SYMMETRIC BANACH MANIFOLDS AND JORDAN C*-ALGEBRAS
NORTH-HOLLAND MATHEMATICS STUDIES Notas de Matematica (96)
Editor:...
120 downloads
387 Views
12MB Size
Report
This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!
Report copyright / DMCA form
SYMMETRIC BANACH MANIFOLDS AND JORDAN C*-ALGEBRAS
NORTH-HOLLAND MATHEMATICS STUDIES Notas de Matematica (96)
Editor: Leopoldo Nachbin Centro Brasileiro de Pesquisas Fisicas, Rio de Janeiro and University of Rochester
NORTH-HOLLAND -AMSTERDAM
NEW YORK
OXFORD
104
SYMMETRIC BANACH MANIFOLDS AND JORDAN C*-ALGEBRAS
Harald UPMEIER Department of Mathematics University of Kansas Lawrence, Kansas 66045 U.S.A.
1985
NORTH-HOLLAND-AMSTERDAM
NEW YORK
0
OXFORD
0 Elsevier Science Publishers B.V.,
1985
All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the copyright owner.
ISBN: 0 444 87651 0
Publishers:
ELSEVIER SCIENCE PUBLISHERS B.V. P.O. BOX 1991 1000 BZ AMSTERDAM THE NETHERLANDS
Sole distributors forthe U.S.A.and Canada:
ELSEVIER SCIENCE PUBLISHING COMPANY, INC. 52 VAN DE R BI LT AVENUE N E W YORK, N.Y. 10017 U.S.A.
Library of Congress Cataloglng in Publication Data
Upmeier, Harald, 1950Symmetric Panach manifolds and Jordan C* -algebras. (lotas de matedtica ; 101) (North-Holland mathematics studies ; 104) Bibliography: p. Includes indexes. 1. Banach manifolds. 2. Jordan algebras. 3. C*-algebras. I. Title. 11. Series: Notas de matedtica (Amsterdam, Netherlands) ; 101. 111. Series: North Holland methematics studies ; 104. gru.N86 no. 101 CQA322.21 510s C515.7'323 84-21122 ISBN 0-444-87651-0 (U.S. )
PRINTED I N THE NETHERLANDS
To Helga and Julia
This Page Intentionally Left Blank
PREFACE
The aim of this book is to give a self contained introduction to the theory of symmetric (complex) Banach manifolds. These infinite dimensional manifolds are natural generalizations of the classical (hermitian) symmetric spaces and have recently received much attention, partly because of their relationship to functional analysis and the theory of operator algebras
(C
*
-algebras and von Neumann algebras). The connection
to functional analysis is of importance in the more general context of infinite dimensional holornorphy on locally convex spaces, whereas the relations to operator algebras depend on the additional structure of symmetric Banach manifolds (i.e., the existence of global symmetries and the homogeneity under biholomorphic transformations). In fact, this additional structure leads to a complete algebraic characterization of symmetric complex Banach manifolds in terms of certain nonassociative generalizations of operator algebras, the so-
*
called Jordan C -algebras (and Jordan triple systems). Using this Jordan algebraic description, many holomorphic and geometric properties of symmetric Banach manifolds can be interpreted algebraically and, conversely, the holomorphic structure associated with (Jordan) operator algebras can be useful for a deeper understanding of these algebras and their automorphism groups. The book is divided into two parts. Part I (Sections 1-13) is devoted to the theory of transformation groups on (real or complex) analytic Banach manifolds (carrying a metric or tangent norm). The main theme is to endow certain groups of bianalytic transformations on Banach manifolds with the analytic structure of a (real) Banach Lie group. In Part I1
vi i
PREFACE
viii
(Section 14-23), these results are applied to a systematic study of the special class of symmetric Banach manifolds and their algebraic characterization in terms of Jordan algebras and Jordan triple systems. In both parts, the general theory is illustrated by many examples, and, in particular, Section 2 3 is entirely devoted to the study of the "classical" symmetric Banach manifolds. Although many of these examples, as well as part of the general theory, are presented for real or complex Banach manifolds simultaneously, it must be emphasized that a fully satisfactory theory using Jordan algebraic structures exists only in the complex case (as can be expected in view of the finite dimensional situation). Throughout, we have tried to make the exposition as self contained as possible. On the analytic side, we require a working knowledge of classical complex analysis (in one variable) and of the principles of functional analysis (HahnBanach Theorem, open mapping theorem) and spectral theory. Beyond this, all proofs are given in full detail, and the first few sections can also be used as an introduction to the theory of (analytic) Banach manifolds and Banach Lie groups, with special emphasis on analytic vector fields .and their integration to one-parameter groups. On the algebraic side, the basic theory of Jordan algebras (and Jordan triple systems) is treated from the beginning, with full proofs of the necessary theorems given. The exposition is relatively elementary, since the exceptional Jordan algebra plays no role in the general argument and the principal results concerning the Jordan theoretic description of symmetric Banach manifolds are proved without using deep theorems about Jordan algebras such as Macdonalds Theorem or the GelfandNeumark type embedding theorem due to Alfsen, Shultz and Stdrmer. For a systematic account of the general theory of Jordan operator algebras, we refer to the recent monograph C521.
*
The term "Jordan C -algebra" in the title has been chosen to indicate that the algebraic systems associated with symmetric Banach manifolds are closely related both to Jordan
PREFACE
ix
x
algebras and C -algebras. In the text, however, this term has * been replaced by "JB -algebra" (i.e., the complexification of a JB-algebra in the sense of C51, cf. C156,1591). This book was written while the author was visiting the University of Pennsylvania during the academic years 1982-84. Thanks are due to my colleagues in Philadelphia, especially Richard V. Kadison, for their interest and encouragement, to Leopoldo Nachbin for his invitation to write this book and to Wilhelm Kaup for invaluable discussions over a period of many years. Finally, special thanks go to Karen Walker and Brigitte Sabrowski for the preparation of the manuscript. Tubingen, August 1984
H. Upmeier
This Page Intentionally Left Blank
TABLE OF CONTENTS
PREFACE
vi i BANACH MANIFOLDS AND TRANSFORMATION GROUPS
1
1.
Analytic Mappings on Banach Spaces
2
Section
2.
3.
Banach Algebras Banach Manifolds
22
Section Section
4.
Analytic Vector Fields
53
Section
5.
Integration of Vector Fields
67
Section
6.
Banach Lie Groups
92
Section
7. 8.
Integration of Lie Algebra Actions
110
Submanifolds and Quotient Manifolds
121
Section
9. Section 10.
Binary Banach Lie Algebras
142
Locally Uniform Transformation Groups
156
Section 11.
Analytic Transformation Groups
172
Section 12. Section 13.
Metric and Norrned Banach Manifolds
187
Groups of Holomorphic Isometries
208
SYMMETRIC MANIFOLDS AND JORDAN ALGEBRAIC
229
PART I. Section
Section
PART 11.
36
STRUCTURES 230
Section 14.
Order Unit Banach Spaces
Section 15.
C -Algebras
247
Section 16.
Tube Domains and Siegel Domains
260
Section 17.
Symmetric Banach Manifolds
280
Section 18.
Jordan Triple Systems
297
Section 19.
Jordan Algebras
Section 2 0 .
Bounded Symmetric Domains and JB -Triples
314 329
Section 21.
Symmetric Siegel Domains
353
Section 22. Section 23.
Jordan Automorphism Groups
372
Classical Banach Manifolds
393
*
*
xi
xii
TABLE O F CONTENTS
REFERENCES
425
S U B J E C T AND SYMBOLS INDEX
435
PART I
BANACH MANIFOLDS AND TRANSFORMATION GROUPS
The first part of this book is devoted to a general study of (real or complex) analytic Banach manifolds. The main objective is to develop a Lie theory for transformation groups on these manifolds. After proving the basic facts about analytic mappings on Banach spaces (Section 1) and recalling the spectral theory for Banach algebras (Section 2 1 , we introduce Banach manifolds in Section 3 and construct, as a typical example, the Grassmann manifold associated with a Banach space. The underlying theme of Sections 4-7 is the study of analytic vector fields (or, more generally, Lie algebra actions) and their "integration" to obtain an analytic flow or a Lie group act ion. Since symmetric Banach manifolds can be represented as homogeneous spaces, submanifolds and quotient manifolds are studied in Section 8. As an application, one can associate a quotient manifold with any "binary" Banach Lie algebra (Section 9) which generalizes the Grassmann manifolds and will be of fundamental importance in later sections. The Lie theory of transformation groups acting on Banach manifolds is developed in Sections 10,ll and 13. The main results show a close connection between topological and analytic properties of transformation groups. Whereas in Sections 10 and 11 we consider transformation groups in general, Section 13 is devoted to a special case of particular importance, namely groups of holomorphic isometries on complex Banach manifolds. The necessary background on metrics and tangent norms on Banach manifolds is provided in Section 12. 1
2
SECTION 1
1.
ANALYTIC MAPPINGS ON RANACH SPACES
A n a l y t i c m a p p i n g s of i n f i n i t e l y many v a r i a b l e s a r e o f b a s i c importance i n t h e following.
We w i l l restrict our a t t e n t i o n
t o a n a l y t i c mappings on ( r e a l or complex) Banach s p a c e s i n s t e a d on more g e n e r a l t y p e s o f l o c a l l y c o n v e x s p a c e s .
This
is j u s t i f i e d by t h e f a c t t h a t R a n a c h s p a c e s i n c l u d e t h e soc a l l e d o p e r a t o r a l g e b r a s and v a r i o u s g e n e r a l i z a t i o n s w h i c h a r e of primary i n t e r e s t i n l a t e r s e c t i o n s .
Further, the
fundamental theorems of d i f f e r e n t i a l c a l c u l u s , e.g., i n v e r s e mapping theorem,
the
a r e s t i l l t r u e f o r Ranach s p a c e s b u t
may f a i l f o r more g e n e r a l t o p o l o g i c a l v e c t o r s p a c e s , e . g . , Fre'chet s p a c e s . Throughout,
w i l l e i t h e r denote t h e f i e l d
K
numbers or t h e f i e l d
C
o f complex numbers.
non-zero
K
w i l l b e d e n o t e d by
elements in
d e n o t e t h e set of a l l non-negative
.
over
KX
.
Let
N
is a v e c t o r s p a c e
K
1.1
endowed w i t h a norm
K
The g r o u p o f a l l
i n t e g e r s and p u t
R+ := { t E R : t > 0 } Recall t h a t a Ranach s p a c e o v e r Z
of r e a l
R
: 2 + R+
(satisfying
Iz+wI < I z I + I w I , ( X z ( = IXI-lzl a n d l z ( = O z=O f o r a l l E Z a n d X E K ) s u c h t h a t Z is c o m p l e t e w i t h r e s p e c t t o t h e m e t r i c d ( z , w ) := (z-wl on Z Two norms [ *Il and on a v e c t o r s p a c e Z o v e r K are c a l l e d e q u i v a l e n t i f z,w
.
t h e y g e n e r a t e t h e same t o p o l o g y o n constants
for all on
z
0
< r
0
. The . The
is defined as the
such that the series
m
1.3.3 1 fi(zl,...,z n 1 n=O converges uniformly for all sequences (Zm)m>l sat is fy ing supmlzml 5 r , whenever r < RHere fnE Ln(Z,W) denotes the symmetric n-linear mapping corresponding to fn
.
1.4 PROPOSITION. The radius of convergence R and the radius of restricted convergence Rare given by the CauchyHadamard formula
ANALYTIC MAPPINGS ON BANACH SPACES
5
1 / ~= lim sup (fnl1/n n+m
1.4.1
and
-
1 / ~ -= lim sup Ifn[l/n n+* The series 1.3.2 converges uniformly in norm for I z ( < whenever r < R The series 1.3.3 converges uniformly norm for all sequences ( z ~ ) ~satisfying > ~ supmlzml < r
l and := supm(zml Suppose first that the series 1; fn(Z) converges uniformly €or < r Then there exists an index m E N such that [?n(y)l< 1 , whenever n > m and < r Hence IFn[< r -n for all n > m , showing that z := z
.
.
lzl
Now suppose 0 Since L < l/s for all n > 0
.
.
,
.
.
r < 1/L Choose s with r < s < 1/L -n there exists c > 0 such that lTnl 6 cs Therefore < r implies
Since r < s and W is complete, the series k; Fn(y) converges uniformly in norm for < r Hence r < For r + 1/L , we get 1/L 5
.
.
IzI
. O.E.D.
f =
For any power series
1;
fn
and
1.5
COROLLARY.
R
and
R-
satisfy
and
r > 0
,
define
6
SECTION 1
{ r > o
R = sup
{ r >
= sup
R-
< +- }
: p(f,r)
o
1 k homogeneous polynomial hk E p ( 2 , V ) , defined by
L
hk(z) :=
nm
l<ml,
,
...,nm >1
satisfies
n 1t...+n m =k nm >I nl>l, m
...,
m
=
1
m=l
IgJ
(
c
n=l
lfnlrn)m
.
