->

(Q

n

->

,....fn((Q)), A

(fl(q)

is a homeomorphism. B is topologically and algebraically isomorphic to P(o(fl.

....f,)).

if B

is moreover a uB-algebra. h

Proof. Let F((Q) =

c(JI).then p(p(fl ,...,fn)) = t#(p(fl ,...,fn)) f o r a l l

polynomials p, hence (Q=$since B is n-generated by fl,

....fn, i.e. P

is injective. h

F is clearly continuous. Hence d f l,...,fn) theorem (1.2.8) and Let z

E

$(f l,...,fn)

f

= ?(M(B)) is compact by

is a homeomorphism.

(the polynomially convex h u l l of o(fl,

...*fn)) be an

arbitrary point, then Ip(z)l

ilP~Io(f,,...,f" 1 = IIp(fl

n

.....fn)

ilM(B)

* IIp(fl ,...* f n N

for a l l polynomials p, where p(f ls...,fn)n denotes the Gelfand transform of p(f l.....fn).

For the l a s t inequality cf. (1.2.3). Hence

(Qz(p(fl,...,fn))

= p(z)

18

Chapter 1

defines a continuous homomorphism on the algebra of a l l polynomials Since B is n-generated by fl,....fn.

in fl,...,fn.

ment

gz of 0(fl

since

p, extends to an ele-

M(B) and

,...,f n1 = 0 (fl ,...,fn) CI

F(gz)= z.

If B is a uB-algebra, then B is isomorphic t o @ by theorem (1.2.10) and it i s easy t o check that h

B, f

P(o(f l,...,fn)) ->

foF

->

i s in fact an isometry. Remark. Iffl,

0

...,fn are arbitrary

general o(f l....,fn)

is not polynomial l y convex.

For example set OD = { x x ->

elements of a B-algebra 8, then in

c

C

:

1x1 = 11, B = C(OD) and

z : OD ->

C.

x. then o ( z ) = OD is not polynomially convex.

(1.3) The Shilov boundary (1.3.1) Let K be a compact Hausdorff space, and l e t B be a pointseparating closed subalgebra of C(K). which contains the constants. By (1.2.4) K can be identified with j(K) tended t o an element

?E @

c

c

M(B) and each f

E

B can be ex-

C(M(B)) by theorem (1.2.10). So M(B)

can be interpreted as the "largest space" t o which a l l elements of

B can be extended in such a way that B i s isomorphic t o the algebra of the extended functions. (This w i l l become clearer in (4.2)). I t is natural t o ask whether there exists a "minimal" compact set

L

c

K such that B is isomorphic to ElL=

{IL:f

E

B).

This question can be answered by introducing the Shiiov boundary. Definition. Let B be a B-algebra. A closed subset E boundary for 8 if each

?I

h

c

M(B) i s a

B assumes i t s maximum modulus on E.

19

Shilov boundary

(1.3.2) Remark. i) In particular M(B) is a boundary for B. ii) Let (B,K) be a uB-algebra. Then

IIfllK = for a l l f

sup{if(x)l

:

x

K) =

Q

ll?llM~B~

B by (1.2.10). Hence j(K) is a boundary f o r B, where j de-

E

notes the map introduced in (1.2.4). the closed unit disc. Then each closed set

Example. Denote by E

c

5 which

contains the set (1x1 = 1) is a boundary for P(E) by the

maximum modulus principle.

(1.3.3) Theorem. The intersection of a l l (closed) boundaries for B is a boundary for B.

(1.3.4) Lemma. Let f l ,....f,

v

= {cp

M(B)

:

Q

B and set

17(cp)i < I , i=1,

...,n).

Then either V meets every boundary for B or E\V is a boundary f o r B f o r every boundary E for B. Proof. Suppose there exists a boundary E for 6 such that E\V is not a boundary f o r 8. Hence there is f

E

B such that

A

ll?llM(B)= 1 > ~~f~I,-\,,.

Replacing f by A

a suitable power fm if necessary, we can assume that ~ ~ f ~ ~ ~ 1, ~ E \ , , j = l,...,n.

If cp

E

E n V , then

IP(cp)i;(cp)l < I. j=l. ...,n, by the definition of V. Since E is a boundary f o r B we get A n

IIf

(*I Let cp

E

= llf-fjllE< 1, j=1, ...,n. A A

M(B) such that If(cp)l = 1. Suppose that cp

there is j

r

(1. ....n) so that

Ifj (cp)i

L

1, hence

Q

M(B)\V. Then

If.?I (cp)l

2

1, and we

20

Chapter 1

get a contradiction t o ~ It follows that tcp E M(B)

A

~ :

< 1.~

~

i f (911 = 1)

f c

~

~

~

~

(

B

V. so V meets every bound-

ary for B.

0

Proof of theorem (1.3.3). Let f

S = {cp

E

M(B)

:

B be an arbitrary element and set

E

= ~~?~~M(B)l.

i?(cp)i

Denote by yB the intersection of a l l boundaries for B. Suppose S E

n yB

=

0. Then for each cp E S, there exists a boundary

for B such that cp

M(B)\EO. Since E is closed we find by cp (1.2.2) (*I an open neighbourhood V of cp of the form described in cp the lemma such that V f l E = 0. Since S is compact there are c p ( Q cpl.....cpr so that (Q

s

c

v,

E

u ... u Vqr

Using the lemma r-times we see that M(B)\(V boundary for B. a contradiction. Hence S

U

91

n yB

*

... U V'r

1 is a

0, i.e. yB

*

0 and

yB is a boundary for 8.

0

(1.3.5) Definition. The set yB is called the Shilov boundary of B. Examples. i) Denote by

5 the

y P 6 ) . More generally l e t K

closed unit disc, then t 1x1 = 1 1 = C be a polynomially convex compact

c

set. Then yP(K) equals the topological boundary of K, which we denote by t3K ([GAM] p.27, [STO] p.277). ii) I f K c C is a compact set. then yR(K) = t3K ([GAM] p.27. [STO]

p.277). (1.3.6) Proposition. Let A be a B-algebra. Then i f f for each neighbourhood U of

(Q

(Q

is contained in yA

in M A ) there exists ,f

that

ll?ullu=

sup tI?,C$)l

:

$

E

U) > ll?ullM(A),u.

E

A so

~

Shilov boundary Proof.1) Let p

E

21

yA. Suppose there exists a neighbourhood U o f p

such that A

IlfllM(A),u

II?II,

for a l l f

t

A.

Then M ( A ) \ U is a boundary for A. hence U

n yA

= 0 , a contradiction.

ill Suppose that for each neighbourhood U of cp there is ,f

t

A so

that

l?,llu > ll?ullM(A),u* Then y A

nU*

0 for a l l neighbourhoods U of p. It follows that

cp c yA. since y A is closed.

0

(1.3.7) Lemma. Let A be a B-algebra. f topologlcal boundary of ?(M(A))

E

A. then ?(yA) contains the

.

Proof. We denote by d?(M(A)) the topological boundary of ?(M(A)). Suppose there exists X c d?(M(A))\?(yA). Since ?(yA) i s a compact subset, there is (2:

Choose Xo

E

E

> 0 such that

it-XI
4 / ~for a l i IJJ

n yA

=

0.

E

S, 0

22

Chapter 1

(1.3.8) Defintion. Let B be a B-algebra. i) A point p

E

M(B) i s called strong boundary point ( o r p-point) for

B if for every neighbourhood V of cp there exists an element f n

ll?ll,,,,~Bl=

such that

A

f(cp) = 1 and fI,l

i?(t))l

< 1 for

JI

E

B

< 1.

ii) p is called a peak point for B iff there is f

and

E

L

B so that ?(p) = 1

M(B)\(p).

Example. Set dD = { x o C : i x i = 1). Each point elt, 0

L

t < 2 x , is a

peak point for P(E) (consider the functions (1 + ~ . e - ~ ~ ) / 2 ) . (1.3.9) We f i r s t show that in general not every point of the Shilov boundary is a peak point. Example. Set K =

x ( 0 ) U ( 0 ) x C0.11 c

4:2 .

Denote by z1,z2 the coordinate functions. We show that K is polynomialiy convex. We prove more generally that if Ki convex compact sets then K = K1 x

... x

Kr

c

.

n1+. .+n,

C

4:

is polynomial l y convex, too. Let x = (xl W.1.o.g.

,...,xr),

x1 c Cnl

C"1 , i = l,...,r, are polynomially

.

n I+. .+n xi o Cni, be a point in 4:

\ K,.

\ K.

Hence there exists a polynomial p so that

ip(xl)i > llpllK,. Set q(y 1

V..

*Yn +

...+nr 1 = P(Y l.....Ynll.

Consider the algebra P(K). We have M(P(K)) = K by (1.2.4) ill. Each point of d 5 x ( 0 ) is a peak point for P(K) (cf. example (1.3.8)

and

replace z by zl). The same Is true fo r a l l points (0.y). O < y r l (consider for example l - ( ~ ~ - y ) Hence ~).

S hi1ov boundary

dD x (01 U (01 x C0,lI

23

yP(K)

c

since yP(K) is a closed subset of K. Now l e t f c P(K) be an arbitrary element. Then flEx

B, b ->

exp(b),

is continuous, since

and the la st expression tends t o zero as llbll tends to zero. (1.4.5) We give some applications of the holomorphic calculus. Theorem (Oka-Well). Let K

c

Cn be a polynomially convex compact

set. Then every function holomorphic in a neighbourhood of K can be approximated uniformly on K by polynomials.

If K

c

Cn is a rationally convex compact set, i.e. M(R(K)) = K. then

FIK c R(K) for each function F which is holomorphic in an open neighbourhood of K. ([GAM] p. 84. tST03 p. 368).

(1.4.6) Shilov idempotent theorem that E

f

L

c

.

Let B be a B-algebra. Suppose

M(B) is an open and closed set. Then there is a unique

B such that f2 = f and

f

is the characteristic function o f E.

Functional calculus

29

([GAM] p. 88, [STOI p. 75). Remark. Let B be a uB-algebra such that M(B) is a totally dlsconnected space. Let x.y, x*y , be two points of M(B). Choose an open and closed neighbourhood U of x such that y c M(B)\U. Then there exists

? c 8 such

functions in

that

?I,

= 1 and

?IH(B),U

= 0, 1.e. the real-valued A

separate the points of M(B) and we get C(M(B)) = B

by the theorem of Stone-Weierstrass. (1.4.7) Theorem. Let B be a B-algebra. f that f=exp(hl. Then there exists g

E

E

B and h c C((M(B)) so

B such that f = exp(g) and h=e.

( C f . the proof of corollary 8.22. CSTOl p. 80). (1.4.8) Theorem (Arens-Royden). Let B be a B-algebra. and l e t h c C(M(B))'l.

Then there exist f c

and g c C(M(B)) such that

h = fexp(g). ([GAM] p. 89). (1.4.9) The next theorem, due to Rossi, is also based on deep methods of the theory of several complex variables.

Definition. Let B be a B-algebra.

1) A closed subset E c M(B) is called local peak set provided there is

?

IPI


=

o

C is holomorphic i f f for j = I ,

....n.

35

General theory of holomorphlc functions For z = ( z l , ...,zn)

C n and r = (rl,...,rn) c

E

DZpr = {(yl,...,yn)

For v = (vl

,...,v n1 e N:,

...-vn!,

v! =vl!*

Cn

E

8

iyl

- zil

z = (zl ,...,zn)

E

zv = zn :1 ;..

.:R

define

< rip i=l,...,d.

Cn and f

c

Hol(G) set

,

dV1+...+vnf

DVf =

"; ... dvn zn

We can now state Cauchy's integral formula: Let w = (wl

,...,wn)

c

G, r = (rl ,...,rn) c

lR: such that the closure of

DWpr Is contained In G. then

for a l l v = (v,

,...,vn)

E

n z = No.

(zl

,...,zn)

E

D and f w .r

E

Hol(G1.

(2.1.4) As a consequence o f Cauchy's integral formula we get the

following estimate: Let K r = (rl

.....rn) w," E

c

G be a compact subset, and l e t

such that the compact set

Kr = t(zl ,...,z 1

E Cn: There exists (yl n lyl-zli s rl for l=l. nl

....

Is contained In G, then IIDVfllK s v!llfll

Kr

/rV

,...,yn)

E

K such that

36

Chapter 2

for a l l v c N ,:

f c Hol(G). where

IIfllKr =

sup{lf(x)l: x

c

Kr).

(2.1.5) As in the case n=l we have a power series representation of holomorphic functions. For x = (x, ,..., xn) c C n set 1x1 = (Ixll Let

c

,...,l x n l ) ,

n a zv be a power series ( a, c 41 for a l l v c No 1

vcN: which converges at x with respect t o a fixed order of summation. Then

c avzv converges absolutely and uniformly on each compact

subset of Do,,xl t o a limit that is Independent of the order of summation.

If 1x1 c RY, then

c avzV defines

an element of H o I ~ D o , l x l ~Vice ,

( x c 43". versa if f is holomorphic on a polydisc D x,r the Taylor series of f

r c R",), then

t o f. converges uniformly on each compact subset K o f D x,r

(2.1.6) Identity Theorem. Let

G

c

C" be a domain. If f c H o l ( G ) van-

ishes on a nonempty open subset o f G then f vanishes identically.

G if f vanishes on a sequence (an)n in G, which converges t o a point a c G.

Remark Note that in the case n,l f must not vanish on

Take for example

G

= C

2 , an = (O.l/n) and f = z,, the f i r s t coordi-

nate function. (2.1.7) We denote by C(G) the algebra of a l l complex-valued continuous functions on G. We have Hol(G)

c

C(G) by (2.1.2).

Theorem. Let f c C(G) and l e t (fmIm c Hol(G) be a sequence which converges compactly t o f, i.e. (fmIm converges uniformly on each

General theory of holomorphic functions compact set K

c

G to f. then f

E

Hol(G).

(2.1.8) Open-mapping Theorem. Let G f

E

37

c

Cn be a domain, and l e t

Hol(G) be a nonconstant function. Then f(G) is an open subset

c.

of

(2.1.9) Maximum principle. Let G

c

Cn be a domain, f

E

Hol(G) and

l e t K c G be a compact subset such that there exists a point a

E

int(K) with I f ( a ) l = 111 f1,

= sup {If(z)l: z

E

K). Then f is constant.

Remark. We conclude from the maximum principle that

Ilfll,

=

11f11,,

for each compact set K

c

G and each f

E

Hol(G). where dK denotes

the boundary of K. (2.1.10) As in the one dimensional case sums, products and complex multiples of holomorphic functions are holomorphic. Hol ( G I is thus a commutative algebra whose unit element is the function with constant value 1. We consider the algebra Hol(G) in greater detail. First we endow Hol(G) with the compact open topology. I f K

c

G is a compact set.

then llf+gIl,

IIfll,

Ilfll,

Ilfgll,

+

11g11.,

11g11,

for a l l f.g

E

Hol(G), A

i.e the mapping

II-II,

:

Hol(G) ->

R, f ->

IIfII,,

defines a submultiplicative seminorm (cf. (3.1.4)).

C.

Chapter 2

30

An open neighbourhoodbasis o f f

E

Hol(G) (wlth respect to the com-

pact open topology) i s given by a l l sets of the f o r m (9 e Hol(G)

:

IIf-gIIK
O.

(2.1.11) We can descrlbe the compact open topology in another way. Let

...Km c Km+l

c

... be a countable compact exhaustion of G such

that Km c int Km+, for a l l m

E

N

(where lnt Km+, denotes the interior

of Km+, 1. Then the sequence of seminorms

(11.11

topology on Hol (GI. A neighbourhoodbasis of f

bm generates a

Km E

Hol (GI i s given

by the sets

It is easily seen that

pology

(II-II Imgenerates Km

just the compact open to-

. Moreover

defines a metric on Hol(G) which generates the compact open topology (cf. tKoT3, p. 205). The operations

C x Hol(G)

Hol(G), (X,f) ->

If,

Hol(G) x Hol(G)-->

Hol(G), (f,g) ->

f+g,

Hol(G) x Hol(G) ->

Hol(G), (f,g) ->

fg,

are continuous. so Hol(G) is a metrizabie topologlcai algebra (cf. (3.1.5) for the definition). Let (fmIm be a Cauchy sequence in Hol(G). that means that f o r each k

E

N

(f,lKk),

Since (C(KC),II.((

is a Cauchy sequence in C(Kk). is a complete vector space and since Kk

39

General theory of holomorphic functions

Kk c int Kk+l

f o r a l l k c N it follows that (fm)m converges com-

pactly to a continuous function f. hence f c Hol(G) by (2.1.7). So

Hol ( G I is a complete metrizable topological algebra. Such algebras are called Frechet algebras ( cf. (3.1.5)).

(2.1.12) Since the coordinate functions are holomorphic. we find f o r each pair of points a.b c G. a+b, an element f c Hol(G) w i t h f(a)*f(b). We say that Hol(G) separates the points of G. Pointseparating subalgebras of C(X), X a a-compact and locally compact space, which are complete with respect t o the compact open topology w i l l be called uniform Frechet algebras, cf. (4.1.2). Hence Hol(G) is a uniform Frechet algebra.

(2.1.13) Using the maximum principle we show that Hol(G1 i s not a Eanach algebra. For simplicity assume that G i s a domain. Recall that a subset E of a topological linear space i s called bounded if f o r each neighbourhood U of zero there exists p>O such that

B c pU = {pf

:

f c U).

Suppose that Hol(G) is a Eanach algebra, i.e. there is a norm

II*II

which generates the compact open topology. Set

E = (f c Hol(G)

:

11f11

(

1).

Then B is a bounded subset o f Hol(G). Since B is a neighbourhood of zero, there exists a compact set K c G and 00 so that

according t o the definition of the compact open topology. Choose a compact set L c G with K c i n t L and l e t f nonconstant function, w.1.o.g. mum principle. For n c N set

Ilfll, =

1. Then

llfll,

)

c

Hol(G) be a

1 by the maxi-

40

Chapter 2

gn = f"

- a/2.

Then gn c B f o r a l l n c N by (*I. Now consider

w

= {f

IIfII,
O such that B C pW. since llgnll, tends t o = for n -> =. Hence B cannot be bounded. (2.1.14) Montel's Theorem. Let F

c

Hol(G) be a bounded set. 1.e. for

each compact set K c G there exists M, sup

{IIfll,

:

f s

FI

6

M .,

> 0 such that

Then F is a relatively compact subset of

Hol (GI. Remark.A Frechet space is called Montel space if each bounded subset is relatively compact. So Hol(G) is a Montel space. Since the unit ball in a normed space

E is relatively compact iff E is finite dimensional, we conclude that there are no infinite dimensional normed Montel spaces. We could have used this observation t o show that Hol(G) i s not a Banach al gebra. (2.1.15)We use Montel's Theorem t o show that Hol(G) Is a Schwartz algebra. We s ta r t w i t h the definition of a Schwartz space. Let E be a Frechet space and l e t (pJn be a sequence of seminorms which defines the topology o f E. W.1.o.g.

we can assume that

pn(a) s pn+,(a) for a l l a c E and a l l n e N, cf. [KOT] p. 205. Set Nk = {a

E

E

:

Pk(a) = 0 )

and l e t Ek be the completion of E/Nk wlth respect to the norm

General theory of holomorphic functions Each Ek is a Banach space. Since pk

41

pk+l we get f o r m > k natural

linear maps @k,:

> -, Ek, a + N

E/N,->

a + Nk '

is continuous we can extend it to Em.

Since @ k,,

We denote the extended map again by @k,. chet Schwartz space i f f f o r each k

E

E w i l l be called a F r 6 -

N there is m(k) > k such that

@m(k).k is a compact operator, i.e. the image of every bounded subset of Em(k) is relatively compact in Ek.

Cn be an open set, Choose a compact exhaustion

Now l e t G

C

... c

Km+l

K,

c

c

defines (II-II Im Km

...

of G so that Kmc int Km+l

for a l l m

a

N. Then

the topology o f Hol(G) by (2.1.11).

CI aim: @ k +,k l : Hol(G)k+l

Hol(GIk i s a compact operator, hence

->

Hol ( G I is a Frechet Schwartz algebra. We f i r s t note that

fl

Int K k + l

6

Hol(int Kk+l) for a1 I f

0

HOl(G)k+l

by (2.1.7). Let (fmIm be an arbitrary bounded sequence in Hol(G)k+l. W.1.o.g. we can assume that each ,f

i s the restriction of an ele-

ment o f Hol (GI. Then a subsequence (fmlllntKk+l 1I converges compactly to f a Hol(int Kk+l)

by Monte1 's theorem. Hence @k+l,k(fml), converges

in Hol(G)k and our claim i s proved.

(2.2)Anal v t ic continuat ion (2.2.1) We now turn to a phenomenon in the theory of several complex variables, which is unknown in the one dimensional theory. As was f i r s t recognized by Hartogs there are domains G in 4:

such that

each holomorphic function on G can be extended t o a holomorphic

42

Chapter 2

function on a larger domain. As for B-algebras we shall introduce the notion of the spectrum of the algebra Hol(G) and discuss the connection with the extension problem. Consider the domain X = C2\{Ol. Let f tion. For (tl,z2)

Do,l = {(z,w)

E

:

E

Hol(X) be an arbitrary func-

i z i < l , i w i < l l set

Then F c Hol(Do,,) since the integration and differentiation with respect t o 7 (j=1.2) may be carried out in any order. By the Cauchy

I

integral formula for holomorphic functions of one variable F = f on the set Do,, \I(z.w)

2-01.

Hence F = f on Do,,\IOl

?(z) =

I

by (2.1.6) and we see that

F(z)

if

z

E

Do,l

f(z)

if

z

E

C2\D0.,

defines an element of Hol(C2 1 such that

31c2\101 = f.

Denote by r the restriction mapping r

:

H~I(C 2 1 ->

HOI(C2\to,),

f ->

flc2\(01.

Then r is a continuous algebra homomorphism. It is surjective, as noticed above, and it is injective by (2.1.6). Hence r is a topological and algebraical isomorphism by the open mapping theorem for Fr6chet spaces (cf. B 1). We say that a domain

G

containing G is a holomorphic extension of

G if the restriction mapping r is a (topological) isomorphism. It is natural t o ask whether t o a given domain G there exists a

"largest" holomorphic extension

G

(the holomorphic envelope of GI,

resp. to characterize those domains G C Cn which admit no proper

43

Holomorphic extension holomorphic extension .Such domains are called domains of holomorphy. Remark. Each domain G

c

C is a domain of holomorphy.

(Consider the set S = f l / ( z - x )

:

x a dGI

c

Hol(G). Let

5

0 G be an

arbitrary domain containing G. Then clearly S is not contained in r(Hol(g)).)

(2.2.2 1 Before we answer the questions posed above we r e c a l l the definition of a complex manifold. Let U

C

Cn be an open set. We say that a mapping

f = (f,,...,f,):

Cm i s holomorphic if fi

U ->

E

,...,m.

Hol(U) for i = 1

Definition. a) Let X be a Hausdorff space. Then X is called a complex manifold of dimension n i f there i s a family (Uiscpi,Vi)iar

is an open covering of X and each Vi is an open sub-

i)(Ui)iaI set o f

such that

cn. Vi is a homeomorphism for a l l i

ii) ‘pi: Ui ->

cp (U 1 -> I ii 0 for i,j

iii) piocp;’:

*

U =U nu ii i i atlas for X.

E

I,

cp (U 1 is a holomorphic map.whenever I il E

1. The system (Ui.ipI,Vi)iEI

is called an

b) A complex structure on X is a maximal atlas A = (Ui,cpi,Vi)iEr. Elements of A are called coordinate systems. c) A function f : X -> each x x

E

E

C is called holomorphic provided that for

X there exists a coordinate system (UI”pi.Vi)

Ui and focpi-’

E

Hol(VI)

such that

. The set o f a l l holomorphic functions on

X w i l l be denoted by Hol(X1. d) Let X resp. Y be an n-dimensional resp. m-dimensional complex

(cjs;j,?j)jaJ.

manifold w i t h atlas (Ui,cpisVi)iaI

resp

A continuous function f : X ->

Y is said t o be holomorphic if

44

Chapter 2

is holomorphlc for a l l i c I, j

L

J whenever f

-1

nd

(Uj)flUi

*

0

.

e l A connected complex manifold X is called Riemann domain over 43" if there exist p1....#pn c Hol(X) so that the map

43", x

p:X-->

-'

(p,(x)

#...#

pn(x))

is a local homeomorphism. I n this case p is called realization map.

Remarks. i) Let U c Cn, W

c

a biholomorphic map f : U ->

Cm be open subsets. Suppose there Is

W, then n = m. It follows that the

dimension of a complex manifold i s well-defined. ii) If (X,p) is a Riemann domain over Cn then the set o f all triples

(U,plu.p(U))

such that plu i s a homeomorphism is an atlas f o r X .

Note that in particular each domain of Cn is a Riemann domain over

a!". iii) We shall only consider complex manifolds X which have a count-

able topology and which are holomorphically separable. i.e. f o r each pair of distinct points x.y c X there is f c Hol(X) so that f ( x ) * f ( y ) . In this case Hol(X) i s again a uniform Fr6chet algebra, when endowed with the compact open topology and (2.1.6)

-

(2.1.15) remain valid.

If X i s connected and holomorphically separable then X has a countable topology, c f . [K/K]

p. 228. In particular each holo-

morphical l y separable Riemann domain has a countable topology. Every domain G

c

Cn i s holomorphically separable since distinct

points o f G can be separated by the coordinate functions. We remark that not every Rlemann domain over Cn Is holomorphlcally separable. iv) By (2.1.8) a l l holomorphic functions on a compact complex mani-

f o l d are necessarily constant, hence Riemann domains over Cn cannot be compact spaces. (2.2.3) We return t o the questlons posed in (2.2.1). As known from

45

Holomorphic extension

the one variable theory there are holomorphic functions on domains

G

c

4: which have no largest domain of definition in C. Take for ex-

ample f i o n

C\{Re(z)aO, Im(z)=Ol. This leads t o the definition of

Riemann surfaces. For n > l it may even happen that every holomorphic function on a domain G

C

Cn can be extended t o a holomorphic func-

tion on a Riemann domain over Cn. which cannot be realized as domain in

c".

For an example cf. for instance CG/Rl. p. 43. Hence it

makes sense to consider the extension problem in the class of Riemann domains over Cn. Definition a) A Riemann domain (Y,p) over Cn is a holomorphic extension of a Riemann domain (X,q) over Cn i f there is a holomorphic map u : X ->

Y so that

is bijective. b) ARiemann domain (E(X1.G) over Cn is called an envelope of holomorphy of (X.p) i) i f there exists u : X ->

E(X), so that (E(X1.G) is a holomorphic

extension of X with respect t o u ; ii) if

B: X

->

Y defines another holomorphic extension of X. there

is a uniquely determined holomorphic map y : Y ->

E(X) so that the

diagram

X

a

is commutative. iii) (X,lr)

is called a domain of holomorphy if X = E(X).

Remarks. i) The envelope of holomorphy E(X) is a holomorphic extension of every other holomorphic extension of X. The uniqueness of y guarantees that E(X) is uniquely determined up to biholomorphism. ii) If Y is a holomorphic extension of X via the map a. then a* is a

46

Chapter 2

topological algebra homomorphism by the open mapping theorem f or Fr6chet spaces. iii) C2 is a holomorphfc extension o f C2\{Ol via the map

a

:

C ~ \ { O I->

~ 2 x,->

x,

as was proved in (2.2.1). i v ) We shall consider in (4.2) the extension problem in the general

setting of uF-algebras.

(2.2.4) It was shown by Thullen that every Riemann domain (X,a) over Cn has an envelope of holomorphy E(X). One is inclined to ask how t o construct E(X). This is the point where the spectrum o f the algebra Hol(X) comes into account. Let Y be a holomorphic extension of the Riemann domain (X.p) over Cn via the map a. Then fo r each y a Y

a

Y

:

Hoi(X1 ->

C , f ->

(a*)-'(f)(y),

defines a continuous nonzero algebra homomorphism on Hol (XI. Hence w e get a map from Y Into M(Hol(X)), the space of a l l nonzero continuous complex-valued algebra homomorphisms on Hol (XI, by Y ->

M(Hol(X)) , y ->

a Y'

and it is reasonable t o conjecture that E(X) = M(Hol(X)) if we can give M(Hol(X)) the structure of a Riemann domain over Cn. Definition. i) Let X be a holomorphically separable complex manifold with a countable topology. Then we denote by M(Hol(X)) the set of a l I nonzero compiex-val ued continuous algebra homomorphisms on Hol(X1. We endow M(Hol(X)) with the relative weak* topology.Note that M(Hol(X)) is a subset of the topological dual space of Hol(X). ii) Let f

E

Hol(X). then

P: M(Hol(X)) ->

C , cp ->

cp(f1,

47

Hol omorphic extension is called the Gelfand transform o f f. The algebra of a l l Gelfand transforms of Hol(X) w i l l be denoted by Hol(XlA.

Remarks. i) The weak* topology on M(Hol(X)) i s the weakest topoh

logy such al I elements of Hol (X)

become continuous functions.

ii) The map j with

(px(f)

j*

:

X ->

M(Hol(X)), x ->

= f ( x ) for a l l f :

Hol (XIA ->

t

9,.

Hol(X) is injective and continuous and

Hol (XI. f ->

A

f 0 j.

is a topological algebra homomorphism, where Hol(X)"

is endowed

with the compact open topology. The inverse mapping is given by Hol(X) ->

P. cf.

A

Hol(X) , f ->

(4.1.3).

(2.2.5) Let (X,p) be a Riemann domain over C" (with countable topology). Recall that if p = (p

,,...,p,).

then pi

E

Hol(X) for i=1,

was shown by Igusa/Remmert/Iwahashi/Forster

...,n.

It

that E(X) is topolo-

gical l y just the spectrum M(Hol(X)). Rossi introduced an analytic structure into M(Hol(X)) making it to a Riemann domain over Cn. The realization map is given by

p^ where

6,

:

M(Hol(X)) ->

Cn.

(p

->

A

(~l((p)....,pn((p)~,

denotes the Gelfand transform of pi (i=l, ...,n).

With this structure we have M(Hol(X)) = E(X). Moreover the holomorphic maps on E(X) are just the Gelfand transforms of Hol(X). So we have answered the f i r s t question o f (2.2.1).

(2.2.6) We turn to the second question: Which properties characterize the Riemann domains (X,rr) which agree with their envelope of hol omorphy?

40

Chapter 2

Defintion. Let X be a complex manifold.

i) X is called holomorphically convex provided that for each compact subset K the holomorphically convex hull RHol ( X I

= {x

E

X

:

If(x)l s

IIfII,

for a l l f c Hol(X)l

is a compact subset of X. ii) X is called Stein (manifold) iff X is holomorphically separable and

holomorphical l y convex. Remarks. 1)Let

Kc

Cn be an arbitrary compact subset.Then R H o l ( C n )

is a closed subset of Cn by the definition. Choose r > O so that

K c Do,r = t(xl and l e t y = (yl

,...,yn)

,...,xn)

E

C : Ixll IIz,II, where z1 denotes the f i r s t coordinate 1 1 c Do,r a 1..' RHoi(X) is a compact subset of function, hence

RHO,

41" and Cn Is holomorphically convex. It was proved by Oka that the envelope of holomorphy o f a Riemann

domain over Cn is Stein. Vice versa, if (X,rr) is Stein then E(X) = X. Even stronger: If (X.A) is holomorphically convex, then (X,lr) is holomorphical l y separable, hence Stein, and E(X) = X. Remark. It follows from this theorem and our earlier observations that C 2 is the envelope of holomorphy o f 4:2 \to}. (2.2.7) Theorem. Let (X,n) be a Riemann domain over Cn. Then the following statements are equivalent: i) X is a domain of holomorphy;

ii) X i s t Stein; iii) X is holomorphical l y convex; iv) the natural map j: X ->

M(Hol(X1). c f . remark (2.2.4111)

Holomorphic extension

49

is a homeomorphism ; v l f o r every sequence (x m Im without accumulation point

there exists f

c

Hol(X) such that sup{lf(xm)l: m

E

IN) = a;

vi) X is holomorphically separable and f o r every finite set of functions fl.....fr

E

Hol(Xl without common zero on X

there exist gl. ...,gr

E

Hol(Xl such that

r

c

i=l

gifi = 1.

I n this case we say that the weak Nullstellensatz is valid f o r Hol(X1. Remark. We shall consider later the properties mentioned in iii) vi) f o r uniform Fr6chet algebras. c f . chapter 4 and (7.3).

Examples.We give t w o important examples of domains of holomorphy. i) We say that a domain G

c

C n is polynomially convex provided that A

the polynomially convex h u l l K (cf. (1.2.4)iil) set K

c

of every compact sub-

G is contained in G. Since each polynomial (restricted t o G)

is a holomorphic function on G, we see that every polynomially convex domain is in particular holomorphical l y convex. ii) A domain G each

c

61" is c a l l e d a complete Reinhardt domain if f o r

( X l....,~nl E G the

{(y,, ...,y

c

set

c":

,...,n)

1y.1 i ix.1. i = 1 I I

is contained in G. A complete Reinhardt domain is c a l l e d logarithmically convex if the set {(logixli

,...,logixnl):

(xl

,...,xn)

E

G. xi

*

....,n)

0 for i = 1

is a convex subset o f Rn.

Let G be a complete Reinhardt domain. Denote by

6

the intersection

of al I logarithmical i y convex complete Reinhardt domains which con-

tain it. Then

is a logarithmically convex complete Reinhardt do-

main and the Taylor series

Chapter 2

50

o f each f c Hol(G) converges uniformly on compact subsets of an element that

5

r"

c Hol(z1. Since

?lG = f by (2.1.5)

5

to

and (2.1.6) we see

is a hoiomorphic extension o f G. I n fact a complete Reinhardt

domain is a domain o f holomorphy iff it is logarithmically convex resp. iff it is polynomially convex. Flnally we state that the domain of convergence of a power series

C avzv is

a logarithmicai ly convex complete Reinhardt domain.