1.10.4
Li
It follows that the power series hk from h o := g(f(o)) , converges and satisfies
Z
to
V
,
with
m
.
for all z in a neighborhood of o E D Hence gof is analytic. If f(o) = 0 , we have ho = g o , and 1.10.4 implies 1.10.2. Further, 1.10.1 is trivial if f(o) # 0 and follows from 1.10.3
if
f(o) = 0
.
O.E.D.
An important global property of analytic mappings is specified by the so-called principle of analytic continuation. Recall that a connected open subset of a Ranach space is called a domain.
11
A N A L Y T I C M A P P I N G S ON BANACH S P A C E S
1.11
THEOREM.
Suppose
D
is a domain in a Ranach space
Z
over K and let f : D + W be an analytic mapping into another Ranach space. Let X be a real subspace of Z satisfying Z K < X > . Assume there exists a non-empty open subset €3 of D A X such that f IB = 0 Then f = 0
-
.
PROOF.
Define a closed subset A : = { o E D :
A
f(")(o) =
of
o
D
.
by
for all
n >
o 1
,
For any o E A , 1.8.2 implies that the power series of f about o vanishes. Hence f vanishes in a neighborhood of o E D It follows that A is open. Now choose o E R and consider the power series 1.6.1. For any x E X , we have
.
fn(x) since Z
=
1 dn f (o+tx 1 t=o = o 5 dtn
.
Hence fnlX = 0 for all n 2 0 is continuous and n-linear over
flB = 0 and fi
.
follows that fn = 0 Hence since D is connected.
o
E
A
.
Since K , it
and therefore
A = D
,
O.E.D.
In case K = C , analytic mappings are also called holomorphic and have certain special properties which are closely related to the Cauchy integral. If y : I + C is a piecewise smooth curve defined on a compact interval I = [a,b] and f : y ( 1 ) + W is a continuous mapping into a complex Ranach space
lb
f(X)dh :=
define
f(y(t))y'(t)dt
I = [0,211] and
In the special case
1 f(X)dX 1 X)=r
f(z) =
,
E
W
a
Y
1.12 THEOREM. and f : D + W subset D of
W
:=
.
y(t) := reit
f2" f(reit)ireitdt 0
,
we write
.
Suppose Z and W are complex Ranach spaces is a holomorphic mapping defined on an open Z For o E D , put R := dist(o,aD) Then
.
1 2ri
.
I IXI=R/r
f (o+X(z-o)) (jX
A-1
1.12.1
12
SECTION 1
.
if 12-01 < r The power series 1.6.1 of radius of convergence > R and satisfies 1 fn (v) = 2ni if
r < R
and
I
IxI=~
Iv( < 1
f
about
f(o+Xv) d X ,,n+1
0
has
1.12.2
.
PROOF. Since the continuous linear forms $ E L (W,C) separate the points of W [15; 33.131, we may assume W = C Now (otherwise, consider the holomorphic function $ o f 1 . Then the function g ( X ) = f(o+h(z-o)) suppose 12-01 < r is holomorphic on an open neighborhood A of { X E C : 1x1 < R/r } Since 1 E A and f z ) = g(1) , the Cauchy integral formula [27; IV.fi.11 applied to g implies 1.12.1. Now put v := 2-0 and assume Iv( < 1 Then g is holomorphic for I h l < R and has the power series expansion I
.
.
.
m
m
.
about 0 E C Applying the iterated Cauchy integral formula [27; IV. 6.21 to g , it follows that f (v) =
n
3 d"g
(0)
dhn has the integral representation 1.12.2 which is independent of
r < R .
O.E.D.
We now deduce some consequences of 1.12. the Cauchy inequalities. 1.13 COROLLARY. for every n 0
Suppose
-n < r
Ifn(
r < R := dist o,aD)
We begin with
.
Then we have
sup If(z) =r
1.13.1
(z-01
and 1.13.2 PROOF.
For any
v
E
2
of norm
< 1
,
1.12.2
implies
13
A N A L Y T I C M A P P I N G S ON BANACH S P A C E S
This proves 1.13.1.
For 1.13.2,
r + R
let
.
O.E.D.
1.14 COROLLARY. The vector space o(D,W) of all holomorphic mappings from D to W is a closed subspace of the vector space C(D,W) of all continuous mappings, with respect to the topology of compact convergence. Let (fj) be a net in i)(D,W) converging to f E C(D,W) uniformly on every compact subset of D Let o E D be arbitrary. Since f is continuous, there exists 0 < r < dist(o,aD) such that
PROOF.
.
sup
For every
n >
I z-o I =r 1 ,
If(z)l
1 Since fn is continuous, 1.2 Now 1.14.1 implies 1.13.1. By 1.4, implies fn E Pn(Z,W the power series 1; fn has a radius of convergence > r Here fo := f(o) For Iv( < r , 1.14.1 and 1.14.2 imply
.
.
.
.
Hence that
fJ(o+v) converges to
1;
fn(v)
=
f(o+v)
,
showing
SECTION 1
14
a
1
f(z) =
1.15
.
< r
1z-01
for
f
n=O
L;
If
COROLLARY.
is holomorphic on
f
Hence
(2-0)
n
,
R
O.E.D.
i s a c o n v e r g e n t power s e r i e s
fn
between complex Banach s p a c e s convergence
.
D
Z
,
and
W
D :=
{ z
w i t h r a d i u s of
then m
1
f ( z ) :=
fn(z)
n=O d e f i n e s a h o l o m o r p h i c mapping o n
1 zI
O (XI=r
If 5 E C \ g(A) satisfies (s-g(O)l < s h(X) := ( g ( X ) - < )-1 is holomorphic on A satisfies
. ,
the function and, by 1.13,
SECTION 1
16
.
sup (h(h)l < (s-ls-g(0)l)-l =r Hence g(h) contains the It follows that \ < - g ( O ) l > s/2 set { 5 E C : 1 j
OrdO(idB-fohj) = OrdO(hj+'-hj)
.
implies foh = idB Applying the same argument to see that h is also a left-inverse for f
h
.
,
we
O.E.D.
As a first type of examples of analytic mappings, we consider the so-kalled evaluation mappings. 1.24 PROPOSITION. For Ranach spaces the evaluation mapping F : pn(ZIW)
x
Z +
Z
and
W
x R +
K
I
,
W
defined by F(f,z) := f(z) , is analytic for every case K = C , the evaluation mapping F : om(B,W)
over
n
.
In
1.24.1
W
is holomorphic for every open subset
R
of
Z
.
PROOF. By considering the complexifications of Z and W in case K = R I we may assume K = c For any r > 0 , the restriction mapping f + f ( B onto B := { z E Z : I Z I < r 1 Hence it defines a topological embedding ?(Z,W) + om(R,W) suffices to consider the mapping 1.24.1. Since F is linear in the first argument, it is enough to show that F has a power series expansion about ( 0 , o ) , where o E A is arbitrary. Put R := dist(o,aR) and endow V = om(B,W) x Z with the compatible norm ((f,z)l := max(lflB,lz() Then
.
.
.
ANALYTIC MAPPINGS ON BANACH SPACES
21
defines a continuous polynomial Fn+l E pn+'(V,W) with associated symmetric (n+l)-linear mapping n 1 zm I . . . , Fn+l((fo,zo),...,(fnIzn) = 1 (n+l) m ' (")(o)(zo I . . . ,h m= 0 where
A
is an omission symbol. IFn+l(f,z)( < R-nlflglzln
Zn)
By 1.13, we have 6
R-"((f,z)l"+'
.
Hence 1.4 implies that the power series
has radius of convergence
> R > 0
.
O.E.D.
NOTES. The basic theory of analytic mapp ngs on Banach spaces can be found in [108,431. For the proof of Theorem ~ 1 . 2 3 , cf. 31,431. Standard references for the differential calculus on Banach spaces are C28,97,201. Throughout, [271 will be used as reference for classical function theory. For self-contained introductions to the theory of holomorphic functions of several complex variables, see C16,1091. The definition of holomorphic functions via convergent power series ("Weierstrass concept") is equivalent to the "Cauchy-Riemann concept" of complex (Frgchet) differentiability C 1 0 8 ; p.17, Remark 13. For Cnf a well known result of Hartogs asserts that these concepts are also equivalent to partial (or "Gateaux") holomorphy. This is no longer true for Banach spaces unless the function is locally bounded C43; Corollary 11.5.51. Also, the radius of convergence for a holomorphic function on an open subset of a Banach space can be smaller than the boundary distance (cf. L108; p.281 for an example). Holomorphic mappings can be studied on more general topological vector spaces. This theory is closely related to functional analysis and studies the underlying spaces in terms of their holomorphic functions (generalizing linear forms). It is also of interest to endow spaces of holomorphic mappings with "natural" topologies generalizing the compact-open topology C1081.
22
2.
SECTION 2
BANACH ALGEBRAS
The analyt c (Ranach) manifolds of nterest in the following are closely related to algebraic structures describing their analytic and geometric properties in algebraic terms. algebraic structures are given by (not necessarily
These
associative) Ranach algebras and various generalizations. Conversely, Ranach algebras provide important examples of analytic mappings via the so-called functional calculus. In this section, we study basic properties of Ranach algebras. The class of Ranach algebras which arise in connection with symmetric Banach manifolds, namely C*-algebras and their generalizations, are treated in more detail in Sect. 1 5 . In the following, let K denote the field R of real numbers or the field C of complex numbers. An algebra over K is a vector space Z over K endowed with a bilinear mapping often denoted by
Z x
,
Z + Z
(z,w)
+
called the product in 2 and Note that associativity is not
.
zw
An algebra A is called unital if there exists a (uniquely determined) unit element e E 2 satisfying
required.
ez = ze = z for all the unitization
z E
Z
.
For a non-unital algebra
Z
,
of Z is an algebra over K , with unit element e = (0,l) For a real algebra X , the complexification
.
xc of
:=
x
OR
c
is a complex algebra in a natural way.
X
A
Banach
algebra over K is an algebra Z endowed with a Ranach space topology such that the product mapping Z x Z + 2 is continuous. The unitization of a non-unital Ranach algebra over
K
is a unital Ranach algebra [17; 3.11,
The
complexification of a real Ranach algebra is a complex Ranach algebra [ 1 7 ; S131. A linear mapping Q : 2 + W between algebras over
K
is called a homomorphism if
BANACH ALGEBRAS
23
.
$(z1z2) = ($zl)($z ) for all z1,z2 E Z If $ is also bijective, then $ : W + Z is a homomorphism and $ is A called an isomorphism (an automorphism if 2 = W ) . homomorphism $ between unital algebras 2 and W is called A bimodule over an algebra Z is a unital if $(e,) = eW vector space W endowed with bilinear mappings 2 x W + W and W x Z + W , denoted by (z,w) + zw and (w,z) + wz , respectively. For example, if W is an algebra and 2 is a subalgebra of W , then the algebra product makes W into a bimodule over 2 A linear mapping 6 : Z + W from an algebra Z into a himodule W over 2 is called a derivation if
-1
.
.
for all
z1,z2
E
2
.
The set of all derivations from
Z
into W forms a vector space under pointwise addition and scalar multiplication. The following property of derivations
is known as Leibniz' rule. 2.1
PROPOSITION.
algebra
Z
.
Let
6 : Z
+ 2
be a der vat on of an
Then the n-th iterate mapping
6" : z + z
satisfies 2.1.2 PROOF. 2.1.1
Assuming 2 . 1 . 2
for
n > 0
,
the derivation property
implies
O.E.D. A
Banach algebra
Z
over
K
is called associative if
2Q
SECTION 2
(xy)z = x(yz) for all x,y,z compatible norm on 2 , then 1x1
:=
for all
x,y
E
2
.
In case
.
SUP
~f
I IYI 161
II*((
is any
I IXYl I
Z satisfying
is a continuous semi-norm on IXYI
z
E
< 1x1 IYI 2
2.2.1
has a unit element
e
, 1.1
is a compatible norm and
lel < 1
2.2.2
In the associative case we (in fact, (el = 1 if 2 # { O } 1. on Z will therefore always assume that the norm 1 - 1 satisfies 2.2.1 and 2.2.2 (if 2 is unital). These properties remain valid under passing to the unitization [17; 3.11 and complexification [17; 13.31. Important examples of associative Ranach algebras are operator algebras and function algebras. In order to describe them it is sometimes convenient to consider also Ranach spaces (but not Ranach algebras) over the division algebra A of all (real) quaternions. Recall that every quaternion has a unique expression c = a + jb , with a,h E C Let a* E C denote the complex conjugate of a Then A becomes an associative unital real division algebra under the product
.