(2.3)Stein spaces (2.3.1) Next we consider complex spaces, which in general have no longer the structure o f a complex manifold. (To keep the text elementary we avoid sheaf theory throughout this book .I Definition. a) Let G c Cn be a domain. A closed subset X of G is called an analytic subset of G i f fo r each x

E

X there exists a neigh-

bourhood U of x in G and a family F of holomorphic functions on U so that

XnU = WF) = { x c U: f(x) = 0 for a i l f b) A continuous function f : X -> x

E

-

E

C is called holomorphic if for each

X there exists a neighbourhood U of x in G and

- flxn".

that fixn"

F).

r"

c

Hol(U) such

We denote by Hol(X) the algebra o f a l l holomorphic functions on X. Let X

c

G

c

f = (f,,...fm):

Cn and X' c G' c Cm be two analytic subsets and l e t X ->

X' be a continuous function.

Then f is called holomorphic if fi c Hol(X). i = 1,

...,m.

c) Let X be a Hausdorff space. Then X is called a reduced complex space if there exists a family (Ui,tpi,Xi)icI

so that

51

Stein spaces

i) (Ul)lcI is an open covering of X and each Xi i s an analytic

subset of a domain GI c C"i

.

Xi i s a homeomorphism for a l l i

ii) pi: Ui ->

iii) cp 00-1: cp (U 1 -> i i 111

E

I,

cp (U 1 i 11 is a holomorphic function whenever U = U l n U ii I 0 for i,j E I. The algebra of a i l holomorphic functions on X is denoted by Hol(X). Remarks. i) Note that if G

c

*

Cn i s a domain then G i t s e l f is an ana-

lytic subset o f G, hence a complex manifold is in particular a r e duced complex space. ill Consider the analytic subset X = {(z.w) c C2: z-w = 01. Then X

looks in most points

-

precisely in a l l points o f X\{(O.O))

-

like a

complex manifold. This statement remains true for arbitrary reduced complex spaces X. A point x

E

X is called regular if there exists a neighbourhood U o f

x in X such that U is biholomorphically equivalent to a domain in

some Cn, otherwise x is called singular. The set of a l l singular points of X is a nowhere dense (analytic) subset of X. iii)As in the case of manifolds we only consider hoiomorphically separable (analogous definition as in remark (2.2.2)iii)) reduced complex spaces which have a countable topology. I f X is holomorphically separable and connected then X has automatically a countable topology, c f . [K/K]

p. 228. Hol(X), when endowed with the compact open

topology, is again a uniform Fr6chet algebra. We note that the identity theorem (2.1.6) does not hold for arbitrary analytic subsets. For example l e t X be as in ii) and consider the hoiomorphic function z:X ->

C . (x.y) ->

x. Then z vanishes on an

open subset o f X but does not vanish identically. However the identity theorem holds for irreducible analytic subsets. (An analytic subset X c G

c

Cn i s called irreducible if whenever X

Chapter 2

52

i s the union of two analytic subsets X1, X2 then either X = X1 or

x

= X2.)

The analogous statements t o (2.1.71, (2.1.8). remark (2.1.9) and

(2.1.13)

- (2.1.15)

remain true. (A proof of the analogous statement

of (2.1.15) w i l l be given in (2.3.8) below.)

(2.3.2) We shortly consider the extension problem for reduced complex spaces. If we replace in (2.2.6) the term

"

complex manifold"

by "reduced complex space", then we get the definitions of a holomorphical l y convex (resp. Stein) reduced complex space. The results differ essentially from that for Riemann domains over Cn. We f i r s t mention that f o r an arbitrary reduced complex space an envelope of holomorphy does not exist in general ( f o r an example cf. [KRA 51). Even if the envelope o f holomorphy exists it must not be a Stein space. The connection between the envelope of holomorphy

E(X) and the spectrum M(Hol(X)) has not been established in the general case. The theorem of Igusa/Remmert/Iwahashi/Forster

however remains

true for arbitrary reduced complex spaces X, 1.e. X is Stein iff the map

j: X ->

M(Hol(X)), x ->

with qx(f) = f ( x ) for a l l f

E

9,

,

Hol(X),is a homeomorphism. If X i s Stein

then E(X) exists and coincides with X. We cannot discuss the importance of the property Stein in detail but state some results, c f . for example [K/K]

5 57.

(2.3.3) Let A be an analytic subset (analogous definition as in (2.3.1)) o f a Stein space X, and l e t f ment. Then there exists

r"

E

a

Hol(A) be an arbitrary ele-

Hol(X) such that

71,

= f.

53

Steln spaces

(2.3.4) The following statements are equivalent for a maximal ideal m c Hol (XI, X a Stein space: i) m is closed in Hol(X); ii) m is finitely generated; iii) there exists x c X so that m = tf

E

Hol(X): f ( x ) = 01.

It was shown by Cartan that each finitely generated ideal I c Hol(X)

is closed, while the converse statement is false ([FOR 31. p. 314). Opposite t o the situation for B-algebras (cf. (1.2.5)) there are "always" maximal ideals m

c

Hol(X), which are not closed. I n con-

nection with this observation there appears the following question: Is every nontrivial algebra homomorphism

'p:

Hol ( X I ->

C automat-

ical l y continuous? This question can be answered in the positive if

X is a finite dimensional Stein space ( f o r a generalization cf. We shall return t o this question for Frechet algebras

-

[EPH]).

known as

Michael's problem - in chapter 10. (2.3.5) Let Y be a reduced complex space, X a Stein space, and l e t cp: Hol(X) ->

Hol(Y) be a continuous algebra homomorphism. Then

there exists a unique holomorphic map f : Y -> for a l l g

c

X so that 'p(g1 = gof

Hol(X1.

(2.3.6) Let X be a Stein space, and l e t Y

c

X be an open subset

h

which i s Hol(X)-convex. i.e. KHo,(X) is contained in Y for each compact subset K

c

Y. then the restriction mapping

Hol(X) ->

Hol(Y), f ->

fly.

has dense range. ((X,Y) is called a Runge pair.)

(2.3.7)We state an approximation result, cf. [FOR 21. Let X be a Stein space and l e t A

- A-separable.

c

Hol(X) be a subalgebra, such that X is i.e. for each pair of distinct points x. y in X

Chapter 2

54 there exists f

-

E

A so that f(x)*f(y),

A-convex, i.e.

RA = ty

E

X: i f ( y ) i

suptif(x)i: x

E

K) f o r a l l f

i s compact for each compact subset K

-

A-regular, 1.e. for each x

E

c

A)

X.

X there are f;,...,f;

E

A, an

open neighbourhood U of x, and an analytic subset Y of an open set in Cr such that

u ->

Y. t ->

(f;(t),...,f;(t)),

is a biholomorphic mapping.

Then A is dense in Hol(X1.

(2.3.8)Finally

- using (2.3.3)and permanence properties of

Schwartz spaces

-

we show that Hol(X1 (X a reduced complex space)

is a Schwartz space.

We mention that subspaces. quotients and products of Schwartz spaces are again Schwartz spaces (cf. [JAR]). First l e t X be an analytic subset of a polycylinder P

c

Cn. One easily

proves that X is holomorphically convex and hence Stein. Set

I = tf

E

Hol(P): fix = 0 )

Then I is a closed subspace of Hol(P) and Hol(P)/I becomes a Fr6chet space when endowed with the quotient topology. Hence Hol(P)/I is a Fr6chet Schwartz space. The map Hol(P)/I ->

fl,

Hol(X), f+I ->

is a continuous and injective linear mapping. It is also surjective by

(2.3.3)and hence a topological isomorphism by the open mapping theorem for Fr6chet spaces. We conclude that Hol(X) i s a FrBchet Schwartz space. Now l e t X be an arbitrary analytic subset of a domain G each x c X choose a polycylinder Px Hol(XnP,)

c

Cn. For

G such that x

E

Px. Since

is a Fr6chet Schwartz space for every x

E

X the same i s

c

55

Stein spaces true for the product

n Hol (XnP,).

Since we can represent Hol(X)

xEX

as a closed subspace of n H o l ( X n P x ) , via the map x EX

Hol(X) is a Fr6chet Schwartz space. too. Let X be a reduced complex space, and l e t (Ui,qJl,Xi)iCl

be as in de-

finition (2.3.1)~). Then Hol(X) can be represented as a closed subspace of the Schwartz space nHol(Xi) via the map id Hol(X)->

nHol(XI), f -> it1

(fa

q~;’),~~

.

Hence Hol(X) is a Fr6chet Schwartz space. Remark. In a similar way one can prove the stronger statement that Hol(X) is a nuclear space (for the definition c f . [JAR]).

This Page Intentionally Left Blank

57

PART 2 GENERAL THEORY OF FRECHET ALGEBRAS

This part is devoted t o the introduction of FrQchet algebras and uniform FrQchet algebras. Important examples of (uniform) FrQchet a l gebras are

-

Hol(X), the algebra of a l l holomorphic functions on a hemicompact

reduced complex space;

- C(X),

the algebra of a l l continuous functions on a hemicompact k-

space. It is an important tool in our theory t o represent each FrQchet alge-

bra as the projectlve limit of a sequence of Banach algebras. This representation enables us to prove many theorems f o r FrQchet algebras which are known for Banach algebras, e.g. the theorem of Gelfand-Mazur, the holomorphic functional calculus, Shilov's idempotent theorem etc.. However the question as whether each mu1tiplicative linear functional on a Fr6chet algebra is necessarily continuous is s t i l l unanswered. Hence in some cases we have to develope new proofs for theorems, which in the Banach algebra case are a consequence of the automatic continuity of characters. For example we mention the proof of the uniqueness o f topology for semisimple Fr6chet algebras. One significant dlfference appears in connection with the fact that the spectrum of a Frechet algebra in general is no longer a compact but a hemicompact space. The topology of the spectrum is not quite well understood. For example there are FrQchet algebras whose spectrum is not a k-space. We also mention that Rossi's maximum modulus principle f a i l s to be true for Frechet algebras.

This Page Intentionally Left Blank

59

CHAPTER 3 THEORY OF FRECHET ALGEBRAS, BASIC RESULTS

We always consider associative and commutative algebras over the field o f complex numbers. Moreover and (3.2)

- we

-

with the exception of (3.1)

shall assume that a l l considered algebras have an

identity 1. (3.1) Fr6chet algebras

I n this section we introduce Fr6chet algebras. By a Frechet algebra A we mean an algebra which space

-

-

considered as a topological vector

is complete metrizable and which has a neighbourhoodbasis

(UiIicN of zero consisting of convex sets Ui such that Ui-Ui all i

t

c

Ui f o r

N. It turns out that the topology o f A can always be defined

by a sequence of multiplicative seminorms. At the end of this section we shall discuss several examples. In particular we shall deal with the question when C(X) - the algebra of a l l continuous functions on a space X. endowed with the compact

open topology

-

is a Fr6chet algebra.

(3.1.1) Recall that by a Banach algebra A we mean a Banach space

A which is also an algebra, in which multiplication and the norm are Iinked by the inequality

IIfgll

Ilfllllgll. f.g

f

A.

Before we generalize this notion in a suitable way for Fr6chet algebras, we r e c a l l some definitions and results which can be found in

Chapter 3

60

any textbook on topological linear spaces, for instance in the books of Kbthe [Kt)TI or Jarchow [JAR]. Definition. Let E be a complex vector space. A subset M c E i s said t o be convex (resp. absolutely convex) if whenever it contains x and

y it contains a l l

TX t

py where t,p

0 and t+p=l (resp. ~ , Ep 4: and

2

i t i t l p l s l ) . T h e absolutely convex h u l l r(M) of a set M c E is the intersection of a l l the absolutely convex sets which contain M. Remark r(M) consists of a l l the terms of the form

n

C lali

1, xi

G

E

cin= lalxi,

al

E

61,

M ([JAR] p. 101, p. 102).

i=l

(3.1.2) Definition. A complex vector space is called topological vect o r space if a Hausdorff topology

is defined on E such that the

t

mappings

E x E ->

E, (x,Y) --+ X+Y,

C x E ->

E, (X,X) ->

and

AX,

are continuous. Remarks. i) If E is a topological vector space, x c E and @ = {U), is a basis of neighbourhoods of zero, then the sets x+Ua. U,

E

@,

form a basis f o r the neighbourhood f i l t e r o f x. ii) A topological vector space is said to be locally convex if it has

a bask of neighbourhoods @ = {U,)

of zero consisting o f convex sets

uciill) A subset M of a vector space E is said t o be absorbent, if for each element x

E

E a suitable multiple px, p>O. lies in M. Note that

every neighbourhood o f zero in a topological vector space is absorbent.

61

Fr6chet algebras (3.1.3)Definition. Let E be a vector space. A map p: E

-*

c0.a~)

is called seminorm if i) p(x+y) s p(x)+p(y) for a l l x,y ii) p(Xx) = IXip(x) for a l l x

Q

E

E, X

E; E

C.

If p i s a seminorm. then the set = t x E E : p(x) s 1) vP is an absolutely convex and absorbent set. Vice versa if U

c

E is an

absolutely convex and absorbent set, then the gauge functional pu(x) = inftp>O: x

pU1

E

defines a seminorm on E ([KOT], p. 182). (3.1.4)Definition. i) Let A be an algebra. A seminorm p: A ->

C0.a~)

is said to be (sub-)multiplicative if p(xy) s p(x)p(y) f o r a l l x.y

E

A.

ii) A subset U c A is called multiplicative if U2 = UU

Remark.Let pl. ...,p,

A->

c

U.

be multiplicative seminorms on A. Then

..

sup{ p, ( x 1 , ..pn (x 11

[ O , O D ) , x ->

defines a seminorm on A which is easily seen t o be multiplicative. The next proposition gives the connection between mu1tip1 icative sets and multiplicative seminorms. Proposition. i) I f p is a multiplicative seminorm on A, then Vp = tx

E

A: p(x)

5

1)

is an absorbent, absolutely convex and multiplicative set. ii) If U is an absolutely convex, absorbent and multiplicative set then

pu(x) = inf{ p>O: x

E

pU)

defines a multiplicative seminorm.

62

Chapter 3

Proof. i) By (3.1.3) we only have to show that V This is the case, since if x.y p(xy) s p(x)p(y)

5

B

V

P

is multiplicative.

then

P'

1.

ii) By (3.1.3) it remains to show that pu is multiplicative.

Let x,y

E

A and a,b>O such that x xy a abU2

aU and y

c

E

bU. Then

abU.

c

hence pu(xy) s ab. and the r e s u l t follows. (3.1.5) Definitlon. i) A topological algebra is an algebra, which is a topological vector space, such that the mu1tiplication

A x A ->

A, (a.b) ->

ab

is a continuous mapping. ii) A topological algebra is called a locally mu1tip1 icatively convex algebra (LMC-algebra) if there is a basis of neighbourhoods o f zero consisting of mu1tiplicative and convex sets. iii) A Frgchet algebra (F-algebra) is an LMC-algebra which is more-

over a complete metrizable topological vector space. Remark.Since we want t o concentrate on F-algebras in this book we do not deal with the more general concept of LMC-algebras. We r e f e r the reader, who is interested in the theory of LMC-algebras, t o the books of Beckenstein e t al. [BNS], Mallios [MALI and Zelazko [ZEL 11. (3.1.6) We now want t o show that the topology o f an F-algebra can be generated by a sequence of mu1tiplicative seminorms. i) Let A be an algebra, and l e t (pnIn be a sequence of multiplicative

seminorms on A such that a) pn(x) L pn+,(x) for a l l n a N, x

a

A;

Fr6chet algebras b) for each x

c

A. x

* 0,there is n

c

63

N such that pn(x)

*

0.

Then (pnIn generates a topology on A in the following way. For x n

E

c

A,

N set Un(x) = {YEA: pn(x-y) < l / n l .

Then @=(Un(x)), is a basis of neighbourhoods of x for a (locally convex) topology on A and A is metrizable ([JAR].

p. 40). By proposition

(3.1.4) each Un(0) i s an absolutely convex and multiplicative set. I t is easy t o show that A is an LMC-algebra. i.e. the multiplication i s continuous, so A becomes an F-algebra i f A is complete. ii) On the other hand every F-algebra arises in this way. Let A be an F-algebra. Since A is metrizable we can choose a basis of neighbourhoods (UnIn of zero consisting of multiplicative and convex sets (cf. [JAR], p. 40). Fix n

E

N. Since A is in particular a lo-

cally convex space we can choose an absolutely convex neighbourhood of zero such that W Uk

c

W. Let x.y

c

c

r(uk) be

Un ([JAR].

p.108). Choose k

c

N so that

two arbitrary elements o f the absolutely

convex hull of Uk. i.e.

I t follows that r ( u k ) i s multiplicative, since

Since r ( U k )

...

3

wk

3

c

W, A has a countable basis of neighbourhoods of zero

Wk+l

3

.... consisting of

absolutely convex and multipli-

64

Chapter 3

cative sets. By proposition (3.1.4) and (3.1.2)iii) every sponds with a multiplicative seminorm p

wk'

because Wk hence p

wk

for a l l k

* P%+,

PWk

3

Wk+,. I f x

E

A, x

E

wk

corre-

We have

N

*

0, there is k such that x

E

A\Wk,

(x) > 0. Finally one sees that the semlnorms (pwk) k ge-

nerate the original topology of A. (3.1.7) Let A be an algebra with identity 1, and l e t (p,), of multiplicative seminorms on A so that pn(l) qJf)

be a sequence

* 0 for a l l n

E

N. Set

= sup{pn(fg): p&g) = 11,

then each q, defines a seminorm on A and qn(l) = 1. Moreover qn(f)

pJf)

and pn(f) c pn(l)qn(f) for a l l n

E

N. f

E

A,

since pn(f.l/pn(l))

* qn(f).

It follows that (qnIn generates the same topology on A as (p,),.

Each qn is multiplicative, since for f,g,h

E

A

pn(fgh) i pn(gh)qn(f) , hence qn(fg) = sup{pn(fgh): pn(h) = 1) i

sup{qn(f)pn(gh): pn(h) = 1)

* qn(f)qn(g).

By remark (3.1.4) q',(f 1 = sup{ql (f1

,...,qJf

1)

defines a multiplicative seminorm on A. Clearly qi

q;l+l,

for n

E

N.

65

Frechet algebras It follows from [KOT]. p. 203 that (q;ln

on A as (q,),.

defines the same topology

Hence the following definition makes sense.

Definition. Let A be an F-algebra. By a generating (or defining) sequence of seminorms for A we mean a sequence of multiplicative seminorms (pnIn on A which generates the topology of A (in the above described way) such that pn s P,+~,

for a l l n

E

El,

and pn(l) = 1, for a l l n

E

N. if A has an identity.

(3.1.8) There is a canonical way of adjoining an identity to an F-algebra A. We denote by A* the direct sum A* = COA = t(X,a): X

E

C, a

E

A)

and define addition and multiplication in A* by

I f (pnIn is a generating sequence of multiplicative seminorms for A then we endow A* with the topology which is generated by the multip l icative seminorms qn(X.a) = pn(a)+lXI, n

E

N.

I t is easily shown that A* becomes an F-algebra and that (1.0) is an

identity. We shall r e f e r to A* as the algebra obtained by adjoining an identity to A. (3.1.9) Examples. i) Of course each B-algebra is an F-algebra. ii) Denote by C"([O,l])

the algebra of a l l infinitely differentiable

functions on the unit interval with pointwise operations. For n define

E

N

Chapter 3

66

pn(f) = 2n-1sup{if(k)(x)i: x

k = 0,1,...,n-1l,

[0,1],

E

where f ( k ) denotes the k-th derivative of f. It can be verified that each pn is a seminorm. By using Lelbniz's r u l e f o r computing the k-th derivative it can be shown that pn is moreover multiplicative. Clearly pl(f) = 0 iff f

*

0. We endow C"([O.l])

with the topology

which is generated by the sequence (p,),. Let (fmIm be a Cauchy sequence in C"([O,l]),

i.e. (fmIm is a Cauchy

sequence with respect to each seminorm pn. It follows for n=l that

(fmImconverges uniformly on [ O , l l to a continuous function g. Since (fmImis a Cauchy sequence with respect t o p.,

the sequence of the

(1) Im derivatives (fm converges uniformly on [O,l]

t o a continuous

function g1 and

-

as is well known

-

g is differentiable and g(')=gl.

Continuing in this manner we see that g converges uniformly on [O,l]

E

C"([O,l])

and (f, (k) Im

t o g(k)s i.e. the topology generated by

the sequence (pnIn is complete, i.e. C"([O,l]) We want to show that C"([O,l])

becomes an F-algebra.

endowed with the above intro-

duced topology is not a B-algebra. Recall f i r s t that a set U in a topological vector space E is called bounded if for each neighbourhood V of zero there is p > O so that U c QV. If

{f

E

E:

(E,II-II)

IIfll
and

l/k~2~k~10~n~k~1~sin~10nx~.

67

Fr6chet algebras i.e. fn

E

U for a l l n

E

N, but

(pl(fn))nEN i s unbounded for every I>k.

Hence U cannot be a bounded neighbourhood of zero, since f o r example there exists no p>O such that

iii) The algebra C(X).

Let X be a Hausdorff space. We denote by C(X) the algebra o f a l l continuous complex-valued functions on X (with pointwise operations).

If K

c

X is a compact subset we define

Ilfll, Obviously

= suptlf(x)i: x

11-11,

E

K).

is a multiplicative seminorm on C(X). We always en-

dow C(X) with the topology which is generated by the seminorms

(ll*llK)K=x compact

*

It is called the compact open topology or topology of uniform con-

vergence on compact sets. A neighbourhoodbasis of zero is given by a l l sets of the form

tf

L

C(X):

IIfII, < €1. E>O, K

c

X compact.

We want to determine a class of spaces such that C(X) becomes an F-algebra. We f i r s t r e c a l l some definitions and results.

A topological space X is called regular if for each point x and each neighbourhood U of x there is a closed neighbourhood V of x such that V

c

U. X is called completely regular if f o r each x

E

X and each

neighbourhood U of x there exists a continuous function f on X t o the closed unit interval such that f ( x ) = 0 and f is identically one on

X\U. A completely regular space is regular.

By a normal space X we mean a space such that for each disjoint pair o f closed sets, A and B, there are disjoint open sets U and V so that A

c

V and B

c

U. By Urysohn’s lemma we find for each dis-

joint pair of closed subsets A,B o f a normal space X a continuous

68

Chapter 3

function f on X t o the unit interval such that f i s zero on A and one on B. Hence normal spaces are completely regular.

By a Lindel6f space we mean a topological space so that each open cover has a countable subcover. Each regular Lindel6f space is normal , in particular compact spaces or topological spaces satisfying the second axiom of countability are normal ([KEL], p. 113). Tietze's Extension Theorem. Let X be a normal space, E be a closed subset, and l e t f be a continuous function on E t o the closed interval [a,b]

g(X)

c

c

R. Then f has a continuous extension g to X such that

[a.b].

([KEL] p. 242).

As a consequence of Tietze's theorem we get another extension theorem for completely regular spaces. Theorem

. Suppose K is a compact subset of the completely regular

space X. If f

E

C(K), then there is g

sup{lg(x)i: x ( Cf. [JAR],

E

e

C(X) extending f such that

X I = llfll,.

p.29 1.

A Hausdorff space is called a k-space i f every subset intersecting each compact subset In a closed set is i t s e l f closed. Note that a complex-valued function f on a k-space X is continuous iff it is continuous on each compact subset of X. Examples of k-spaces are locally compact and f i r s t countable spaces ([KEL], p. 231). Definition. A Hausdorff space X is called hemicompact if there is a countable compact exhaustion each compact subset K

c

... Kn

c

X there is n

Kn+l E

c

... of

N so that K

X such that for c

such an exhaustion (Knln admissible. Remark. Each hemicompact space is a Lindel6f space.

Kn. We c a l l

69

Frechet algebras

Theorem. Let X be a completely regular space. Then C(X) i s an F-algebra iff X i s a hemicompact k-space. Proof. i) Suppose X is a hemicompact k-space. Let (KnIn be an admissible exhaustion of X. The compact open topology is generated by the sequence of seminorms ([Kt)T],

(II-IIK n1n,

i.e. C(X) is metrizable

p. 2 0 5 ) . Let (fn),, be a Cauchy sequence in C(X), i.e. (fnIn

is a Cauchy sequence with respect to each seminorm

11-11

C(Km) is complete with respect t o the norm

Km

I.11 . Since nKm

N

verges uniformly on Km t o a continuous function fm N

f: X ->

C . x ->

fm(x). if x

Then f is well-defined and f l m c

N. Since X

c

Incon-

. Set

Km.

.v

Km

I

, (f

Km

=fm, i.e. fl

i s continuous for each

Km

is a k-space this implies that f c C(X).

ii) Now l e t C(X) be an F-algebra.

Since C(X) is metrizable. there is a countable neighbourhoodbasis o f zero, hence there are a sequence of positive numbers (cnIn and a sequence of compact subsets (KnIn such that

(*I

{f

c

C(X):

IIfllKn< cn)

is a neighbourhoodbasis of zero. We claim that (KnIn i s an admis-

sible exhaustion of X. i.e. X is a hemicompact space.

To see this choose an arbitrary compact subset K

c

X. By (*I there

i s n c N such that

v

= tf

6

C(X): IIfllKn < c,)

Suppose there is a point x there is g g

E

c

E

K\K,.

c

w=

{f

E

C(X):

IIfIIK
1/2 for a l l i

E

2

so that l>a,' ai>ai+l

N.

Since K1 is a normal space we find fl

1 2 fl

K1. Choose a de-

E

E

0, fl(x) = 0 and fliEnK,

C(K1) such that

=

1.

Suppose that we have already constructed fl

E

C(Ki) ( 1 4 ,

...,n-1)

such

that

a) fiaO; filEnKl a

y) filKl-,

ai;

- f,-l

for i=2

,...,n-1. fn-l

According to Tietze's theorem we can choose an extension ?o ,f ry

ru

t o Kn such that fn 2 0 . If fn(x)aan f o r a l l x

ru

E

EnKn then set fn=fn.

ry

Otherwise Kn-l

and EnKnfl{xrKn: fn (x)san) are disjoint closed sub-

Frechet algebras

71

sets o f Kn (recall that an-)>an). Hence we find g

C(Kn). 1 2 g

E

2

0.

such that glKn-l

Set fn =

Tn + fn

2

50



C , x ->

fn(x) if x

Then f is well-defined and f n

E

P

E

C(X) by

Kn.

(**I. since fl

Kn

= fn f o r each

N.

Moreover f ( x ) = 0 and f(y)

2

1/2 for y

E

E. Hence {y

E

X: f(y) < 1/41

i s an open neighbourhood o f x which does not intersect E. i.e. E is closed since x was chosen arbitrarily.

0

(3.1.101 Remarks, i) Note that the hypothesis "completely regular space" was only needed t o prove that X i s hemicompact if C(X) is an F-algebra. In particular we have shown that C(X) is an F-algebra if

X is a hemicompact k-space. ii) Let X be a hemicompact k-space. l e t x

E

X be an arbitrary point,

and l e t U be an open neighbourhood of x. Repeating the l a s t part o f the preceeding proof we find f

P

C(X) so that f k O , f(x)=O and f(y)21/2

for a l l yrX\U. Hence (yoX: f(y)r1/41 is a closed neighbourhood of x which i s contained in U, i.e. X is regular, hence X is a normal space since it is also a Lindelbf space ([KEL] p. 113). ill) For a detailed study of the algebra C(X)

-

X a regular space

-

we r e f e r the reader to Beckenstein et al. [BNS].

(3.2) The soectrum o f a Fr6chet alaebra As for B-algebras we define the spectrum M(A) of an F-algebra A t o be the set o f a l l nonzero complex-valued and continuous algebra

Chapter 3

72

hornomorphlsms endowed wlth the Gelfand topology. Recall that the spectrum of a B-algebra (with identity) is a nonempty compact space. The maln re su l t of thls sectlon shows that In general the spectrum of an F-algebra Is not a compact but a nonempty hemicompact space. As a consequence we get Gelfand-Mazur's theorem, 1.e. we prove that an F-algebra which Is a fleld Is lsomorphlc t o C. This enables us to exhibit a one-to-one correspondence between closed maximal

Ideals and the elements of M(A). Finally we determine the spectrum of A*, the algebra obtained by adjoining an Identity.

(3.2.1)Deflnltion.

Let A be an F-algebra.

I) We denote by S(A) the set of a l l nonzero complex-valued algebra hornomorphlsms and by M(A) the set o f a l l continuous members of S(A). We ca ll M(A) the spectrum o f A.

Ill For f

E

A, we define the functlon

P: S(A) ->

C , cp ->

cp(f),

A

f is called the Gelfand transform o f f. We set

111) We always endow S(A) resp. M(A) wlth the coarsest topology

such that a l l Gelfand transforms are continuous functions on S(A) resp. M(A).This topology is called the Gelfand topology. (3.2.2) Remarks. I ) If cp

E

S(A) (resp. cp

E

M(A)), then sets of the

form { @ E S(A): Ig(fl)-cp(fl)l

A* ->

9:,

A.

MA*).

E

S(A) define

E

$: A* and for cp

S ( A * ) \ { ( P ~ ) set

E

cp': A ->

C . f ->

cp(0.f).

It can be easily checked that

for a l l J,

S(A) (resp. cp

E

X+J,(f),

C , (X.f) ->

->

E

$

E

S(A*)\{cpo)

S(A*)\{cpo))

(cp')= cp), i.e. the map ry

J, ->

S(A*)\{cpo).

J,,

is bijective. Let cp

S(A). and l e t n

E

E

N so that

Icp(f)l s pn(f) for a l l f (1.e. cp

E

E

A,

M(A) by (3.2.2)ii)) then

for a l l (X,f)

E

A*. i.e.

Vice versa i f cp Icp(X,f)l

E

3E

M(A*)\{p,).

S(A*)\{cpo) and

< qn(X.f) for a l l (1.f)

E

then Icp'(f)l = lcp(O,f)i

for all f

E

A. hence

s qJ0.f)

E

S(A))

and that ($1' = J, (resp.

m

I: S(A) ->

(resp. cp'

= pJf)

A*,

76

Chapter 3

Finally choose arbitrary fl

,...,fr

I t follows from this and (3.2.211)

E

A, X1

....'1, c 4: and c~ E S(A).

then

that

I M P O R T A N T REMARK. I n the sequel we only consider F-algebras with identity 1. (3.2.5) Let A,B be F-algebras, and l e t T: A ->

B be a continuous

nontrivial algebra homomorphism, i.e. T(1) = 1. The adjoint spectral

Lemma. Let A,B and T be as above. Then i) T* is continuous; ii) T* is injective if T has dense range; iii) ( S o TI* = T*o S* i f

C is another F-algebra and S: B ->

continuous nontrivial algebra homomorphism;

i v l T* is a homeomorphism if T is bijective.

C is a

77

Spectra of Frechet algebras

*

The l a s t set is open in M(B). hence (T 1-1 (U) is open for every open set U

c

M(A) ( cf. remark (3.2.2)i)).

ii) Let ~ * ( c p ) = T*(+) for

cp,g

M(B),

E

then cpo

T(f) =

T(f) for a l l f

+o

E

A.

hence q(g) = +(g) for a l l g

E

B.

since T(A) is dense in B, thus cp =

+.

iii) This is obvious.

iv)By the open mapping theorem for Frechet spaces T is a homeomorphism. We have (T-l)*o T* = so T* is bijective and

(3.2.6)We shall need lemma (3.2.5) to determine the spectrum of an F-algebra. Let A be an F-algebra with defining sequence of seminorms (pnIn ( r e c a l l that pn

pn+l for a l l n

E

N). For each n

E

N denote by In

the ideal

In = ker pn = {arA: pn(a) = 01, and denote by An the completion of the algebra A/In with respect to the norm p;(a+In)

= pn(a),

so An is a B-algebra by definition. Note that An has an identity

since A has an identity by our general assumption on F-algebras. We set K~

A ->

An. a ->

a+$.,

Then nn is a continuous homomorphism which has dense range. hence

Chapter 3

78

n,

* is continuous and injective by lemma (3.2.5).

This implies

-

since

M(An), as the spectrum of a 9-algebra w i t h identity, is compact that

*

*

nn (M(An))

nn: M(An) ->

c

-

M(A)

is a homeomorphism. Let cp be an arbitrary element of M(A), then there exists n

E

N such

that icp(f)i s pn(f) fo r a l l f

E

A (cf. (3.2.2Jii)).

I f we define

3: A/In then

C, f+In ->

->

3 is a well-defined

continuous homomorphism on A/In and if

we denote the unique extension of

*

Itn ($1

Conversely, if

cp(f),

3

to A,

again by

3 then

= cp. e

M(An), then

inn*(cp)(f)i=

icpo

nn(f)i

p;l(f+In) = pn(f).

(for the inequality cf. (1.2.1)) hence

(*I

= { c p ~ ~ icp(f)i ) : s pn(f) for all f

~;(M(A,)I

E

A)

so

*

nn (M(An))

c

*

nm (M(Am)) for man

and

MA) =

u I~;(M(A,,)). neN

(3.2.7) In the sequel we always identify M(An) with n,*(M(A,,)). (3.2.6) we already know that

...M(An)

c

M(Ant,)

c

By

... i s a countable

compact exhaustion of M(A), 1.e. M(A) is a-compact. I n fact we can improve this result. Recall that a countable compact exhaustion

...K,

c

K,

c...

of a

79

Spectra of Fr6chet algebras

Hausdorff space X is called admissible provided that for each compact set K

c

X there is n

E

N such that K

c

Kn. A space with an ad-

missible exhaustion is called hemicompact. Remark. There are a-compact spaces which are not hemicompact. Take f o r instance Q . the set of a l l rational numbers, endowed with the induced euclidean topology. Q is clearly a-compact and satisfies the f i r s t axiom of countability. Since hemicompact spaces which satisfy the f i r s t axiom of countability are locally compact. cf. (4.4.11, Q cannot be hemicompact. We want to prove that (M(An))n is an admissible exhaustion of M A ) . To show this l e t K cp

E

K the set { f

E

c

M(A) be an arbitrary compact set. For each

A: itp(f)i s 1) is closed, hence

is closed, too. Since

-

by the definition of the Gelfand topology

transform

?

is continuous on M A ) ,

?

- each Gelfand

is bounded on the compact set

K. hence W

A = U nKo. n=l By Bake's theorem (cf. 8.3) there is n KO

E

N such that nKo

- has nonempty interior. So we can choose

such that

hence

From this follows

k

E

-

and thus

N, E>O and g

E KO

80

Chapter 3

We can now summarize our considerations above. (Recall once more that we consider only algebras w i t h identity.) (3.2.8) Theorem ([MICI). Let A be an F-algebra. and l e t An be defined as above. Then (in a natural) sense M(A) = U M(An) ncN and (M(An)In is an admissible exhaustion of M(A), 1.e. M(A) Is a hemicompact space. Corollary. The spectrum of an F-algebra is never empty. Proof. M(A,) is not empty by (1.2.8).

0

(3.2.9) As an immediate consequence we get the theorem of GelfandMazur f o r F-algebras. Theorem. If A is an F-algebra which is a field, then A = C. Proof. The complex multiples of the identity form a subalgebra of A which i s isomorphic t o C. It suffices to show that every f

E

A is a

complex multiple of the identity. Let f

E

A. By the corollary M(A) is not empty. Choose an arbitrary

(Q I

M.(A), then f-cp(fl.1 is not invertible. (Otherwise there exists

g

A such that

E

Spectra of Frechet algebras

81

This i s absurd.) Since A i s a field we have f-cp(fl.1 = 0.

0

(3.2.10) Let A be an F-algebra, whose topology is defined by the Let 1 be a proper closed ideal of A.

sequence of seminorms (p,),.

Then A / I endowed with the quotient topology is a Frgchet space and the topology is defined by the seminorms qn(f+I) = inf{pn(f+g): g

E

1) (cf. [JAR], p. 76).

It is easy to show that multiplication is continuous, i.e. A / I is a

topological algebra. Let f,g p,(f+hl)pn(g+h2)

2

E

A, hl,h2

E

I be arbitrary elements. Then

pn(fg+fh2+ghl+hlh2).

It follows that

qn(f+I)qn(g+I) k qn(fg+I). since fh2+ghl+hlh2

E

1.

i.e. each qn is multiplicative and A / I becomes an F-algebra. Next we want t o determine the spectrum of A/I. Set

V(I) = {cp

E

M(A): cp(f) = 0 for a l l f

E

I).

Since the natural map j: A ->

A/I, f ->

f+I,

is a surjective and continuous homomorphism, the map j*: M(A/I) ->

M(A). cp ->

cpo

j

i s continuous and injective (lemma (3.2.5)). We have

j*(y)(f) = 0 for each f hence j*(M(A/I)) c V(1). For cp

E

V(1) define

E

1 and each y

E

M(A/I)

82

Chapter 3

p: A / I

C, f+l ->

->

3 i s a well-defined

Then

cp(f).

homomorphism. By (3.2.2)ii) there is n

c

N

so that icp(f)i s pn(f) for a l l f

Let h

E

E

A.

1 be an arbitrary element. Then iq(f+I)i = icp(f)i = icp(f+h)i s pn(f+h),

f or each f c A, hence i@f+I)i s qn(f+I) f o r a l l f thus

3 E M(A/I)

E

I,

by (3.2.2)ii). Clearly j*@) = cp, hence

j*(M(A/I)) = V(I). Using (3.2.211) It is easy to show that

j*:M(A/I) ->

V(I)

is in fact a homeomorphism, V(I) endowed with the relatlve Gelfand topology. (3.2.11) An ideal I of A is called maximal If I * A and I is contained In no other proper ideal of A. I is maximal iff A / I Is a field. Hence

- by

the theorem of Gelfand-Mazur

is an F-algebra and I

c

-

A/] is isomorphic t o C. if A

A Is a closed maximal ideal. So we can inter-

pret the natural projection j: A ->

A / I as an element of M(A) with

kernel I. On the other hand ker cp is a maximal closed ideal of A if cp

I

M(A).

This proves the following theorem. Theorem

. The correspondence cp ->

ker

(9.

Is a one-to-one corres-

pondence between the elements o f M A ) and the maximal closed ideals In A. Remark. Every maximal ideal in a B-algebra is closed (1.2.51, hence

Spectra of Frbchet algebras

83

there is a one-to-one correspondence between the maximal ideals of A and the elements of M(A) in this case (1.2.6). Opposite t o this result maximal ideals of F-algebras are in general not closed, as we shall see next. Example. We have M(C(R)) = R by example (3.2.2). 1.e. each maximal closed ideal m

I , = {f for some x

E

E

c

C(R) is of the form

C(R): f ( x ) = 0 )

R by the theorem above. Consider the following idea

I = {f

E

C(R): There exists k

E

N so that f(x) = 0 for xkk

By Zorn's lemma 1 i s contained in a maximal ideal m. Since there exists no x

E

R so that I

c

I,, the same is true for m. i.e.

m is not closed.

(3.2.12) We close this section with a consideration o f

2. the algebra

of the Gelfand transforms. Let A be an F-algebra with a defining sequence of seminorms (p,),. By definition

^A is a subalgebra of

r: A ->

c(M(A)). f ->

C(M(A)) and n

f,

is an algebra homomorphism. We endow C(M(A)) with the compact open topology. Recall from (3.2.8) that (M(An)In is an admissible exhaustion of M A ) . I t follows that the compact open topology is generated by the seminorms

For f

hence

E

A we have

r is continuous.

Chapter 3

84

(3.3) Projective limits I t is an important tool in our theory t o represent each F-algebra as

the projective limit of a sequence o f B-algebras. This representation enables us to carry over many results which are known for B-algebras as we shall see later.

(3.3.1) Let (En1dnln be a sequence o f metric spaces and assume that for each n

E

N a continuous map

Pn: En+, ->

En

is given. We say that this constitutes a projective system of metric spaces

... ->

Tn En+1->

En ->

...

I f P,,(E~+~) is dense in En for a l l n

E

N then we speak o f a dense

projective system. Definition. The subset

n

l -i m En = {(fnIn E En: qn(fn+l) = f, < ncN

for a l l n

E

N)

endowed with the relative product topology is called the projective limit of the projective system.

(3.3.2)Lemma [ARE 11. Let

Qn

En+1 ->

En

-> ...

of complete metric spaces. Then the map "k

:

I i m En ->

has dense range for all k c N.

f k'

85

Projective limits

Proof. It is sufficient t o prove our assertion for k=l. We shall denote the composite o f any m successive mappings in Let el

E

E

&&En

such that


O. We shall construct an

element (cnIn

c

(*I by

E.

of positive numbers such that

(E,)

< d 2 . Since p has dense range we find e2


Bk'

(3.3.4)Theorem. Let

be a dense projective system of B-algebras. Then A = I&Bn

is an

F-algebra. M(A) can be identified in a natural way w i t h UM(Bn) and (M(B,,)),,

is an admissible exhaustion of M(A).

Proof. Use the proof of theorem (3.2.8)and the considerations above. 0

(3.3.5)Example. For k

E

N denote by Ck([O,l])

the algebra o f a l l

functions on [0.1] which are k-times continuously differentiable.

Ck ([0.1])

becomes a B-algebra with respect to the norm Pk+l(f) = 2k SUp{if 0 ) ( X ) i :

X E

[0.1], i=o,

...,k},

where f(') denotes the i-th derivative of f. We have

M(Ck([O,ll)) = [0.11. k more precisely: Each element of M(C ([O.l])) homomorphism at some point x

E

[0,1].

is the point evaluation

cf. [GAM], p. 6.

We claim that Gelfand topology and induced euclidean topology coin-

88

Chapter 3 Set z: [0,1] ->

cide on [O,l].

C, x > x .

Let U be an open subset of [0,1] with respect t o the euclidean topology, and l e t x c U. Then there is E>O such that {y

6

iZ(X)-Z(y)i


ck+l(cogll)

Ck"0,lI)

...

->

constitutes a dense projective system of B-algebras with respect t o the natural restriction mapplngs rk: Ck+1([0.11) ->

c k ([0,13),

Recall the definition o f C"([O,l])

C"CCO,II)

f ->

f.

from (3.1.91. We have

-

= qm C~(CO,II)

and hence M(C'"([O,l]))

= [0,1] by (3.3.4).

The Gelfand and the euclidean topology coincide on [O.l] identify C"([O,l])

with Cm([O,l])".

and we can

the algebra of a l l Gelfand trans-

forms o f C"([O,ll).

-

(3.3.6) We have proved above that l i m B, is an F-algebra if it is the projective limit of a dense projective system. We now show that each F-algebra arises in this way, i.e. if A is an F-algebra then A is the projective limit of a dense projective system of B-algebras. Let A be an F-algebra with defining sequence of seminorms (pnlnv

89

Projective limits cf. (3.1.7). As in (3.2.6)we denote by A,

the completion of the algebra

A/ker p, with respect to the norm p;l(f

+

= pn(f)

ker p,)

and by r r , the canonical mapping

A ->

A.,

f ->

f+ker p,.

For m m we define

x

m.n

: Am ->

A,

t o be theextension of the mapping A/ker pm ->

f + ker pm ->

A/ker p,

f + ker p.,

rr, (resp. xm,,,) are (we1I-defined) continuous algebra homomorphisms which have dense range, and

(*I

p ; ( ~ ~ , ~ ( f )s) p h ( f ) for a i l f

E

Am,

since p, s pm. Note that

x m,n

- "n+l.n

O

"n+2,n+1

O

.*.

O

"m,m-1.

We consider the dense projective system of B-algebras

... ->

An+1

"n+l.n > A, ->

...

Let T:A ->

i-i m An, f ->
A

(A,

lldlM(A))

A

A. f ->

n

f , is a topological homomorphism, i.e.

is a uB-algebra (cf. (1.2.10)).

Since M(A) is a compact nonempty space (cf. (1.2.8)) and rates the points of M(A) we see that each &-algebra

^A

sepa-

is topologi-

cally and algebraically isomorphic t o a closed pointseparating subalgebra of C(K), K a compact nonempty Hausdorff space. Vice versa each closed pointseparating subalgebra A of C(K) is a uB-algebra, since clearly Ilf 211, = IIfll,2 for each f c A. (4.1.2) Definition. Let A be an F-algebra. Then A is called uniform

Chapter 4

92

Fr6chet algebra (uF-algebra) if there is a defining sequence of seminorms (pnIn such that 2 2 pn(f 1 = pn(f) for a l l f

A, n

E

E

N.

(4.1.3) Theorem .a) The following statements are equivalent for an algebra A: i) A is a uF-algebra; ii) A is the projective l i m i t of a dense projective system of uB-alge-

bras; iii) A contains the constants and is topologically and algebraically

isomorphic t o a pointseparating and complete subalgebra of C(X), where X is a hemicompact space and C(X) Is endowed with the compact open topology

.

b) If A is a uF-algebra, then

I': A->

h

A. f->

P,

defines a topological and a1gebraical isomorphism. Proof. I)

4

111) By theorem (3.2.8) M(A) is a hemicompact space. We

shall show that h

A

I':A->A.f->f

is a topological and algebraical homomorphism. Let (pnIn be a defining sequence o f seminorms f o r A so that pn(f 2 1 = pn(f) 2 for a l l f

A, n

E

E

N.

Recall from (3.2.6) the definition of An, rrn and p;l. Note that pk(g2 1 = pk(g) 2 for a l l g

E

An,

hence each An is a uB-algebra and

118'1M A "1 = pA(g) f o r So we have f o r every f

E

A. n

all g E

N

E

An (cf. (1.2.10)).

Uniform Fr6chet algebras

93

Here we have identified M(An) and R,,*(M(A~)) (cf. (3.2.7)) and denote by (nn(f)In the Gelfand transform of nn(f). A

Since the topology o f A is generated by the seminorms (cf. (3.2.12)) Clearly

I'

ll-llM(An)

is a topological algebra homomorphism.

3 separates the

points o f M(A) and contains the constants,

since we consider only algebras with identity. iii) 4 ii) Let (K,),

be an admissible exhaustion of X. By our hypo-

thesis A is an F-algebra with defining sequence of seminorms

T: A ->



(.rrn(f)),

defines a topological homomorphism. Since

each A,

is a uB-algebra.

ii)4 t) Let

... ->

->'h B, ->

,B ,,

...

be a dense projective sytem of uB-algebras. By (3.3.3)J&Bn

is an

F-algebra with defining sequence of seminorms p;((fn),,)

= maxi llfjlll, ]sk 1. k

P

N.

Since

i IIf1I2. i for

llf211 = we have

*

pk(((f,),)

2

all f

E

B (cf. 4.1.1)

i

* 2 1 = pk((fn)n) for a l l k

E

N, (fnIn P

Y such that the map

Chapter 4

100

induces a topological a gebra isomorphism between B and A. ii) A morphic extension (B,Y) of (A.X) is called morphic h u l l of (A,X)

if for any other morphi extension (C,Z) of (A.X) there i s a unique continuous function h: Z ->

Y so that the diagram

is commutative , i.e. (B,Y) is a morphic extension of (C,Z) via the map h and hog = f. Remarks. i) If (B.Y) is a morphic extension of (A,X) then f is necessarily injective, since A separates the points of X. So we can interpret the elements of B as extended functions of A . Part ill guarantees that Y is the largest space t o which a l l elements of A can be extended. ii) If (B,Y) is a morphic extension of (A,X) via f: X ->

is a morphic extension of (B.Y) via g: Y ->

T (c) = TfoTg(c) for a l l c 9.f ill) The structure map j: X ->

3 ->

Z. then (C,Z) is a morphic

Z, since

extension via go f : X ->

TI:

Y and (C.Z)

n

A, f ->

E

C.

M ( A ) is continuous and n

f

o j

=f

is just the inverse of the mapping

r

(cf. 3.2.12) and hence a topo-

logical algebra isomorphism by (4.1.31). so ( 2 , M ( A ) ) is a morphic extension of (A.X). Before we show that ($,M(A))

is the morphic h u l l of (A,X) we prove

two lemmata. Recall from (3.2.5)iv)

that

Extension o f uf-algebras

101

is a homeomorphism, i f B i s a morphic extension of A via f.

(4.2.3) Lemma. I f (B.Y) is a morphic extension of (A.X) via the continuous map f. then the diagram f

X

> Y

is commutative, where ,j

: Proof. Since T e:T

(resp. j,)

denote the structure maps.

is a homeomorphism it suffices to show that

j, =

f.

Let z c X. and l e t b

E

B be arbitrary elements, then A

= b(CpZmTf) = Cp,(Tf(b))

hTf*"j,(z))

= cp,(bof)

=

= bof(z) = 6 ( j y o f ( z ) ) . Our assertion follows. since (4.2.4)Lemma.

separates the points of M(B)

Let A,B be two uF-algebras, and l e t T:B ->

A be a

h

topological algebra isomorphism, then ( B .M(B)) is a morphic extension of (j\.M(A)) via the adjoint spectral map T*. Proof. The map A

S:

B ->

A

h

A, b ->

I;oT*,

i s a we1 I-defined injective algebra homomorphism. It is continuous, since T* is a homeomorphism. Let b

E

B such that T(b) = a. then

^a

A

E

A be an arbitrary element. Let

102

Chapter 4

hence s is a topological isomorphism by the open mapping theorem. 0

(4.2.5)Theorem. (s,M(A)) is the morphic hull of (A,X). Proof. We have already remarked that (hA,M(A)) i s a morphic extension of (A.X). Let (8.Y) be another morphic extension with respect to the map f:X->

Y.

Define

h: Y ->

*

(Tf 1-1 o

M ( A ) , y ->

then the diagram

) ~ .

is commutative by (4.2.3) and (s,M(A)) is a morphlc extension of (B,Y) via h by lemma (4.2.4) and remark (4.2.2)ii)-lii),

since

( T i ) - ' = (Tf -1 1*

by the proof of lemma (3.2.5)iv). Let h,, h2

:

Y ->

M(A) be two maps such that the diagram

is commutative for 14.2. Then A

A

Tf(?ohl) = aohiof = aajx = a for a l l a

E

A, 14.2.

Extension of uF-algebras

103

hence n

aohl = Tf-'(a)

for a l l a

E

A, b1.2,

h

and we have hl = h2 since A separates the points of M A ) .

0

(4.3) Convexity for uF-alaebras Recall from (2.2.6) that a domain G

c

Cn (or more general a re-

duced complex space X I is said to be holomorphically convex if the holomorphically convex h u l l of each compact subset K

G is again

c

a compact subset of G. As noted in (2.2.7) G is holomorphically convex iff M(Hol(G)) = G. In this section we introduce convexity f or uF-algebras in an analo-

gous way. (4.3.1) Definition. Let (A.X) be a uF-algebra, and l e t K

X be a com-

c

pact set. We c a l l the set

R,,,

= {x s X: I f ( x ) l

g

IIftt,.

f or a l l f

E

A)

the A-convex h u l l of K in X. We say that a subset S

if

R(,,,

c

S for each compact set K

we use the abbreviation

c

S. I f K

c

c

X is A-convex

M(A) is compact

RA = R(2,M(A)).

(4.3.2) Example. If K c X is compact then in general

R(,,,)

is not

compact. For example l e t C* = C\{Ol and set A = Hol(C)lc,.

It is

easy to show that A is a complete subalgebra o f C(C*). hence (A.C*) is a uF-algebra by (4.1.3). Set dD = {x

h(AmCO) = { x c*: 1x1 E

since each f

E

E

C*: 1x1 = 1). Then

1).

(A,C*) is the restriction of an element

r"

E

Hol(C).

Definition. Let (A.X) be a uf-algebra. We say that X is A-morphically h

convex if Kc,,,

is compact for every compact set K

c

X.

Chapter 4

104

G

Remark. As noted above a domain

iff M(Hol(G)) =

c

Cn is holomorphically convex

G. Of course this result

is not valid in our general

setting. Take fo r example (P(dD),dD), then dD is P(dD)-morphical l y convex, but M(P(dD)) = (x

6

C : 1x1

1) (cf. (1.2.4)ii)).

S

A

However we can prove that M(A) is A-morphically convex.

(4.3.3) Theorem. Let A be a uF-algebra, and l e t K pact set. Then M(2,)

c

RA and R, convex .

can be identified with A

set. In particular M(A) is A-morphically Proof. Since A is a uF-algebra

h

r: A ->

M A ) be a com-

A, f ->

is a compact

A

f , i s a topological

algebra isomorphism, cf. (4.1.3). hence we can naturally identify

M(A) and M(%. Since the restriction mapping

has dense range the adjoint spectral map

r*:

A

~ ( 2 , )->

MA) = MA)

is continuous and injective (cf. (3.2.5)ii)). M(S), of a uB-algebra

- is compact,

r*: M(hA,)

->

-

as the spectrum

hence

r*(M@,))

i s a homeomorphism. Our assertions w i l l follow if we can show that

r*(M(%,)) = Let cp

E

R,.

M(2,) and

? E ^A

be arbitrary elements, then

iP(r*(cp))t = tcp(r(f))l

PI .,I

for the inequality cf. (3.2.2)ii). On the other hand, if cp e R A then

7: 21K->

C,?I, ->

is a well-defined mapping, since

A

f(cp1,

?I, a 0 implies that

hence in particular ?(cp) = 0. Clearly

3

t

0 and

A

is an algebra homomorphism.

105

Convexity f o r uF-al gebras It is continuous since

i?(PIK)i Hence

= iP(tp)i s

IIPIIR = IPI,

for a l l f

6

A.

A

A

3 can

be extended t o an element of M(AK) and

r*(g) = 9.

(4.3.4) Remarks. i) It follows from the theorem that we can always choose an admissible exhaustion (Knln of M(A). which is moreover A

(PnlAf o r

A-convex, i.e. Kn =

all n

ii) If (A,X) is a uF-algebra and K

easy t o show that A, A

,, to ,,A

( j:

X ->

c

c

N.

X is a compact subset, then it i s

is topologically and algebraically isomorphic

M(A) the structure map, c f . (4.2.1)). Hence we

can identify M(A,) ill) Let L

E

with j(K)",.

the 2-convex h u l l of j(K).

X be a hemicompact set, and l e t (Knln be an admissible from (4.1 5 ) .Then

exhaustion of L. Recall the definition of A, A

= U j(KnIA,

M(A,) and (j(Kn):ln

is an admissible exhaustion of M(A,)

by theorem

(4.1.6) and ill. In particular i f L c M(A) is a hemicompact and 2-convex set we can identify M&,)

and L as sets. A

Let L c M(A) be a hemicompact and A-convex subset, and l e t K

c

L

be a compact subset. Identify L and M(3,) as sets. We claim that on A

K the relative topology (inherited from M(A,))

and the original topo-

logy (inherited from M(A)) coincide. A

We interpret the elements of A,

as complex-valued functions on L

which can be approximated uniformly on each compact set o f L by A

elements of A. cf. (4.1.5). Since

PI,

A

E

A,

for each

? E

2, we

see that

the original topology on K is coarser. To prove that both topologies coincide we have only to show by (3.2.2111 that sets o f the form

106

Chapter 4 U

fls

?l,...,?r

- ...fr

A

c A c,p

E

.P

= (9

E

,...,r~

K: i?+cp)-?pi < I, i=1

K, are a neighbourhood of cp with respect to the

To see this choose

original topology.

II?i-tiIIK


ii) Let K be a compact neighbourhood of

E

109 M(A)\U f 1..

llGllK f or (9.

...fr -

g = fl-P1(tp).

Choose fl.

....fr

E

A by

(3.2.211) so that

Then U

n

is A-convex and relatively compact. It i s moreover

.....f,.9

f1

hemicompact, since n '

= {J, MA):

i$(~,)-?~(tp)i

1-(1/n)), n

E

N,

is an admissible exhaustion. Example. The closure of an A-convex set need not be A-convex.

For instance consider the uF-algebra (Hol(C).C) and define

G = {z

E

C : i z +l i < 2 and i z l > 1).

Then G is a simply connected domain, hence G is polynomially convex ([NAR 13, p. 151). hence Hol(C)-convex. but

-

by the remark above

-G

is also

a - the closure of G - is not polynomially con-

vex, since it contains {z

E

C : iz+11 = 2).

(4.4) Uniform Fr6chet algebras with locally compact spectrum

Many of our l a te r results w i l l be valid only f or uF-algebras with local ly compact spectrum. Although the hypothesis "locally compact spectrum" is quite elementary, it would be important t o find criterions for A which imply this hypothesis. No such crlterions seem t o be known, besides very special situations, cf. for instance (18.2.8). I n [H/V 11 Hayes and Vigue constructed even a reduced complex space X with countable topology such that M(Hol(X)) is not locally compact at any point.

110

Chapter 4

(4.4.11 We f i r s t state a sufficient topological condition. Proposition. If a hemicompact space X satisfies the f i r s t axiom of countability then X is locally compact. Proof. Let (Knln be an admissible exhaustion of X. Assume there is a point x c X without compact neighbourhood. Let (UnIn be a countable basis for the neighbourhood system of x. Choose f o r every n

E

N a point xn

e Un\Kn. then

K = {xn: n

e

NI U {XI

is compact. but there is no n

E

N such that K

c

Kn. This is a contra

diction.

0

With regard to the problem described above it would be enough t o find criterions fo r A which imply the axiom of f i r s t countability f or the spectrum M(A). Unfortunately in the next theorem we have to assume a priori that M(A) is locally compact. (4.4.2) Theorem [KRA

11. Let

spectrum M(A), and l e t cp

E

A be a uF-algebra with locally compact

M A ) . Suppose there is a sequence

(fnIn in A so that the ideal (fl.f 2....) lies dense in the maximal ideal ker cp = { f e A: cp(f) = 0 ) . then there exists a countable basis f or the neighbourhood system o f cp.

For the proof we use a proposition. Proposition. Let X be a locally compact Hausdorff space. Suppose that x c X is a Gg-point, i.e. there exists a sequence of open neighbourhoods (Un),,

such that {XI = n Un. nEN

Then x has a countable neighbourhoodbasis.

111

uF-algebras with locally compact spectra Proof. Let K be a compact neighbourhood of x. Set L1 = (int KInU,. Choose a compact neighbourhood Q, of x such that Q,

c

L1, and set

L2 = (int Ql)nU2. Continuing in this manner we get a sequence (LnIn of open neighbourhoods such that

... where

3

Ln

3

Ln+1 3 L,+,

i n denotes the

3

Ln+2 D

...

closure of Ln. By our assumption we have

n ?In= {XI.

(*I

nEN

Now l e t W be an arbitrary open neighbourhood of x. Let y an arbitrary point. By

(*I there is n

compactness of K\W there is m Lm

c

N such that y

E

K\W be

X\cn.By the

N so that L,n(K\W)=0.

i.e.

W, and we are done.

Proof o f (4.4.2). For n

Then

E

E

E

E

0

N set

nun = ( 9 ) .

0

(4.4.3) We say that a uF-algebra is topologically countably generated if there is a sequence (fnIn in A such that C[fl.f 2,...], the ring of polynomials in fl.f 2.... lies dense in A. Note that in this case A is separable. Corollary. Let A be a separable uf-algebra with locally compact spectrum M A ) . Then M(A) satisfies the f i r s t axiom of countabillty. Proof. Let (gnIn be a countable dense subset o f A, and l e t cp

E

M(A)

112

Chapter 4

be an arbitrary point, then the ideal (gl-01(p),g2-02(p),...)

lies

dense in ker 9. Remarks.1) Both results

0

- even for

finitely generated uF-algebras

-

are In general not valid If we drop the assumption "locally compact spectrum". Otherwise the spectrum of every finitely generated uFalgebra would be automatically locally compact by (4.4.1). This is false as we shall see in (5.1.12). ill On the other hand l e t K be a compact Hausdorff space which does

not satisfy the f i r s t axlom o f countabllity. Then (C(K1.K) is an example of a uB-algebra with (locally) compact spectrum, which does not satisfy the f i r s t axlom of countablllty. Ill) I n (5.1.12) we shall give a simple example of a uF-algebra (A,X)

on a locally compact space X such that M(A) Is not locally compact (see also the example of Hayes and Vigue, mentioned above).

113

CHAPTER 5 FINITELY GENERATED F-ALGEBRAS

In this chapter we deal with a class of F-algebras which admit a finite set of topological generators. In this case the problem of determining the spectrum is equivalent t o the problem o f determining the polynomially convex h u l l of compact subsets o f C". (5.1 .l)Definition. We c a l l an F-algebra A (topologically) n-generared provided there exist elements fl.....fn

in A such that A is the

closure of the polynomials in fl.....fn. (5.1.2) Examples. i) Let G

c

Cm be a polynomially convex open set,

i.e. there exists an admissible exhaustion (KnIn of G consisting of polynomially convex compact subsets. Let f

E

Hol(G). then f can be

approximated uniformly on each Kn by polynomials (cf. (1.4.5)). So

Hol (GI is n-generated by the coordinate functions zl,

...,zn.

ill Using the theorem of Stone-Weierstra6 we see that C(R) is singly generated by the function z : R -> iii) Let f

E

C"([O.l])

C , x ->

x.

be an arbitrary element, and l e t n e

N. By

the

theorem of Stone-WeierstraB we can choose a sequence of polynomials (Pk)k such that

11 f(")-Pkll[O,

11 ->

0, as k ->

Q),

where f(") denotes the n-th derivative o f f. Consider on [ O , l ] the polynomials

Chapter 5

114

Then

J f (n) (t)-Pk(t) dt

=

if("')(x)-p"k(x)i

1

0

~ ~ f ( n ) - p k ~ ~ [ O->, l ] 0, as k ->

m.

Continuing in this manner we get a sequence of polynomlals (qk)k so that

'If

(1)-

(1) qk "[0,1]

->

I t follows that CQ)([O,l])

z:R->

0, as k ->

Q),f or i=O,

...,n.

is singly generated by the function

x.

C. x->

For more information about the algebra CaD(M),M a differentiable manifold, we r e fe r the reader t o the book of Mallios [MALI. (5.1.3) Remarks. i) Generating elements are not uniquely determined.

For example every real-valued function f

c

C(R). which separates

the points of R, generates C(R) by the theorem of Stone-Weierstra6. ii) We do not assume that the number n i s minimally chosen, so if A

i s m-generated, then A i s also n-generated for nam. Next we want t o reformulate our main theorems for finitely-generated uF-algebras. First we consider the joint spectrum of the generating elements. (5.1.4) Let A be an n-generated F-algebra with generating elements

fl,...,fn.

We denote by

f

I?: M A ) ->

Cn.

Clearly Pis continuous. n

p(flD".Dfn)(p)

the mapping

,...,fn(cp)). If F ( I ~= )P~JI), then

= ^p

~p->

(flp.*.Dfn)($)

(?l(~)

n

Finitely generated F-algebras for a l l polynomials p, where ^p(f,....,f,)

115

denotes the Gelfand trans-

A

form o f p(f ls...,fn). Hence @(cp) = g(Q) f or a l l g cp =

Q, i.e.

f

E

A. and we have

i s injective.

(5.1.5) Proposition. Let A be a uF-algebra with generating elements

fl,...,fn,

A

and l e t K c M(A) be an A-convex compact set. Then P ( K ) h

is a polynomially convex compact set and A,

is topologically and

algebraically isomorphic t o P@(K)). h

Proof. By our assumptions A,

is n-generated by

fll ,,...,f A

A

nlK Since M(hA,)= K by (4.3.31, the r e s u l t follows from (1.2.13).

.

(5.1.6) Let (Knln be a sequence o f compact sets in Cm such that Kn Kn+l for a l l n uB-algebras

... ->

E

N. Consider the dense projective system o f

P(Kn+,)->‘n

P(Kn)

-*

...

with the restriction mappings ‘n

:

P(Kn+l) ->

P(Kn). f ->

f lK n.

-

Then A = l i m P(Kn) is a uF-algebra by (4.1.3), the spectrum of A can be identified as a set with U Qn , where nomiaiiy convex hull o f Kn. and

pn denotes the poiy-

(RnIn is an admissible exhaustion

with respect t o the Gelfand topology (cf. theorem (4.1.6) and (1.2.4)ii)). Clearly A is m-generated by the coordinate functions zl.

....zm.

(5.1.7) Remarks. i) Let A be as above. Then we can interpret each

2 as a complex-valued function on U Rn. which can be approximated uniformly on each Rn by polynomials. ii) On every pn the induced Gelfand topology and the induced eu-

element?

E

clidean topology are equivalent ( cf. (1.2.4)ii)).

Chapter 5

116

Rn is finer

In genera the Gelfand topology on U

than the induced

euclidean topology, since the coordinate functions z1,....zm elements

A by i) and for each x z U

3f

Rn , E>O the

define

set

u Un: Izl(x)-zi(y)i < E, i=1,...,m~ = {y o u Un: Ixi-yil < E, i=1,...,m~ {y

E

is Gelfand open (cf. (3.2.2)i)). Both topologies are equivalent iff each element with respect to the euclidean topology on U

A

f

E

A is continuous

Rn.

Suppose that &,,In is an admissible exhaustion of U spect to the euclidean topology. Let and l e t (xnIn be a sequence In U

Rn

f 3

2,

with re-

be an arbitrary element,

L

h

which converges t o xo z U Kn

.(euclidean topology). Then K = { x n : nkO) is a compact set (euclidean topology), hence there exists I

E:

N such that K

both topologies coincide, we see that tf(xn)), and it follows that

f

it,. Since on R, A

converges to f (xo)

Is continuous with respect to the euclidean is an admissible

topology. Hence both topologies coincide iff exhaustion of U

c

Rn with respect to the euclidean topology.

We give a simple example, where both topologies do not coincide.

For n

o

N set Kn = {exp(it): 2 n - ( l / n )

2

t

0).

Then each Kn i s polynomially convex, since C\Kn is connected (cf. (1.2.4)ii)) and U Kn = {z

E

C : l z l = 1) = dD. Clearly dD is a compact

set (euclidean topology) but there is no I Example. Let

G

c

E

N so that aD

c

KI.

Cn be a polynomially convex domain, and l e t (Knln

be an admissible polynomially convex exhaustion of G. Then

T: Hol(G) ->

&&P(Kn),

f ->

(flK,In.

defines a topological and algebraical isomorphism by (1.4.5). Hence

Finitely generated F-algebras

117

M(Hol(G)) = U Kn = G as sets. Since (KJn is an admissible exhaustion of G with respect to the euclidean topology we see that M(Hol(G)) = G as topological spaces by (5.1.7)ii). (We could have also derived this res ul t from (2.2.7). since G is holomorphical l y convex.)

.

(5.1.8) Structure theorem fo r finitely generated uF-algebras Let

... ->

An+, ->

...

An ->

be a dense projective system of &-algebras. A = ) L A n is n-generated by fl1...'f,.

Suppose that

Then

i) A is topologically and algebraically isomorphic t o

f?(M(A)) is a

homeomorphism then M(A) i s metrizable. (5.1.10)

Recall that fo r finitely generated uB-algebras B the map

f?

is always a homeomorphism and f(M(B)) is polynomially convex (cf. (1.2.13)). The next examples

-

due to Brooks [BRO 31

- show that

these results are not valid fo r F-algebras. Example. We construct a finitely generated uF-algebra A. such that fo r no generating elements fl.....fr

the map f is a homeomorphism.