.
(al+jb )(a2+jb2) = (a1a 2-b*b 1 2 1 + j(bla2+a!b2) 1 The symbol D will always denote one of the division algebras R , C or H The center of D will be denoted by K Thus K = R if D # C The division algebra D carries a canonical involution a + a* , satisfying (a*)* = a and This involution is the (ab)* = b*a* for all a,b E D
.
.
.
.
identity if D = R For D = H , put
and the complex conjugation if
(atjb)* = a*
-
jb
D = C
.
BANACH ALGEBRAS
.
25
€or all a,h E C In all cases, we have R = { a E D : a* = a } For a E D , define the real part
.
Re(a) := (a+a*)/2
E
(a1 := (a*al1/2
R,
R
and the absolute value E
.
Since a*a belongs to the center of D , we have lab1 2 = (ab)*(ab) = b*a*ab = (a*a)(b*b) Hence lab1 = la(*Ib for all a,b E D A right vector space E over E endowed with a comp ete norm * I : E + R, is called a Ranach space over H if
.
.
I
lual = lul-lal
for all
u
E E
and
a
E
H
.
2.3 EXAMPLE. For D E {R,C,H} , the set B(S,D) of all bounded functions z : S + D on a set S is a Ranach space over D with respect to the pointwise algebraic operations and the supremum norm
The "function product" zw)(s) := z(s)w(s) satisfies lzwls < lzlslwls and eS( 1 , where es(s) := 1 for all s E S Hence B ( S , D ) is an associative unital Ranach algebra over the center K of D In case S is a measure space, the set L a ( S , D ) of all (equivalence classes of) essentially bounded measurable Lbvalued functions on S is a In case S is a closed unital subalgebra of B(S,D) topological space, the set C - ( S , D ) of all bounded continuous D-valued functions on S is a closed unital subalgebra of B(S,D) In case S is a locally compact Hausdorff space, the set C , ( S , D ) of all continuous D-valued functions on S vanishing at infinity is a closed subalgebra of C = ( S , D ) which is unital if and only if S is compact. In the special case that S = {l,.,.,n} is finite, we obtain the algebra
.
.
.
.
Dn =
([)
26
SECTION 2
of all n-tuples over D , endowed with the component-wise algebraic operations and the supremum norm
for
2.4 EXAMPLE. Let E,F,L be Ranach spaces over D E {R,C,R} Then the set L ( E , F ) of all continuous D-linear mappings z : E + F is a Ranach space over the
.
center
K
of
with respect to the operator norm
D
:= sup Izxl/lzI =
.
sup 12x1 2.4.1 lxl 0
. .
Since i.e.,
,
.
g
is injective, the polynomial g ( z ) = az for some a E cX
.
3.16 COROLLARY. Every automorphism of collineation, i.e., the homomorphism r isomorphism Aut(P1(C)) PROOF.
-
It suffices to show that
g
has degree
O.E.D.
P1(C) is a
induces an
GL2(C)/G(C) r
1
.
is surjective.
For any
SECTION 3
50
g
there exists y E GL2(C) such that g ( m ) = y#(m) Hence we may assume g ( m ) = m Therefore g(C) = C For every b E C I the translation E
Aut(P1(C))
.
.
.
(i
:)#(z)
=
z+b
leaves m Therefore
fixed. Hence we may further assume g(Cx) = C x Since
for every
a
3.17
E
Cx
COROLLARY.
the assertion follows from 3.15. For the Gauss plane
C
.
g(0) = 0
.
O.E.D.
we have
where the semi-direct product of the multiplicative (homothety) group G ( C ) and the additive (translation) group C is defined by the action a * b = aba-1
.
PROOF.
Let
g
E
Aut(C)
. .
B y considering translations of
we may assume g(0) = 0 Hence assertion follows from 3.15.
g(Cx) = C x
C
and the O.E.D.
In order to describe the automorphisms of the "hyperbolic" simply-connected Riemann surfaces equivalent to the open unit disc, we first consider "circles" on the Riemann sphere. Suppose S = S* E L 2 ( C ) is a self-adjoint matrix with Let Det(S) < 0
.
be a non-zero vector and define V*SV = ( v ~ , v $ ) S ( ~ ~E )R
.
.
a* is the complex conjugate of a E C Let z = Cv E P1(C) be the complex line generated by v
Here
.
Then
51
BANACH MANIFOLDS
the signum Z*SZ
is well-defined.
For
u
su
s g n ( v * ~ v )E {0,1,-1}
:=
,
{0,1,-1}
E
:= { z
E
P1(C)
define : z*sz
= a
}
.
-
Then S o is a circle in P1(C) (i.e., a proper circle in C or an affine real line through ) and every circle in P,(C) arises in this way. The complement P,(C) \ S o is the disjoint union of the open connected subsets S1 and S
-1 Now suppose
g
GL2(C)
E
.
Then
g*Sg
E
L2(C) is also
self-adjoint and satisfies Det(g*Sg) = [Det g12-Det(S) < 0 It is clear that the collineation satisfies
.
r
3.18.1
U
for
u
E
{0,1,-1}
.
For example, th
s
:=
(-;
matrix
0 1) So = a A
represents the unit circle
and
S1 = A
.
Define the
group U!i
1,1
{ g
( C ) :=
GL2(C) : g*Sg = S }
E
.
and consider the 1-torus U(C) := { a E Cx : a*a = 1 } For every g E UL (C) , 3.18.1 implies that the collineation 1,1 r(g) leaves A invariant. 3.18
PROPOSITION.
The homomorphism
r : ULII1(C)
induces an isomorphism Aut(A) PROOF.
For every
w
E
A
,
-
UL
1,1
(C) / U(C)
the transformation
.
+
Aut(A)
SECTION 3
52 gw(z) = (,*1 belongs to
Aut(A)
Y ) # ( Z )
= (z+w)(w*z+l)-1
since
.
Now suppose g E Aut(A) Since gw(0) = w g ( 0 ) = 0 , By Schwarz' Lemma [ 2 7 ; VI.2.11, a E U(C) such that
It follows that the homomorphism kernel
G(C)n Ual,l(C) = U(C)
By 3.18.1,
r
.
,
we may assume there exists
is surjective and has the O.E.D.
the Cayley transformation 1
1
maps the open unit disc
onto the riqht half-plane
Similarly, the Cayley transformation h(z) = (i+zl(l+iz)-l = maps
A
(i1
il)#(z)
3.18.2
onto the upper half-plane
NOTES. The construction of the Grassmann manifold associated with a Banach space can be found in C311. This paper, using Banach analytic spaces (with singularities) for the solution of a deep problem in finite dimensional complex analysis, was quite influential for the development of infinite dimensional holomorphy. Linear fractional transformations were introduced in operator theory by Potapov (cf. [17; p.209, Remark]).
ANALYTIC VECTOR FIELDS
4.
53
ANALYTIC VECTOR FIELDS
Tangent vectors and vector fields are basic algebraic objects associated with a Banach manifold. Of particular importance is the commutator product for analytic vector fields, giving rise to a Lie algebra structure. In later sections, many "deep" geometric and analytic properties of certain Ranach manifolds will be obtained by studying Lie algebras of analytic vector fields. As in the finite-dimensional case, tangent vectors for Banach manifolds can be defined using the concept of "germ" of an analytic mapping. For a given Ranach manifold M over K and any Ranach space A over K , consider analytic mappings f : Q +. A defined on an open neighborhood Q of m E M (depending on f 1. Two such mappings f l and f 2 are called equivalent at m E M if f l and f 2 coincide on an open neighborhood of m An equivalence class defined by
.
this relation is called a germ of analytic A-valued mappings A at m The set om of all these germs is a vector space A over K in a natural way. Let fm E om denote the germ of f at m and define the linear evaluation mapping A := f(m) In case A is a (unital) r : om + A by r (f,) m is a (unital) algebra and rm is a Banach algebra, m:o
.
.
(unital) homomorphism. This applies in particular to Now let (P,p,Z) be a chart of M For m E P A f E om define
A = K
.
and
Note that fop-' is well-defined and analytic in an open neighborhood of p(m) E Z and that the derivative (fop 1 (pm) depends only on the germ of f at m Any vector h E Z defines a linear mapping
.
a af := -(m)h aP (hap)f
.
.
SECTION 4
54
If (Q,q,W) is another chart of rule implies
M
and
m
E
PnQ
,
the chain
where
is an isomorphism.
It follows that the vector space :=
T,(M)
{ h-a aP
: h
E
Z }
of linear mappings from 0; + A (for any Ranach space A ) is independent of the chart (P,p,Z) of M Endowed with its canonical topology, the Ranach space T,(M) (isomorphic to 2 ) is called the tangent space of M at m , and its elements are called tangent vectors of M at m In case A is a Ranach algebra, the product rule [ 2 8 ; 8.1.41 implies
.
.
.
A for all f , g E Om It follows that tangent vectors are A-valued derivations of the algebra 0; , with respect to the A O,-bimodule structure on A induced by rm Unlike the finite-dimensional case, tangent vectors can in general not be K characterized as the derivations of om , since not every K derivation of 0, corresponds to a tangent vector. In the special case that 2 = Kn is finite-dimensional and
.
defines a chart T,(M) , for m
of M , the tangent vectors in have the form
(P,p,Z) E
P
,
n
a
j=1 where h E K and a/apj denotes the partial derivative with 1 respect to the j-th "coordinate" Pj As in the finite-dimensional case, tangent vectors can also be defined as directional derivatives along smooth
ANALYTIC VECTOR FIELDS
curves.
c : I
is a smooth curve defined on an Then containing 0 and put m := c(0)
Suppose
open interval
55
I
+
M
.
6cf := - f(c(t))t=O dt * OA + A which is a derivation &c * m is a Ranach algebra. For any chart (P,p,Z) of M
defines a linear mapping if A and m
E
M
,
we have d c = (poc ' ( 0 ) - a
aP
where
h
M
(poc)'(O)
Z , with E
L(K,Z) = Z -1 c(t) := p (p(m)+th c(0) = in such that E
.
Tm(M)
E
,
Conversely, for every defines an analytic curve in
6c = h-a ap Now suppose g : M + N is an analytic mapping between Banach manifolds. For any m E M , there exists a linear mapp i ng
.
(fog)m for all f E O A If A is a gm (unital) Ranach algebra, g i is a (unital) homomorphism. For
defined by
v E Tm(M) putting
g;(f)
,
:=
define a linear mapping
Tm(g)v(f) for all
f
E
O gm A
.
If
along the smooth curve
v
+
by
A
v(gif) = v((fog)m)
:=
4.0.1
is the directional derivative
= 6
C
c : I
Tm(g)v : O t m
+
M
satisfying
,
c(0) = m
then
is the directional derivative along the Tm(g)6c = 6 goc smooth curve goc : I + N It follows that The linear mapping Tm(g)Tm(M) C Tgm(N)
.
.
is called the differential of another analytic mapping, then
g
at
m
.
If
f : N
+
P
is
SECTION 4
56
If M is an open subset of a Ranach manifold N and i : M + N is the inclusion mapping, then the differential Tm(i) : Tm(M) + T,(N) is an isomorphism. We will always identify Tm(M) and Tm(N) by means of this isomorphism. For any Banach space 2 with "coordinate" z , the tangent vectors at z E Z have the form
v = h-aaz
4.0.3
'
.
An associated smooth curve is given by where h E Z In the following, TZ(2) will he identified c(t) = z+th with Z via 4.0.3. Using these identifications, the
.
differential Tm(p) : Tm(M) (P,p,Z) of M and a point isomorphism defined by
+ Z
m
E
associated with a chart P is the canonical
.
for all h E 2 For any analytic mapping g : M + N between Ranach manifolds, the differential Tm(g) at m E M has the following local representation: consider charts (P,p,Z) of M and (Q,q,W) of N exists a commuting diagram
g(P) C Q
such that
P
L
P 1
4.0.4
r
q(Q) g#
where
is the loca
g#
Then there
Q ./4
p(P)-
.
representation of
g
.
Then
Tm for all
m
E
P
and
h
E
Z
,
where 4.0.5
It follows that Tm(g) is continuous. Taking differentials in 4.0.4 and using the identifications mentioned above, we get a commuting diagram
ANALYTIC VECTOR FIELDS
57
- 3. $Jg)
Tm(M)
1z
Tm(P)
T (N) gm Tgm(q)
-w
gpPm) In the special case that D C Z and B C W are open subsets of Banach spaces, the differential of an analytic mapping g : D
+
B
at
z
D
E
a
Tz(g)(hz)
has the form
a
= g'(z)hG
and can therefore be identified with the derivative g'(z) E L ( 2 , W ) dimensional and
.