Let ($,I,be , a decreasing sequence of positive numbers such that t1 < a / 2 and I i m t,

the plane and set n

= 0. Let Ln denote the segment [O,exp(itn)] in

Finitely generated F-algebras

119

Each Kn is polynomially convex (cf. (1.2.4)ii)). By (5.1.6) the spectrum o f the singly generated uf-algebra A = I&P(Kn)

can be iden-

tified with U K n and (KnIn is an admissible exhaustion with respect to the Geifand topology on U Kn. We shall show that M(A) Is not metrizable. which yields our r e s u l t by proposition (5.1.9). Suppose there is a metric d on U Kn which defines the Gelfand topology. Since Gelfand topology and euclidean topology coincide on each Kn by (5.1.7)ii). we can choose for every n xn c Ln\{O)

6

N an element

< 2-".

c Kn so that d(xn,O)

Hence K = ( 0 ) U {xn: n

E

N) is a compact set with respect t o the

topology defined by d on U Kn. but K is not contained in any Kn, therefore K cannot be Gelfand compact. This is a contradiction. (5.1.11) Example. We construct a singly generated uF-algebra such that h l ( A ) ) is not polynomial l y convex. Recall that a set M

c

Cn i s called polynomially convex provided that

the polynomially convex hul I

R of

each compact set K

c

M is con-

tained in M. Let K,

= [0,2rc-(l/n)],

then Kn is poiynomiaiiy convex (cf. (1.2.4)ii)).

and l e t A = < l -i m P(Kn). Then M(A) = C0.2.r~)and Gelfand topology and euclidean topology are equivalent on [0,2rt) by (5.1.7)il). We have P(Kn) = C(Kn) by the theorem of Stone-WeierstraB. thus

A = C([Ov2.rr)). Equally the function g: [0,2r)->

C , t ->

exp(it1

generates A. But 6([0,2%)) = {z c C : Izi=l) is not polynomially convex. Moreover

8

i s not a homeomorphism, while

5?

is a homeomor-

phism f o r the generating element

?: [0,2rr) -' c, t ->

t.

So we have seen that there are singly generated uF-algebras such

Chapter 5

120 that

?

is a homeomorphism fo r one generating element. while

?

fails

t o be a homeomorphism fo r another generating element. (5.1.12) Example. We construct a doubly generated uF-algebra (A,X) on a locally compact space X such that M ( A ) is not locally compact (cf. remark (4.4.3)iii)).

For n

E

N define the following subsets of C2

Kn = ((z.0): 1zlr:l) U ((0.w): w

E

C0.231 U { ( Z ~ W )iz : l = l, l wlr: l-(l/ n)~

Then Kn may be illustrated in the following way

i) Let

Ln = {(z,w): Izlr:l, iwir:1-(1/n)) U ((0.w): w

E

C0.23)

Claim. Ln is the polynomlally convex hul l of Kn.

We f i r s t show that Ln is polynomially convex. Let (x,y) be an arbitrary point of C 2 \Ln. I f I x i > l , then Izl(x.y)l

> ~~zl~~Ln. where z1 denotes the f i r s t coordinate

function.

I f Ol-(l/n), hence Ip(x*y)l >

IIPllLn

.

for the polynomial p(zlsz2) = ~ ~ ( z ~ / ( l - ( l / n ) )k) sufficiently ~ large.

If x=O, then y

E

C\M with M =

{WE

C : Iwl r: l -(l/ n)1 U CO.21.

Since M is a polynomially convex compact subset of the plane (cf. (1.2.4)ii)). we find a polynomial p so that Ip(y)l > llpllM

Finitely generated F-algebras

121

and so IpX,yN >

llplLn

for the polynomial ~ ( z 1 , z 2 ) = p(z2), hence Ln is polynomially convex and the polynomially convex hull

f?,

of Kn is contained in Ln.

Now l e t p be an arbitrary polynomial, and l e t (x,y)

E

Ln\Kn. Then

applying the maximum principle t o the polynomial q(z) = p(z,y)

-

-

we get ip(x,y)l = I q o d l s l l q l l { i z p l }

so Ln

c

- llpll{(z~y) I z l = l }

llPllK".

:

2,.

ii) Set X = U Kn. Then X, endowed with the induced euclidean topo-

logy, is a locally compact space and (Knln is an admissible exhaustion of X. Set

A = {f

t

C(X): f can be approximated uniformly on each

K,

by polynomials}. Then (A.X) is a uF-algebra by theorem (4.1.3)iii) and (A.X) is doubly generated by the coordinate functions. We have A = l L P ( K n ) by theorem (4.1.6) and h

M(A) = U Kn = U Ln as sets and (LnIn i s an admissible exhaustion of M A ) with respect t o the Gelfand topology by (5.1.6). ill) Assertion. The point (0.1)

E

M A ) has no compact neighbourhood

(Gelfand topology). We assume the contrary. Then there is n

E

N so that Ln is a compact

neighbourhood of (0,l). By (3.2.211) there exist fl.....fr

(*I

{(y1,y2) A

A

E

The functions fl....,fr

A

M A ) : Ifl(y1,y2)-fl(0,1)i A

< 1, i=l,...,r}

E

A such that

c

Ln.

are continuous on every Ln (euclidean topo-

logy) by (5.1.7)i). So we can choose m c N and w l - ( l / m ) > w > l - ( l / n ) and Iq(0,w) -?,(O,l)l

< 1/4 for i=l

,...,r.

E

R such that

122

Chapter 5

Agaln because of the contlnulty of the ?,Is we can choose u = (ul,u2)

c

Lm\Ln such that

< 1/4 for l=l,...,r,

I?l(0,w)-?l(ul,u2)i hence

i?i~~.1~-?l~ul,u2~~ < 1 for 1=1, ...*r. But thls 1s a contradlctlon t o (*I.

(5.2) Ratlonal l y finitely generated F-algebras (5.2.1)Deflnltlon.

We say that an F-algebra A Is rationally n-gener-

ated provided there exist elements fl....*fn

in A such that a l l func-

tlons of the form p.q-l* p and q polynomlals in fl,...,fn,

q lnvertlble

in A, lie dense In A. (5.2.2)Example. Let G

c

C be a domain, and l e t (Knln be an ad-

mlsslble exhaustlon of G such that for a l l n

c

N there Is no bounded

component of C\Kn, which is relatlvely compact In G. Fix n

c

N. Let V1,V 2,... be the bounded components of C\Kn. For

each k we choose a point Xk

E

Vk\G. Then R(Kn) Is doubly generated

by the functions Z:Kn

->

c, X

X,

and f:Kn ->

c, X ->

k

Ek/o(-Xk)

where 8k are sultable chosen constants (cf. the proof of (24.4) In CSTOI). R(Kn) 1s slngly generated by z if there Is no bounded compo-

Rational l y finitely generated F-al gebras nent. Let g

E

123

Hol(G) be an arbitrary element. Then g1

Kn

E

R(Kn) by

(1.4.5). Hence g can be approximated uniformly on Kn by polynomials in f and z. Since a l l xk

E

C\G, the partial sums of f are of the form

p/q. p.q polynomials, q invertible in Hol(G). I t foiiows from this that

Hol(G) is singly rationally generated by Z:

G ->

C , x ->

X.

(5.2.3) We note that n-generated F-algebras are in particular rational l y n-generated. I n the same way as for n-generated F-algebras we can show that the map A

F : M(A) ->

Cn. cp ->

(?l(p)s...,?n(cp))

is injective and continuous. if A i s rationally n-generated by fl,...,fn.

(5.2.4) Recall that a compact set K

c

Cn is called rationally con-

vex, if K = M(R(K)). Proposition. i) Let A be a rationally n-generated uF-algebra w i t h generating elements f,

.....fn

E

A and l e t K

A

c

M(A) be an A-convex comA

pact subset, then P ( K ) is rational i y convex and A, is topologically and algebraically isomorphic to R(P(K)), where

f

is defined as in

(5.2.3). ii) If (Knln is an admissible $-convex exhaustion of M(A). then A is topologically and algebraically isomorphic t o Proof.

PI,:

K ->

I_imR(f(Kn))

P(K)

is a homeomorphism by (5.2.3) and it is easy t o show that -1 A T:A,-> f->

R(P(K)),

is a we1 i-defined algebra homomorphism. Since

l l T ( g ) I l ~ ~=, ~11g11,

for a l l g

n E

A .,

.

Chapter 5

124 h

A, is isomorphic to a closed subalgebra of R(P(K)). Let g

E

R(P(K))

be an arbitrary element. Using (1.4.21 we can show that there is h c $K such that h = g a p (note that M(R(K)) = K by (4.3.3)). i.e. T(h) = g and T is surjective, hence

2Kis isomorphic to R(P(K)).

The adjoint spectral map

(T-$*: MQ

->

M(R(P(K)))

is a homeomorphism by (3.2.511~). Let cp c K = M(2,)

and f

E

R(P(K))

be arbitrary elements, then (T-’)*(p)(f)

= cp(T-’(f)) = p(feP) = f(P(cp1).

so

(T-ll*(cpI = p(p), and (T-~)*(K) =

RK).

hence

P(K)= M(R(P(K))). ill For a l l n

E

N the diagram

commutes, where rn resp. rk denote the natural restriction mappings and Tn+l

resp. Tn denote the isomorphisms of 1).

Hence

... ->

R(P(K,+~ r

R(P(K,N ->

...

constitutes a dense projective system of uB-algebras and A is topologically and algebraically isomorphic t o LimR(e(Kn))

.

0

In the.case of ii) of the theorem we can identify M(&imR(P(K,))) with U p(Kn)= f ( M ( A ) ) and @(KnN,,

is an admissible exhaustion of

Rationally finitely generated F-al gebras

125

U P(Kn) with respect t o the Gelfand topology (cf. (4.1.6)). (5.2.5) We close this chapter considering the projective system

(*I where

... ->

R(Kn++

... c Kn

c

Kn+,

c

rn

> R(Kn) ->

...

,

... is a sequence of compact

subsets of Cm

and rn denotes the restriction mapping. We f i r s t remark that in general

(*I is not a dense projective system,

hence we can not apply our previous results in this case. We give an example.

For x a C, r,O set Dx,r = tz a C: I z - x i < r) and

b = {Izlsl).

For

n c N set

Then the restriction mappings r

I .n

:

R(KI)

-*

R(Kn). I > n,

don't have dense range, since for example l / ( z - ( l / n ) ) cannot be approximated uniformly on Kn by elements of R(KI). Each element f c&&R(Kn)

can be interpreted as complex-valued

function on U Kn such that fl of llm_R(K,)

Kn

c

R(Kn) for a l l n

E

N.The topology

is generated by the seminormsystem 11.11

to see that )&R(Kn)

Kn

and it i s easy

is in fact a uF-algebra.

(5.2.6) Even if (*)in (5.2.5) constitutes a dense projective system it is in general not true that & L R ( K n ) i s rationally m-generated.

while

9,

deflnes a continuous and Injective algebra homomorphlsm. We claim that T is surjective, too. Let h

E

A be an arbltrary element. Choose polynomials qn, pn so that

iItn(f)/Gn(f)-tiiKn < l/n ,

(*I

where ^pn(f) denotes the Gelfand transform of pn(f). Since $,(f) Invertible in 3, ^q,(f) has no zero on D, hence tn(f)/$,(f) for a l l n

E

N. Set D' = {z: lzl

E

Is

Hol(D)

1/21. It follows by (*I and the max-

imum principle that

is a Cauchy sequence wlth respect to the norm

.

11-1ID. hence this

sequence converges on {iz1 Cn.

cp ->

n

(fl(q)

E

A. We de-

n

....,fn(cp)).

The set P(M(A)) is c a l l e d the joint spectrum of fl.....fn. (6.1.2)Theoret-n [ZAM]. Let A be an F-algebra. and l e t f ,,...,f, Let U c Cn be an open set which contains the joint spectrum of

fl.

...,fn. Then there i s a unique continuous homomorphism 8,: Hol(U) ->

A

such that i) 8F(zi) = fi, i = l . ...,n. and

ii) 8,(gIn

h

= goF f o r a l l g

E

Hol(U).

E

A.

130 where z1

Chapter 6

,...,zn

denote the coordinate functions and 8,(gIA

denotes

the Gelfand transform of 8,(g). Proof. Represent A as the projective limit of a dense projective system of B-algebras

... -'

Ak+l ->

...

Ak ->

Ak, resp. x,,~:

(Cf. (3.3.7)).Denote by % k ' A ->

Am ->

Ak

(mkk), the canonical maps (cf. (3.3.6)). As usual we identify M A k ) and Z;(M(Ak))

(Cf.

(3.2.7)).

We have a(nm(fl),...,~m(fn))

a(xS(f,),...,xs(f,,))

for a l l m,s ( s k m ) . where a(xs(fl),...,xs(fn)) trum of the elements xs(fl),...,%s(fn) By (1.4.2) there exists for each m

t

c

P(M(A))c u,

denotes the joint spec-

As.

c

N a unique continuous algebra

homomorphism

am: Hol(U) ->

Am,

such that

(*I

i)

...,

= xm(fi), i=l, n, and

ii) e m ( g f = g.clM(Am) for a l l g

E

Hol(U).

(Note that cp(xm(fi)) = x&(tp)(fi), for i=l,...,n

and for a l l

'Q E

M(Am).)

For Sam the homomorphism

x

s .m

0 8 ~ : H o l ( U->)

A ,

is continuous and satisfies i) and ii) of

(*I. Hence

by the uniqueness assertion of (1.4.2). I t follows that

8,: Hol(U) ->

A =

(8m(g))m

i s a we1 i-defined continuous homomorphism which has the required properties 1) and ii).

Holomorphic functional calculus

131

A be another continuous homomorphism which f u l -

Let t: Hol(U) ->

f i l l s i) and ii).Then x m - t = 8" f o r a l l m c N. by the uniqueness o f the homomorphisms

(6.1.3) Let S

c

em and it

follows that t = 8.,

0

Cn be an arbitrary set. Recall from (1.4.1) the defini-

tlon of the algebra H(S). Corollary. Let A be an F-algebra. and l e t fl,

...,fn

E

A. There exists

a unique continuous homomorphism

8F: H(P(M(A))) ->

A

such that

1) 8,(Zi)

= fi, i=l, ...,n and

ii) 8,(glA

5-P for

=

all

c

H(PMA))),

denote the germs o f the coordinate functions.

where

Proof. Let U. W c Cn be open sets so that f?((M(A)) c U c W. Denote by rU,Wthe natural restriction mapping Hol(W) ->

Hol(U), f ->

and denote by 8;: Hol(U) ->

flu

A the unique homomorphism o f (6.1.2).

Then

(*I

8F*ru,w = 8:

by the uniqueness assertion of (6.1.2). Now l e t

9" E

H(C(M(A1)) be an arbitrary element, and l e t g

be a representative of

9"

c

Hol(U)

in an open neighbourhood U of P(M(A)).

Set

8&)= It follows from

8$g).

(*I that 8,(5) is a well-defined mapping and it is

easy to check that 8, is in fact a homomorphism, which has the required properties i) and ii). (Recat 1 that

9"

E

H(F(M(A))) and x

E~A(A)).)

g ( ~is)

well-defined for

132

Chapter 6

To see that 8, is continuous l e t U be an arbitrary open neighbourhood of p (M(A I) and denote by T, the natural mapping H ~ I ( U )->

T,,:

Then 8,.

H ( ~ I ( A ) ) ) , g ->

5.

= 8,U by the definition o f 8,. 1.e. 8 , o r U

T,

is continuous

for each open neighbourhood U, so 8, is continuous by (1.4.1). Let 0 : H@(M(A))) ->

A be another continuous homomorphism with

the required properties, then

= 8; f or a l l open neighbourhoods

POT,,

U of P(M(A)) by (6.1.21, hence p = 8.,

0

Remark. A weaker form of the holomorphlc functional calculus for F-algebras was f i r s t proved by Arens [ARE 21 and Waelbroeck [WAE] , cf. also Rosenfeld [ROS].

(6.1.4) As an Immediate consequence of the holomorphic functional calculus we can describe the invertible elements of an F-algebra. Corollary. I f A is an F-algebra, then A - ~= {f

Proof. i) I f f

E

E

A: cp(f)

* 0 ,for a i l cp

A - l there exists g

1 = cp(f)cp(g) for a l l cp ii) If cp(f) 9 0 for a l l cp

af: Hol(C*)

->

E

E

E

MA)).

A so that fg = 1, hence

MA).

M A ) . then ?(M(A))

c

C* = C\{Ol. Let

A be as in (6.1.21, then

1 = 8f(z*(1/2)) = 8f(2)8f(1/z) = f . B f ( l / t ) . where z : C*->

C . x->

x.

0

(6.1.5) Shilov's idempotent theorem. Let A be an F-algebra and l e t

E

c

M(A) be an open and closed set. Then there is a unique idem-

potent f

E

A such that

?

Is the characteristic function of E.

Holomorphic functional calculus

133

Proof. Let A = ) L A n (3.3.7). Then

... c

M(An)

c

M(An+l)

c

...

i s an admissible exhaustion of M(A) by (3.3.4) and M(An) f l E is a closed and open subset of M(An) for each n

By (1.4.6) there is a unique idempotent fn characteristic function of M(An) (man) the canonical maps A ,

n E for

c

c

N.

An. so that?,,

is the

each n c N. Denote by xmSn

An (cf. (3.3.6)). then

->

is an idempotent element of An and Gm,,(fm)

=

fn, where

$m,n(fm)

denotes the Gelfand transform of It follows from the uniqueness assertion of (1.4.6) that

xm,n(fm) = fn. Hence f = (fnIn

E

A -, where N

f(x) =

N

A. f ->

{

f,

f(x) if x

€

u

if x

E

M(A)\U

0

A

= U and A,

Holomorphic functional calculus

135

Proof. By our hypothesis U is a hemicompact space, hence

2,

is a

uF-al gebra by (4.1.5). A

We interpret each element of A,

as complex-valued function on U.

which can be approximated uniformly on each compact subset of U by elements of

2.

By Shilov's theorem

xu

h

h

is contained in A. Let f

E

h

A

A,

element. Choose a sequence (fnIn in A such that A

t o f in A .,

Then

(? - x 1 u

be an arbitrary

?Ju converges

N

converges t o f with respect to the com-

n

A

N

pact open topology, hence f

E

A and the map incl is a well-defined

injective and continuous homomorphism. We have

h

A

h

so incl(Au) is a closed subalgebra of A, which is isomorphic to A ., Let K

c

U be a compact subset, then we can show as in the proof o f

(6.1.7) that

RA c

U. hence

by (4.3.4)iii) and it follows from (*I and (3.2.211) that the Gelfand A

topology on M(A,)

coincides with the original topology on U. h

A

Remark. Note that we have proved that A,

c

C(U), i.e. (A,.U)

0

is a

uF-algebra. (6.1.9) In (1.2.6) we established a one-to-one correspondence between the maximal ideals and the elements of M(A) for B-algebras A. As an easy consequence we proved that the ideal, which is generated by fl,...,fr

s A. is proper iff

{cp E M A ) : q ( f l ) = 0 . i=1.

...,r 1 *

0

(cf.(1.2.9)).

The analogous r e s u l t for F-algebras is true (Corollary (6.1.10) below) but requires much more effort. It was f i r s t proved by Arens

[ARE 11. We give a generalization of this r e s u l t due to Brooks [BRO 21.

136

Chapter 6

Theorem [BRO 21. Let A be an F-algebra and l e t (a,),

be a se-

quence In A such that

Then there exists a sequence (bnIn In A such that

E

aibi = 1,

i=1 i.e. the sequence of the partial sums converges t o 1. Proof. Let (p,),

be a defining sequence o f seminorms f o r A (cf.

(3.1.7) and r e c a l l in particular that p, s P,+~ (3.2.6) the definitions of the algebras A,

f o r a l l n). Recall from

and of the norms p" , resp.

from (3.3.6) the definitions of the maps nn,k (nrk). As usual ( cf. (3.2.7))

we identify M A n ) and r:hl(An)).

We divide the proof into

several steps. i) Set 1,

= 1 and choose for each n

(*I

M(An)

n {cp

E

E

N an integer in so that

M(A): q(al) = cp(a2) =

and in>$,-,, hence in>nfor a l l nc

... = p(al,)

N.

Suppose this choice is not possible. Then, for each k cpk

E

= 0) = 0

c

N, there i s

M(An) such that qk(a1) = qk(a2) =

= qk(ak) = 0.

Since M(An) is compact, a subsequence of (9k)k converges t o an element cp c M(An). Hence by continuity cp(ai) = 0 for a l l i c N. and we get a contradiction.

11) Let (&,,In be a sequence of posltive numbers such that We f i r s t prove by induction that for every n X~~),....X(~) c

in

a)

k

i=l

A,

so that

xn(ai)x/n) = 1,

E

c 8,

< ao.

N there exist elements

Holomorphic functional calculus

Note that

137

(*I implies that

{cp

E

6 n c e each A,

M A n ) : cp(xn(al))

=

... = p ( x n(ain1)

is a B-algebra, we always find zl.

a) is f u l f i l l e d (cf. (1.2.9)). For n=l the conditions B) and y) are vacuous. Now l e t n > l and suppose that x1(n-1) ,....x (n-1) in-1

been chosen. Let zl, ...,z trary t i ,...

= 01 = 0.

in

c

c

....zin c A,

so that

An-1 have already

A, so that a ) is fulfilled. Choose arbi-

E A and set t;=O for in-l 0 iff f c (An)-' by (1.1.5). We f i r s t choose bl c (A1)-', g1 c C(M(A1)) so that f l M( A l )

(Recall that f

Q

= 61exp(g,)

(cf. (1.4.8)).

C(kM(A1) iff flM(An) t C(M(A,))

for a l l n

t

N.)

We now use the proposition to choose inductively sequences (bnIn and (gnIn such that for a l l n c N 1) b,

E

(An)-',

gn c C(M(A,)); A

ii) flM(An) = bnexp(gn); ill) p i (xn+l,,,(bn+l) iv)

Now fix n

t

IIgn+J M ( A n )

-

b,)

- gn'lM(An)

N, and l e t k > l k n . Then

.

< 2 -("+')- m i d l ,dl( b, 1,. .,dn( bn)l; < 2-(n+1)

Chapter 6

144

It follows that (%ksn(bk))kanis a Cauchy sequence in A.,

Denote

the limit by an. I n particular we have by (*I

f o r a l l k c N. So p;l(an-bn) s (1/2)dn(bn), i.e. a,

E

(A,) -1

.

I t follows from iv) that (gklM(An))kan is a Cauchy sequence In

C(M(A,)),

hence converges t o an element h, c C(M(A,)).

We have

flM(An) = %k,n (bk)exp(gklMcAn)) for a1 I k m by ii), A

where %k,n(bk) denotes the Gelfand transform of "k,,,(bk).

Hence

for each p c M(An) f(p ) = i i m % k,n(bk)(p)eXp(gk('p)) = $,(p)exp(h,(p)), -> aD flM(An)

n

= anexp(hn). I t i s easily seen that

I

= h, fo r a l l n

and hn+l Mw,)

ments a

E

a

= a.,

aIM(A,,)

A and h A

E

= a,

N. So (an),, and (hnIn define eien

c

C(kM(A)) so that f = aexp(h1. Since

A

a never vanishes on M(A), hence a

E

A - l by (6.1.4).,,

Corollary. Let A be an F-algebra. Suppose that M(A) i s a k-space. e.g. a locally compact space or a space which satisfies the f i r s t axiom of countability. Then for each f c C(M(A))-' there exist g

c

A-l

and h c C(M(A)) so that f = Gexp(h). Remarks. I ) The theorem was stated f i r s t

-

without proof

- by Arens

[ARE 31. The proof above is due to Brooks [BRO 11. 1 ii) One can use the Arens-Royden theorem to describe H (kM(A).Z).

the f i r s t Cech cohomology group of kM(A) with coefficients in 2. H'(kM(A)) is isomorphic to A-'/exp(A)

for each F-algebra A [BRO 11.

145

Arens-Royden theorem

(Note that exp(A) is a subgroup of A’lJ

Hence, if M ( A ) Is a k-space.

then H1(M(A)) is isomorphic to A-’/t?xp(A). iii)

In the next chapter we shall construct an F-algebra A such that 1

M(A) is not a k-space. It is unproved whether H (M(A)) = H’(kM(A)) in this example.

This Page Intentionally Left Blank

147

CHAPTER 7 AN F-ALGEBRA WHOSE SPECTRUM I S NOT A K-SPACE

The results of (6.2) provide interest to the question whether the spectrum of an F-algebra is always a k-space. The following example due to Dors [DOR] shows that it is not always the case. Furthermore we consider this example also in connection with the following problems: i) Given an F-algebra A and a subalgebra B. When can a l l elements

of M(B) be extended to elements of M A ) ? ii) Give conditions such that the weak Nullstellensatz is valld for a

uF-a1 gebra (A,X). (7.1) The example of Dors (7.1.1) We f i r s t introduce the concept of dimension for complex spaces. Let X, Y be topological spaces. We say that a map

I[:

X

-*

Y is

discrete i f x-'(y) is a discrete subset of X for each y c Y. Definition. Let A be an analytic subset of an open subset U let x

E

c

Cn. and

A. Then

dim,A

= min{k c N: There exists an open neighbourhood V o f

x in A and a discrete holomorphic map

I[:

V ->

Ck I .

i s called the dimension of A at x. If dim,A=k

for a l l x

E

A we say that A is of pure dimension k.

Remarks. i) Let dim,A=k.

then there i s a neighbourhood U of x in A

Chapter 7

148

such that dim Ask for a l l y a U. Y ill Note that dimxAsn if A is an analytic subset o f a domain in Cn. ill) I f x i s a regular point, 1.e. if there exists a neighbourhood U of x

in A such that U is biholomorphically equivalent t o some domain in Ck , then dlmxA=k. One can show that dimxA = supik a N: I n every open neighbourhood of x In A there is a regular point y such that dim A=k). Y (Recall from remark (2.3.1)ii) that the set of a i l regular points lies dense in A,) iv) Let X be a reduced complex space, and l e t x

X. Let (UI,pI,XI)

E

be

a chart around x (cf. definition ( 2 . 3 . 1 ) ~ ) )then ~ set dimxX = dimpi(x)Xi. It can be shown that the definition i s independent of the chosen

charts. v) As a consequence o f the maximum principle we see that a com-

pact analytic subset A c U c Cn consists of finitely many points. Hence dimxA=O for a l l XLA.

vi) Let A

c

G

c

Cn (n22) be an analytic subset such that dim,Agn-2

f o r a l l x a A. Then every f

E

o f Hol(G), i.e. there exists

f" E

Hol(G\A) can be extended to an element .u

Hol(G) such that f IG,*=f

(Riemann

extension theorem, second form). vii) Let A be a proper analytic subset of G. Then f

led locally bounded if each a

c

E

Hol(G\A) is cal-

A has a neighbourhood U in G so that

sup{if(x)i: x a U\A) < a.

If f

a

Hoi(G\A) is locally bounded, then f can be extended to a hoio-

morphic function

viii) Let U

c

r" on G (Riemann extension

theorem, f i r s t form).

Cn (n22) be an open set, and l e t f

V(f)={x

E

u: f(x)

E

Hol(U). I f

= 0)

is not empty, then V ( f ) is an analytic subset of U such that dimxV(f)

2

n-1 for a l l x a V(f).

An F-algebra whose spectrum is not a k-space In fact the following is true: Let A that dimxA2k for a point x o f x, and l e t f A' = A n { y and dimxA'

c

U be an analytic subset such

A. Let W

E

149

c

U be an open neighbourhood

Hol(W) such that f(x)=O. Then

E

W: f(y) = 0) is an analytic subset o f a suitable open set

E

k-1. cf. [G/R].

2

p. 115.

i x ) Now consider the following situation: Let fl.....fr

E

Hol(Ck), and

I et A = (x

E

be nonempty. For x

A we have dimxV(flPn-l

E

dimxV(fl)nV(f2bn-2 dimxA

2

...= fr(x)=Ol

Ck: fl(x)=

and

by viii). Continuing in this manner we see that

maxh-r.01.

Hence, if n > r, A cannot be compact. (7.1.2)Example.

Set CN =

n

4: (countable many copies with the nrN

product topology),

wk

= ((xnIn

xn= 0 for j>k)

o CN:

and

Kk =

((Xnln

E

wk:

lXil

S

k for l

C,

(XJn

xk '

->

Define

P(Kk) = i f : Kk ->

C , f can be approximated uniformly on

Kk by polynomials in tl,...,zk). Then P(Kk) is a uB-algebra and

(*I

M(P(Kk)) = Kk,

since Kk (interpreted as a subset of C

k is poIynomiaIIy convex (cf.

(1.2.4)ii)). Set A = < l -i m P(Kk). Then A is a uF-algebra. M(A) = U Kk

(as sets)

Chapter 7

150

and ( K k l k is an admlssible exhaustion o f M ( A ) with respect t o the Gelfand topology, cf. (3.3.4). Set

X = {(xnIn Clearly X

n Kk

c

U K k : I x , l < l for a l l i

c

N).

i s an open s e t (with respect t o the induced Gelfand

topology on Kk, which equals the original topology on Kk by each k

E

(*I) f o r

N.

Claim. X is not Gelfand open, i.e. M(A) is not a k-space. Suppose X is Gelfand open. We have 0 = (0.0.....1 a r e f,

.....fr

(**I

ty

c

E

X. hence there

A so that

M(A): IPi(y)-P,(o)l
r . Hence g

,,..,,

A

{Y

A

w r + 1 ' fiIwr+,(y) =fi(0). i=l, ...r)

is unbounded, a contradiction t o

(**I.

Remarks.1) Another example is due t o Hayes and Vigue [H/V 21. They constructed even a reduced complex space X with countable topology, such that M(HoI(X)) i s not a k-space. ii) It was proved by Warner [WAR] that the spectrum M(A) o f an F-

algebra i s a k-space Iff C(M(A)) is an F-algebra.

(7.2) Surjectlvity of the transpose map

Let A,B be F-algebras, and l e t T: A ->

8 be a continuous algebra

homomorphism. We showed in (3.2.5) that the transpose map

Surjectivity of the transpose map

T'

M(A),

M(B) ->

~p

->

151

TOT,

is injective f T has dense range. I n this section we deal w i t h the surjectivity of T*. (7.2.1) For n

E

N set

= {(f,

U,(A)

,...,f,)

E

A": There is (gl

,...,9),

A" so that

E

n-

c

i=l For every n

E

T:,

figi = 1).

N we have a natural mapping

A" ->

Note that Tn(Un(A))

,...

B", (fl ,fn) -> c

(T(fl)

,...,T(fn)).

Un(B). since we always assume that T(1) = 1.

Further denote by Uw(A) the set of a l l sequences (fnInin A so that there is another sequence (gnIn in A such that

c 9ifi = 1.

iEN

Set

Theorem. Let A, B be F-algebras, and l e t T: A ->

B be a continuous

algebra homomorphism with T(l)=l. Then T*(M(B)) = M(A) iff

(T~)-~(u~(= B )u)~ ( A ) . Proof. First note that (f,,),, each cp

E

E

M(A) there exists i

1

Uw(A) (resp. ( fl.....fn E

N (resp. i

E

E

U,(A))

i f f for

{l, ...,n l ) so that cp(fi)

* 0,

cf. (6.1.9).

We always have U,(A)

c

(Tw)-l(Uw(B)).

I ) Suppose that T*(M(B)) = M(A). Let (fn)n cp

E

E

(

n

A)\Uw(A) be an arbitrary element. Then there exists naN

M(A) so that cp(fn) = 0 for a l l n

E

N. Choose

9

E

M(B) such that

152

Chapter 7

T*(J,) = cp, i.e. J,oT = cp, then J,(T(fn)) = 0 for a l l n

ii) Suppose there Is cp

E

N, i.e.

M(A)\T*(M(B)).

Then we flnd for each J, c M(B) an element f J,(T(fJ,))

E

* 0 and cp(fJ,) = 0.

J,

E

A such that

By continuity there exists a neighbourhood U of J, in M(B) such that $(T(f,,,))

J,

*0

for a i l

6

E

U6.

Since M(B) is hemlcompact there exists a sequence

In M(B) so

that

.

M(B) = U U 1 6 N $1 Hence we have

= 0 for a l l I c N and

a) P(fJ,i)

B) for each It follows that (T(f

5E

M(B) there is n

N such that $(T(f

E

)Iic UoD(B), while (f Ii E $1

Qi

JIn

1)

*

0.

n A \ UoD(A).

ieN

0

(7.2.2) We remark that (Tm)-'(Um(B)) = U,(A)

(*I

(T,,)-~(u~(B)) = u,(A)

for a i l n

E

implies that

N.

This leads t o the question whether the weaker condition (*I already guarantees the surjectivity of T*. We shall show below that in general this is not the case. However we have the following result. Corollary. Let A, B be F-algebras, and l e t T: A ->

B be a continu-

ous algebra homomorphism such that T(l)=l. Suppose that (*I is satisfied. Then T* is surjective If M(B) is compact, e.g. B is a B-algebra, or i f there is (fl,

...,fn) E A"

such that the map

153

Surjectivity of the transpose map

f: M(A) ->

A

(fl(cp)

Cn. cp ->

....,?,,(cp)),

is injective, e.g. A i s rationally finitely generated. Proof. i) Let M(B) be compact. Suppose that T* is not surjective. We use the same notations as in the proof of ii) of theorem (7.2.1). Repeating this proof we see that there are finitely many points $, ,....$, r

so that

.

M(B) = U U i=l

6

I t f o l ows that ( T ( f ),...,T(f 1) $1 Qr (f,,,,...,f,,+) ii) Let

c

f

E

Ur(B), while

Ar\Ur(A), a contradiction to

(*I.

be injective.

Suppose there is cp

E

M(A)\T*(M(B)). Then

M A ) : cp(fi) = Jl(fi). i=1,

{cp) = {$

E

(fl-p(fl),

...,fn-cp(fn))

...,n).

Hence

A"\u,(A)

I

but (T(f l-cp(f 1 ) ) ,...,T(fn-cp(fn)))

E

Un(B)

I

a contradiction to (*I. (Actually we have proved a stronger result. Namely if then T* is surjective if (Tn)-'(Bn) = An.)

f

is injective, 0

(7.2.3) Finally we use the example of Dors (7.1.2) to show that condition ( * ) o f (7.2.2) in general does not imply the surjectivity of T*. We use the same notations as in (7.1.2). Set Dk = {(x,,),,

z

Kk: l ~ ~ 1 < 1 / 2 i=l, . ...,k}

and

Ek = Kk\Dk.