If
2 =
Kn
defines a chart (P,p,Z) of an analytic mapping 4 =
W
and
M
,
=
Kk
are finite-
the differential
Tm(g)
of
[?I k'
on M at m E P Jacobi matrix
is the linear mapping corresponding to the
The following inverse mapping theorem follows directly from 1.23. 4.1 THEOREM. Let g : M + N be an analytic mapping between Banach manifolds such that To(g) : To(M) + Tg(o)( N ) is an isomorphism for some point o E M Then there exist open neighborhoods P of o E N and Q of g(o) E N such that
.
g : P + Q is bianalytic. For any Banach manifold
M
,
the disjoint union
58
SECTION 4
T(M) :=
U [m}
x T,(M) mEM is a bundle of Ranach spaces, with canonical projection T~ : T(M) + M defined by ~ ~ ( m , v:=) m , called the tangent
.
bundle of M Any analytic mapping g : M + N Banach manifolds induces a commutative diagram
between
T(M) T(g)h T(N)
1
TM
1
M-N where the differential
T(g)
of
‘IN
I
4
g
is defined b y
If P C M is an open subset, then T(P) can be identified with a subset of T(M) via the differential T(i) : T(P) + T(M) of the inclusion mapping i : P + M F o r any open subset D C 2 of a Ranach space 2 , the tangent bundle T(D) can be identified with 0 x 2 via the mapping
.
a
T(D) 3 ( z , h x )
+
(z,h)
E
DxZ
.
4.2.1
Using this identification, the differential T(p) : T(P) + p(P)xZ can be written as
for all m of M and
P
E
m
E
and PnQ
h
,
.
E 2 If (Q,q,W) is another chart the chain rule implies that the
transition mapping
has the form
for all m E PnQ and h E 2 and is therefore analytic. Hence T(M) carries a unique topology such that T~ is continuous and the collection { (T(P),T(p),ZxZ) } , for all charts (P,p,Z) of M , is an analytic atlas on T(M) Since this topology is Hausdorff, T(M) is a Banach manifold
.
ANALYTIC VECTOR FIELDS
.
over K analytic.
59
The canonical projection 'I : T(M) M Further, for every analytic mapping
+
M
is
g :
M +
N
between Banach manifolds, 4.0.5 implies that the differential T(g) : T(M) + T(N) is analytic and, by 4.0.2, we have
whenever
g : M + N and f : N + P are analytic mappings between Banach manifolds. Let M be a Banach manifold over K E {R,C} with tangent bundle -cM : T(M) + M An analytic vector field on
.
M
is an analytic mapping T ~ O X= id , i.e.,
X : M
Xm := X(m)
+
T(M)
satisfying
Tm(M)
E
.
The vector space of a 1 analytic vector m E M for a1 fields on M , under pointwise addition and scalar multiplication, will be denoted by T(M) Vector fields can be regarded as differential
.
operators: M and let space W
.
Let Q be an open subset of a Ranach manifold f : Q + W be an analytic mapping into a Banach Given X E T(M) , define a mapping Xf : Q -+ W by 4.3.1
for m E Q , where Then the diagram
fm
E
0;
denotes the germ of
f
at
m
.
commutes, where pr2 denotes the projection from T(W) = WxW onto the second factor (cf. 4.2.1). Hence Xf is analytic. The local representation of tangent vectors in terms of a chart (P,p,Z) of M carries over to analytic vector fields: If X E T ( M ) and m E P , we have Xm = h(m)-a
aP
where
E
T,(M)
,
SECTION 4
60
Hence the vector field as
X
,
restricted to
P
,
can be written
X = h-a ap where h = Xp : P + Z is analytic. For any analytic mapping f : P + W , the analyti mapping Xf : P + W is given by m) = -(m)h(m) af Xf(m) = (h5)f a
aP In the special case that
2 =
Kn
defines a chart (P,p,Z) X on P has the form
of
M
where
hj = Xpj : P
+
.
4.3.2
is finite-dimensional and
,
an analytic vector field
are analytic functions.
K
PROPOSITION. Suppose M is a Banach manifold and X,Y E T ( M ) Then there exists a unique analytic vector field [X,Y] on M , called the commutator of X and Y such that 4.3
.
[X,Ylf = Y(Xf )-X(Yf 1
4.3.3
for all analytic mappings f : Q + W from open subsets of M into a Banach space W The vector space T ( M ) becomes a Lie algebra under the commutator product.
.
PROOF.
Given
m
E M
,
define a linear mapping
,
Q
6m
by 6,fm
:= Ym(Xf Im-Xm(Yf )m
whenever f : Q + W is analytic on an open neighborhood of m E M Let (P,p,Z) be a chart of M and put h := Xp and k := Yp Then 6m is given by
.
.
Q
ANALYTIC
VECTOR FIELDS
61
for all m E P , since the second derivative (fop-1 )"(pm) E L 2 (2,W) is a symmetric bilinear mapping [28; 8.12.21. It follows that g m E Tm(M) for all m E M and that M 3 m
+
[X,Y], := g m
E
T(M)
defines an analytic vector field on M satisfying 4.3.3. The Jacobi identity 2.14.1 for T ( M ) follows from 4.3.3. O.E.D. 4.4 LEMMA. Suppose g : M -t N is an analytic mapping between Ranach manifolds, and let Xk E T(M) and Yk E T ( N ) be vector fields such that T(g)OX k = Yk og for k = 1,2 Let s,t E K Then
.
.
T(g)o(sX 1+tX 2 ) = (sY1+tY 2 ) o g and
PROOF. Since Tm(g) is linear for every m E M , the first assertion is trivial. Now let f : 0 + W be an analytic mapping defined on an open subset Q of N Then
.
(Yk fog)(m) = (ykf)
gm
k = (Tm(g)Xm)f
for all
m
E
P := g -l(Q)
1 2 Tm(g)[X ,X l m f g m
- 2 1 - Xm(X (fog))m 2
1
gm
=
-
= (Y (Y f)og)(m)
=
Yk f
gm gm
k = Xm(fog)m
.
=
k X (fog)(m)
Therefore 4.3.3
implies
1 2 [X ,X lm(fog)m
1 2 2 1 Xm(X (fog)), = Xm(Y fog)m
-
1
2
1
2
-
1 X,(Y
( Y (Y f)og)(m) = [Y ,Y Igmfgm
2
fog)m
.
O.E.D.
4.5 COROLLARY. Every bianalytic mapping g : M + N induces a Lie algebra isomorphism g , : T(M) -t T(N) such that the diagram
62
SECTION 4
4.5.1
commutes for every whenever
g
and
X
h
E
T(M)
.
Further,
g*oh, = (goh),
are composable bianalytic mappings.
is PROOF. Since g is bianalytic, the vector field g,X well-defined by 4.5.1 and is analytic. By 4.4, g, is a Lie algebra homomorphism. Since T(g0h) = T(g)oT(h) , it follows -1 Applying this to h := g , it that (goh), = g*oh,
.
follows that If
g*
is an isomorphism.
(P,p,Z) and
for all 4.6
m
E
P
EXAMPLE.
space Z over have the form
are charts of
(Q,q,W)
respectively, such that
O.E.D.
g(P) C Q
,
M
K
.
D
,
N
we have
and every analytic function Suppose
and
h : P
+
.
Z
is an open subset of a Ranach
Then the analytic vector fields on
D
where h = X(idD) : D + Z is analytic. Here z denotes the "coordinate" (canonical chart) of D and we sometimes write h(z) for the mapping h For any analytic mapping f : D + W , 4.3.2 implies
.
4.6.1 where f'(z) E L(Z,W) is the derivative of f The commutator product in T(D) is given by
at
z
E
D ,
4.6.2 g : D + B is a bianalytic mapping onto an open subset B of a Ranach space W , the isomorphism g, : T(D) + T ( R ) is given by If
ANALYTIC VECTOR FIELDS
-
a
9 * h ( z ) E ) = g'(g
'w)h(g
-
63
a
'w)- aw
4.7 EXAMPLE Let $2 : = { s E C : Re(s) > 0 } denote the right half-plane and define an := { re it E C : r > 0 , It1 < a / 2 n } whenever n is a positive integer. Then y ( s ) := sn defines a hiholomorphic
.
NOW let z he an associative unital mapping y : $2" + ii complex Banach algebra and define domains Q z and $2; in
Z
as in 2.10.
implies that mapping g :
for m imply
E
2
Then the holomorphic functional calculus g(z)
Qi
:=
+ Qz
defines a biholomorphic The vector fields
y z ( z ) = z"
.
are analytic on
and
51;
QZ
,
and 4.6.1
and 1.7.1
Many important Lie algebras of vector fields admit natural gradations with respect to the commutator product. 4.8 S
DEFINITION. Suppose g is a Lie algebra over is a subset of C A direct sum decomposition
K
.
and
4.8.1
is called an additive gradation of g all m,n E s (putting g n := { O } if suppose
S C Cx
.
n
c g,+,
[g,,g,]
if E
c
\
s
.
for
NOW
A direct sum decomposition
g =
@"g
4.8.2
nEs
is called a multiplicative gradation of g if n fmg,"g] c mn g for all m,n e s (putting g := n E
4.9
{o}
if
c x \ s 1. EXAMPLE.
Let
6
be a derivation of
g
splits into a direct sum of eigenspaces
g,
:=
{
x
E
g: 6X
= nX
}
such that
g
SECTION 4
64
for n belonging to the "spectrum" 4.8.1 is an additive gradation of g
whenever X E g m and gradation 4.8.1 of g 6 of g defined by
S
C K of since
Then
.
Y E gn Conversely, every additive with S C R determines a derivation 619, := noid
.
4.10 EXAMPLE. Let $ be an automorphism of g splits into a direct sum of eigenspaces ng := { X
.
6
g
such that
E : ~$ x = n x ]
for n belonging to the "spectrum" S C K x of 4.8.2 is a multiplicative gradation of g since
$
.
Then
.
whenever X E "g and Y E ng Conversely, every multiplicative gradation 4.8.2 of g with S C K x determines an automorphism 4 of g defined by $Ing := n-id
.
4.11
EXAMPLE.
any integer
Suppose
n > -1
2
is a Banach space over
.
K
For
let Tn(Z) = { h ( Z )aE : h
E
Pn+l ( 2 , Z )
}
denote the vector space of all polynomial vector fields on 2 which are homogeneous of degree n+l Then
.
a
T-l(Z) = { b z : b
2
E
]
consists of all constant vector fields,
a
To(Z) = { a ( z ) z : a
E
L(2)
}
consists of all linear vector fields and
consists of all quadratic vector fields. For X = h- a E T,(Z) and f E Pn(2,2) , 1.7.1 implies az
ANALYTIC VECTOR FIELDS
Xf(z) = n f-(h(z),z,
65
...,z )
,
4.11.1
where f - E Ln(2,2) denotes the symmetric n-linear mapping associated with f by 1.1.2. It follows that +n+l Xf E 6" (Z,Z) By 4.5.2, this implies that the algebraic direct sum
.
P ( 2 ) :=
Jn(Z)
@
n >-1 of all polynomial vector fields on 2 is a subalgebra of T(Z) endowed with an additive gradation. The associated derivation 6 of P ( Z ) is given by 6X =
a
[X,Z--l
I
az
since J,(z)
{x
=
P ( Z ) : [ X , aZ ~ ]=
E
nX }
4.11.2
.
for all n > -1 In particular, the space J o ( Z ) of all As a linear vector fields on 2 is a subalgebra of P ( 2 ) Banach Lie algebra over K , T o ( Z ) is isomorphic to g a ( Z ) via the mapping ga(z)
3 a
.
+
a
az-a z
E
To(Z)
.
We will often identify g E ( Z ) and T o ( Z ) , regarding (continuous) linear operators on as linear vector fields. In particular, the linear vector field I := z-a
az
corresponds to the identity operator
idZ
.
4.12 PROPOSITION. Consider the Ranach algebra R associated with a Banach space Z over K via 3.5.1 and let g be a subspace of R which is closed under the commutator product [X,Y] := XY-YX and satisfies the associativity condition 3.6.1. Then g is a Lie algebra and
a (c
d)#
:=
a
(az+b-zcz-zd)-a z
SECTION 4
66
d e f i n e s a L i e a l g e b r a homomorphism
4.13
EXAMPLE.
L
Let
b e a Ranach s p a c e o v e r
a n d c o n s i d e r a s p l i t t i n g 3.8.1
of
a Ranach s p a c e o v e r t h e c e n t e r Ranach a l g e b r a
R
K
associated with
closed suhalgehra.
By 4 . 1 2 ,
.
L
of Z
Then D
D Z
{R,C,H}
E
:=
L(E,F)
a n d , by 3 . 1 1 ,
contains
L(L)
is
the as a
it follows t h a t
b d ] # := ( a z + b - z c z - z d 1-aaz
4.13.1
d e f i n e s a L i e a l g e b r a homomorphism
NOTES.