Ek is a compact set for each k

E

N and

Chapter 7

154

...->

C(Ek+l)

-> C(Ek) Ik

_*

...

constitutes a dense projective system of uB-algebras with respect t o the natural restriction mappings

Set B = & L C ( E n ) . Then B is a uF-algebra by (4.1.3) and M(B) = U M(C(En)) = U En neN nE N

8 Is the algebra of

as sets by (1.2.4111 and (3.3.4).

ued functions f on U En such that

fl

a l l complex-val-

is continuous f or a l l k

E

N.

Ek

Let A = f L P ( K n ) be the algebra o f Dors example. Interpret each ? €

^A as complex-valued

function on U Kn, which can be approximated

uniformly on each Kn by polynomials. Then n n

A

T: A ->

B,f ->

1"

n

f

defines a continuous homomorphism and

T*(M(@)) =

u E,

* u K,

= M&.

i.e. T* is not surjective. (Note that we can naturally identify M(A) and M(^A).) Now f i x k x

E

E

*k A N, and l e t (il,...,tk) E A \Uk(A). Hence there exists

U Kn such that i l ( x ) =

v

= {y

6

... = gko() = 0,w.1.o.g.

wk+1: B l ( y ) =

A

x

c

K,.

The set

A

..a

= gk(y) = 0)

is not empty and we conclude as in (7.1.2) that V (interpreted as a subset o f Ck+') is unbounded, i.e. there exists

Weak Nullstellensatz

155

Remark. Corach and Suarez [C/S] proved that the condition

(*I is

equivalent t o the surjectivity of T* in the case that A and B are Balgebras. The proof o f theorem (7.2.1) is a suitable modification of their proof. (7.3) The weak Nullstellensatz (7.3.1) Definition. The weak Nullstellensatz is valid for a uF-algebra

(A,X) if for every finite set of functions fl.....fm zero in X there exist functions gl,...,gm m fig1 = 1. i=l

E

a

A with no common

A such that

c

(7.3.2) Remarks. i) We use the terminology of Hayes [HAY 21. This terminology is motivated by a theorem in algebraic geometry referred t o as the weak Nullstellensatz. It states that every proper ideal in in a polynomial ring K[T l,...,Tn]

in n variables with coefficients in

an algebraically closed field K has at least one zero in Kn. Since K[T1,...,T,,]

is Noetherian, this is equivalent t o saying that finitely

many elements in K[T

l... ..Tn]

without common zero never generate

a proper ideal. ii) Recall from (2.2.7) that for domains G

c

Cn the following is true:

The weak Nullstellensatz is valid for (Hoi(G).G) if G is a domain of holomorphy resp. iff M(Hol(G))=G. This result does not hold in our general setting as we shall see below. iii) I t follows from (6.1.10) that the weak Nullstellensatz is valid for

&.M(A)) i f A is a uF-algebra.

156

Chapter 7

(7.3.3) Theorem. The weak Nullstellensatz is valid f or a uF-algebra

iff

F(X) = P(M(A)) fo r a l l F c Am and a l l m

...,),f

(For (f,,

E

PMA))

E

N.

Am we set F(X) = ((f,(x),...,fm(x))

= {tPl(x)

Proof. The inclusion

"c"

c

Cm: x

X I and

E

....,fm(x)) E cm:x E M(A)).) A

i s always true.

i) Suppose that the weak Nullstellensatz is valid for (A.X). Assume there is m

E

N and f r Am such that

P(M(A))\F(x)* 0 . We choose a point (A,, ...,) ,A fm-Am

in this set. The functions f 1- A 1'""

have no common zero In X, hence there are gl,..,gm

that

m

c

i=l

E

A so

(fi-Ai)gi = 1.

This implies m (?i-Ai'^al i=l

c

= 1,

a contradiction to the fact that ?l-X1s...s?m-Xm

have a common zero

in M A ) .

ill Now letF(X) = h ~ l ( A ) fo ) r all f

,...,,g

Let g1

c

E

Am.

A be without common zero in X. Then

8,. ...,g, A

have

(7.3.4)Corollary. i) The weak Nullstellensatz Is valid for a uF-algebra (A,X) if the structure map j: X ->

M(A). x ->

cpx, (cf. (4.2.1))

is surjective.

ill If (A,X) is a uF-algebra and there is F

E

Am such that

Weak Nu1I stel lensatz

F: MA)

ern

->

157

is injective, e.g. if A is rationally finitely gener-

ated, then the weak Nullstellensatz is valid iff j is surjective. Proof. i) This is a direct consequence o f theorem (7.3.3). ii) If j is not surjective we find a point x

Q

M(A)\j(X). Since

e

is in-

jective we have

P(M(A))\F(x),

A

F (XI

and our assertion follows from (7.3.3).

El

(7.3.5) We use the example of Dors (7.1.2) to show that in general the converse of statement i) is false. We use the notations of (7.1.2) and (7.2.3): wk =

{(Xnln

t

Kk =((Xn),, Dk =

E

{(Xnln

E

CN:

X1

wk:

lXil

Kk:

lXil

= 0,j>k),

...,k), i=1, ...,k)

k. i=1,

< 1/2,

and Ek = Kk\Dk. Let A = &&P(Kn)

be the algebra in the example of Dors. Recall that

M(A) = U Kk as sets, that (Kk)k is an admissible exhaustion of M(A) with respect to the Gelfand topology. and that

^A

is the algebra of a l l complex-valued

functions on U Kk which can be approximated uniformly on each Kn by polynomials. Set X = U Ek, and endow X with the relative Gelfand topology. Note that (EkIk is an admissible exhaustion o f X. i.e.

X is a hemicompact space. Consider the restriction mapping h

r : A ->

h

Al,f

A

->

A

fl,.

Clearly r is a surjective algebra homomorphtsm. Let r(?) = 0. Then A

1

0

f o r every k

L

N.

158

Chapter 7 A

Since we can interpret ,fl have

?Iwk

k

k as a holomorphic function on 41 , we

m 0 by the identity theorem, i.e.

? * 0,

and r i s injective.

Moreover it follows from the maximum modulus principle for hoiomorphic functions that

A

1.e. r is a topological homomorphism. So (AI,.X)

is a uF-algebra.

which is isomorphic to ( h l ( A ) ) and we see that j:

M(AI,) A

x ->

= M(A)

is not surjective. A

A

A

NOW

l e t gl(x,...,gklx

E

be arbitrary elements without common

zero on X.Then we see as in example (7.2.3) that tj,.....gk A

a common zero on M(A)=M(A),

e "h;gl

i-1

hence there are

6 , ,...,fik

A

c

don't have

2 so

that

= 1,

by (6.1.10). i.e. the weak Nuilstellensatz is valid for (hA),,X).

(7.3.6) Remarks. i) Nevertheless the validity of the weak Nullsteliensatz implies that the structure map j has dense range in M A ) . Otherwise there would be a point x

E

M ( A ) and a Gelfand open neigh-

bourhood U of x,

u = {y

E

M(A): i?p)-?,(y)i

< 1, i = i ,...,r i ,

(cf. (3.2.2111) such that

U

c

M(A)\j(X).

But this would imply that

Pw =

(?,(xi.

....fn(x))

?(M(A))\F(x),

and we get a contradiction to theorem (7.3.3).

f,

...,f r

E

A,

Weak Nu1l s t e l lensatz

159

ii) The converse of remark i) is false.

For instance l e t

5 = (1x1 i 1).

X =

B\

(01,and l e t A be the algebra

of a l l functions on X which can be approximated uniformly by polynomials on each compact subset of X. Then (A,X) is a uF-algebra which is algebraically and topologically isomorphic t o the algebra

P ( 5 ) . We have M(P(B)) = j: X ->

by (1.2.4)iI). hence

-

M(A) = D,

has dense range, but Z(X)

* ~(M(A))

for the map Z:

-D ->

C . x ->

X.

So the weak Nullstellensatz is not valid for (A.X) by (7.3.3). iii) I n [HAY 23 Hayes constructed a hemicompact reduced complex

space X so that the weak Nullstellensatz is valid for (Hol(X),X) but j is not surjective.

This Page Intentionally Left Blank

161

CHAPTER 8 SEMIS 1MPLE F- ALGEB RAS

An F-algebra A is called semisimple if the Gelfand map

f ->

r: A ->

h

A.

A

f . is injective. I n particular each uF-algebra is semisimple by

the proof of (4.1.3). I n this chapter we prove two results for semisimple F-algebras. Firstly we show that each semisimple F-algebra has a unique topology as an F-algebra. This extends the analogous r e s u l t for semisimple B-algebras (1.2.12). Secondly we prove that each derivation D on A is automatically continuous. (By a derivation we mean a linear mapping D: A -> f o r a l l f.g

E

A so that D(fg) = f D(g)+gD(f)

A.) We remark that, opposite t o the situation for semi-

simple 8-algebras, there exist nontrivial derivations on semisimple F-algebras. Both results are due t o Carpenter [CAR

11 and

[CAR 53.

(8.1) Uniqueness of topology (8.1.1)Definition. The radical of an algebra A is the intersection of a l l maximal ideals in A . It is denoted by rad A. A is called semi simple i f rad A = (0). (8.1.2) Proposition. An F-algebra A is semisimple i f f the Gelfand map

r: A ->

h

A, f ->

A

f,

i s injective. Proof. I t suffices to show that rad A = fl qtM(A)

ker C Q .

162

Chapter 8

Certainly rad A Now l e t x

(*I

L

c

n ker

cp.

ker cp for a l l p

E

M A ) . Then

cp(l-xy) = 1 for each cp

E

M(A). and a l l y

hence l - x y is invertible by (6.1.4) for each y

E

L

A.

A.

Suppose there is a maximal ideal M in A so that x

L

A\M.

The smallest ideal which contains M and x is A. Hence there is y and m

E

M. so that 1 = xy + m, i.e. 1

diction t o

- xy

E

E

A

M, and we get a contra-

(*I.

0

(8.1.3) Remarks. i) Each uF-algebra is semisimple by the proof of (4.1.3). ii) Let A = CaD([O,l])

(cf. (3.3.5)). Then M ( A ) = [0,1] by (3.3.5) and

A is not a uF-algebra by (4.1.4)iv). Let f x

c

M A ) , hence f(x)=O for a l l x

E

E

rad A, 1.e. ?(x)=O for a l l

So fmO and A is semisimple

[0,1].

by (8.1.2). (8.1.4) Proposition. Let A be an F-algebra. Let of distinct points in M A ) . Then there exists h

be a sequence E

A so that (cpi(h))l

i s a sequence of distinct points in the plane. Proof. For i*j set

Aij = t f

E

A: cpl(f) = cpj(f)}.

Obviously A

is a closed subset o f A. Let (pnIn be a defining seii quence of seminorms for A. cf. (3.1.7). Suppose that A has nonil empty interior. Then there are f E A n E N such that li ' tg Choose u

E

E

A : pn(f-g) < l / n }

A

-

ii A such that cpi(u) = 0 and cp (u) = 1 by (3.2.3). If pn(u)=O

then pn(f-(f+u)) be a derivation. Let

be a sequence o f distinct points in M(A) so

('pill

that the functionals 'pIoD: A -> Then there exists c

E

D ( d . is unbounded on

A

C are discontinuous for a l l i

c

N.

A such that 6 ( c ) , the Gelfand transform of ('pi)i.

Proof. First note that D(c-1) = 0 for a l l c

E

C.

Let (pill be a defining sequence of seminorms f o r A (cf. (3.1.7) and r e c a l l in particular that pnspn+l we can assume that each

(*I

I'pi(f)i

P

'pi

and pn(l)=l for a l l n

E

N). W.I.0.g.

is continuous with respect to pi, i.e.

pi(f) for a l l f c A, i E N.

According to (8.1.4) we can choose a sequence (fk)k in A so that cp(f = 0 for j ll&(Ak),"-

I t follows that U

n M(AP1 contains

a local peak set for A

By P' Rossi's maximum principle (1.4.9) V contains a peak set for A so P' V n yA 0 . Since V was taken t o be arbitrary and yA Is closed, P P hence we see that Jl c yA P'

*

Repeating the l a s t argument for W and r A we conclude that

'Q E

FA

as desired.

0

(9.1.4)Remark and notation.Let A be a B-algebra. I t follows from (9.1.3) that r A = yA in thls case. So we shall denote henceforth the Shilov boundary of an F-algebra A by yA, too. (9.1.5) Example. Proposition (9.1.3) fails t o be true if M(A) is not locally compact.

For n

L

N define the following sets in C2: 'n = {(t,t/n): t e C0.13), n K n = {(z,w): n ( w - ( z / i ) ) = 0 , l z l i=1

1, i wi

1)

and

K;I = Kn U In+l. i) We claim that Kn and K;I are polynomially convex compact sets.

This is clearly true for Kn. Let (x.y) Case 1: (x,y) L C 2\Kn+l. Then, since Kn+,

Q

2

C \KA

.

is polynomially convex, we find a polynomial q

Shilov boundary f o r Fr6chet algebras

175

such that Iq(x,y)l > llqllKk. Case 2: (x.y) Then x

E

E

Kn+,.

C\[O.l]

and we find a polynomial q in one variable so that

Iq(xN = 1 > llqll[o,,]*

since [O,l]

is polynomially convex, cf. (1.2.4).

Set r(2.w) = q(z). then

Furthermore consider

m sufficiently large. ill We note that

(*I

... c

Kn

Set A = l&P(K',).

c

K;1

c

Kn+l

c

K"+l

c

... .

Then A is a doubly generated uF-algebra. Note

that a defining sequence of seminorms is given by

(ll-llKh)n and

that

We have M(A) = U K', (as sets) and (KInIn is an admissible exhaustion of M(A) by (5.1.6). ill) Claim: (0.0)E yP(K',)

Let (x,y)

E

for all n

N.

In+1 \C(O.O)> be an arbitrary point, and l e t U be an open

neighbourhood of (x,y) in K;1. Set

and

E

176 Hence

Chapter 9

- for

sufficiently large m

I t follows that yP(K;l)nU

*

-

0. Since U was chosen t o be arbitrary,

and since yP(K;I) is closed, we see that (x.y)

In+l\{(O,O)l

c

E

yP(K;I), hence

yP(Kh) and finally (0,O) E yP(KA).

iv) Define

U = {(z,w)

M(A) = U K;I: lzi I?,(Y)I

I?,(x)I

for a l l y c K\U.

Thus I(efu)A(x)i > I(ef,)^(y)i

f o r a l l y c M(A)\U,

where (efu)^ denotes the Gelfand transform of ef,. this observation that x

E

It follows from

yA.

0

(9.1.7) Proposition. Let A be an F-algebra such that y A = 0. Then A

each element of A satisfies the maximum principle on M(A), 1.e.

Ilf I,

(*I for a l l

A

=

Ilf ,I

? c 2 and a l l

compact subsets K

M A ) . (Here dK denotes

c

the boundary o f K.) Vice versa, if M ( A ) i s locally compact and

(*I is satisfied, then

y A = 0. Proof. i) Let y A = 0 . By (9.1.6) dK

K

c

+ 0 for

a l l compact subsets

M(A).

Suppose there exists a compact set K c M ( A ) and

? c ^A

so that

M ( A 1. Then by Rossi's P maximum principle (1.4.9) - there exists cp E int M ( A 1 n y A where P P' int M ( A 1 denotes the interior of M ( A 1 with respect t o the Gelfand P P topology on M A ) . Repeating the proof of proposition (9.1.3)ii) (re-

Choose a seminorm p c P(A) such that K

c

place Ak by A ) one sees that cp c yA. This is impossible. P ii) Let (pnIn be a defining sequence of seminorms. Assume

(*I, then

178

Chapter 9

YAP,

c

dM(A

1 for a l l n

Pn

c

N.

Since M(A) is locally compact we have f or each k

n MA

nkk

1=

Pn

E

N

0,

so

n YA~,= 0,

nkk

hence yA = 0 by (9.1.3).

0

(9.1.8) Example. We construct a uF-algebra A such that

(*I of

(9.1.7) is satisfied but yA 8 0 (hence M A ) is not locally compact). For n c N set

Every Kn is polynomially convex (cf. (1.2.4)ii)).

Set A =

C , continu-

ous or not. We endow S(A) with the Gelfand topology, that is the h

coarsest topology such that a l l Gelfand transforms ? c A are contlnuous functions on S(A). A basis for the neighbourhood system of a point cp

E

S(A) is given by a l l sets of the form

cf. (3.2.2111. We know by (1.2.1) that S(A) = M(A) for B-algebras. Michael [MIC] posed the question as whether the same is true for F-algebras. This question is s t i l l unanswered. Arens [ARE 11 proved that S(A) = M A ) i f A is a rationally finitely generated F-algebra. Besides some other results we sketch in this chapter an approach t o this problem due to Dixon and Esterle [D/E],

which leads t o an

interesting problem in the theory of several complex variables.

(10.1) Results on the automatic contlnuitv o f characters (10.1.1)Lemma. M(A) is a dense subset of S(A) for each F-algebra A. Proof. Suppose the lemma is false. Then there are cp fl*....,f n c A such that

E

S(A).

186

Chapter 10

Set Xi = p(fi) and gi = fl-Xi,

i=1,

...,n. Then g1,....gn have no common n

n

zero on M(A), hence by (6.1.10) there are hl,...,hn

E

A such that

n

higl = 1. i=1 But this leads t o a contradiction, since 9,. ...,gn

E

ker p. which i s a

proper ideal in A.

0

(10.1.2) Remark. One easily sees that if X is a topological space and

Y Is a k-space, then every continuous map f: X ->

Y which i s prop-

er is closed. Now consider the map

M A ) ->

S(A),

->

p.

I t is continuous and it is also proper. (Otherwise there would be a

compact set K

c

S(A) such that K f M(A) l is not compact. Copying

the proof of (4.3.6) bounded on K

-

-

using (3.2.6)

we find f

n M(A), a contradiction to

E

A so that

?

is un-

the compactness of K.)

Hence, if S(A) is a k-space, then the map is closed and we have M(A) = S(A) by (10.1.1). (10.1.3) Let (fl

,...,fn) E An Then set

P: S(A) ->

Cn , cp ->

n

(fl(p)

,....fn(p)).

Proposition. Let A be an F-algebra, and l e t (fl,....fn)

E

An. Then

PMAN = PMA)). Proof. Suppose there exists X E P(S(A))\P(M(A)). Set g1 = fi-Xi.l (i=l,

....n)

and continue as in the proof of (10.1.11.

(10.1.4) Theorem. Let A be an F-algebra. Suppose there are

(f,.....f,,)

E

An such that for each p

E

M(A) the set

f-V(p,)n M(A) is at most countable, then S(A) = M(A).

0

Michael's problem Proof. Let

187

IJ 6 S(A) be an arbitrary element. Choose

(10.1.3) such that

p(cpo)= f($). The

cpo c M(A) by

set

is at most countable by our hypothesis, hence

- by

(8.1.4)

-

we

n

find fn+l c A so that fn+l i s injective on S. (If f is finite use (3.2.31.1 So, using again (10.1.31,

there is exactly one element p1 E M ( A ) such

that

Let g c A be an arbitrary element. Then pl(g) = $(g) by (10.1.3) and

(*I. But this implies

p1 = @.

0

Remark. The theorem is due to Zelazko [ZEL 23, the proof i s as in [GOL

31.

(10.1.5) Theorem [MIC],[ARE 11. Let A be an F-algebra. Suppose k k there are ( f l , fk ) c A such that for each z c C the set

....

W Z )

n M(A)

is compact. Then M ( A ) = S(A). Proof. Let $ c S(A) then, by the hypothesis,

K

={Q E

MA):

p($)= p(p)l

is compact. By (10.1.3) K i s not empty.

For each b

E

A the set

u b = {cp c M(A): cp(b)

* IJ(b)l

is open. Suppose that U Ub is an open covering of K. then there are brA bl,

(*I

...,br c A such that K

C

U

bl

U

U Ub;

188

Chapter 10

By (10.1.3) there exists cpo c M(A) so that I) vo(bl) = $(bl)

So po

E

(i=l,...,r),

and

K by ill. which is a contradiction t o 1) and (*I.

Hence there exists

1.e. cp =

JI.

0

(10.1.6) Corollary [ARE 11. I f A is a rationally finitely generated Falgebra then M(A) = S(A). Proof. Let fl,...,fn

be rational generators f or A, then f or each z

E

Cn

the set M(A) n f - ' ( z ) is either empty or,consists of a single point by (5.2.3). So we get the desired r e s ul t from the theorem above. (10.1.7) Examples. i) Let A = C"([O.l])

M(A) = [0,1]

(cf. (3.3.5)). Then

by (3.3.5). Hence M(A) = S(A) by (10.1.41, since

2 : [0,13

_*

defines an element of

c, t

->

t,

2.

ii) Let X be a hemicompact k-space. Then C(X) i s a uF-algebra by

(4.1.4)ii). We claim that

The f i r s t equality was already established in (4.1.7)i). Let (Knln be an admissible exhaustion of X and suppose that there is

J,

c S(C(X))\X.

S(C(X)) is a completely regular space, since it carries the initial topology generated by all Gelfand transforms

?.

f

c

C(X) (cf. [ENS]

(0.2.2).(0.2.3)). Hence we find for each nc N an element

Michael's problem

189

fn c C(S(C(X))) such that I) fn(J,) = 0,

ii) fn(S(C(X))) = co,2-nl,

Note that fnl, f

=

c

C(X). By ill)

c fnl,

nc N

defines an element of C(X) and f(cp) Consider the Gelfand transform

?E

i?(J,)l > E > 0. Choose n E N so that

* 0 for a l l cp

E

X by ii) and 111).

C(S(C(X))). Suppose that

c

2-i < ~ / 4 .Let i>n

Then U is an open neighbourhood of J, and we can choose a point cp

P

M(A) flU by (10.1.11. Hence

a contradiction to cp

E

U. So

f(cp) * 0,

f or a l l cp

E

X, and

= 0,

and we get a contradiction to (10.1.3). ill) A uF-algebra A i s called Stein algebra if there exists a reduced

Stein space X ( c f. (2.3.2)) such that A i s topologically and aigebraically isomorphic to Hol(X), the algebra of a l l holomorphic functions on X endowed with the compact open topology.

I f X is a connected finite dimensional reduced Stein space, then by Remmert's embedding theorem there exists an injective holomorphic mapping into some Cn. Since X = M(Hol(X)) by (2.3.2) we have M(Hol(X)) = S(Hol(X)) by theorem (10.1.4) in this case. (We remark for the sake o f completeness that this res ult remains true f or not

necessarily reduced Stein spaces (X.0) and the algebra Ho(X,O).

190

Chapter 10

For a generalization of this result cf. Ephraim [€PHI.) We give a simple proof rem

-

-

not depending on the deep embedding theo-

for M(Hol(X)) = S(Hol(X)) if (X,p) is a holomorphically se-

parable Riemann domain over Cn. Note that in this case pi (i=l,...,n) Since

6

Hol(X)

cf. (2.2.2). Recall from (2.2.5) that

if p=(p1,...,p,,),

(M(Hol(X)).;)

E

is again a Riemann domain over CnDwhere ;=(p,

A

A

,...,p ,I.

i s a local homeomorphism and M(Hol(X)) is a hemicompact

space we conclude that for each z

c

Cn, fi-l(z) is either empty or

at most countable.(Suppose that this is false. Then ;-l(z) is not countable for some z

c

Cn. Let (KnIn be an admissible exhaustion

of M(Hol(X)), then there is k K~

E

N so that

n 6%)

is not countable, i.e. f-'(z)

has an accumulation point in M(A). and

we get a contradiction.) Now our assertion follows from (10.1.4). (10.21 The aDDroach of Dixon and Esterle

Next we want t o sketch the approach of Dixon and Esterle t o Michael's problem. Their work leads to an interesting problem in several complex variables. Recall that we always consider commutative F-algebras with unit.

(10.2.1) Proposition. Let f

E

Hol(Cn), and l e t al,

...,an be elements

of an F-algebra A. Let f(Zl

c

,z& =

I...

il

,. .,iaO

xI1,...Dinzl

11'.."Zn in

*

be the power series expansion of f (cf. (2.1.51). then

defines an element of A and the map

191

Michael's problem

A" ->

A, (al

,....an

f(al

->

,...,an)

is continuous. Proof. That f(a l.....an)

c

A follows from (6.1.2) (use the same argu-

ment as in (1.4.4)i)). Let (pI II be a sequence o f defining seminorms for A (cf. (3.1.71). and l e t (al(k) ,...,a(k))k be a sequence in An which converges t o n (bl, bn) E An.

...,

Fix arbitrary I

E

N and E>O. Choose r > O so that

pl(ai(k)) < r , i=1,

...,n.

k

E

N

and pl(bi) < r , i=1,

...,n.

Then pI( f (a! ),

+ 2.

...

c

$+...+I n >s

1-f ( bl,. ..,bn1 1 s

I X i1,

...,in

Ir

il+...+ in

For s large enough the value of the second sum i s less than e/2 and the f i r s t sum tends to zero for k -> So we have proved that f(a!k)s...,a(k)) n

(10.2.2) Remark. If F = (f,,...,f k )

E

-.

converges to f(b l.....bn).

HOI(C".Q:

k

1, i.e. fi z HOI(C") for

...,

i=l. k. then the map

An ->

A k , (al

,...,an1 ->

F(al

,...Dan) =

192

Chapter 10

,...,an), ...,fk(al ,...,a,))

= (fl(al

,

is continuous. This is an easy consequence of the proposition.

(10.2.3) Proposition. Let F = (flD....fk) F-algebra. Then, for each a

E

E

Hol(C",C k 1, and l e t A be an

An,

F(a)^ = Fog , where F(a)^ denotes the Gelfand transform of F(a). More precisely: Let cp

E

S(A), then tp(fi(al

Proof. Let cp

E

....,an)) = fi(cp(al) ,...,cp(a,)).

...,k.

S(A). Denote by B the closed subalgebra of A gener-

...,an.

ated by al,

14,

Since cp restricted to B is continuous by (10.1.6)

we get the desired r e s u l t from the definition of fl(a l,...,an).

0

(10.2.4) Theorem. Let (snIn be a sequence of positive numbers. If there is a discontinuous character on an F-algebra A, then the projective limit of every projective system

... -* where F,

E

CS"+1

F , > CS"

->

...

H O I ( C ~ " ~ , C ~ " )is not empty.

Proof. Let cp be a discontinuous character on A. i) Since the closure of ker cp is an ideal and ker cp is a maximal

ideal we see that ker cp is dense in A, cf. (3.2.11). ii) Set

q1 = 0, q, = S~+..,+S,,-~for n > l . If we endow ker cp with the discrete topology then

En = A

S

x (ker pIqn

is a complete metrizable space for a l l n Tn: En+l ->

En, (al

,...,asn+l,xl,

E

N. We consider the map

...,x qn+11 ->

Michael's problem

,...,aSn+l ) + ( x ~ ~..., + xQn+, ~ . ),x p...1x 1. qn

(Fn(al T,

193

is continuous f o r each n

E

N

by (10.2.2) and since ker cp Is en-

dowed with the discrete topology. Moreover Tn(En+l) by 1). Hence we can choose an element 8 =(Bn),

z

is dense in En

lh(En.Tn)

by (3.3.2). For each n

c

N we can write

B, = (bn,xn), b,

E

AS". x, c (ker cp) 4n

.

Since Tn(bn+lsxn+l) = (bn,xn) we have Fn(bn+l)-bn

c

(ker plSn,for a l l n

E

N,

hence by (10.2.3)

(10.2.5) Corollary. Let p

c

N. I f

there exists a sequence (Fnln of

holomorphic mappings from Cp into i t s e l f such that

n F,'

...o

ncN

F,(cP)

= 0 ,

then S(A) = M(A) f o r every F-algebra A.

(10.2.6) Remarks. i) Although the statement of the corollary is very simple, the question of the existence of such a sequence (Fnln

E

Hol(CP,CP) seems t o be a difficult problem. First note that

if one mapping Fi is constant, then

n F~~

...o

F,(cP)

194

Chapter 10

is never empty. So we consider only the case when no Fi is constant.

If p=1, the l i t t l e Picard theorem ([NAR l],p. 94) asserts that each nonconstant entire function attains each value with one possible exception. I t follows that F1o ...* Fn(C) is a dense open subset of C for each n e N. hence

n F,"

...o

F,(c)

is a nonempty subset o f C by Baire's theorem.

If p > l it is known that there are injective entire functions

F: Cp ->

Cp, whose jacobian identically equals one but whose range

is not dense in Cp. So for p > l a sequence with the desired property might exlst. For a detailed discussion of this question we ref er the reader t o the paper o f Dixon and Esterle [DIE]. 11) Clayton [CLA 13, Schottenloher [SCH],

Dixon and Esterle [D/E]

and Craw [CRA 21 constructed test algebras f o r Michael's problem. More precisely they constructed F-algebras A with the property that Michael's problem can be solved iff it can be solved f or A.

195

PART 3 ANALYTIC STRUCTURE IN THE SPECTRUM OF AN F-ALGEBRA

An important subject in the theory of uB-algebras is the question o f the existence of analytic structure in spectra, cf. f o r example chapter I11 of Stout's book [STO]. One seeks for conditions which ensure that parts of the spectrum of a uB-algebra can be endowed with the structure of a complex space in such a way that a l l Gelfand transforms become holomorphic functions on it. As proved in (2.1.13) algebras of holomorphic functions are not uBalgebras but uF-algebras (cf. (2.3.1)). Hence algebras

- we

-

in the case o f uF-

are moreover interested in the question when a given

uF-algebra is topologically and algebraically isomorphic to Hol (XI,

X a suitable complex space. In some cases we shall obtain a function algebraic characterization of certain classes of holomorphic functions, for instance we shall characterize Hol(X) in the case that

X = C. X

c

41 a domain, X a logarithmically convex complete Rein-

hardt domain, X

c

Cn a polynomially convex domain etc.

.

There are results f o r uB-algebras, in particular Rossi's maximum principle (1.4.91, which reminds one of the behavior o f holomorphic functions. This led t o the question whether these results depend on the existence of analytic structure in the spectrum. I n general this turned out to be false. In this part of the book we adopt this problem for uF-algebras and ask to what extend properties of Hol(X) (for example the maximum principle, Liouvil ie's theorem, identity theorem, reflexivity, Montel's theorem, etc. 1 characterize this a l gebra within the class of uF-algebras. I n our setting we shall prove that a uF-algebra has analytic struc-

196

Part 3

ture in every point of i t s spectrum iff it is a Stein algebra, 1.e. iff it is topologically and algebraically isomorphic t o the algebra of al I holomorphic functions on a (reduced) Stein space.( For the definition of Stein spaces cf. (2.31.) Hence, in connection with the problem of finding analytic structure in the spectrum of a uF-algebra. we are interested in characterizing Stein algebras by intrinsic properties within the class of uF-algebras. This project is also interesting, since Stein algebras play an important r o l e in the theory of several complex variables.

197

CHAPTER 11 STEIN ALGEBRAS

For the definition and properties o f Stein spaces cf. (2.2) and (2.3). As mentioned in the introduction t o part 3 of the book we shall prove that a uF-algebra is a Stein algebra iff it has analytic structure in each point of i t s spectrum (11.1) Analytic structure in spectra of uF-algebras (11.1.1)

Definition. A uF-algebra A is called Stein algebra if there

exists a (reduced) Stein space X such that A i s topologically and a l gebraical l y isomorphic to Hol ( X I . Remark. Let A be a Stein algebra, and l e t T: A ->

Hol(X) be an iso-

morphism, X a Stein space. Recall from (2.3.2) that X = M(Hol(X)). more precisely the structure map j: X -> (cf. (2.3.2)) T*: X ->

M(Hol(X)), x ->

cp,

defines a homeomorphism. So the adjoint spectral map

M(A) is a homeomorphism by (3.2.5). Note in particular

that the spectrum of a Stein algebra is always locally compact.Furthermore the map (2.M(A)) ->

(Hol(X)^ ,M(Hol(X))) = (Hal (X),X).

P->

A

faT*,

where Hol(X)^ denotes the algebra of a l l Gelfand transforms, defines a topological homomorphism by (4.2.4). (11.1.2)Definition. Let A be a uF-algebra, and l e t cp z M(A). We say h

that A has analytic structure in cp provided there are an open neighbourhood U of cp in M(A), an analytic subset Y o f a domain

G

c

C"

198

Chapter 11

and a homeomorphism p: Y

-*

?

U such that

0

p

L

Hol(Y) for al I

2.

? €

Remark.Let

(9,

U, Y, p and G be as above. Note that U is locally

compact. Hence there exists an open, hemicompact, re1atively comh

pact and A-convex neighbourhood V of cp such that V

c

U by (4.3.7).

Then p-’(V) is an analytic subset o f G\p-l(U\V). Let K c p-’(V) be a compact subset. Then

where $KIA denotes the 2-convex hull o f p(K). is a compact subset h

o f P - ~ ( V ) ,since p is a homeomorphism and V is A-convex. Let

x

L

p-’(V)\L.

then p(x)

E

M(A)\p(KIA. i.e. we find

? ^A 6

so that

It follows that

{y since

? op

E

L

p - l ( ~ ) ; ig(y)i

11g11,

for a l l g

6

H ~ I ( ~ - ’ ( v ) ) Ic L.

H o I ( ~ - ~ ( V ) So ) . p-’(V) is holomorphically convex, hence

Stein, since the coordinate functions separate the points of p-’(V). Thus we can w.1.o.g. assume In the definition above that Y is a Stein analytic subset and that U is a hemicompact, relatively compact and A

A-convex subset. (11.1.3) We need a r e s u l t of Rossi [ROI],

which can be stated in the

following form: Theorem. Let X be a reduced complex space, and l e t (A.X) be a uFalgebra such that X is A-morphically convex (cf. (4.3.2)) and A is a subalgebra of Hol(X). Then M(A) can be given the structure of a Stein space such that

^A

= Hol(M(A)).

Stein algebras

61.

(11.1.4) Theorem [KRA

199

Let A be a uF-algebra. Then the following

statements are equivalent: i) A i s a Stein algebra; ii)

2 has analytic structure

in each point cp

E

MA);

iii) There exists an open cover (Ui)lcI of M(A) consisting o f hemicomh

pact, relatively compact and A-convex subsets such that each algebra

2UI is a Stein algebra.

Proof. i) 4 ill) Let A be a Stein algebra, and l e t T: A -> an isomorphism, X a Stein space. Let cp

E

Hol(X) be

M A ) be an arbitrary point.

Since M(A) is locally compact there exists an open, relatively compact, hemicompact and 2-convex neighbourhood U c M(A) of cp. Then is an open and Hol(X)-convex subset of X, hence a Stein

(T*)-'(U)

space. The map

i s well-defined and defines a topological and algebraical isomorphism. By (2.3.6) and since HoI((T*)-~(U)) is complete we have

iii) +ii) Let cp

E

M(A) be an arbitrary point, and l e t cp

Ui. By defini-

tion there exist a Stein space Xi and a topological algebra homomorphism

We can identify M(Hol(Xi)) with Xi by (2.3.2).By (4.3.4)iii) we can identify

M(2"I

1 with Ui as topological spaces. Thus Ti

*: Xi

->

Ui

defines a homeomorphism. By the definition of a reduced complex space (cf. (2.3.1)) there exist an analytic subset Y in a domain G of some Cn. an open neighbourhood V of (Ti 1-1 (9) in Xi and a homeo-

*

morphism p: Y -> Trap: Y ->

V so that fop Tr(W

c

Hol(Y) for a l l f

E

Hol(XI). Then

200

Chapter 11

is a homeomorphism onto the open neighbourhood TrW) of cp. Let

9 c 2 be an arbitrary

element. Then

~l,;~ Hol(Xi)

by remark (11.1.1).

E

Thus G*T:ep

E

91UI E 2UI and

2 has analytic structure

Hol(Y). i.e.

in cp.

ill -b i) For each cp c M(A) choose an analytic subset Y

9

In a domain

of some Cng, an open neighbourhood U of cp in M(A) and a homeocp U so that ?*p E Hol(Y 1 for a l l ? E 2. By morphism p 1 Y -> c p c p cp cp cp remark (11.1.2) we can assume that each U is a hemicompact and cp h A-convex set. Endow U with the complex structure so that p be9 cp comes a biholomorphic mapping, i.e.

Hol(U ) = (9: Ucp-> cp Then

21" 9 c

Hol(U 1, thus cp

C : gep

2uQ c

cp

c

Hol(Yp)).

Hol(U 1, since Hol(U 1 is complete cp cp

with respect t o the compact open topology. A

h

-morphically convex, since it is A-convex and by (4.3.3).

U9 is A so

(2UQ,U

Up = M 6

1 satisfies the hypothesis of theorem (11.1.3). Hence (cf. (4.3.4)iii)) can be given the structure of a Stein

UP A

space so that A

= Hol(U

cp

,a),where

Hol(Ucp,O) denotes the alge-

bra of a l l holomorphic functions with respect t o this new structure. Claim. These new structures coincide on their intersection. Let cp, $

E

M(A) such that U '

n U,

i 0, and l e t cpo

E

Ucpt l U.,

Let

A

V be an open,hemicompact and A-convex neighbourhood of cpo so

.

U flU Denote by Hol(V) resp. Hol(V) the algebra of c p Q cp 51 a l l holomorphic functions on V with respect to the complex struc-

that V

c

ture which is induced from (U,,O)

resp. (U,,,,O).

Then V is

vex, thus Hol (U ,a)-convex, and the restriction mapping cp Hol(U9,0) -> Hol(V1 has dense range by (2.3.6). So cp

2UP-con-

201

Stein algebras

JI ,a) we

Repeating these arguments for H o l W

A

= A,

A

=

(A

1

uJ,

get

by (4.1.5).

Since po was an arbitrary point o f U

V

f l U

J

,

and since every point of A

M(A) has a neighbourhood consisting of open. hemicompact and Aconvex subsets (cf. (4.3.7)ii)), we get our assertion. Hence we have equipped M(A) with the structure of a complex space so that

2c

Hol(M(A)). Since M(A) is 2-morphically convex we get

our r e s u l t from (11.1.3).

0

(11.1.5) The next example shows that subalgebras of Stein algebras need not be Stein algebras. Example. Set

A = {f Then A

-

Hol(C): f(0) = f(i) f o r all i=1,2D...1.

as a closed subalgebra of the Stein algebra Hol(C)

- is a

uF-algebra. Set po: A ->

then po

c

C . f ->

f(0).

M(A).

Claim. M(A) = C\{1,2 i) For each x c C\{1,2,

,...1 as

sets.

...1 the evaluation homomorphism at x

an element of M A ) . I f x, y o C\{1,2,

defines

...1 are distinct points there is

f o A, by the theorem of Weierstra6. such that f(x) 4 f(y). 1.e. C\{1,2,

...1 c

MA).

ii) Let p 4 po be an arbitrary element of M(A). We show that p can

be extended t o an element of M(Hol(C)). Choose f c A such that p ( f - f ( O ) )

= 1 and define

202

Chapter 11

Then

7 is well-defined,

g(f-f(O)) c A. Let g, h

since f(i) = f(0) f or i=1,2,...,

E

1.e.

Hol(C) be arbitrary elements, then

g(gh) = p(gh(f-f(O)l) = p(gh(f-f(O)))p(f-f(O))

= p(gh(f-f(0)I2) = c9(g(f-f(O)))cp(h(f-f(O)))

= g(g)F(h).

defines an algebra homomorphism. Moreover +lA=cp.

It follows that

Since p is continuous there is a compact set K c 4: such that for a l l h

A,

ip (h )i

llhll,,

ig(g)i

~ ~ g ~ ~ K ~ ~ f -for f ( 0a)l~l lgKE, Hol(C),

E

hence

so

x

E

M(Hol(C)), 1.e.

c

3 is

the evaluation homomorphism at a point

+ 0.1, ... , since p t po, and our claim is proved.

C. We have x

Claim. po has no compact neighbourhood, hence A is not a Stein al gebra. ill) For n c N set Kn = (1x1

g

(II.IIK n1n generate

nl. The seminorms

topology of A. hence

?i ,

= {cp

E

M(A):itp(f)i s IlfllKn for all f

E

A), na No

is an admissible exhaustion of M(A) by (3.2.6)(*), Let x

c

C\{O,l,

(3.2.8). ry

...1, ixl>n. We claim that

x is not contained in Kn.

To prove this choose h c Hol(C), such that {y c C: h(y) = 01 = {O,l 1.

...

Choose k c N. so that (ixl/n l k > llhll Then g = zk- h

E

Kn

/ih(x)i.

A and ig(x)i > llgllKn.

iv) Now suppose that po has a compact neighbourhood K

Choose n

E

N. so that K

U = {p

c

MA).

.u

c

Kn and choose fl.....fr

c

A so that ry

E

M(A): ip(fi)-po(fi)i < 1, i=l,,.,,rl c K n

(cf. (3.2.2)i)). Since a l l fl are continuous functions on the plane and fi(0) = f,(]) for j=1,2

,... , we find x

E

C\{1.2

,...1, Ixi>n, so

that

the

Stein algebras

203

N

x

E

U, i.e. x

E

K n , and we get a contradiction t o ill).

(11.1.6) Using (11.1.3) we get a criterion when a subalgebra of a Stein algebra is i t s e l f a Stein algebra. Lemma. Let X be a Stein space, and l e t (A,X) be a closed subalgebra of Hol(X) such that the adjoint spectral map o f the inclusion homomorphism I: A -->

Hol(X). f ->

f, is a proper mapping. Then A is

a Stein algebra. Proof. By (2.3.2) the structure map

j: X ->

M(Hol(X)), x ->

qx,

where q x ( f ) = f ( x ) for a l l f s Hol(X), i s a homeomorphism. It follows that i * o j : X ->

M(A)

is a proper mapping. Let x

E

X. Then

i*oj(x)(f) = f ( x ) f o r a l l f e A.

i.e. the structure map

j,: X ->

M(A). x ->

9,.

is a proper mapping, hence X is A-morphically convex by (4.3.5) and our assertion follows from (11.1.3).

0

This Page Intentionally Left Blank

205

CHAPTER 12 CHARACTERIZING SOME PARTICULAR STEIN ALGEBRAS

We s t a r t with a function algebraic characterization o f Hol(G) in the case that G is

- an

open subset of the plane,

- a polynomially convex open subset o f -

Cn.

a logarithmically convex complete Reinhardt domain, a domain o f holomorphy in some Cn.

(12.1) Polynomial Iy convex analytic subsets (12.1.1) Many results of this chapter are based on the following theorem, which has been used in a similar form by various authors. Theorem. Let A be a rationally n-generated resp. n-generated semisimple F-algebra with generating elements fl.

...,fn.

Suppose that

the map

is a homeomorphism onto an open subset G

c

Cn. Then G is a ration-

ally convex resp. polynomially convex open subset and A i s topologically and algebraically isomorphic t o Hol (GI. Proof. Recall that a subset S

c

Cn is called rationally convex resp.

polynomially convex if the rationally convex h u l l resp. the polynomially convex h u l l of each compact subset o f S is contained in S. Since

P is a homeomorphism,

G is rationally resp. polynomially con-

206

Chapter 12

vex by (5.2.4111 resp. (5.1.5). Denote by 8 the (unique) continuous homomorphism 8 : Hol(G) ->

A

such that I) 8(z1) = fl

, i=1, ...,n.

ii) k h ) = h o f for a l l h

E

Hol(G1,

where &h) denotes the Gelfand transform of 8(h) (cf. (6.1.2)). We see that 8 i s injective by ill. If we can show that 8 is surjective, our assertion w i l l follow from the open mapping theorem for Fr6chet spaces. Let g c A be an arbitrary element. Since A is rationally n-generated there exist polynomials pk, qk (k

E

N) so that qk(fl#".mfn) is inver-

tible in A and gk = Pk(fl,".#f,)/qk(fl"...f,) k ->

converges t o g as

Q).It follows from (6.1.4) that Gk(fl#..#,fn)

transform of qk(f1,' ..,fn)

-

the Gelfand

- has no zero on M(A). so Gk(fl,..'.f,)oe-l

has no zero on G. 1.e.

&OF-' Hol(G) for a l l k N, in fact a k o F - l is a rational function which is analytic on G. BY (3.2.12) $k converges t o 8, hence converges compactly t o t o e - 1 on G, since f is a homeomorphism. So :OF-' Hol(G) by (2.1.7). Set h = OF-^, then &h) = 3 by ii). Hence 8(h) = g, since A E

E

Bkof-'

E

is semisimple and we have proved that 8 is surjective.

0

(12.1.2) Corollary. Let A be a rationally n-generated semisimple Falgebra with generating elements fll....fn.

A

Suppose that the map F

defines a homeomorphism onto an analytic subset of a domain of holomorphy G c Cn. Then A i s topologically and algebraically isomorphic t o Hol(X). Proof. a) Denote by 8 the continuous homomorphism from Hol(G) to

A such that

207

Characterization of Stein algebras

Let g

E

Hol(X). Choose h

-*

A. g

8,: Hol(X) ->

We show that 8,

Hol(G) so that hl,

E

= g by (2.3.3) and set

8(h).

is well-defined. Let h

E

Hol(G) so that hlx

BI

0.

Then &h) = h o e = 0 by ii), hence 8(h) = 0, since A i s semisimple. It follows that 8,(g). extension h

E

g

E

Hoi(X), is independent of the choice of the

Hol(G). Moreover we see that 8,

is an injective algebra

homomorphism. Let g

E

A be an arbitrary element. Repeating the proof of (12.1.1) we

get a sequence of rational functions ( r k ) k . analytic on X, such that

rk converges compactly on k

E

N, we have h

x

to h =

6oP-l.

Since rk

E

H ~ I ( X ) for all

Hol(X). since Hol(X) is complete with respect to

E

the compact open topology. Moreover 8,(h) 1.e. 8,(h)

= hop =

6,

= g since A is assumed to be semisimple. Hence 8, is an

algebra isomorphism. Consider the map

8: A Then

8

"-1 9.F A

Hol(X), g ->

->

.

is well-defined by a), and it is easily seen

hypothesis "semisimple" - that

r: A ->

h

8

-

using ill and the

= 8;'. The map

A

f ,

A. f ->

Is continuous by (3.2.12) and the map T:

3 ->

is continuous since

Hol(X),

6

->

"-1 goF ,

A

? is a homeomorphism.

Hence

8

= T o r is contin-

uous, and A is topologically isomorphic t o Hol(X) by the open mapping theorem.

0

208

Chapter 12

(12.1.3) Corollary. Let A be an n-generated uF-algebra with generating elements fl,...,fn.

Suppose that

an analytic subset X of a domain G

c

? is

a homeomorphism onto

Cn, then A Is topologically and

algebraically isomorphic t o Hol ( X I . Proof. Recall f i r s t that A is isomorphic t o

2. Since P is a homeo-

morphism we see that X is polynomially convex (cf. (5.1.8)). Using similar arguments as above we show that A n "-1 T: ^A -> Hol(X), f -> f*F , defines an injective and continuous algebra homomorphism. We show that T is surjective. Let f

E

Hol(X) be an arbitrary element. We shall show that there is

a sequence of polynomials (Pk)k so 'that pk converges compactly t o f. Then Pk'P converges t o an element

t e ^A , since

A

A is complete. and

T ( t ) = f. Now l e t K

c

X be a polynomially convex compact subset. Choose an

open relatively compact subset U

c

G which contains K, and a closed

polycylinder P which contains U. For each x

E

P\U we find a poly-

such that ipx(x)l > 1 > ilpxllK. By the compactness of P\U

nomial p,

we find polynomials pl,...,pr

K

c

Q = {x

E

such that

P: Ipl(x)l < 1, i=l

,...,r).

and Q c U. Hence Q flX is an analytic subset of the polynomially convex open set Q. Extend

fl,

Q (cf. (2.3.3)).

r" -

By (1.4.5)

to a holomorphic function

and hence f

-

r"

on

can be approximated

uniformly on K by polynomials.

0

(12.2) Polynomiall y convex open subsets of Cn (12.2.1) Let G c 4: be a polynomially convex open subset. Then Hol(G) i s a singly generated uF-algebra with generating element

Polynomial ly convex open subsets

209

by (5.1.2)i). The map

D: Hol(G) ->

Hol(G), f ->

f',

where f' denotes the derivative of f, defines a derivation on Hol(G) (cf. (8.2)) such that D(z) = 1 is invertible in Hol(G). Our f i r s t theorem

-

due t o Carpenter [CAR 6 1

-

shows that these

properties characterize in fact the algebra Hol(G). G

c

4: a poly-

nomially convex open set. Theorem [CAR 61. Let A be a singly generated uF-algebra with generating element f. Suppose there is a derivation D: A ->

A such

that D(f) is invertible in A. Then ?(M(A)) is a polynomially convex open subset of the plane and A is topologically and algebraically isomorphic to Hol (?(M(A)). Proof. By (5.1.8) it suffices t o consider the algebra A = /&P(Kn), where

... c

Kn

c

Kn+l

c

... is a sequence

of polynomially convex com-

pact subsets of the plane. Recall from (5.1.6) that (KnIn is an admissible exhaustion of M(A). Moreover A is singly generated by the polynomial z: U K n --+

C,

x. By our hypothesis there exists a derivation D on A so that

x ->

D(z) Is Invertible in A, 1.e. fi(z1, the Gelfand transform of D(z), has no zero on U K n = M(A) by (6.1.4).

By (5.1.4) that

2

2

i s an injective and continuous mapping. We want to show

is an open mapping, i.e.

2

is a homeomorphism.

Suppose the contrary. Let U be an open subset of M(A) (Gelfand topology) such that P(U) = U is not an open subset of the plane. Let y be a boundary point of ?(U) such that y

Q

s(U). W.1.o.g. y

E

Since D is continuous by (8.2.3). and since A is topologically and A

algebraically isomorphic to A there is I s N and c>O such that

K,.

210

Chapter 12

where $(f) denotes the Gelfand transform of D(f), hence

(*I

i$(f)(y)i

I;

CII?I,~

Since d is continuous by derivation on ^A

KI

fo r a l l

(0)

n ?E

A.

it can be extended t o a continuous point

= P(K,) at y (cf. (1.3.14)). The point y i s a boundary

point of KI (euclidean topology) since on KI the (relative) Gelfand topology and the euclidean topology coincide by (5.1.7)ii). Hence d

=0

by (1.3.14). This is a contradiction, since

= $(z)(y).

d(?l,l) and

6(z)

has no zero on UK,.

Our assertion follows from (12.1.1).

a

(12.2.2) Remarks. i) Theorem (12.2.1) becomes false if we replace "uF-algebra"

by "semisimple F-al gebra".

The algebra C"([O,l])

is a semisimple F-algebra by (8.1.3)ii). It is

singly generated by the map

z: [0,13 ->

c, x

->

_*

C"([O,l]),

x.

(cf. (5.1 .2)iIi)), and D: C"([O,l])

f ->

f',

where f' denotes the derivative of f, is a derivation on C"C[O,l]). Clearly D(z) is invertible in C"([O.l]). 11) We can sharpen theorem (12.2.1) in the following way (see

[CAR

61): Let A

xl, ...,xn

c

be a uF-algebra. Suppose there are f

C and a derivation D: A

-*

A such that

E

A,

211

Polynomial ly convex open subsets

ad f-xi.l i s invertible in A for i=l,...,n, 8) The polynomials in f. (f-xl~l)-l,...s(f-xn.l)

-1 are dense

in A, y) D(f) is invertible in A.

Then ?(M(A)) is a finitely connected open set in C and A is algebraical ly and topologically isomorphic to Hot (?(M(A))). The proof of this theorem runs along the lines of the proof of (12.2.11,

using the fact, that for each 2-convex compact subset K o f

M(A) our assumptions imply that every boundary point of

?(K)

is a

peak point f o r R(?(K)). In general it is false that each boundary point of a compact subset

K

c

C is a peak point for the algebra R(K). Hence we cannot copy the

proof of (12.2.1) for rationally singly generated uF-algebras. and it is unknown whether an analogous result holds f o r these algebras. (12.2.3) Let G

c

Cn be a polynomlally convex open subset. Then

Hol(G) is n-generated by the coordinate functions zl,...,zn

2:

M(Hol(G1) ->

Cn,

(Q

->

and

A

(21(p)....,zn('pl),

is a homeomorphism onto G by example (5.1.7). Hence Hol(G) is a n-generated semisimple F-algebra and M(Ho1(GI)is a 2n-dimensional topol ogicai manifold

.

We use this l a s t observation for a function algebralc characterization of Hol(G), G as above.

Theorem [H/W].

Let A be an n-generated semisimple F-algebra with

generating elements fl,...,fn. h

F : M(A) ->

Cn,

(Q

Let ->

A

(fl(9)

....,fn(p)). A

Suppose that M(A) i s a topological 2n-dimensional manifold (without boundary), then %A(A))

is a polynomially convex open subset of Cn

and A is topologically and algebraical l y isomorphic to Hot @(M(A))).

212

Chapter 12

Proof.? is an injective and continuous map by (5.1.4). If we can show that

P is an open mapping, hence a homeomorphism, our assertion

w i l l follow from theorem (12.1.1). Let V be an arbitrary open subset o f M(A). Let cp c V. Choose a chart (U,h,W) around cp (1.e. U is an open neighbourhood of cp, W is an open subset of R2n and h: W ->

-

that U c V and Since

PIC

U is a homeomorphism) such

-

the closure of U

is a compact subset of M(A).

is a homeomorphism we see that

from W onto the subset h U )

c

P o

h is a homeomorphism

Cn = R2n. By Brouwer's domain of in-

variance theorem (cf. [DUG]) f ( U ) is an open subset of Cn. Hence

P

is an open mapping. (12.2.4) Corollary [H/W]. ted by f,....,f,.

Let A be a semisimple F-algebra genera-

Suppose that M(A) i s a topological 2n-dimensional

manifold (without boundary) such that for each c>O the set

K, = {cp c M(A):

ifi(cp)i

C,

i=i,...,n}

is compact in M A ) . Then A i s topologically and algebraically isomorphic t o Hol(Cn). Proof. By (12.2.3) f?(M(A)) is an open subset of Cn and A is topologically and algebraically isomorphic t o Hol(P(M(A))). We show that

P(M(A)) is a closed subset of Cn. Then FS(M(A)) = 41" and we are done. Let (Xk)k be a sequence in

P(M(A)) which

converges to y . Choose c > o

so that

{y,x1,x2

,...I

c

{(w,

,...,wn) c c": IW,IO, i=1, ...,n).

I f t=(tl

,....t n1

E

Cn. r=(rl,...,rn) E R:.

D ~ = ,{(w, ~

,...,wn) c c":

define

< r i, i=1,...,n).

Iwi-tii

Let A be an F-algebra and l e t D1.....Dn be derivations on A (cf. n (8.2)). I f v = (vl ,...,v), c No and f E A set

--

D V ( f ) = Dno ...a Dno

...a

D,o

vn- tlmes

...o

Dl(f),

vl- tlmes

D0( f ) = f for 0 = (0. ....0). Theorem. Let A be a semisimple F-algebra with connected spectrum

M(A), and l e t (p,),

be a defining sequence of seminorms for A (cf.

(3.1.7)). Suppose there are elements f,

.....fn c A and derivations

D1, ....Dn on A such that I)

P:M(A) ->

c". cp ->

A

(fl(cp)

....,f n ( p ) ) , i s an injective A

map. ii) D ( f 1 = 6 .1, for i,j=l, ...,n, where 6 denotes the Kronecker 1 j ii 11 symbol,

ill) there is a sequence r1,r2,... in R: such that

pk(DV(f))

pk+l(f)vl/ri

for a l l f

A. k

t

N. v

c

Po.

Then G = f(M( A ) ) is a domain of holomorphy In Cn and A is topological l y and algebraical ly isomorphic to Hol (GI.

216

Chapter 12

Proof. As usual denote by Ak the completion of the algebra A/{pk=O) wlth respect to the norm p i (g+{pk=o)) = pk(g). By (3.2.8) (M(Ak))k is an admlsslble exhaustion of M A ) . a ) Claim. t(M(Ak))

f?(M(Ak+l))

1s contained in int P(M(Ak+l))

- for each k

E

-

the interior of

N.

Let 'k = (r\k),...,r~k)). Choose an arbitrary t=(tl,...,tn) Let g

c

E

D,~.

A be an arbltrary element. Then

by Ill). It follows that

defines an element In Ak, and that the series converges In any order of summatlon. We note that

is a llnear map, slnce a l l Di are linear maps. I n fact rule

- we see that Tt

-

using Lelbniz'

Is even an algebra homomorphism. This homo-

morphism Is continuous by (*), hence can be extended to Ak+l. denote this map again by Tt. We remark that

(**I Tt(fi + {pk+l'o))

' fi

by 11) and since D (1)=0 (j=l

I

+ ti'l + {pk'o)

,...,n).

Conslder the spectral map T:

M(Ak) ->

M(Ak+l),

cf. (3.2.5). By (**I we have for cp A

fl(T:(p))

= cp(f,) + ti,

E

M(Ak)

for 11' .

...,n,

We

Domains of holomorphy

217

and our claim is proved.

B) Since

h

F I M ( A ~ )is

from a) that

a homeomorphism for each k

f: M(A) ->

P

N. we learn

G is a homeomorphism onto an open sub-

set G c Cn. Since M(A) is connected the same is true for G.

8

y) Claim.

W.1.o.g. 0

c 1-101 (GIfo r a! I g E A.

o f - '

P

G. Let po

A

P

M(A) such that F(cpo) = 0, w.1.o.g.

po P M(A1).

Let t

E

Do,rl

be an arbitrary point and define Tt: A2 ->

A1 as in a).

Then

P-'(t) = T:(cpo) by (***I,

so we have for g

E

A

8tP-'(t)) = G(T:(cpo))

=

c

= Po( Dv(g)tv/vl + tpl=O)) v EN:

aoe-ll

i.e.

show that x

Do.rl

E

=

cpo(DV(g))tV/v!, V E N ~

Hol(D 1 by (2.1.5). In the same way we can o,rl

8 0 f - l is holomorphic in

a neighbourhood of any point

G and our claim is proved.

I

8 ) Denote by 8 the continuous homomorphism from Hol(G) t o A such that i)8(zi)=fi fo r i=l,

....n.

and ii) &h) = h o p for a l l h

E

Hol(G).

cf. (6.1.21. Let 8(h) = 0. Then 0 = &h) = hoe, hence h is injective. Now l e t g

P

A be an arbitrary element. Then

0. i.e. 8

Chapter 12

218

a mP-l c Hol(G) by y ) , hence 8(amP-') = a. where &a *?-'I denotes the Gelfand transform of 8 ( $m f - l ) .

and it

follows that 8($*P-') = g since A is semisimple, i.e. 8 is surjective. So 8 is a topological isomorphism by the open mapping theorem. 6)

Let x c G\P(M(Ak)) be an arbitrary point. Then cp = P - l ( x ) c M(A)\M(Ak).

By (3.2.6)(*) M(Ak) = {Cp

0

M(A)' icp(f)i

Pk(f) for a l l f

6

A).

Hence there Is g c A so that

The last inequality follows by (1.2.3). So

A

for the holomorphic function h = gap-'.

convex, since

Hence G is holomorphically

( P ( M ( A ~is) )an~ admissible exhaustion of G,

is a domain of holomorphy by (2.2.7).

thus G 0

(12.4.2) Remarks. I) Theorem (12.4.1) is based on an idea of Arens [ARE 13 who considered rationally singly generated (semisimple) Falgebras which satisfy hypothesis ii) and 111). ill Let

G

c

Cn be a domain of holomorphy. Denote by zl,...,zn

the co-

ordinate functions. By (2.2.7) we can identify G and M(Hol(G)), hence

2:

M(Hol(G)) ->

@"*

~p

->

A

(41(9),*..szn(cp)),

is injective. Set for i=l ,...*n DI: Hol(G) ->

Hol(G), f ->

df .

dzi

Then D1,...*D,

are derivations on Hol(G) which satisfy ii) and ill) (cf.

Reinhardll domains

219

(2.1.4)). Hence theorem (12.4.1) characterizes Hol (GI. G a domain o f holomorphy among the semisimple F-algebras. (12.5) Logarithmical l y convex complete Reinhardt domains (12.5.1) Finally we consider F-algebras A which have a special basis. We say that flD....fn generate a basis in A. provided each g

E

A has

a unique representation g = c (v,

....,v n

E

aq..

'1 'n f 1 ...-fn , n

-

n 'v1.....v )€No

C f o r a l l (vl,...,vn)

E

No. n

...v n

This means that the series converges with respect to a fixed order o f summation in the topology of A. We use the notation

g =

C

a? ,.

Theorem [S/W].

fl.....fn.

Let A be an F-algebra with basis generated by

Suppose that

G=

{(?l(cp)D...D?n(cp))

E

is an open subset of Cn. Then

Cn: cp

E

M(A))

G is a logarithmically convex complete

Reinhardt domain (cf. example (2.2.7)ii)) and A i s topologically and algebraically isomorphic t o Hol (GI. Proof. We f i r s t note that A is n-generated by f,,...,f,

in the usual

sense (cf. (5.1.1)). So

f: M(A) ->

Cn, cp ->

A

(fl(p)

.....fn(cp)). A

i s an injective and continuous map by (5.1.4). We claim that Let g =

P is a homeomorphism and that A is semisimple.

c avfv be an arbitrary element of A.

such that i t i l > O (i=l ,...,n) Then

and set it1 = (itll

Let t = (tl....,tn)

r

G

....,itnl). Let cp = C-'(t).

220

Chapter 12

G oP-'(t)

= c p ( ~a v f v ) =

c avtv. C

It follows from (2.1.5) that the series

function on the polycylinder Do,ltl

8 oP-l(x) Hence

=

c avxv

avzv defines a holomorphic

Moreover

for a I x

c

DOlltlnG.

Z avzv

hgM =

defines a holomorphic function on m

G = ((zl,

...,zn)

E

C?: There exists (tl,...,tn)

c

G so that

izll < ltil for i=l ,...,nl,

and

Note that

G" is a complete Reinhardt domain.

Now choose an arbitrary element g g =

c

rad A, i.e.

Gm

0 by (8.1.2). Let

C avfv. Then

for a l l v c t$ Let U

c

h = 0 by the identity theorem (2.1.6). Hence av=O 9 and g = 0. Thus A is semisimple.

M(A) be an open subset, and l e t cp

ment, Set zo =

P(cp1. Choose gl,...,gr

W = (J,

E

c

c

U be an arbitrary ele-

A such that

...

M(A): iGi(J,)-~i(tp)i < 1, i=l, ,rI

c

U,

cf. (3.2.2111. Then

P(w) = {z E G: i ~ ~ O f ? - I z ) - ~ ~ F -< ~I.( zi=l. ~ ).... i r). Hence P(W) is an open subset of G. since

GloP-'

E

Hol(G) for i=l,...,r.

So P is an open mapping, hence a homeomorphism. I t follows from (12.1.1) that G is polynomially convex and that A is

topologically and algebraically isomorphic to Hoi (GI. We see from the proof of (12.1.1) that the map A ->

HoI(G), g ->

GoF-'.

defines a topological algebra isomorphism. By the proof above each element

of Hol(G) can be extended to a holomorphic function

Reinhardt domains

221

G.

h on Since G is polynomially convex this implies that G = GS 9 hence G is a logarithmically convex complete Reinhardt domain (cf. example (2.2.7) ii)1.

0

(12.5.2) Remarks. i) There are B-algebras A with a basis generated by n elements ( see [S/Wl). In this case G = {(?,(cp)

.....f,(cp)) A

c

Cn: cp

E

M(A1)

is of course not an open subset. Hence we cannot drop the hypothesis "G is open" in (12.5.1). ill Let G c Cn be a logarithmically convex complete Reinhardt domain.

The coordinate functions zls...,zn

generate a basis for Hol(G) by ex-

ample (2.2.7)ii). Since G i s a domain of holomorphy we can natur a l l y identify M(Hol(G)) and G by (2.2.7). Hence A

{($,(cp)*...*zn(cp)): cp

c

M(Hol(G))) = G

is an open subset o f Cn. So (12.5.1) is again a characterization theorem.

This Page Intentionally Left Blank

223

CHAPTER 13 LIOUVILLE ALGEBRAS

(13.1) Liouville algebras Liouvllle's theorem f o r entire functions s-ates tha each bounded function f

E

Hol(C) is constant. Since we can naturally identify

M(Hol(C)) and C (cf. (4.1.7)ii)) resp.

Hol(C) and Hol(C)^. the alge-

bra of a l l Gelfand transforms, we can rephrase Liouville's theorem in the following way: If f

E

Hol(C) is an element so that ?(M(Hol(C))

is a bounded subset of C , then f must be a scalar multiple of 1 c Hol(C). The l a s t formulation motivates the following definition. (13.1.1) Definition. An F-algebra A is called Liouville algebra if a

f(M(A)) Is an unbounded subset of C f o r each element f cA\{c-l:crCl. (13.1.2) Remarks. i) Note that C((xl), the algebra of a l l (continuous) complex-valued functions on a point x i s a Liouville algebra. ii) A nontrivial Liouville algebra was constructed in example (9.2.2).

iii) Each Liouville algebra is automatically semisimple, 1.e. the map

A ->

A

A, f ->

an element g

?,

i s injective (cf. (8.1.2)). Otherwise there would be

* 0 in A such that G(M(A)) = ( 0 ) . but g c A\{c.l:

c c Cl.

i v ) Using Shilov's idempotent theorem (6.1.5) one sees that the spec-

trum o f each Liouville algebra is connected. v) Let A be a Liouville algebra, f

G

A\{c.l: c

c

Cl.

Suppose that C\?(M(A)) contains an open disc { z : I z - z o l < e l . Then f-zo.l is invertible by (6.1.4). We have l/(f-z;l) since the same is true for f. but

c

A\{c.l: c r C l

224

Chapter 13

l/d,

(l/(f-zo-l))n(M(A)) c {z: i z i

where (l/(f-zo-l))n denotes the Gelfand transform of l / ( f - z o - l ) ,

and

we get a contradiction. It follows that ?(M(A)) is a dense subset of C.

(13.1.3) We can give a more detailed description of the spectrum of an unbounded element o f a Liouville algebra. Proposition [DAL]. Let A be a Liouville algebra, f

t

A\{c.l: c c C ) .

Then C\f(M(A)) contains no closed connected subsets other than single points. Proof. Let K be a closed connected subset of C which is contained in c\?(M(A)). i) Assume that K is unbounded.

Consider K as a subset of S, the extended plane, and set U= S

\ (K U

{a)).U is an open subset of the plane. U is connected.

since f(M(A)) is a dense subset o f U by (13.1.2)~) and since i?(M(A))

Is connected by (13.1.2)iv). S\U = K U {a)is connected, hence U is a simply connected domain of the plane (cf. [NAR l],p.

151). Since K

there exists a biholomorphic map g

E

*

0 we have U

* C . hence

Hol(U) which maps U onto the

open unit disc D by the Riemann mapping theorem. Denote by 8 the continuous homomorphism from Hol(U) to A such that 8 ( z ) = f and 8(h)^ = ha? for a l l h

E

Hol(U), cf. (6.1.2). Then 8 ( g )

8(g)^(M(A)) = goi?(M(A))

c

1.e. @(g)^(M(A)) is bounded but B(g)

E

A,

Do A\{c.l: c

t

C), since

B(g)^(M(A)) is a dense subset of D. Hence we get a contradiction. ill Assume that K is bounded.