The d e f i n i t i o n 4 . 3 . 3
of t h e c o m m u t a t o r p r o d u c t of
a n a l y t i c vector f i e l d s i s c o n v e n i e n t t o i d e n t i f y l i n e a r v e c t o r f i e l d s a n d ( l i n e a r ) o p e r a t o r s . Most a u t h o r s [ 6 2 , 9 7 1 u s e a n o t h e r d e f i n i t i o n d i f f e r i n g from o u r s by s i g n . For t h e t h e o r y o f t a n g e n t v e c t o r s a n d vector f i e l d s on d i f f e r e n t i a b l e Banach manifolds,
see C 2 0 , 9 7 1
.
The c a l c u l u s o f p o l y n o m i a l v e c t o r f i e l d s w i l l b e o n e o f t h e basic technical tools for t h e study of manifolds.
( s y m m e t r i c ) Banach
I n t h e f i n i t e dimensional s e t t i n g , t h i s approach
w a s i n t r o d u c e d b y M.
K o e c h e r [ 9 5 1 . I t w i l l be shown i n P a r t I1
t h a t f o r s e v e r a l i m p o r t a n t c l a s s e s o f c o m p l e x Banach m a n i f o l d s
(e.g., c i r c u l a r bounded domains, S i e g e 1 domains and symmetric m a n i f o l d s ) e v e r y a n a l y t i c vector f i e l d ( g e n e r a t i n g a 1-parameter group o f automorphisms) is a c t u a l l y a polynomial vector f i e l d (of degree $2) with respect t o a s u i t a b l e "canonical" c h a r t and i s t h e r e f o r e "algebraic" i n character. T h i s c o r r e s p o n d s t o t h e w e l l known f a c t t h a t a n a l y t i c a u t o morphisms o f complex m a n i f o l d s are o f t e n g i v e n by algebraic e x p r e s s i o n s ( c f . Chow's Lemma f o r p r o j e c t i v e v a r i e t i e s 8.31).
C103;
For a more p r e c i s e s t a t e m e n t i n t h e i n f i n i t e d i m e n s i o -
n a l c a s e , see C82,831 a n d S e c t i o n 2 3 .
67
INTEGRATION OF VECTOR FIELDS
5.
INTEGRATION OF VECTOR FIELDS
Analytic vector fields on Ranach manifolds can be regarded as a global form of (analytic) differential equations. example, every analytic vector field
For
on an open subset of a Ranach space gives rise to the timeindependent ordinary differential equation
-dz_ dt
with analytic coefficients.
h(z)
The solutions of these
differential equations are called analytic flows.
As
a
principal example, we are interested in analytic flows which can be expressed in terms of linear fractional transformations. Let M be an open subset of a Ranach manifold N over K and consider an analytic mapping r : il + N defined on an open subset
c
il
RxM
such that for every
m
E
M the following
condition holds:
{ t
ilm :=
R : (t,m) interval containing 0 Then
E
E
n }
,
and
is an open r(0,m) = m
is called a local analytic flow on
r
M
5.0.1
.
if
whenever t E ilm , r(t,m) E M and s E R r(t,m)* The analytic mapping rm : ilm + N , defined by
(‘met)
rm(t) := r(t,m) , is called the evaluation mapping at m E M By 4 . 0 . 3 , we identify T o ( B m ) with R and define a
.
vector field
X
on
M
by putting
Xm := To(rm)
for all
m
E
M
.
Then
X
E
T(M)
E
Tm(M) since the diagram
68
SECTION 5
.
commutes, where $(m) := (0,m) The analytic vector field X on M is called the infinitesimal generator (or If differential) of the local analytic flow r on M f : Q + W is a Ranach space valued analytic mapping defined on an open subset Q C M , we have
.
5.0.3
.
for all m E Q In particular, for any chart M , the infinitesimal generator is given by
(P,p,Z) of
5.0.4
.
for all m E P In case D i s ’ a n open subset of a Banach space 2 , the infinitesimal generator
of a local analytic flow
r : 51
+
Z
on
D
is defined by
It will be shown in the following that a local analytic flow is essentially uniquely determined by its infinitesimal generator X and can, in fact, he expressed in terms of an exponential series in X Recall (4.3.1) that every analytic vector field X E T ( M ) on a Ranach manifold M can be regarded as a differential operator, acting on Ranach space valued analytic mappings f on (open subsets of) M For n E N , define the n-th power Xn of X (as a differential operator) inductively, by putting X 0 f := f and
.
.
.
for all f Note that, in general, field if n # 1 5.1
THEOREM.
.
Suppose
M
Xn
is not a Vector
is an open subset of a Banach
INTEGRATION OF VECTOR FIELDS
69
manifold N and let r : il + N be an analytic mapping defined on an open subset Q of R x M satisfying 5.0.1. Then the following conditions are equivalent: r is a local analytic flow on generator X E T(M) For all
(t,m)
with
il
E
with infinitesimal
M
.
r(t,m)
M
E
,
we have
Tt(rm) = Xr(t,m) F o r every chart
a at whenever
(P,p,Z)
p(r(t,m))
(t,m)
=
and
R
E
of
,
M
we have
h(p(r(t,m))) r(t,m)
P
E
.
Here
.
a
p*X = h(~)az
F o r every Ranach space valued analytic mapping
f : Q + W
on an open subset
a f(r(t,m)) at whenever
(t,m)
and
D
E
=
Q
of
M
,
we have
(Xf)(r(t,m)) r(t,m)
E
0
.
In this case there is a convergent power series 5.1.1
f (r if
m
E
Q
and
is small.
(tl
PROOF. It is clear tha the conditions (ii) , i i i ) and iv) are equivalent. Now assume (i), and let (t,m E D sat S fY if n := r(t,m) E M Then 5.0.2 implies r(s,n) = r(s+t,m Is( is small. It follows that
.
r'
(t
Now assume (ii), and let m := r ( s , o ) E M Then
.
,rn)
= To(rn)
Tt (rm)
Satisfy I := D m n (Q,-s) is an open (s,o)
E Q .
interval containing 0 , and the analytic mappings $ : I + N , defined by
SECTION 5
70
5.1.2
$(t) := r(t,m) and $(t) := r(t+s,o) respectively, satisfy It1
is small.
$(O)
=
m
M
E
,
5.1.3
and
Tt($) = X4(t)
if
By induction, it follows that
for all n E N and all analytic mappings f : Q + W defined on an open neighborhood Q of m E M I n particular,
.
.
a" f(g(t))t=O
= (xnf)(m) 5.1.4 atn By 3.1, the analytic mappings 5.1.2 and 5.1.3 coincide on I Hence r is a local analytic flow. Further, if It1 is small, we have t E , r(t,m) E Q and there is a convergent power series
.
m
1 3 tn a n=O Applying 5.1.4,
f(r(t,m)) = with coefficients
5.2
space
an
E
w
.
it follows that
Suppose €3 c C are open subsets of a Ranach and let r : I x R + C be an analytic mapping, for
COROLLARY.
Z
.
some open interval I containing 0 Then r is a local analytic flow on B with infinitesimal generator
if and only if
r
satisfies the differential equation
a r(t,z) at
= h(r(t,z))
5.2.1
.
for all (t,z) E I x B , with initial condition r(0,z) = z In this case, the derivatives g;(z) E L(Z) of the analytic mappings gt(z) := r(t,z) on R satisfy the differential
71
INTEGRATION OF VECTOR F I E L D S
equation
for all (t,z) E I x R , with initial condition Further, for every z E B the power series r(t,z) =
tn 1 n!
g;)(z) = idz
.
(Xnid)(z)
n =O
converges if PROOF.
It1
is small.
Differentiating 5.2.1
a g'(z) at t -_ - a
=
and using [28; 8.12.21, we get
a a r(t,z) at az
=
aaz aat r(t,z)
h(r(t,z)) = h'(r(t,z)) g;(z)
az
.
O.E.D.
5.3 COROLLARY. Suppose rl : ill + N and r 2 : i12 + N are local analytic flows on M C N having the same infinitesimal 1 2 generator X Then rl and r2 coincide on il := il n il , and the restriction is a local analytic flow on M with infinitesimal generator X
.
.
PROOF. For every m E M , Qm = il m1 n ili is an open interval 1 2 containing 0 Since 5.1.1 implies that r (t,m) = r (t,m) if It\ is small, the assertion follows from 3.1.
.
O.E.D.
COROLLARY. Suppose rj : ilj + N is a family of local analytic flows on M C N having a common infinitesimal generator X Then there exists a local analytic flow r : il + N on M , defined on il i l j I such that 5.4
.
r J Q J = rJ
for all
j
.
:=u. 3
= V. ilj is an open interval PROOF. For every m E M m' 3 m containing 0 By 5.3, ri and rj coincide on Hence there exists an analytic mapping r : il + N Qinilj such that rlnj = rj for all j If (t,m) E il and r(t,m) E M , we have r(s,r(t,m)) = r(s+t,m) if I s ( is small. Since nrr(t,m) A (am+) is an open interval containing 0 I the assertion follows from 3 . 1 . O.E.D.
.
.
.
SECTION 5
72
Since local analytic flows can be characterized by an ordinary differential equation, the existence of a flow with a given infinitesimal generator could he deduced from general existence theorems for solutions of ordinary differential equations with analytic coefficients. However, since the infinite-dimensional case is not easily accessible in the textbook literature, we prefer to prove the existence theorem for flows by a more direct method using exponential series. For a subset R of a Ranach space Z and s > 0 , the s-neighborhood of B ,
is an open subset of
Z
[28; 3.4.21
containing
R
.
We have
and
whenever s,t > 0 convex. For every
.
IT,:= { t
If
R is convex, 0 , define
T
>
E
R : It1
0 For 1 < k < n and hk E Om(C,Z) , consider the vector fields
.
T(C)
.
is a complex Ranach space and
f
Xk := hk(z) Suppose
W
a
E
.
E
For 1 < k < n PROOF. Put R o := A and d := R/n open subsets A k := Ukd(B) of C satisfy dist(Bk-lraRk) > d Hence 1.13 implies for all
.
g
O,(Ak?W)
.
Om(C,W)
,
Then
the
INTEGRATION O F VECTOR F I E L D S
An induction argument, applied to
73
g = Xk+l... Xnf
,
completes
the proof. 5.6 COROLLARY. Banach space Z R := dist(B,aC) neighborhood T
0 . E. D.
Let R C C be open subsets of a complex such that C is bounded and > O Then there exists an open of 0 E O,(C,Z) such that
.
defines holomorphic mappings F : TxB+C.
F : T
+
Om(B,Z)
and
PROOF. By 5.5, the n-homogeneous polynomial Fn : Om(C,Z) + (),(BIZ) , defined by
,
Fn(h) := (h&)"id
is continuous with norm IFn I < (n/R)"I id1 1.5.5 imply that the power series
.
Hence 1.4 and
m
has a radius of convergence > R/e and thus defines a holomorphic mapping F : T + O,(B,Z) By 1.24, the corresponding evaluation mapping F : TxR + Z is holomorphic. Since F(0) = idB , we may assume F(Txi3) C C
.
.
Q.E.D.
5.7 COROLLARY. Banach space Z
Let B C C be open subsets of a complex such that C is bounded and Then for every h 6 Om(C,Z) there exists
.
dist(R,aC) > 0 T > 0 such that
de f nes an analytic mapping
r : I~
+
O m ( ~ , Z ) satisfy
SECTION 5
I4
and inducing a local analytic flow infinitesimal generator X := h(z)=a
r : ITxR
t
C
with
.
.
Put fn := X"id Then fl = h and By induction on n+l (z) = fA(z)h(z) for all z E C n > 1 , it follows that
PROOF.
.
m k >1
m
1f..
.+m
k =n
k
Here h(k)(z) E L ( 2 , Z ) denotes the k-th derivative of z E A It follows that
.
h
at
m
h(r(t,z))
=
h(z) +
2'
h(k)(z)(r(t,z)-z)/k!
k=l
. ...
1+. .+mk t = h(z) + k=l 1 -1k! 1ml>l ml! mk! h(k)(z)(fm (z), 1 m
. ..,f m k (z))
mk >1
Now the assertion follows from 5.1.
O.E.D.
An analytic flow on a Banach manifold M is a local M = N The set of all analytic flow r : fi + M , i.e., analytic flows on M is (partially) ordered by defining (rl,nl) < (r2,Q2) if and only if Q1 C Q 2 and The maximal elements with respect to this r1 = r 2 ( a 1
.
.
.
ordering are called maximal analytic flows on M By 5.4, a maximal analytic flow is uniquely determined by its infinitesimal generator. We will now show that, conversely, every analytic vector field X is the infinitesimal generator of a maximal analytic flow obtained by "integration" of X We first show that every local analytic flow can be modified to yield an analytic flow.
.
INTEGRATION OF VECTOR F I E L D S
5.8
LEMMA.