Let x c K. Consider the map

225

Liouville algebras

Then h is a homeomorphism, thus h(K) is a connected compact sub-

s. Note

set of

there is y

E

that h ( K ) \ t 4 is a closed subset of C. Suppose that

K \ t x } . Let L be the component of h(K)\tm) which con-

tains h(y). Then L is unbounded, since otherwise L would be a component of h(K) such that L 9 h(K). Now we get a contradiction as in I), replacing f by l / ( f - x - l ) . and K by L.

0

(13.1.4) Let G

c

U! be a polynomially convex domain, i.e. G is a

simply connected domain by (1.2.4)ii). If G 9 4: then there exists a biholomorphic map from G onto the unit disc D by the Riemann mapping theorem. This shows in particular that the Liouville property characterizes Hol (4:) among the algebras Hol (GI.G a polynomial ly convex domain of the plane. These algebras are singly generated nontrivial uF-algebras and one i s inclined to ask whether Hol(C) is characterized by the Liouville

property among the singly generated nontrivial uF-algebras. The next example shows that this is not the case. (13.1.5) Example [B/L].

We construct a singly generated nontrivial

Liouville algebra A, which is not isomorphic to Hol(C). For z

c

U! denote by Re z resp. I m z the real resp. imaginary part o f

z. For n

c

N set = tz

E

C : -n s Re z c n. -n c I m z

En = t z

E

41: I m z > 0. l / ( n + l ) < Re z < l / n )

Ln

S

n),

and

Each Kn is a polynomialiy convex compact set, since C\Kn is connected, cf. (1.2.4)ii). Set A = & L P ( K n ) , then A i s a singly generated uF-algebra.

Chapter 13

226

1

..... . . . . . . .

By (5.1.6) M(A) = U K n = C as sets and (Knln is an admissible exhaustion of M(A) with respect to the Gelfand topology. Note that

(Kn),,

i s not an admissible exhaustion of C with respect t o the eu-

clidean topology, since f o r example the set {z: OrRe zrl, I m z=1) i s not contained in any Kn. I n the same way as In example (5.1.12)we can show that M(A) i s not locally compact. More precisely, no point of the set

(2:

O=Re z, I m z a 0 ) has a compact neighbourhood (Gel-

fand topology). Hence A is not a Stein algebra (cf. (11.1.1)).

in par-

ticular A is not isomorphic t o Hol(C). Claim. A is a Liouville algebra. Let f c A be an arbitrary element. Interpret ? a s a complex-valued function on C which can be approximated uniformly on every Kn by polynomials, c f . (5.1.7)i). Then

?lC\l. We give an example of a doubly generated Liouville algebra A with locally compact spectrum, which has an empty Shilov boundary but which is not a Stein algebra. Since this example is a slight modification of example (9.2.2) we omit most of the proofs of the assertions. Define the following sets in C 2

229

Liouville algebras

x1 = {lIXC. x2 = CX{O). Xn = {l-l/n)xC for n>2, x = u xn. nkl

KI = X f l U z l 4 , i w i 4 I for 1

N.

E

Each KI is a polynomially convex compact set. So A =

c. w->

?(t.w)

is a holomorphic function for every t a R. Hence (s,M(A)) is a m.m.a.. iii) Let A be an F-algebra and denote by yA the Shilov boundary o f A

(cf. (9.1)). Then (s,M(A)) is a m.m.a.(+) yA

by (9.1.7)i) and (9.1.6) if

+ 0.

iv) It follows from Rossi's maximum principle (1.4.9) that

(e,M(B)\yB) is a m.m.a.(*)

for each B-algebra B.

v) We shortly sketch an example due t o Wermer [WER 11. We shall

use it to show that there exists a m.m.a. (A,X) such that X does not admit analytic structure, in fact X does not even contain an analytic disc. (We say that X contains an analytic disc if there are a disc D in the plane and a nonconstant continuous function f: D -> that gof

E

X so

Hol(D) f o r a l l g a A.)

We denote by al.a2,

... the points in the disc {Izl

< 1/21 with rational

real and imaginary part. For each j a N denote by B the algebraic j function

Bj(z) = (z-al)(z-a2)

...(2-a i -1-1

1

and set

where cl,

...,ci

are positive constants. Denote by

C(Cl,...,cn)

the sub-

set of the Riemann surface of gn which lies in t i x l ~ l / 2 1 , i.e. C(C1*

...,c n1 = t(2.W)'

where w(")(z). j=1.....2n

i

l Z i ~ l / 2 ,w=w(")(z), j=l, j

...*2")

are the values of gn a t z.

For a sequence of positive numbers (cnIn define X(cn) to be the set of a l l points ( 2 . ~ 1E C2 such that i) I z l s 112 and

ii) there is a sequence (z.wn) c C(c l....cn)

such that

Chapter 14

234

wn --+ w as n---> QD. It was proved by Wermer that there exists a sequence (cn),,

of pos-

itive constants so that a ) X = X(cn) is a polynomially convex compact set in 4:2 which contains no analytic disc,

C) X is the poiynomially convex h u l l of the compact set Y =

x n {(z,w):

i z i = 1/21.

y) zl(X) = {lzlSl/21, where z1 denotes the f i r s t coordinate

function.

For la te r use we remark that

8 ) C(c l.....cn) X

n ({al)xC)

fl({allxC) = {(al,O)l for a l l n e N. hence

= {(al,O))

by the construction.

We now use Wermer's example to construct a m.m.a..

Kn = X

n {(z,w):

izl

For n> 2 set

(1/2)-(1/n)1.

Each K n is a polynomially convex compact subset of C2. since the same is true fo r X. Note that Kn is not empty by y ) . Set A =

log IIgIIf-l(A),

R U {-a),X ->

is a subharmonic function for each g

Q

A.

(Note that f-l(dD(X0)) and f - l ( X ) are compact subsets of X by i).) Proof. a) By I) X is a locally compact space. Hence

zg: u ->

R, 1->

l l g i f- 1 ( ~ )D

is upper semi-continuous by (14.2.1) and the same is true for log Z fl)Let D c U be a closed disc centered at A,.

f"(D)

Fix xo

Q

f-'(X0).

is a compact subset of X we see that log Z

9' Since

is bounded from 9 above on D, hence we can find a sequence of continuous functions (uI), on D such that uls log Z

9

, i.e. u1

2

u2

2...2

jog Z

9

and

Maximum modulus algebras un(x) -> each k

E

239

log Z (XI as n -> Q) for each x c D, cf. appendix A.2. For 9 N choose a polynomial pk such that IRe(pk)-ukl < l / k On dD.

The choice o f pk i s possible by A S . Then

(*I Fix k

log E

z9

s Re(Pk) + l / k on dD.

N. Let (qnIn be a sequence of polynomials which converges

uniformly on D to exp(-pk-l/k).

Since g.qn(f) z A there is yncf-'(dl))

so that

Ig(xo)qn(f(xo))l s Ig(yn)qn(f(yn))l by hypothesis 11). Hence we find Xk Ig(Xo)eXp(-(pk(f)(Xo)+l/k))l

E

f-l(dD) such that

s Ig(xk)exp(-(Pk(f)(xk)+l/k))l.

we get

For the equality cf. A.4. For k -> log Z ( X 1 9 0

5

1/2n

I

dD

(log Z ) ( t ) d t . 9

Since this inequality holds for each closed disc D subharmonic on U by A.7.

c

U. log 2

9

is 0

240

Chapter 14

(14.2.3)Corollary.

Let (A,X) be a m.m.a.(*)

resp. a m.m.a.. Suppose

there are an open subset U c C and an element f

f: X ->

U is a surjective and proper mapping. Then log Z

9

RU

U ->

I

{--),

X ->

is a subharmonic function for each g

E

log IIgIIf-l(X), A.

Proof. Let D c U be a closed disc centered at A,,

xo c f-'(A0). df-l(D)

A so that

E

E

and l e t

Then f-l(D) c X i s a compact neighbourhood of xo and

f - b D ) . Hence we have ig(xo)l

llglldf-1(D)

* llgllf-1(dD)

for every g c A by the definition of a m.m.a.(*),

and our assertion

.follows from (14.2.2).

0

Remark.This theorem was also obtained by Senichkin [SEN 11. (14.2.4) We give a f i r s t application o f this theorem. Definition. Let U c C be an open set. A subset E set if there exists a subharmonic function p E c {Z

I

U:

P(Z)

*

--a0

U is called polar

c

on U such that

= -=I.

We remark that a polar set has Lebesgue measure zero ( cf. A.14 1, in particular an open subset W

c

U is not a polar set. We are now

able to prove an identity theorem for maximum modulus algebras. Theorem. Let A,X,f,U connected. Let E

c

be as in theorem (14.2.2). Suppose that X is

U be a nonpolar set and l e t g

vanishes on f-l(E). Then g vanishes on X. Proof. First note that U is connected. By (14.2.2)

E

A such that g

Maximum modulus algebras log Z

9

:

U ->

R U {-a),X

-*

241

IIgIIf-l(X)S

is a subharmonic function and log Z (1)= -m f o r a l l X 6 E. Since E 9 Is a nonpolar set we see that log Z 9 -a on U, i.e. g 8 0 on X. 9 (14.2.5) Corollary. Let X,U,f.A be as in (14.2.4). Let x g c A vanishes in an open neighbourhood V of f-'(x).

E

U. Suppose

then g vanishes

identically on X. Proof. We show that there exists an open neighbourhood W c U of x so that g vanishes on f-l(W). Then our assertion follows by (14.2.4).

Suppose W does not exist. Then there exists a sequence (xn),,

in

X\V such that f(xn) converges to x. Since

nrN) U {x))\V

f-'((f(x,):

is a compact subset of X a subsequence of (xnIn converges to a

point y

E

X\V. But f(y) = x, and we get a contradiction.

0

Example. Let A.X.Y be as in example (14.1.3)~). We remarked that (using the notations of (14.1.3)~)): i) M ( A ) = X\Y contains no analytic disc, ii) ?l(M(A))

= {z

E

C: lz1O and Flu: U -> L=

F(U) is a homeomorphism. W.I.0.g. t(Zl

I...,

F(U) = Do,r. Let

zn): z2= ...=z n= O ) .

then

L

n F(U) = DX~OI,

where D = {z

E

C: I z l t r l and 0

E

Cn-l. By Wermer's subharmonicity

theorem (14.2.3). applied to the m.m.a.(*) (AIxLnU,XLflU)

and to

fllxmu E Alxmu* the function D ->

R U {-a),X ->

is subharmonic for a l l h

c

AIxLnu

loglho(f,IxLnu)

(note that fl: XLnU ->

homeomorphism), in particular for gIxLnu-a, Then

(*I

- by a theorem of h: D ->

Hartogs (cf. A.8 1

C, X ->

-1 ( X I I .

go(fllxLnu

a

E

D is a

C.

- either

)-l(M

or the conjugate of this function, i s holomorphic. Repeating the arguments for the function g.flIxLnu

(**I

D ->

C . X ->

X-ga(fll,

E

AIxLnu

we see that either

nu) -1 ( X I

L

or the conjugate of this function is holomorphic. We conclude easily from

(*I and (**I that h is holomorphic.

Repeating these arguments for an arbitrary complex line L which is parallel to a coordinate axis and for which L

n F(U) + 0 we

see that

g O F-' ILnFCU, is holomorphic. By Hartogs' theorem on separate analyticity (2.1.2)

Riemann domains is a holomorphic map and thus g

^A

ii) We have

c

E

247

Hol(X).

Hol(X) by I). The holomorphicaliy convex hull h

(cf. (2.2.6)) is compact for every compact subset K c X. since the Aconvex h u l l of K is already compact,cf. (4.3.3). Hol(X) separates the h

points of X. since A does. Hence X is a Stein manifold (cf. (2.2.6)). Moreover F forms a coordinate system at every point z

T1...Tn

E

2 we

see that

h

^A

X. Since

E

is a dense subalgebra of Hol(X) by (2.3.71, h

thus A = Hoi(X). because A is complete.

a

Remarksel) Theorem (15.1.2) is based on an idea of Rusek [RUS]. ii) Let (X.n) be a Riemann domain over Cn. Then II

(2.2.2). I f L is a complex line in Cn such that L

X,

E

Hoi(X)", cf.

n n(X)

= 0

is an analytic subset of (x.111. It follows that (Hol(X)I

XL

i s a m.m.a.(*).

.

then

X , ),

cf. (2.1.9) and remark (2.3.1). Thus theorem (15.1.2)

gives a characterization of the algebra o f al I holomorphic functions on a Riemann domain over Cn within the class o f uF-algebras. h

(15.1.3) Corollary. 1) Let (A,M(A)) be a rationally n-generated uFalgebra with locally compact spectrum M(A) and generating elements h

fl....,fn

such that (AIM(A)L.M(A)L) is a m.m.a.(*)

plex lines L

c

for a l l affine com-

Cn which are parallel to a coordinate axis and for

f(M(A)) which L fl

+ 0 (P=(?l.....f,)).

n

Then C(M(A)) is a rationally convex open set in Cn and A is topological ly and algebraically isomorphic t o Hol &M(A))). h

li) I f (A,M(A)) is n-generated by fl.....fn

then

-

under the assumptions

h

of part i) - F(M(A)) i s moreover a polynomially convex open set. Proof. By (15.1.2)

f

is an open mapping, hence a homeomorphism by

(5.1.4) resp. (5.2.3) and our assertion follows from (12.1.1). Example. Consider example (14.1.3)ii). In this example A is doubly

Chapter 15

248

A

generated, M(A) is locally compact and ( A I M ~ A ~ L , M ( A ) Lis) a m.m.a. for a l l affine complex lines which are parallel to the second coordinate axis. But A is not a Stein algebra. Suppose the contrary. Then there exists a Stein space X and a topological algebra homomorphism T: A ->

Hol(X). By remark (11.1.1)

SloT* i s a nonconstant holomorphic function on X. hence S1oT* i s an open subset o f the plane. But

'i,*T*(X) = ?l(M(A)) = R by (14.1.3)ii). I n particular A is not isomorphic t o the algebra of a l l holomorphic functions on a polynomially convex domain, although A is a m.m.a.(*) "on many lines". (15.2) Maximum modulus algebras and finite mappings Let (A.X) be a &-algebra.

f

c

A and W be a component of

f(M(A))\f(X). Suppose that there exists a set o f positive plane measure

G

c

W so that f o r a l l X c

G the set

f-'(X)

c

M(A) is finite.

Then - due to a classical r e s u l t of Bishop (cf. [WER each point p

c

?-'(W)

41,~.6 5 ) -

has a neighbourhood in M(A) which is a finite

union of analytic discs. Basener [BAS] and independently Sibony [SIB], introducing the concept of higher Shilov boundaries, obtained a r e s u l t on analytic structure of dimension n > l in the spectrum of uB-algebras if certain functions have finite fibers. Among others Wermer [WER 31, Senichkin [SEN 21. Rusek [RUS] and Kumagai [KUM 11 put these results in the setting o f maximum modulus algebras. We follow the proof o f Rusek and shall obtain the results of Basener and Sibony as a corollary in the next chapter. (15.2.1) Definition. We say that a t r i p l e (X,F.Y) i s an analytic cover with critical set S i f

Finite mappings

249

i) X is a locally compact Hausdorff space, Y

c

Cn i s a domain. and

F i s a proper, discrete and continuous mapping of X onto Y; ii) S i s an negligible set in Y. i.e. S is nowhere dense and f o r every

domain D

c

Y and every f

c

Hol(D\S) which is locally bounded on D

there exists a unique holomorphic extension ill) there is k

X\F-'(S) 8

E

r"

to a l l of D;

N such that F is a k-sheeted covering map from

onto Y\S. i.e. Flx,F-l(s)

is a local homeomorphism and

F-'(y) = k for a l l y c Y\S;

iv) X\F-'(S)

is dense in X.

(15.2.2) We shall deal with algebras o f the following type: Let X be a locally compact Hausdorff space. and l e t A be a subalgebra of C(X)

- not necessarily closed - which contains the constants

and separates the points o f X. Let fl,...,fn i) F: X ->

W c Cn. x ->

E

A such that

,...,fn(x))

(fl(X)

is a proper mapping onto a domain W ii)

(AI,.X,)

Cn;

is a m.m.a.(*) for a l l affine complex lines f o r

* 6.

which L f l W

(X,

c

is defined as in (15.1.2). for the definition of a m.m.a.(*)

cf.

(14.1.2)iii)J (15.2.3) Theorem [RUS]. Let A. X, F, W. as in (15.2.2). Suppose that there exists a nonpluripolar subset E o f W such that number of points in F-l(X) wk =

E

-

is finite for every X

w: #F-l(X)

E

8

F-l(X)

-

E. Set for k

ii) the set

EL

N such that

S = W, U

N

= k).

Then i) there is k

E

the

w

=

w1 u ... u

... U w k - 1

w k and w k

*

#,

is a proper analytic sub-

set of W. ill) the triple (X.F,W) is an analytic cover with critical set S,

iv) there exists an analytic space structure of pure dimen-

Chapter 15

250

sion n on X such that A c Hol(X), if we endow X with this structure. (For the definition of a pluripolar set cf. A.14, for the definition of the dimension of a complex space cf. (7.1 1.1 (15.2.4) We divide the proof of theorem (15.2.3) into several steps. We f i r s t prove a lemma due t o Senichkin [SEN 21. Notation. Let (A,X) be a m.m.a.(*),

and l e t K

c

X be a compact set,

then A, denotes the uB-algebra on K which is obtained by completing AtK with respect to the norm

11-11,.

Lemma [SEN 21. Let (A,X) be a m.m.a.(*). an open set such that f: X ->

Let f

E

A. and l e t G

C be

G is a proper mapping onto G. Let

a e G, and l e t D be an open disc centered at a such that sure of D

c

- the clo-

- is contained in G.

Let m c f-'(a),

and l e t p be a representing measure for m concen-

trated on f"(dD)

with respect to the algebra A f - 1 ( ~ ) . For g

E

A

define qg

1,

C, z ->

= q: D ->

f

(f-a)g/(f-z)

dp.

(aD)

Then i) q is a bounded analytic function and

max{lq(z)l: t ii) if

c

D I s llgllf -1 (dD)'

c E dD and if a nontangential

then iq(Cl1 s

limit q(c) = l i m q(t) exists, t->c

llgllf-l(c,,

iii) q(a) = g(m).

(Here dD denotes the boundary of D.) Before we prove this lemma we give some explanations. Note that

f - h is a compact subset o f X since f is a proper .mapping. Denote by yAf-i(E) the Shilov boundary of Af-1(5). Then yAf-l(El

is contained

Finite mappings in f-’(dD) since (A.X) Is a m.m.a.(*).

251

By (1.3.10) there exists a repre-

senting measure p f o r m concentrated on yAf-1(5).

i.e. a positive

measure p such that

I

dp = 1

yA -1f

(D)

and

I

yA -1f

9 dP = g(m). for a l l gE

Af-1(5).

(0)

Hence the existence of a measure p in lemma (15.2.4) is always guaranteed. Proof of (15.2.4). i) For simplicity we assume that D is the open unit disc, hence a = 0 and

It is easy t o show that q i s complex differentiable and hence analytic

on 0. Let

M(Af-l(E)), then by (1.2.1)

‘QE

1R’Q)I g l l f l l f - l ( ~ ) 1. Hence t

E

(l-Tf)^ -

the Gelfand transform of 1-Tf

-

has no zero f o r

D , w h e r e T denotes the complex conjugate o f t, i.e. 1 - t f is inver-

tible in

A,-I(~)

by (1.2.9). We have f g / ( l - T f ) dp = f g / ( l - T f ) ( m ) = 0,

tl-’(aD)

since f(m) = 0. We multiply the l a s t equation b y 7 and add it to (*), then

For the l a s t equation note that for

C

E

dD

252

Chapter 15

c/ ((I-t 1(l-TC

t->

Since hn(yn) = h(y) we get from (*I

This inequality holds for almost a l l

E

dD. Since q i s a bounded

holomorphic function on D we have Ih(m)l = Iq(a)l

5

llhllx-l(aD).

0

(15.2.6) Remark. Let (A,X), f, G be as in (15.2.4) and use the notations o f (15.2.5). Let g

c

A.

I t follows by (14.2.2) and (15.2.5) that the map

G ->

R U {--I,

X ->

log IIhIIn-l(l),

256

Chapter 15

is a subharmonic function for each h

E

This observation i s of par-

A,'.

ticular interest in the case that h(xl

,...,xn)

= n(,g(xl)-g(xj)). 1

Note that h L Ah, since h is the sum of terms of the f orm kl k g (x1)*...*g "(x,), 0~k1,...,kn.

(15.2.7) Notation. Let A. X. F. W be as in (15.2.2). Fix k22. Denote by xk the subset of the k-fold Cartesian product of X consisting of a l l points

(Xi

,...,X k )

such that F(X1)'

...' F(Xk). Endow Xk

with the sub-

space topology and set w

X:

X ->

W. (xl,

....x k

->

F(xl).

n"

Then xk is a locally compact space and W. Fix g

E

9":

is a proper mapping onto

A. Define xk ->

c.

(X1*...8Xk) ->

n(g(xi)-g(xj)) i

and $k,g: W ->

RU

{--OD),

IOg

X ->

IIgII$-I(l)-

We remark that J, (X)=-m i f rrF-'(l)

is continuous. hence

R, X ->

II~II~-l(~),

is upper semi-continuous by (14.2.1). Hence the same i s true for log

zg = *k.g'

ill Let L be an arbitrary affine complex line such that L f l W 4 0.

Recall that (AIXL.XL) is a m.m.a.(*)

by our assumption. Identify L

Finite mappings

257

with C and denote by p the orthogonal projection from Cn onto L. Then poF

E

A and poFIxLr AIXL. Since poFI

(x)=F(x) for a l l x r X,

XL

we have FI

XL

E

A

. Then FIX, is a proper mapping from X, onto

1%

L n W . Furthermore G-~(XI =

X-~(X).

fo r

x

E

Lnw.

where we use the notation o f (15.2.5) replacing f by F(

xi

(AX .,(),

(A,X) by

and G by L n w . Denote by c k the intersection of x k with

the k-fold product of X,

g ~c:k

c,

->

and set (xl,...,xk)

31zk*

->

Then R U {--I,

L n W ->

X ->

log

is subharmonic by (15.2.6). So $k,g(Lnw

(15.2.8) Proof of (15.2.3). i) For i r

IIgJIn-l(x,. is subharmonic. since

N set

Ei = { A r E: #F-l(X) = i). Then

E = U El. Since a countable union

of piuripolar sets is pluri-

icN

polar, there i s k

E

N such that Ek is nonpluripolar. Assume there i s

X r W such that F-l(X) contains k + l different points ~ ~ , . . . , x k + ~Since .

A separates the points of X and contains the constants, there is grA so that g(xl) 8 g(x 1 for i 9 j (cf. the proof of (3.2.311, hence j

$k+l,g(X) But $k+l.g

*

is plurisubharmonic on W by (15.2.7) and $k+l,g

Ek, and we get a contradiction. Hence

=

-a on

258

Chapter 15

Suppose that there exists an open set U

c

w\wk=w1u...uwk-1'

Then there is I t k so that WI is nonpluripolar, since sets which have positive Lebesgue measure are nonpluripolar. Repeating the argument above this implies that W -0 for j > l . But this is a contradiction to

* 6, thus w k

wk

1-

iS

dense in

w.

ii) We f i r s t show that F is an open mapping.

FIX xleX and set y=F(xl). By I) there is I r k such that F~'~y~=~xl,...,xll. Let K

c

X be a compact neighbourhood of xl,

and l e t U be a compact

such that K n U = 0.

neighbourhood of {x2....,xI)

We claim that there i s a closed polycylinder that F ' b

c

Cn centered at y so

KUU.

Otherwise there would be a sequence (z,), y as v ->

F(zJ ->

c

F-~({F(Z,,):

in X\(KUU) so that

Since F Is proper

QD.

{y))

V~NIU

is a compact subset of X and a subsequence of (zJV converges to a point z zv

E

X. Then z

E

E

{x,,....xI)

X\(KUU) for a l l v

Next we claim that D

E

c

and we get a contradiction to

N.

F(K).

Suppose the contrary. Then there is an affine complex line L so that a ) L n D n F( K )

+ 0 and

*

8) L ~ ( D \ F ( K ) )

0.

Since K Is compact it follows by a ) and 8) that F(K)nLfID contains a boundary point (in the relative topology of L). But (A1

XL

m.m.a.(*),

,XL) is a

and FlxLc AtXL by the proof of (15.2.7). hence F(F-l(LnD)nK) = F( K) n L n D

i s an open set by lemma (14.1.4)ii) (in the relative topology of L).

Thus we get a contradiction, hence F is an open mapping. Let k be as in I), and l e t x

w k . Then

F - ~ X )

= (x l.....xk).

Choose

259

Finite mappings

....Uk of

disjoint open and relatively compact neighbourhoods U1, x

,,...,xk

In X. Then k V = f l F(Ui) l=1

i s a neighbourhood of x. since F is open, and V

w k ' Hence w k is

c

open and

s

=

w1u ...uWk-1

Is closed. Let x,

= w\wk

Ul,....Uk be as above. Then Ui fl F'l(V)

is an open

neighbourhood of xi and F i s injective on UI fl F-l(V) f o r i=l,

. F: F-'(Wk) ->

...,k,

i.e.

wk

Is a local homeomorphism. We can now follow the proof of theorem

(15.1.2) t o show that there is a uniquely determined structure of an n-dimensional manifold on F-'(Wk) such that glF-l(wkl

is a holo-

morphic map (with respect t o this structure) for a l l g

e

A.

Next we show that S Is an analytic subset of W. FIX g

L

A and define if z

0 where x;

....,x;

the value of

9"

if z

wk t

w\wk

are the points of F - b ) in some ordering. Note that is independent of which ordering is chosen. Let Ui be

open nelghbourhoods of x;

so that FI

UI

is a homeomorphism ( 1 4 .

....k).

Then 9" F- lIF(uI)

Q

Hoi(F(Ui)) for i=l,

as observed above, and tions o f this form. thus We claim that

9"

....k

9" is locally just the 9" i s analytic on w k '

sum of products of func-

Is continuous on W.

To prove this l e t b

f

( d W k ) n w be an arbitrary point. Let (zvIv be a

260

Chapter 15

sequence in w k which converges t o b. Suppose there exists 8 > O and a subsequence (zVili such that Ig(x;v')-g(x;v91

4

2

s

if i

* j.

i

c

N.

z

Let yI be an accumulation point of

{z

vi

1

i

(Such a point exists since

N)U{b) is a compact set and F is a proper mapping.) Then

E

yI c F-l(b) and

lg(yl)-g(yj)12s But b

E

If j 4 i.

w \ w k , so *F-'(b)" is trivial. ii) Assertion 1. Let F=(fl....,fn)

E

An. cp

E

V h . and l e t U be an open

neighbourhood of V h i n M(A). Then there is G a Bn such that cp and V(%) Let a

E

c

E

V(&

U.

C\{Ol. Obviously V t R = VtA) for H=(af l.....afn).

and by the compactness of M(A)\U x

a

M(A)\U there is i

Now choose gl. ...,gn

a

(1.

-

-

we can assume that for each

....n l such that

l?l(x)l>l.

....

A

a

Therefore

B such that ~ ~ ~ i - f l ~ ~ M ~ fAo r) n V ( R so that

E

A

2 71

Higher Shllov boundaries n

g(5) =

l1311v(finK = 1,

and

I'$1'

< 1/2.

V (fin( K \ int L)

The set

S = (8

t

V t R n K : c(8) = 1) (consider h = (g+1)/2).

is a peak set of Av(finK

A

Let (Knln be an admissible A-convex exhaustion of M(A). w.1.o.g.

K

c

K1. For each n

principle (1.4.9) '('V

E

N S is a local and hence - by Rossi's maximum A

-

a global peak set of Av(finK

n

. (Note that

(finKn 1 = VtfiflK, by (4.3.31.)

Suppose that for each 8

E

S there exists n(8)

E

N such that 8 is not

A

contained in yoAv(p,nKnte). Then there exists an open nelghbourhood N of 8 in V ( f i such that

Since S Is compact there i s m yO'V(%Kn

=

E

N such that

0 for a l l n

2

m.

This is impossible. A

By (9.1.3) S fl yoA,(p,

* 0, hence K n yo%,,(p,*

0. A

Since there exists a neighbourhoodbasis of cp consisting of A-convex compact subsets by (4.3.7)ii).

we are done.

The case n=O can be proved in a similar way.

0

(16.2.3) Corollary. Let A be a uF-algebra with locally compact spectrum M(A1. and l e t n t (0,l.

...). Then the following statements are

equivalent: i) ynA = 0; A

A

ii) ynAU = 0, for each hemlcompact A-convex open subset

U of M(A);

Chapter 16

272 h

ill) ynAK c dK

-

the boundary of K

- f or

A

each A-convex com-

pact subset K of M A ) . Proof. i)osil) This follows from (16.2.2). A

i i i b i ) Suppose there exists p

c

ynA. Choose a compact A-convex

neighbourhood K of 9 . Then p

c

y n 2 K by part i) of the proof of

(16.2.2). This is impossible. ii)=+iii)Suppose there are a compact 2-convex set K p

c

M(A) and

A

E

(int K) f l ynAK.

Choose an open, hemicompact and 3-convex neighbourhood U c K of p by (4.3.7)ii). Then, repeating the proof of part 1) in (16.2.21, we h

see that p c ynAU. This is a contradiction.

O

273

CHAPTER 17 LOCAL ANALYTIC STRUCTURE IN THE SPECTRUM OF A UF-ALGEBRA

In the theory of several complex variables one is not always interested in the global behavior of a function on i t s entire domain of definition. There are situations in which one considers the behavior of a function in arbitrary small neighbourhoods of a fixed point. This leads t o the notion of germs of holomorphic functions at a point p

€

cn.

This notion can be easily carried over t o uF-algebras. Let A be a uFalgebra with locally compact spectrum M(A). and l e t cp denote by

^Acp

E

M(A). We

A

the algebra of germs of functions of A at cp. I n this

chapter we show that A has analytic structure in 9, if there are a l A

gebraic relations between A holomorphic functions at 0

(Q

E

-

and

the algebra o f germs of

Cn.

Most results of this chapter are due t o Brooks. (17.1) Local rings of functions of uF-algebras (17.1.1) Throughout this chapter we always consider uF-algebras with locally compact spectrum M(A). If U

c

M(A) is an open and hemicompact subset, then C(U) i s a uFA

algebra by (4.1.4)ii). hence A, Let cp

Q

c

C(U) by (4.1.5).

M(A). Let U and W be open and hemicompact neighbourhoods

of cp in M(A). We c a l l f

h

E

A,

A

and g E,,,A ,

there exists an open neighbourhood V

c

equivalent at cp. provided U f l W of cp such that fl,=glv. ' A

,, This gives an equivalence relation on the disjoint union U A

where

the union is taken over a l l open and hemicompact neighbourhoods U

274

Chapter 17

of cp. A

An equivalence class w i l l be called a germ of a function of A at cp. We denote by f A

P

the equivalence class of an element f

A

A,

E

and by

Ap the set of a l l equivalence classes. h

h

A

Let gp, fcpc Alp. Choose representative functions f c A, A

Then f+&

and g E. ,A

V

resp. fglv are w e l l defined elements of .A ,

U n W an

c

open and hemicompact neighbourhood of cp. We set

flp + go = (f+gIp, f+gcp = (fgIcp. It is easy to see that these operations are well-defined, i.e. they are

independent of the choice of representatives of fcpand g

A

9'

Hence Acp

is a commutative algebra with unit. Let f

h

cp

c

Alp. Choose a representative f

A

E

A,

of f

cp'

Set fcp(cp)= f(cp).

A

then f (cp) is well-defined. A is a local algebra, i.e. there exists cp cp exactly one maximal ideal M In and

glp,

M = {f, (If f (cp)

c

gcp:fcp(cp)= 0)

* 0, there exist

A

an open, hemicompact and A-convex neigh-

cp A bourhood W of cp and a representative f c ,A

f($)

* 0 for a l l

$

E h

we see that l / f c ,A

W (use (4.3.7)il)). Since M(S,)

6

= W by (4.3.4)iii) A

by (6.1.41, hence fcpis invertible in AT.)

Let U, W be open neighbourhoods of p g

of flp such that

c

Cn. Then f c Hol(U),

Hol(W) are called equivalent at p if there is an open neighbour-

hood V c U n W of p such that fl,

= 9.1,

The algebra of germs of

holomorphic functions a t p. defined as above, w i l l be denoted by

0

n P'

(17.1.2) Theorem [BRO 51. Let A be a uf-algebra with locally compact spectrum M(A), and l e t cp be a Gg-point in M(A)\yn-,A. Then the following statements are equivalent: i) There exists a surjective algebra homomorphism

Local analytic structure ii)

2 has

275

analytic structure in cp. More precisely. there are

a neighbourhood W of cp in M(A), an open set D homeomorphism F A

Cn and a

D, such that the map -1 foF

W ->

:

Hol(D). f ->

>-,,A ,,

c

,

is a (we1I-defined) topological algebra isomorphism. (Here Y,-~A

denotes the (n-1)-th Shilov boundary of A, cf. chapter

16. Recall that cp is a Gg-point if it is the intersection of countable many open subsets o f M(AI.1 We divide the proof of the theorem into several steps. (17.1.3) Lemma [BRO

61.

[CLA 21. Let A be a uF-algebra with lo-

cally compact spectrum M(A), and l e t cp

E

pose that there are a neighbourhood U

M(A) of cp, a topological

space Y and a map F: U -> For each f

E

c

M(A) be a Gg-point. Sup-

Y with the following property:

A there exists a complex-valued function G defined in

some neighbourhood of F(p) so that

? = GmF in

some neighbourhood

of p. Then F is injective in some neighbourhood of p. Proof. 1) By proposition (4.4.2) there exists a countable descending open neighbourhoodbasis V1

3

V2

3

... o f cp.

ii) We may assume that V1 is contained In Us the domain of definition

of F. We shall prove the lemma by contradiction. Suppose the lemma i s false. Then there is a sequence of pairs (xi. yiIi such that for a l l i r N

i) xi and yi are elements of Vi\{cp), ii) xi

W.1.o.g.