Suppose
M
75
is a topological space and
is a
T
locally compact, locally path-connected space. Let R be an open neighborhood of { o } x M in T x M , for some point
.
o E T For any m E M r let denote the connected component of the open set am := { t E T : (t,m) E a ] Then containing o
.
is an open subset of
T x M
. .
PROOF. Suppose (t,m) E ao , i.e., t E ao since T is m locally connected, f i x is open. Hence there exists a pathconnected compact neighborhood S of t E T such that S C We may assume that o E S (otherwise take the joining S and o ) . Since union of S with a path in :R is open and S is compact, there exists a finite open
.
covering {TLr...rTj} of S such that TixNi C a for suitable neighborhoods Ni of m E M and 1 6 i < j N = T N i is a neighborhood of m E M satisfying Since S is connected and contains o , it SxN C Q
.
.
follows that
SxN C Qo
.
Then
0.E. D.
N on an open subset M of a Ranach manifold N can be modified to yield an analytic flow on M with the same infinitesimal generator. 5.9
COROLLARY.
PROOF.
Since
M C N
Then
rlQo
+
is open,
is an open neighborhood of as in 5.8.
r : A
Every local analytic flow
{O}
x M
in
R
x
M
is an analytic flow on
. M
Define
.
no O.E.D.
LEMMA. Suppose M is a Ranach manifold over K and X E T(M) Let (P,p,Z) be a chart of M about o and let C be a domain in
5.10
.
76
SECTION 5
containing
.
0
Let
h
O(C,Z
E
C
satisfy
)
5.10.1
hop = Xp on
p-l(C)
.
Then the vector field
a X # := h az
E
T(C)
satisfies (X:flop
5.10.2
= X"(f0p)
on p-l(C) , whenever n E N and f : C mapping into a complex Banach space W
.
PROOF.
Let
m
E
p-l(c
+
and assume 5.10.2
T (Xif)Xp(m) = T (X:f)Tm(p)Xm Pm Pm n n+l (fop)(m) = Tm(X (fop))Xm = X
=
W
is a holomorphic
for some
n
.
Then
= Tm(X#fop)Xm n
.
O.E.D.
5.11 THEOREM. Every analytic vector field X on a Ranach manifold M over K is the infinitesimal generator of a unique maximal analytic flow r on M
.
PROOF. For any o E M , there exist a chart (P,p,Z) of about o , a bounded domain C C Zc containing 0 and a mapping h E Om(C,ZC) such that Xp = hop on p-l(C) Choose an open neighborhood R of 0 E C with dist(R,aC) > 0 By 5.7, there exists -c > 0 such that
.
.
defines a local analytic flow infinitesimal generator
By 5.10, we have
r#
:
I xB 'I
+ C
with
M
INTEGRATION OF VECTOR FIELDS
if
t
E
IT
and
m
E
p-'(J3)
C p(P)
r#(ITxp(p-'B)) diagram
.
.
77
Hence we may assume
It follows that the commuting
defines a local analytic flow on pq1(R) with infinitesimal generator X The preceding arguments show that there exists an open covering {Qj} of M such that for each j there is a local analytic flow rj : Q j + M on O j with infinitesimal
. .
Ry 5.3 and 5.4, ri and ilin.Qj and the analytic mapping r : s2 Q Q j by r ( Q j := rj for all j , on M with infinitesimal generator X the maximal flow, apply 5.4 to the family flows on M with infinitesimal generator generator
X
:=q
.
rj +
coincide on M , defined on
is an analytic flow In order to obtain of all analytic X O.E.D.
.
The fundamental algebraic operations on vector fields, namely addition and the commutator product, can be derived from the corresponding local analytic flows. 5.12 PROPOSITION. Suppose D is an open subset of a Ranach space Z and let rX : fix + 2 be a local analytic flow on D with infinitesimal generator X E T ( D ) Then we have for all X,Y E T ( D ) and every z E D :
.
(X+Y)id(z) = lim (rx(t,ry(t,z))-z)/t t+O
5.12.1
and
.
2 5.12.2 [X,Y]id( z ) = lim (rx(t ,ry( t ,rX(-t ,ry(-t , z ) 1 ) 1-2 )/t t+O PROOF. For any z E D , there exist an open neighborhood R of z E D and 'I > 0 such that IrxB C Q x nQy For ( t , z ) E ITxR , define +,(z) := rx(t,z) and $t(z) := ry(t,z) In order to prove 5.12.1, put
.
.
SECTION 5
78
wt
:= $ , ( z )
and
zt := @ t ( w t )
c h a i n r u l e a n d 5.2.1
if
It1
is s m a l l .
Then t h e
imply 5.12.3
where
h X := X ( i d )
.
For
t = 0
,
dz t ( 0 ) = hX ( z )
dt
it follows t h a t
+
hy(z)
.
a n d a n o t h e r a p p l i c a t i o n of t h e c h a i n rule i m p l i e s
For
t = 0
Using 5.12.3
,
we g e t
and t h e p r o d u c t a n d c h a i n r u l e s , w e get
79
INTEGRATION OF VECTOR FIELDS
5.12.5
= 2(hi(~)h,(z)-h;(~)h,(~))
=
O.E.D.
2h [x,Yl(z)
5.13 COROLLARY. Suppose M is a Ranach manifold and let rx : ax + M be the maximal analytic flow on M with Let (P,p,Z) be a chart infinitesimal generator X E T ( M ) of M about o Then
.
.
(x+y)p(o) = lim p(rx(t,ry(t,o)))/t t+O and [X,YIP(O) = lim p(rx(t,ry(t,rx(-t,ry(-t,o)))))/tL t+O 5.14 PROPOSITION. Suppose g : M + N is a bianalytic mapping between Banach manifolds. Let g , : T ( M ) T(N) be the associated Lie algebra isomorphism. Consider the maximal analytic flows (Bx,rx) on M and (Qy,ry) on N , -f
generated by X E T ( M ) and Y := g,X E T ( N ) , respectively. Then Q Y = { (t,gm) : (t,m) E $2X } and there is a commuting
.
SECTION 5
80
diagram Lx
idxgJ QX
.M s.9
Q N -
rY We can now introduce the class of analytic vector fields which is of primary importance in the following. 5.15 DEFINITION. An analytic vector field X on a Ranach manifold M is called complete if it is the infinitesimal generator of a qlobal analytic flow rX : RxM + M defined on fix = RxM , The set of all complete analytic vector fields on M will be denoted by aut(M) For X E aut(M) and t E R , define an analytic mapping exp(tX) : M + M by exp(tX)(m) := rx(t,m) for all m E M Then exp(sX)exp(tX) = exp((s+t)X) for all s,t E R Since exp(0X) = idM , it follows that exp(tX) E Aut(M) for all t E R and exp(tX)-’ = exp(-tX) The homomorphism
.
.
.
.
R 3 t
+
exp(tX)
E
Aut(M)
is called the 1-parameter qroup associated with X E aut(M) Since tX E aut(M) whenever t E R and X E aut(M) , it suffices to consider the mapping exp : aut(M) defined by
exp(X) := exp(1.X)
+
. By
.
Aut(M) 5.14, we have
5.16 PROPOSITION. Suppose g : M + N is a bianalytic mapping between Ranach manifolds Let g,: T(M) + T(N) denote the associated Lie algebra isomorphism. Then g,(aut(M)) = aut(N) and every X E aut(M) satisfies
5.17 LEMMA. Let K be a compact subset of a Banach manifold M and let G C Aut(M) be a subgroup such that G(K) = M Suppose X E T(M) satisfies
.
g*x =
x
5.17.1
INTEGRATION OF VECTOR FIELDS €or all
g
E
.
G
Then
X
E
aut(M)
81
.
PROOF. Let (Qx,rx) be the maximal analytic flow on Then 5.17.1 and 5.14 imply generated by X
.
.
M
Since K is compact, there exists T > 0 for all g E G such that ITxK C Qx Since G(K) = M , 5.17.2 implies Define g(t)(m) := rx(t,m) for all ITxM C R X (t,m) E I x M Then g(t) E Aut(M) and
.
.
.
g(s+t) = g(s)g(t)
5.17.3
.
whenever s,t and s + t belong to I T Given any t E R , choose a non-zero integer n such that t/n E I T and define This definition is independent of n g(t) := g(t/n)" since 5.17.3 implies
.
g(t/n)"
=
g(t/kn)kn
= g(t/k) k
.
.
whenever t/k E I T Now suppose s,t E R Choose n such Since the that s/n , t/n and (s+t)/n belong to I T automorphisms g(t) for t E IT commute by 5.17.3, we get
.
The mapping r : RxM + M defined by r(t,m) := g(t)(m) analytic since rlIrxM = rX and r(t+s,m) = r(t,r(s,m))
.
all t,s E R and m E M global analytic flow on M 5.18
COROLLARY.
manifold. PROOF.
Then
It follows that
.
Apply 5.17 to
is a O.E.D.
M is a compact analytic aut(M)
Suppose T(M) =
r = rx
is for
K := M
.
and
G := {idM}
.
O.E.D.
Note that a compact analytic manifold is finitedimensional, being locally compact and having a finite number of connected components.
82
5.19
SECTION 5
PROPOSITION.
Suppose
.
M
is a Banach manifold,
X E aut(M) and Y E T(M) Let f : Q .+ W he a Ranach space valued analytic mapping on an open subset Q of M
.
Then the mapping RxQ 3 (t,m)
(exp(tX),Y)f(m)
+
is analytic and satisfies
a -(exp(tX).Y)f(m) at Put
PROOF.
gt := exp(tX)
t (g,Y)f(m) is analytic i n
-
(g,Y)f t
-
(Y(hogs)
# 0
s
a t -(g,Y)f at
RxQ
E
= (Y(hogS -h)
Dividing by
E
Aut(M)
and
)f(m)
h := fogt
. .
Then
t -t Y(f0g ) ( g m) = Yh(g-tm)
=
(t,m)
(g:+SY)f =
= (exp(tX),[X,Yl
and for all
E
= Y(fogt+s)og-t-s
R
-
we have Yhog-t
Yhogs)og-t-s
-
= Y(Xh)og -t
-t
.
(Yhogs-Yh))og-t-s
and letting
= [X,Y]hog
s
s + 0
-
,
we get
X(Yh)og -t
.
t
= g,[X,Ylf
O.E.D.
Interesting examples of analytic flows and complete analytic vector fields are given by Moebius transformations. Let 2 be a Ranach space over K and consider the Ranach algebra B defined in 3.5.1. 5.20
LEMMA.
Suppose
x
:=
(Z
b d)
E
R
satisfies the associativity condition, i.e., (aulcv ((ucv)cw for all
U,V,W
E
2
(au) d a(ucv) (ucvld) = (uc(vcw
.
Then
a(ud) uc ( v d 1
5.20.1
INTEGRATION OF VECTOR FIELDS
(trz)
+
exp(tX)#(z)
defines an analytic flow on
with infinitesimal generator
Z
.
X# = (az+b-zcz-zd)-- a
az
PROOF.
t
For
consider the matrices
R
E
83
By assumption, the unital subalgebra of B generated by X satisfies the associativity condition. Hence 3 . 7 implies exp(sX)# exp(tX)#(z) = exp((s+t)X)#(z) for all s,t well-defined.
E
R and all z E It follows that
(trz)
+
exp(tX)#(z)
Z
such that both sides are
(atz+bt)(ctz+dt)
=
-1
5.20.2
.
By differentiation, it defines an analy ic flow on Z follows that the infinitesimal generator of 5.20.2 has the form dat
(&
0 z
+ Fdbt (o)
- z &dct O)z
-
ddt z -(0) dt
a
(az+b-zcz-zd)-a z = X #
=
a az
C).E.D.
'
2 is an associative 5.21 PROPOSITION. Suppose c E L ( Z , Z ) algebra product on Z and b E 2 Then the local analytic flow generated by the vector field
.
X# := (b-zczl-aaz has the form
for all
z
E
Z
with
caz+d
E
GR Z ) m
a := cosh(bc) 112 =
1
where
a4
SECTION 5
and m
d := cosh(cb)lI2 =
n=O
(cb)"
o !
L(Z)right
'
PROOF. Since (ucv)cw = uc(vcw) for all u,v,w E 2 by assumption, the unital subalgebra of R generated by
x = ( cO is associative.
b 0)
Therefore an induction argument shows that
and
n
f o r all
Since 5 . 2 0 follows 5.22
E
N
.
Hence
exp(X#)(z) = exp(X)#(z)
implies
.
COROLLARY.
If
lbl
ICI
is small, we have
and
and
the assertion O.E.D.
where
PROOF.
,
Using the notation of 5.21, we have
I N T E G R A T I O N O F VECTOR F I E L D S
85
.
Therefore B = a -1 ab = abd-’ = exp(X#)(O) -1 y = caa-l = d c a NOW
.
1/2
2
a2 - abca = cosh (bc)
-
sinh(bc)
.
and sinh(bc) 1/2
bc
(bc)lI2
(hc) 1/2
- sinh implies
idz
2 2 112 d2 - ca b = cosh ( c b )
sinh (bc) 1/2
=
.
a -1 (a2-abca)a-l = a - 2
=
-By
Similarly,
sinh( bc) 1/2 1/2 (bc1
2 (cb) cosh (cb)1/2 - sinh(cb) l i 2 cb (:;Ah (cb) (cb)lI2 - sinh
imp1 ies
idz -
d-l (d -ca b)d-l = d-2
yB =
=
a(~+B)(yz+id~)-’d-~= (az+ag)(dyz+d)-’
=
(az+ab)(ca+d)-’
5.23
PROOF.
exp(X#)(z)
If
COROLLARY.
exp(X#
=
)
Ibl
’ ( 0 )v
= (
Ic(
.
Hence
.
O.E.D.
is small, we have
idz-By )avd-’
By 3 . 5 . 3 , exp(X#) ‘(0)v = (av-abd -1 cav)d-l = (av-Byav)d -1 O.E.D.
5.24
COROLLARY.
If
c
E
L2(Z,Z)
is associative, the vector
field X # := zcz
a az
generates the local analytic flow exp(X#)(z) = z(idZ-cz) -1
86
SECTION 5
COROLLARY. vector field
For
5.25
a
X # :=
is complete on
PROOF.
and
L(Z)
E
b
E
Z
I
the affine
(az+b) aG
and
Z
The matrix
trivially satisfies the associativity condition 5 . 6 . 1 exp(a) ah exp(X) = ( 1 , 0 id where
exp(ta)dt
a =
L(Z)
E
0
.
Now apply 5.20.
and
0 . E. D.
Although the set aut(M) of all complete analytic vector fields on a Ranach manifold M is in general not a subalgebra of T(M) , certain subalgebras of T(M) contained in aut(M) will play an important role in the sequel. 5.26 DEFINITION. Suppose M is a Ranach manifold over K and g is a Ranach Lie algebra over R or C A n action of g on M is a real-linear Lie algebra homomorphism
.
p(g) C
such that p
aut(M)
: gxM +
.
T(M)
defined by p(X,m) := (pX), associated with the action
The mapping
,
5.26.2
p
is called the evaluation mapping Let
.
5.26.3
denote the evaluation mapping at : = p(X,m) An action p p,(X) defines an injective mapping. An algebra g is called analytic if
.
m
E M I defined by is called faithful if 5.26.1 action p of a Ranach Lie 5.26.2 defines an analytic
INTEGRATION OF VECTOR FIELDS
mapping. It is often convenient to study Lie algebra actions locally
.
is an action of a Banach Lie algebra g on a Ranach manifold M A local representation of p with respect to a chart (P,p,Z) of M is a continuous mapping DEFINITION.
5.27
Suppose
p#
:
p
g
.
Om(C,Z
-+
C
)
,
5.27.1
is a bounded domain in Zc := ZBKC such that for all X E g the identity
where
C
containing
0
5.27.2
holds on
p-l(C)
Note that C
. p#X
is uniquely determined by 5 . 2 7 . 2 ,
since
is a domain.
5.28 LEMMA. Suppose representation of p
.
P#
:
For
g
+
X
E
a
X# : = p#(X)= Then PROOF.
=
X + X
#
Om(C,Z 1 is a local g , define E
T(C)
.
is a homomorphism of real Lie algebras.
By 5 . 1 0 ,
the identities
[PX,PYl(fop) = p[X,Yl(fop) = [X,YI#fOP -1
hold on p (C) , whenever f is a Ranach space valued analytic mapping. Since C is a domain, the assertion Q.E.D. f0 1 lows. 5.29 DEFINITION. An action p of a Ranach Lie algebra 9 on a Ranach manifold M is called locally uniform if for
SECTION 5
88
every point o E M there exists a chart (P,p,Z) of M about o such that p has a local representation p # with respect to this chart. If dim(g) < uniform.
+a
,
every action
p
of
g
is locally
PROPOSITION. An action p of a Ranach Lie algebra g on a Ranach manifold M is analytic if and only if p is locally uniform and the mapping p : g + T ( M ) is linear. 5.30
PROOF.
Suppose first that
p
is an analytic action.
Then
for every m E M , the real-linear evaluation mapping Pm is analytic and therefore linear. Hence 5.26.1 defines a linear mapping. To prove the existence of local representations, choose a chart (P,p,Z) of M about o , a convex open neighborhood T of 0 E g and a bounded domain C in Z c containing 0 , such that there exists an analytic mapping F : T ~ C+ zC satisfying F X,pm) = (pX)p(m) for all (X,m) E Txp-l(C) and I SUP lD1F X,Z)l < +m X ET z EC where D1 denotes the first partial derivative. Since p is linear, it follows that there exists a continuous linear C ) such that (p#X)(z) = F(X,z) for mapping p # : g + O,(C,Z Hence p is a locally uniform action. all X E T Conversely, suppose that p is linear and defines a locally uniform action. Then any local representation is linear and 1.24 implies that the P # : g + Om(C,Z ) mapping
.
p#
: g x c + zC
defined by p#(X,z) := (p#X)(z) is analytic. B y 5.27.2 we have p#(X,pm) E Z for all m E p-l(C) and the diagram
gxc idxp
7
'
"iZ
T(p) gxP-l(C) p - T ( P )
89
INTEGRATION OF VECTOR FIELDS
commutes, where
.
$(X,z) := (z,p#(X,z))
Hence
is
p
analytic.
O.E.D.
5.31
COROLLARY. An action g on a Banach manifold M locally uniform.
of a real Ranach Lie algebra is analytic if and only if it is p
5.32 THEOREM. Suppose p is an action of a Ranach Lie algebra g on a Ranach manifold M Then p is analytic if and only if the mapping
.
F : RxgxM
M
, ,
F(t,X,m) := exp(t*pX)(m)
defined by PROOF.
-t
For any chart T(P)(PX),
=
(P,p,Z)
at a
of
M
is analytic.
,
5.0.4 implies
p(F(t,X,m))t=O
5.32.1
.
for all m E P Since T(p) is bianalytic, it follows that analytic if p is F is analytic. Conversely, suppose p is ana ytic. Since F ( t+s, X,m for a1 +
(0,o)
E
F(l,sX,F(t,X,m))
t suffices to show that the mapping exp(pX)(m) is analytic on an open neighborhood of gxM , where o E M is arbitrary. Ry 5.30, there s,t
(X,m)
=
E
R
t
C
exists a local representation p i t : g + O-(C,Z 1 respect to a chart (P,p,Z) of M about o and continuous mapping p # is linear. Let R be an neighborhood of 0 E C such that dist(B,aC) > 0 there exists a starlike open neighborhood T of that
defines an analytic mapping m
E
p-'(~)
,
5.10 implies
r# : TxR
-f
C
.
For
of p with the open By 5.6, 0 E 9 such
.
X
E
T
and
SECTION 5
90
Hence we may assume r#(Txp(p-lB)) C p(P) is a commuting diagram -1 r ( B ) P-
.
Therefore there
ID
Tjp
idxp Txp( p-lB )-Z
I
r# where r is analytic, and 5 . 7 implies that (t,m) + r(tX,m) -1 defines an analytic flow on p ( R ) with infinitesimal generator pX Since r(X,m) = F(l,X,m , it follows that
.
F
is analytic.
Q.E.D.
COROLLARY. Let p be an analytic action of a Ranach Then the evaluation Lie algebra g on a Ranach manifold M mapping rm : g + M I defined by rm(X) := exp(pX)(m) I has the differential 5.33
defined by PROOF.
p,(x)
(px),
:=
Apply 5 . 3 2 . 1 .
PROPOSITION. Banach Lie algebra 5.34
for all
X,Y
E
g
,
O.E.D.
Let
g
be a locally uniform action of a on a Ranach manifold M Then p
.
exp ( pX)* pY =
p
where
L(g)
adX
E
(exp(adX)Y 1
(adX)Y := [X,Y]
is defined by
.
5.34.1
C
PROOF. Let p # : g + U m ( C , z ) be a local representation of p with respect to a chart (P,p,Z) of M By 5 . 1 9 , the mapping t + (exp(t*pX),pY)p(m) is analytic for every -1 m c p ( C ) and satisfies
.
a
z ( e x p (t px) * pY)p(m =
Put
6
:=
(exp(t*pX),p[X,Yl
adX
.
An induction argument shows
91
INTEGRATION OF VECTOR FIELDS
for every
n
E
N
.
Hence there exists a power series
expansion
about
t = 0
analytic in NOTES.
. t
Since E
R
,
p exp(tS)Y)p(rn) = p#(exp(t&)Y)(prn) is O.E.D. the assertion follows from 3 . 1 .
For analytic spa1 e s of finite dimension the integra-
tion theory for analytic vector fields can be found in [SO]. Our presentation, with Proposition 5.5 playing a central role, follows ~137,1381.Analytic flows given by Moebius transformations (cf. 5.20 - 5.27) are of primary interest in later sections. Proposition 5.21 appears in C841, and the analytic transformations described in Corollary 5.22 were first considered in [581. The notion of complete analytic vector field is of fundamental importance throughout the book. In general, the set aut(M) manifold
of all complete analytic vector fields on an analytic M
is not closed under addition or the commutator
product. See [1101 for counterexamples on
R2
.
The results
of [110] concerning finite dimensional subalgebras of T ( M ) contained in aut(M) (for finite dimensional M ) play an important role in modern differential geometry (cf. C93; Ch. I, § 31). For Banach manifolds M , we will be interested in Banach Lie algebras of vector fields contained in aut(M) In certain cases (cf. Section 13) it is even possible to endow aut(M) with the structure of a Banach Lie algebra.
.
SECTION 6
92
6.
BANACH LIE GROUPS
Symmetric Ranach manifolds are homogeneous in the sense that they admit transitive groups of analytic automorphisms. More precisely, these groups of automorphisms can be endowed with the analytic structure of a Ranach Lie group, In the finitedimensional case, it is even possible to characterize and classify symmetric manifolds in terms of (semi-simple) Lie groups 1621. Although such a complete Lie-theory does not exist for symmetric Ranach manifolds, it will be shown later that Ranach Lie groups and Lie algebras play an important role in the infinite-dimensional case and, in fact, lead to another kind of algebraic structure (Jordan algebras and Jordan triple systems) forming our principal algebraic tool for studying symmetric Ranach manifolds. A Aanach Lie group over R E {R,C} is a group G which is also a Banach manifold over K such that the multiplication mapping GxG 3 (g,h) + gh E G is analytic. The unit element of G will be denoted by e The left and right translations L h := gh and R h := hg define g 9 bianalytic automorphisms Lg and Rg of the underlying Banach manifold M := G
.
.
6.1 LEMMA. Let G be a Banach Lie group with tangent bundle T(G) Then the mappings
.
and
are analytic. PROOF. Since the product mapping r : GxG + G is analytic, the differential T(r) : T(G)xT(G) + T(G) is also analytic. Let i : T(G) + G denote the canonical projection. Then
BANACH LIE GROUPS
for all
v,w
93
.
Since the "zero vector field" G 3 g + Og E T(G) is analytic and satisfies T(r)(O , w ) = T(L )w and T(r)(v,O ) = T(R )v , the assertion 9 9 4 9 O.E.D. follows. Let
E
T(G)
T(G)
denote the Lie algebra of all analytic vector
fields on a Banach Lie group :=
{
x
E
G
T(G) : (R
9
.
The subalgebra
)*x =
x
for all
g E G
of all "right-invariant" analytic vector fields on called the Lie algebra of G
.
g
The vector fields in g
LEMMA.
6.2
c
aut(G)
PROOF.
{ Rg
.
Apply 5.17 : g
G }
E
to
M :=
G
,
G
} is
are complete, i.e.,
K : = {e}
and the group
of all right translations of
G
.
O.E.D.
The bianalytic mappings exp(X) on G associated with X E 9 commute with all right translations, since 5.16 implies R
9
exp(X) R-l = exp((R ),XI 9
9
=
exp(X)
.
is a left translation, uniquely determined by the value exp(X) := exp(X)(e) E G The mapping exp : + G defined in this way is called the exponential mappinq of G It follows that
exp(X)
.
.
6.3
9
.
LEMMA. Let G be a Banach Lie group with Lie algebra Then the evaluation mapping 6.3.1
defined by
pe(X) := Xe
defined by
(X,)g
,
is a linear isomorphism with inverse
:= Te(R9)v
for all
PROOF. By 6 . 1 , the vector field
Xv
g
on
E
G
G
. is analytic and
SECTION 6
94
.
clearly right invariant. Thus Xv E g for all v E Te(G) Since the vector fields in g are right invariant, the linear mappings 6.3.1 and 6.3.2 are inverse to each other. O.E.D. It follows from 6.3, that space topology such that 6.4
G
pe
g
carries a unique Ranach
becomes a homeomorphism.