* yi

but F(xi) = F(yi).

we can assume that xi, yi, xl, yj are pairwise disjoint points

for i*j. By (8.1.4) we can choose an element f

B

A such that P i s injective

276

Chapter 17

on the union of a l l points xi, yi.

?

Now choose G defined in a neighbourhood of F(p) such that GoF = in a neighbourhood W of p, w.1.o.g. W = V1. Then for a l l i E N,

?(xi) = GoF(xi) = GoF(yi) = ?(yi)

and we get a contradiction.

0

(17.1.4) Let A be a uF-algebra with locally compact spectrum M(A), cp

E

M(A). Let U

and l e t F

c

M(A) be an open, hemicompact neighbourhood of cp. such that F(cp) = 0. Set

E

A

F*: nOo ->

*

Acp. g ->

where g represents

9"

(goF)cp,

in a neighbourhood W of 0

E

Cn. We claim that

F* is a well-defined algebra homomorphism. Since M(A) Is locally compact and F: U ->

Cn is continuous there

A

exists an open, hemicompact and A-convex neighbourhood V of CQ in M(A) such that V c U and F(V) A

by (6.1.21, since M(AJ

c

W, cf. (4.3.7)ii). We have geF

A

B

A,

= V by (4.3.4)iii). Using ii) of (6.1.2) one

sees that F* is independent of the choice of g, and our claim is proved. Let f be a representative of

7 c ,,ao.Set

?iO) = f(O), then 7(0)is

well-defined, and

cl =

(3

E

,O0' 7CO) = 0)

is the only maximal ideal in

Lemma [BRO 51, [CLA 21.

Let A be a uF-algebra with locally com-

pact spectrum M(A), and l e t cp finite ,O0-algebra f

,,...,fr

h

E

A

E

M(A) be a point such that Alp is a

via some algebra homomorphism q, i.e. there are

A,+, such that

^A cp

= q(,O0)f,

+

... + q(,O0)fr. A

Then there exist an open, hemicompact and A-convex

neighbour-

Local analytic structure

hood U o f cp and F

(3Jn so that

E

Proof. As above denote by

M resp.

277

F(cp)=O and q = F*.

G the maximal

A

ideal of A

n00' i) We f i r s t endow

2cp

,ao with the m-adic

and

resp.

9

topology and show

that both spaces are Hausdorff spaces.

-

A fundamental system of neighbourhoods of

E

,ao resp. gcp

i s given by sets of the form h + -k M resp. g +Mk, k cp

E

E3cp

N, where

Gk = (9" E ,,ao:There exist I E N. elements r"ii E G , lsisk, I k lsjsl. such that 9" = c n q,}. j=l i = l

We have

n Gk= 0 (cf.

[KIK].

p. 70). hence

kc N

,ao is a Hausdorff

space. Since

,ao is a noetherian

q(nOo) and A

A

-

([NAG],

cp

ring ([K/K],

p. 801, the same is true for

- for

since it is a finitely generated q(nOo)-algebra

p. 9). So

^A

cp

is a noetherian local ring, hence

n

Mk = 0

k tN A

by Krull's intersection theorem, and we see that A

cp

is a Hausdorff

space, too. ii) We claim that the germs of the polynomials lie dense in

,,ao.

9" E ,ao be an arbitrary element, and l e t g be a representative 9". Considering the power series expansion of g at 0. we may write

Let of

OD

9 =

c Pi' i=o

where the Pi's are homogeneous polynomials of degree i. Set

then

9" -

rw

(?o+...+Pi-,)

cu

= Qj

and our assertion follows.

E

Gj. for a l l j

E

IN.

278

Chapter 17

....zn the coordinate functions and set

iii) Denote by zl.

(fiIp = q(Zi)

, i=1,...,n.

Choose representatives fi of (f 1 in an open, hemicompact and %icp “ n convex neighbourhood U of cp, i=l ,n. Set F = ( fl,...,fn) c (Au)

,...

We claim that q(% Let

5

E

0

.

M.

c

be an element such that q(5) c 29\Ms i.e. q(5)(cp)=cSO.

OIv

Then q(5-c-1 )(cp)=O, hence gh-c-i* is not invertible in 1.e.

5 e ,0,\3.

“ “k 1

It follows that q(h+M

k for every k

c q(C)+M

E

,o0, so t(O)=c,

N. hence q is con-

tinuous.

,....

iv) We have (f 1 (cp) = 0 fo r i=1 n. i.e. F(cp) = 0. by iii). icp ‘v Let P(zl,...,zn) be a polynomial and denote by P ( z l.....~n) the germ

of P(z

,,...,zn)

in

,,ao.Then

Iv

q(P(z

= (P(f,

=P(qq

2,))

,#””

#...)

q(i;l)l = P((fl)cp #...,(f 1 1 = ncp

,...,f n1)c p = F*(P(z,, ...,zn)). N

Hence q and F* coincide on the polynomials. v ) Since q and F

* coincide on a dense subset of

by ii) and i v )

we see that F * = q by 1) and iii).

0

(17.1.5) Proof of theorem (17.1.2). Assume 1). Since yn-lA

is a closed

subset of M(A) and M(A) is locally compact, there exists an open, hemicompact and %-convex neighbourhood U of cp in M(A). such that yn-lAnU = 8 , by (4.3.7)ii).

It follows from (16.2.2) that

A

Since q is surjective, A

,ao

is a finite algebra (via q). Hence there cp A exists an open, hemicompact and A-convex neighbourhood W c U of cp such that there exists F e (%,,,,I with n q = F* by (17.1.4).

279

Local analytic structure

Let h

E

A be an arbitrary element. Choose

5 c ,ao such that

n

hp = F*(9").

Hence there exist a neighbourhood V of 0 in Cn and a representative

g

E

Hol(V) of

5

such that h = goF in a neighbourhood of cp. Thus F is

injective in a neighbourhood % o f cp by (17.1.31, w.1.o.g.

W=%. Note

that M(A,)=W. Now l e t L be an affine complex line in Cn. parallel t o a coordinate axis, so that LnF(W)

* 0. Set

W, = F-'(F(W)nL). Then M($

= W,

WL h

=

from yn-lAw

(cf. (4.3.4)iii))

is locally compact and it follows A

0

(cf. (16.2.2)) that ( AJWW ., ),

is a m.m.a.(*),

using (9.1.6) and (9.1.7). We can now apply theorem (15.1.2) t o A

(Aw.W) and F. By (15.1.2)i) F is a homeomorphism onto the open subset F(W)

E

Cn, and g0F-l

E

Hol(F(W)) for each g

A

E

,A

by the proof

of (15.1.2)i) p. 246/247. Using (15.1.2)ii) we see that the map A

->

Hol(F(W)). g

_*

9.F -1

I

defines in fact a topological algebra isomorphism. The inverse statement is evident.

0

(17.1.6) Corollary [BRO 51. Let A be a separable uF-algebra with locally compact spectrum, and l e t Y,-~A = 0. Then the following statements are equivalent: i) For every cp z M(A) there exists a surjective algebra homo-

morphism q

0

A

A (9' ( 9 n o ii) M(A) can be equipped with the structure of an n-dimen:

->

sional Stein manifold such that M(A) with this structure.

3 = Hol(M(A)).

if we endow

280

Chapter 17

Proof. Assume 1). Let cp

E

M(A) be an arbitrary point. By (4.4.3) cp is

a G8-point. h

For each cp c M(A) there exist an open, hemicompact and A-convex nelghbourhood U of cp in M A ) , an open set Wp cp -> W , so that the map morphism F : cp cp ucp

c

Cn and a homeo-

defines a topological algebra isomorphism (note that F c cp by (17.1.2)). Let U nu c p Q

(2

In

"9

* 0. We conclude easily that

,is a biholomorphlc mapping, i.e. M(A) can be equipped with the struch

ture of an n-dimensional complex manifold such that A

c

Hol(M(A)).

Since M A ) is 2-morphically convex we can endow M(A) with the h

structure of a Stein space such that A = Hol(M(A),O). where Hol (M(A),O) denotes the algebra of al I holomorphic functions with respect to this structure, cf. (11.1.3). Let f be a holomorphic map on M(A) with respect to the manifold structure, 1.e. f0F-l cp flUp E

E

Hol(U 1 for a l l cp c M(A). Then 9

2up = H~I(M(A),o)

for a l i cp

E

MA).

and it follows that both structures coincide on M(A), since a l l elements of Hol(M(A),O)

are holomorphic on U with respect t o the cp

new structure, because Hol(U ,O) is complete with respect to the cp compact open topology. 0 (17.1.7) The next theorem shows that In cp c M(A) if

^A cp

^A has s t i l l

analvtic structure

is only a finite ,Oo-algebra.

Theorem [BRO 51. Let A be a uF-algebra with locally compact spec-

28 1

Local analytic structure trum M A ) , and l e t cp

M(A)\yn,lA

E

be a G&-point. Then the following

statements are equivalent:

^A '0

i)

is a finite ,O0-algebra

morphism

(via some algebra homo-

r));

ii) There exists a neighbourhood U of cp which can be en-

dowed with an analytic space structure of pure dimension n

= Hol(U) with respect t o this structure.

n such that A,

Proof. Assume i). Choose a neighbourhood that

r) N

Let hl, hl,

U of

cp and F

(3Jn such

E

= F*, according to (17.1.4). Recall that F(cp) = 0.

"

n

...,hr

...,hr

A

in,,,,A, A:

over F*(,,o0), and choose representatives cp an open neighbourhood of cp. Set

generate A W

c

W ->

U

Cn+r,

JI ->

(F($),H(JI)),

where H = (hl,...,hr). a ) Claim.

A

is injective in some neighbourhood o f cp.

Set n(cp) = p. Let g n

= i=l

g~

E

A be an arbitrary element. We have

F*(Si)Ci

for some

Choose a polydisc D centered at 0 sented by a function si cp such that F(V) A

g =

c

E

Zl,..., E

'r "

n 0o *

Cn so that each Ci is repre-

Hol(D). Choose a neighbourhood V

c

W of

D and

C (sioF)hi r

on V.

i=l Define

P

G : DxCr -> Then G

E

C, ( 2 . ~ 1->

f: si(z)wi.

i=l

Hoi(DxCr) and A

g =

Golr

on V.

Our assertion follows now from lemma (17.1.3). W.i.0.g. that

A

is injective on V.

we assume

282

Chapter 17 hl

8) Claim. There exists a neighbourhood V of cp such that


A / J , (f+I) + J

f+J,

defines a bijective algebra homomorphism. Fix n

E

N, and l e t f

E

A,

then inf{pn(f+g): g by

(*I.

E

J") = inf {pn(f+g+h): g E J", h E I)

I t follows from this equality that T is a topological iso-

morphism. Since A/? is a uF-algebra by our hypothesis the same i s true fo r (A/I)/J and it follows that A / I is a u*F-algebra.

(19.2.3) Proposition. Let A be a uF-algebra. Then A is a u*F-algebra iff the restriction map

is surjective for each kernel ideal I. A

(More precisely: Let (KnIn be an admissible A-convex exhaustion of M(A), set K;1 = V(I)nKn. Then (K;In

A

is an admissible A-convex ex-

haustion of the closed subset V(I), and we consider the map

Proof. i) Suppose the restriction map is surjective. The map

T: A / I ->

A

A,(

f+I

-' (?IKhIn,

is an injective and continuous homomorphism. By our hypothesis T is even surjective, hence a homeomorphism by the open mapping theoA

,(, rem for Frechet spaces. Our assertion follows, since A

i s a uF-

algebra. 11) Suppose that A / I Is a uF-algebra. By (3.2.10) M(A/I) = V(1). Let

(Knln and (K,'),

be as in i). Then the topology of (A/I)*, the algebra A

o f the Gelfand transforms of A/I, and the topology of A ,(, fined by the same sequence of seminorms

(ll-llKh)n. Hence

defines a topological homomorphism onto it s image.

is dethe map

Chapter 19

308

Since (A/IIA is a uF-algebra by our hypothesis and since

T"

has

h

dense range we see that (A/1IA is isomorphic t o Avo,. A

Let f

E

A,(,,

be an arbitrary element. Choose g

A such that

E

?((g+I)^ 1 = f. Then i(g) = f.

0

(19.2.4) Remark. Let A be a u*F-algebra. It follows from the proof of ill that in this case A / I i s topologically and algebraically ison

morphic t o Avo,

n

for each kernel ideal I A. Hence M(AV(,+ can be

identified as a topological space with V(1). (In the case that M(A) is locally compact this follows by (4.3.4)iii)J A

If A is moreover a u*FS-algebra, then A,(,,

is a u*FS-algebra by

remark (19.1.1)ii) and (19.2.2). (19.2.5) Examp1es.i) Let X be a hemicompact k-space. Then (C(X).X) is a uF-algebra by (4.1.4)ii). Let Y

c

X be a closed subset, and l e t (KnIn be an admissible ex-

haustion of X. Set K;I=YnKn, then (K;In of Y. Let (fnIn e C(X),

=)kC(K;)

i s an admissible exhaustion

be an arbitrary element.

Using the extension theorem of Tietze (3.1.9) we find F1 P C(K,) so that FIK,=

fl. Set

g2: K i U K 1 ->

C , x ->

I

F1(x) if x f2(x)

6

K1

if x is not contained in K,.

Then g2 is continuous. Using again Tietze's theorem we find F2

E

C(K2) such that F2lKkUK,= g2. Continuing in this manner we get

a complex-valued function F on X. such that FI

Kil

F is continuous, since FI

Kn

= fn for a l l n

= Fn is continuous f or all n

E

E

N.

N and X is

a k-space. Hence (C(X),X) is a u*F-algebra by (19.2.3). ii) Let A be a Stein algebra, i.e. A is isomorphic to Hol(X1. X a hemi-

compact reduced Stein space. Let I

c

Hol(X) be a kernel ideal. Then

Strongly uniform Fr6chet algebras

309

each element of Hol(V(1)) can be extended t o an element of Hol(X) by (2.3.31, hence A is a u*F-algebra by (19.2.3). iii) In [KRA

51 Kramm

gives an example of a uFS-algebra which is

not strongly uniform. (19.2.6) Finite sets are the only compact analytic subsets of a domain in

c".

cf. (7.1.1)

. We prove an analogue

Definition. Let A be a uF-algebra, and l e t S

c

f o r u*FS-algebras.

A be an arbitrary set.

The set WS) =

((QE

M ( A ) : q(f) = 0 for a l l f

E

S)

is called the h u l l of S. Theorem [KRA

11.

Let A be a u*FS-algebra. and l e t Y be a compact

hull in M A ) . Then Y is a finite set. Proof. Let I algebra

E

A be the kernel with respect t o Y. Then A / I is a uFS-

. We have M ( A / I ) = V ( I ) = Y.

A / I is isomorphic t o ( A / I I A , since it is a uF-algebra. The topology of ( A / I I A is defined by the norm

11-11,,

i.e. (A/IIn and hence A / I is a

uB-algebra. By remark (19.l.l)iv) A / I is finite dimensional and our assertion follows.

0

Remarks. i) We need the strong uniformity to conclude from the compactness of M(A/I)=Y that A / I is a uB-algebra. cf. (19.1.2)ii). ill Let A be a u*FS-algebra, and l e t Y

c

M ( A ) be a hul I . Suppose

that X i s an open and closed subset of Y (with respect to the relah

h

tive topology). Then A, is a u*FS-algebra. Y=M(A,),

by (19.2.4).

and

X is a h u l l in M(2,) by Shilov's ldempotent theorem (6.1.5). Hence

if X is moreover compact then X is a finite set by (19.2.6).

310

Chapter 19

(19.3) Cheval lev dimension fo r uF-alaebras (19.3.1) We introduce the notion of dimension for uF-algebras. I t is motivated by the Chevalley dimension in complex analysis (cf. (7.1.1)) or commutative algebra.

Definition. i) Let A be a uF-algebra, and l e t cp

I

M(A). Define d(cp)

....fn e ker cp

t o be the minimum of a l l n c N such that there exist fl,

-

and a neighbourhood U of cp such that the map

P: u

c", Q ->

tf;(Q)

A

fJQ)),

I...,

has finite fibers, 1.e. the set u n F - V c $ ) ) is finite for each Q

I

U.

If the minimum does not exist set d(cp)=aD. The Chevalley dimension of cp in M(A) is defined by dlmpM(A) =

I

0

if cp is an isolated point in M(A),

d(cp1 otherwise.

-

ill Let A be a u*F-algebra, Y c M(A) a hull, and l e t cp

3: A/k(Y)

->

C , f+k(Y)

e

Y. Then

cp(f),

defines an element o f M(A/k(Y)) by (3.2.10). We set dim Y = dim#vl(A/k(Y)) 9

and c a l l it the dimension of cp w i t h respect to the hull Y. Remarks. 1) We can identify M(A/k(Y)) with V(k(Y))=Y as topological spaces by (3.2.101, hence we can interpret cp as an element of M(A/k(Y)) and shall use henceforth the more suggestive formulation dim Y = dimpM(A/k(Y)). 9 ill I f X is a hemicompact Stein space, i.e. X = M(Hol(X)) by (2.3.21,

the above defined dimension equals the usual notibn o f dimension, cf.

311

Cheval ley dimension (7.1.1).

(19.3.2) Definition (19.3.1) immediately yields the following remarks. Remarks. I) Let A be a u*F-algebra, and l e t Y

c

M(A) be a hul l.

Then the mapping

N U {OD-),

Y ->

dim Y. cp

cp ->

is upper semi-continuous. i.e. for every cp a Y there exists a nelghbourhood U

c

-

Y of cp such that

dim Y 5 dim Y fo r a l l @ e U. cp 4 If dim Y < f o r a l l cp E Y. this mapping i s bounded on any relatively cp compact subset o f Y. ii) The sets {cp a Y: dim Y

cp

n),{cp

a

Y: dim Y < n), cp

are open, the sets {cp

e

Y: dim Y > n). {cp cp

a

Y: dim Y cp

5

n),

are closed for a l l n a N. (19.3.3) Next we prove a theorem which is an analogue t o the semicontinuity of fiber dimensions of holomorphic mappings (cf. [G/R]. p. 114). I n complex analysis this theorem is proved by the classical Weierstra6 theorems, which are not available in our setting. Theorem [KRA 11. Let A be a u*FS-algebra with locally compact spectrum M(A). and l e t F = (f l,...,fn)

M(A) ->

NU{O,-I, cp ->

An. Then the map A- 1 dimPF E

(F(cp)).

is upper semi-continuous. i.e. for every cp bourhood U of cp such that "-1 A dimpF (F(cp)) 2 dirn,f-'(~(@))

c

M(A) there exists a neigh-

for a l l

"-1 A (Note that F (F(cp)) is a h u l l in M(A) f or a l l cp

JI c U. E

M(A1.1

312

Chapter 19

We shall deduce theorem (19.3.3) from part i) of the next theorem. Part ii) w i l l be needed later. (19.3.4) Theorem. Let A be a u*FS-algebra with locally compact spectrum M(A), and l e t F = (fl,

...,fn) e An.

1) Suppose that cp c M(A) is an isolated point of the fiber

Then there is a neighbourhood W c M(A) of cp such that fibers on W, 1.e. dim M(A) c n fo r a l l $

JI

E

P-l(f(cp)).

c has finite

W.

ii) Let U be an open, hemicompact and %-convex subset of M(A). and A

l e t cp e U. (By (4.3.4)iii) we can identify M(AJ as a topological space h

with U, hence we can interpret cp as an element of M(A,). dim M(A) = dim M($J 9 cp

for a l l cp

E

Then

U.

Proof. I) Denote by [cp] the Gleason part which contains cp. By remark (19.1.l)iii),(l8.2.3)

[cp] is an open subset of M(A). If [cp] is

compact then cp is an isolated point of M(A) and our assertion becomes trivial.

If [cp] is not compact we can choose a compact neighbourhood K of cp with 6K

* 0 and

(*I

KnC-'(P(cp))= (91.

W.1.o.g. we can assume that t ions ?l-?l(cp

1,...,fn-?,, ( cp) . Set

P(cp) = 0 . otherwise consider the func-

A

S: M(A) ->

R, J, ->

max{i?l($)l,

...,i?"($)i).

Then S is continuous and we have

by the compactness o f 6K and by

W =

(*I. Set

{JI E K: S($) < 8/21.

Then W is an open neighbourhood o f cp. For an arbitrary point $, the set

E

W

Cheval ley dimension

is compact and relatively open in

C-l(C(+o))nd~ = by (**I. Since ?l(?(@o))

313

P-l(P(@o))s since

0

is a h u l l in M(A). YJlo is a finite set by r e -

mark (19.2.6) ii). h

ii) By our assumptions (A,,U)

A

is a uF-algebra and M(A,)

= U by

(4.3.4)iii). Clearly dim M(A) cp

2

dm i pM$ (.),

To show the inverse inequality choose an arbitrary point cp e U and set A

r = dim M(A,). 9

The assertion is trivial if r=m or r=O. Let O

h

Avo,

. We shall

show that the restriction map

is surjective. This yields our assertion by (19.2.3).

We endow Xi always with he induced euclidean topology. If V(l)nX, has an accumulation point, then Xi V(I) = U X I IRJ

c

V(I) by the identity theorem. So

u s.

where J is a subset of NU{O) (possibly empty) and S is a discrete subset of M A ) \ U X i with respect t o the Gelfand topology (possibly IRJ

Chapter 19

320

empty). This follows by (*I and since the induced Gelfand topology and the induced euclidean topology are equivalent on X and on every xi* A A Let f E A,(*)

be an arbitrary element.

Case 1: J is a finite set. Then U X i is an analytic subset of C2, hence by (2.3.3) there exists IaJ

f'

Hol(C2) such that

f'lu x, -- fl u

ItJ

IrJ

We note that

XI'

= f'IMCA)c 3.

r"

Case 11. J is not finite. It follows by the identity theorem that 0 c J. Set

ax) =

Then

?Ix

E

I

if x

f(x)

c

uxi,

IrJ

f(l/i,O) if x c Xi, i not contained in J.

Hol(X) and

r"I(D,{O,lx{O, is bounded, hence r"

I n both cases we have found an element il"X,

r" c ^A

c

^A

by ill).

such that

- flux, * ItJ

IrJ

* 0 , then SnX, consists of a t most countable many points without an accumulation point In Xi. If SnXo * 0 , then J is finite or

If SnXi

empty and SU{(l/i,O): IEJ) is a discrete subset of Xo. Choose go c Hol(Xo) such that g,(y)=f(y)-?(y)

for a l l y

E

SnX,,

and go(l/i,O)=O for i

E

by the theorems of Weierstra6 and Mittag-Leffler. The function

go(x)

=

I

go(1/i,O) If x c Xi' go(x)

defines an element of

^A

if x

E

xo

by lli). If SnXo = 0 we set

go = 0.

J.

Characterization of Stein algebras Set J' = (i

321

N: sncx,\cci/i,o)~)*:rs~.

6

For i e J' we use again the theorems of Weierstra6 and Mittag-Leffl e r t o find g1 E Hol(Xi) so that gi(y)=f(y)- fz(y)-g",(y)

if y

E

Sn(Xi\((l/i),O)))

and gi(l/i,O)=O.

By iii) the function

1

if x is not contained in Xi

I 0

i&(X)

lies in

=

gi(x) if x

0

xi

2. Hence

vi) M(A) is purely one dimensional.

The map p: M(A) ->

C. (z.w) ->

defines an element of

2. For x

Z-w

,

X it is easy t o find a (Gelfand open)

Q

neighbourhood U of x such that p has finite fibers on U. Now set f(z,w) =

Then f

E

^A

n w if (z.w)

E

Xn, n r l .

0

€

xo.

i2

if (2.w)

by Hi). Set

U = ( ( 2 . ~ 1a M(A): if(z.w)l < 1) Then U is a (Gelfand open) neighbourhood of ( 0 . 0 ) by (3.2.211).

.. .. .... . . U ......... .... *

. :. .

.-.. . . :. .: -... . . -*. . .

. ..

\

Chapter 19

322 Choose an arbitrary point zo

C\COI. For rial set

S, = p-l(zO)nunxn = {(l/n,w): (l/n)-w=zo, I w i < l / n2I. Hence Sn = 0 for a l l n

L

N such that (l/n)+(l/n 2 )

ker p, g ->

fg.

Clearly i Is a continuous and surjective linear mapplng. If 0 = i(g)=fg,

then BIM(A)\W

by

(**I, hence g

=o

0, since {p) is not an Isolated point of M(A). This

proves that i Is Injective, hence a homeomorphism by the open mapping theorem. By the proof of (4.4.2) there exists a countable open nelghbourhoodbasis

...

3

Un

3

Un+l

3

... of p. Let (KnIn

be an admissible exhaus-

tion of M A ) , w.l.o.g. we can assume that U1 c K1.

327

Riemonn algebras Suppose that cp

E

yg

and since the set of

21Kn Is dense in 3K n, the strong boundary points of ^A lies dense in Kn

Kn

for all n

E

N. Since

A

yAKn by (1.3.12) we find for each n c N an element gn

Hence i(gn)=fgn tends t o zero for n ->

E

A such that

Q).Since i is a linear homeo-

morphism this implies that gn tends t o zero f o r n + Q),a contradiction t o

ll$nllKn = 1 for

all n

E

N. Hence there is m

c

N such that

A E m

Set Kn = Kn+,

Km\yAK,. for a l l n

c

N.

0

(19.5.3) Theorem. Let A be a uF-algebra, and l e t cp

c

M(A) be a

point which is not isolated in M(A). Suppose that cp has a compact neighbourhood in M(A), and that ker cp is a principal ideal, generated by f

E

A. Then there exist an open set V

hood U of cp in M(A) such that

?:

U ->

c

C and an open neighbour-

V is a homeomorphism and

the map A

Hol(V) ->

,A ,

h ->

n

hof ,

defines a topological a1gebra homomorphism. Proof. According t o (19.5.2) there is a compact neighbourhood K of cp in M(A)

cp

E

- w.1.o.g.

A

we can assume that K is A-convex

K\Y%~.

This implies

(*I

vc?)nyhAK=O.

Clearly the ideal

?lK.hAK lies dense in

(ker tpIK = tg

A E

A :,

cp(g)=Ol.

By (*I and lemma (19.5.1) we have in fact

- such that

320

Chapter 19

A

So we can apply Gleason's theorem (1.5.5) t o the uB-algebra A ,,

.,1

A

and f

cp

It follows that there are an open set V c C and an open

neighbourhood U of cp in M($,)=K 1)

?:

such that

V Is a homeomorphism

U ->

and ii) t o ? - '

a

H ~ I ( V )for a l l g

A

A, ->

A .,

assume that U i s a hemlcompact and 3-

By (4.3.7) we can w.1.o.g. convex set. 1.e.

A

M(3,) = U by (4.3.4)iii). Hol (V), g ->

It follows that the map

CO?-'.

is a continuous and lnjectlve algebra homomorphlsm. By (6.1.2)

this

map Is even surjective and our assertion follows from the open mapping theorem.

0

Remark. Note that in general (19.5.3) becomes false if cp has no compact neighbowhood. In example (19.4.3) ker cp = (zl) for p=(O,O) and the f i r s t coordinate function, cf. (19.4.3)vii) neighbourhood U of o, such that

tl

, but there Is no

is injective on U or such that

2,

is a Stein algebra. (19.5.4) Theorem. Let A

$

C({pl) be a uF-algebra with locally com-

pact and connected spectrum M(A). Then the following statements are equivalent: I) A is algebraically and topologically Isomorphic to Hol(X). X a Rlemann surface; ill ker p is a principal Ideal for a l l cp c M(A).

Proof. 11)

4

1) Let cp s M(A) be a point with ker cp

= (f 1. Choose an P

open neighbourhood U of p in M(A) and an open set V

c C accordcp ing t o (19.5.3). Now we conclude as in the proof of (17.1.6) that the

cp

329

Rlemann algebras family (Up.fp.Vp

defines an atlas on M(A) such that M(A)

becomes a Riemann surface and such that

ill

4

I) This follows by [FOR 13, Satz 5.4.

^A = Hol(M(A1). 0

This Page Intentionally Left Blank

331

APPENDIX A SUBHARMONIC FUNCTIONS, POISSON INTEGRAL

We c o l l e c t some results which are needed in the t e x t . For the proofs and further results cf. for example the books of Narasimhan [NAR

11

and Garnett [GAR]. A.l Definition. Let X be a metric space, and l e t u: X ->

RU{-m) be

a map. Then u is called upper semi-continuous (usc.) if. for any

X

c

R. the set tx

c

x: u(x)

< XI

is open in X. A.2 Let X be a metric space, and l e t u be an

USC.

function on X.

Suppose that u is bounded above. Then there exists a sequence of continuous functions on X such that i un(x) f o r a l l nkl, x 'n+l (XI

6

X

and un(x) ->

A.3 Let G

c

u(x) as n -*

a.

C be an open set. Denote by C 2( G I the set of a l l twice

continuously differentiable complex-valued functions on G. Definition. We say that u

-d2U + dx2

d2u dy2

We remark that i f f

E

E

C 2 (GI is harmonic if

=OonG.

Hoi(G), then real- and imaginary part of f are

Appendix A

332

harmonic functions on G. Vice versa if f is a harmonic function on an open disc D c C, then there exists g

E

Hol(D) such that Re g = f on

D, where Re g denotes the real part of g.

A.4 Harmonic functions satisfy the mean value property: If f is harmonic on the open set G and the disc

{z: ia-zirR1, R>O, i s contained

in G, then f(a+Rexp(it)) dt.

r is a closed disc in C and f is a continuous function on dn, the boundary of n,then there exists a continuous function h on 16, A S If

which is harmonic on the open disc D, such that h l a E = f. Combining this result with A.3 it is easy t o show that f o r each realvalued function f

E

CCdn and each k

L

N there exists a polynomial pk

such that Ilf-Re PkllaE < l / k .

A.6 Definition. Let u be an

USC.

function on an open set G

c

C. Then

u i s called subharmonic on G provided that the following condition is satisfied: Let U be a relatively compact open subset of G, and l e t h be a continuous real-valued function on u(z)rh(z) for a l l z

E

uswhich i s harmonic on U. Then i f

dU we have u(z)rh(z) for a l l z

A.7 A theorem o f Littlewood states that if u is an an open set G

c

E

U.

USC.

C then u is subharmonic, if for every a

function on E

G, there

exists a sequence (rnInof positive numbers such that rn -> n ->

= and u(a) r 1/2x r u ( a + r n e x p ( i t ) ) d t f o r a l l nkl.

0 as

333

Subharmonic functions

A.8 The next theorem is due to Hartogs, for a proof see [AUP 2 1 , ~ .

174: Let D

c

C be an open disc, and l e t f: D ->

C be a bounded function

such that logif-ai is subharmonic on D for every a

c

C. Then either

f or the conjugate o f f is holomorphic on D. A.9 We now consider the Poisson kernel for the open unit disc D. Definition. We define the Poisson kernel for D t o be the function

D ->

(1-r 2 )/(l+r2 -2rcos(t)) =

R, rexp(it1 ->

A.10 We collect some properties of P. First note that P(z)>O for a l l

z

c

D. Furthermore we have dt = 1. Osr

E

h( t o ) as z ->

exp(ito).

Hol(D) be a bounded function. Then for almost a l l

[ 0 , 2 x ] the non-tangential limit

Appendix A

334

f*(exp(it)) =

Ii m f(z) z->exp ( it 1

exists, i.e. the limit exists if z approaches exp(it1 through the angle between two rays emanating from exp(it) which are not tangential to the unit circle and which approach exp(it) through D. A.13 Let f, f* be as In A.12, then

A.14 Definition. I) Let G c Cn be an open set, and l e t f:G -> be an

USC.

function. We say that f i s plurisubharmonic if for each

complex line L = {az+b)

{z

RU{-mI

E

c

C". the function

C: az+b s G I ->

RU{-mI, z ->

f(az+b).

is subharmonic. ill A set E

c

G is called pluripolar i f there exists a plurisubharmonic

function f on G. f

*

-QD,

such that f ( x ) = -a for a l l x

Q

E.

Remark. I f E i s a pluripolar set, then X2n(E) = 0, where X2n denotes the Lebesgue maesure In R2n, cf. [G/L].

p. 234.

335

APPENDIX B FUNCTIONAL ANALYSIS

We state three theorems which we often used in this book. 8.1 Open mapping theorem. If E, F are Frechet spaces and

T: E ->

F i s a continuous, linear and surjective mapping, then T is

an open mapping, 1.e. T(U) is open in F if U is open in E. I n particular T is a homeomorphism if T is bijective. 8.2 Closed graph theorem. If E, F are Frechet spaces and

T: E ->

F is a linear mapping, then T is continuous if its graph

{(x,T(x)): x c E l is closed in ExF. 8.3 A subset E of a topological space X is called nowhere dense i f Interior

= 0 , where

denotes the closure o f E.

Baire category theorem. Let X be a complete metrizable space. Then the countable union of nowhere dense subsets of X contains no interior point. Equlvalentlyi Let (Ul)i be a sequence of dense open subsets of X. Then

nui Is

a dense subset of X.

336

LIST OF S Y M B O L S C , R, N, complex, real, natural numbers

No = NU{O) Re z, Im z denote the real resp. the imaglnary part of a point z

dK denotes the boundary,

denotes the closure of a set,

Ck([O,ll),

C(K), P(K1, R(K), 4

d b ) , B'l,

5

r(bL 7

M(B), ker cp,

6.8,9

R, 11 n(B), 13 (B,K), 15 rad A, 15 O(fl,..fr),

17

YB, 20 Hol(G),H(S), 26

dz, a 'dzj , 3 4

Ilm En, 8 4