PROPOSITION. The Lie algebra g of a Ranach Lie group is a Ranach Lie algebra acting analytically on G
.
PROOF. Let (P,p,Te(G)) be a chart of G about e such that T,(p) = id Put Z := T,(G) Then $(Z,W) := p(p-l (z)p-l(w)) defines an analytic mapping $ : $2 + Z on an open neighborhood $2 of ( 0 , O ) E 2x2 For u E Z , we have
.
.
.
where D1 denotes the first partial derivative. 4.6.2 implies
Therefore
it follows that
.
is a continuous bilinear form on 2 Hence g is a Ranach Lie algebra. The canonical action p of g on G satisfies p(X,g) = Te(R )Xe for all (X,g) E gxG and is therefore 4 analytic by 6.1. O.E.D. 6.5 COROLLARY. The exponential mapping analytic and the differential
exp : g
+
G
is
To(exp) = pe : g + Te(G)
is the evaluation mapping
pe(X) := Xe
at
e
E
G
.
There
BANACH LIE
exists a "canonical" chart that
GROUPS
(P,p,g) of
95
G
about
e
such
p(exp X) = X for all
X
in a neighborhood of
0
E
g
.
PROOF. Since g acts analytically on G by 6.4, 5 . 3 2 implies that exp is analytic and 6.5.1 follows from 5.33. Since pe is a Banach space isomorphism, 4.1 implies that there exist open neighborhoods P of e E G and T of 0 E g such that exp : T + P is hianalytic. Put -1 p := exp : P + T O.E.D.
.
6.6 COROLLARY. The inversion mapping j(g) := g-l of a Banach Lie group G is analytic and satisfies For every X E g , j*X is the unique left Te( j ) = -id invariant analytic vector field on G satisfying
. (j*XIe = -Xe .
PROOF. Since j o L = R j(g)oj for all g E G , it suffices to show that j is analytic in a neighborhood of e E G Since the diagram
.
G
j
.G
4-9
-id commutes, 6.5 implies the assertion.
0.E.D.
6.7 PROPOSITION. Let G he a Ranach Lie group with Lie algebra g Then we have for all X,Y E g
.
exp(X+Y) = lim (exp(X/n)exp(Y/n)) n n+and 2
exp [x,YI = 1im n+-
( exp ( x/n
exp ( Y/n ) exp ( -x/n exp ( -Y/n
PROOF. For the canonical chart 5.13 implies
(P,p,g) of
G
about
e
. ,
96
SECTION 6
X + Y = lim p(exp(tX1 exp(tY))/t t+O and t +o Now put t = l/n and use the fact that exp is continuous and satisfies exp(kX) = for all integers k
.
O.E.D.
6.8
COROLLARY.
group
G
k
.
:=
Let
K
be a closed subgroup of a Ranach Lie
Then
{ X
E
g
: exp(tX)
E
for all
K
is a closed real subalgebra of g PROOF.
By 6 . 7 ,
closed since
h
exp
t
E
R }
6.8.1
.
is a real subalgebra of
g
which is
is continuous.
Q.E.D.
The basic examples of Ranach Lie groups are provided by associative unital Ranach algebras. EXAMPLE. Let A be an associative unital Ranach algebra By 2 . 7 , the group G := G(A) of all invertible over K elements of A is open and is therefore a Ranach manifold. Since the product in G is the restriction of a continuous bilinear mapping, G is a Ranach Lie group over K The tangent space Te(G) at the unit element e E G can be identified with A Let g denote the Lie algebra of G By 6 . 3 , the evaluation mapping pe : g + A is a Ranach space isomorphism. Let z denote the "coordinate" of A Then the right-invariant analytic vector field Xa on G satisfying pe(Xa) = a E A is given by 6.9
.
.
.
.
.
a Xa = az By 4 . 6 . 2 ,
this implies for
az
'
a,b
E
A
a = X [Xa,Xb] = [a,blz az
where
[a,b] := ab
-
ba
[arb]
denotes the commutator in
A
.
Let
BANACH LIE GROUPS
97
g(A) denote the Ranach Lie algebra associated with A (cf. 2.14) endowed with the commutator product. Then there exists a commuting diagram
where
pe
is an isomorphism of Ranach Lie algebras.
It
follows that the Lie algebra of G(A) can be identified with g(A) In particular, for any Ranach space L over
.
D E { R , C , E } , the group GI1(L) of all invertible D-linear operators on L , associated with the associative unital Banach algebra A := L(L) over the center R of D , is a Banach Lie group over K whose Lie algebra can be identified with the Banach Lie algebra gll(L) over K , endowed with the commutator product. In case L = Dn is finite-dimensional, we write Gtn(D) := GR(Dn) and gEn(D) := gI1(Dn)
.
6.10 LEMMA. Suppose G is a connected Ranach Lie group with Lie algebra g Let n : G" + G denote the universal covering of G and choose e E. G" such that n(e) = e Then there exists a unique Ranach Lie group structure on G" with unit element e , such that n is a locally bianalytic homomorphism. The Lie algebra of G- can be identified with g such that there is a commuting diagram
.
.
6.10.1
PROOF. By 3 . 3 , G" is a Banach manifold such that n is locally bianalytic. Let r : GxG + G and j : C, + G denote the product mapping and inversion mapping respectively. By 3 . 4 , there exist commuting diagrsms r G" XG" G* nxm
/,
SECTION 6
98
and
G
>G
j
where r" and j" are analytic mappings satisfying r"(e,e) = e and j"(e) = e , respectively. Applying 3 . 4 to the pairs of analytic mappings
G'+ G-
x + r"(x,e) 3 x + r"(e,x) 3
E
G-
E
G"
,
and
it follows that G" is a Ranach Lie group with unit element e such that IT is a homomorphism, Since II is locally bianalytic, the Lie algebra of G* can be identified with g such that the diagram
rr
commutes, where p and p denote the canonical actions of g on G and G- , respectively. Then 6.10.1 is also a commuting diagram. O.E.D. Many important Ranach Lie groups arise naturally as transformat ion groups I' act i ng on Banach man i fo1d s
.
is a group and M is a Ranach on M is a homomorphism
6.11 DEFINITION. Suppose manifold. An action of G r : G
G
+
Aut(M)
6.11.1
BANACH LIE GROUPS
99
into the group of all bianalytic automorphisms of mapping r : GxM
+ M
M
.
,
The
6.11.2
defined by r(g,m) := r(g)(m) , is called the evaluation Let mappinq associated with the action r
.
rm : G + M
6.11.3
denote the evaluation mapping at m E M , defined by rm(g) := r(g,m) An action r is called faithful if 6.11.1 defines an injective mapping. An action r of a topological group G on M is called continuous if 6.11.2 defines a continuous mapping. An action r of a Banach Lie group G on M is called analytic if 6.11.2 defines an analytic
.
mapping. 6.12 PROPOSITION. Suppose r is an analytic action of a Banach Lie group G on a Banach manifold M Then there exists a unique analytic action r* of the Lie algebra g of G on M such that there is a commuting diagram
.
6.12.1
PROOF. For every X E g , rx(t,m) := r(exp(tX ,m) def nes be the a global analytic flow on M Let r,X E aut(M corresponding infinitesimal generator. Then the diagram 6.12.1 commutes. The evaluation mapping rm : G + M at m E M is analytic and, by 5.33, satisfies
.
(r*XIm = at a rx(t,m)t,O Since of X
E
r OR = r m g r(g,m) g implies
for all
= Te(rm)Xe
g
E
G
,
6.12.2
the right-invariance
100
SECTION 6
r,(X+Y)orm
=
(r,X+r,Y)or rn
r,[X,Ylorm
=
[r,X,r,Ylor m
and
.
Evaluation at e E G shows that r* : g + T(M) algebra homomorphism. The differential T(r) : T(G)xT(M) + T(M) of 6.11.2 satisfies
for all
v
E
T (G) and 9
w
E
Tm(M)
(r*X)m = T(r)(Xe30,,,) the action
r,
of
g
on
.
Since 6.12.2 implies
9
is analytic.
g
The action r, of action r of G on M By 6.12.2, we have 6.13 COROLLARY. Let action of G on M commuting diagram
M
.
is a Lie
O.E.D.
associated with an analytic is called the differential of r
.
r* be the differential of an analytic Then for every m E M there is a
The next results concern "lifting properties" of analytic actions of Ranach Lie groups and Banach Lie algebras. 6.14 PROPOSITION. Suppose r is an analytic action of a connected Banach Lie group G on a connected Banach manifold M Let nG : G" + G and n M M" + M denote the
.
universal coverings of G exists an analytic action is a commuting diagram
.-
and M r" of
,
respectively. Then there G" on M" such that there
BANACH L I E G R O U P S
101
.
r
G- x -M"
n xn G
M-
I
M
I*M
6.14.1
'
GxM A M
r PROOF. By 6.10, Gis a Ranach Lie group such that n is Ga locally bianalytic homomorphism. Choose a point o E M .. By 3.4, there exists a commuting diagram 6.14.1, where r is an analytic mapping satisfying r-(e,o) = o Applying 3.4 to the analytic mappings
.
.
and E
it follows that M-
.
M"
,
defines an ana ytic act on of
r-
G-
on O.E.D.
6.15 COROLLARY. Suppose p is an analytic action of a Banach Lie algebra g on a connected Ranach manifold M Let n : M- + M denote the universal covering of M Then there exists an analytic action p- of g on M- such that the diagram
.
.
g x MA
1
idxn
4XM
commutes.
For all
X
E
g
,
P
-
,T( M - ) T(n) T(M)
-1
6.15.1
we have 6.15.2
PROOF.
For any
X
E
g
,
the assignment
.
defines an analytic action of R on M By 6.14, there exists a unique vector field p - X E aut(M-) such that
102
SECTION 6
.
for all t E R By differentiation, it follows that the diagram 6.15.1, for the respective evaluation mappings p and p- , commutes. Since T(n) is locally bianalytic, i t follows that the mapping p A : gxM" + T(M-) is analytic. Since T ( T ) O ~ - X= pxon , 4 . 4 implies
and
Since Tm(n) is an isomorphism for all m E MA that pis an analytic action of g on M-
.
,
it follows O.E.D.
6.16 COROLLARY. Suppose r is an analytic action of a connected Banach Lie group G on a connected Ranach manifold M Let p denote the differential of r Then the lifted action p- of g on M" can be identified with the differential (r")* of the lifted action r- of G- on M-
.
.
.
PROOF. By 6.10, the Lie algebra of G- can be identified with 4 . By 6.14.1 and 6.15.2, we have for all X E g
Since
g
is simply connected, it follows that
r"(exp XI = exp(p-x)
.
9.E.D.
6.17 EXAMPLE. Suppose M is a Ranach manifold and X E aut(M) Then the global analytic flow r,(t,m) := exp(tX)(m) on M generated by X defines an analytic action rx : R + Aut(M) of the additive Lie group R on M We can identify the Lie algebra of R with R , in which case the exponential mapping is the identity. Then the differential p x : R + aut(M) of rx is given by px(t) = tX
.
.
.
6.18 EXAMPLE. Let G be a Banach Lie group with Lie algebra g Then r(g,m) := L m defines an analytic action 9
.
BANACH
LIE GROUPS
103
of G on the Ranach manifold M := G by left translations. The corresponding evaluation mapping is just the product in G Since
.
Lexp(x)
= exp(X) E Aut(M)
X E g , the differential r, : g inclusion mapping.
for all
6.19 Then
EXAMPLE.
Let
+
be a Banach Lie group and
G
is the
aut(M)
g
E
G
.
Int(g)h := ghg-’ defines a bianalytic group automorphism of G called the inner automorphism induced by g The homomorphism Int : G + Aut(G) defines an analytic action of G on M := G Unlike the left translation action 6.18, the differential Int, : g + aut(M) does not yield rightinvariant vector fields, but rather
.
.
Int,(X) for all
X
E
g
, where
j
= X
+ j,x
is the inversion mapping (cf. 6.6).
EXAMPLE. Suppose L is a Banach space over D E {R,C,H} and let M = ME(L) be the Grassmann manifold
6.20
(over the center K of D ) of all split subspaces of L , The homomorphism r : GE(L) + Aut(M) , associating with each g E GE(L) the collineation r(g) defined by r(g)H := g ( H ) for all H E M , defines an action of GL(L) on M which is analytic by 3.14.1. This action is called the collineation action. Now fix a splitting 3.8.1 of L and put Z := L(E,F) Then the Lie algebra g := g E ( L ) of GE(L) has an additive gradation g = g,l fB go @ g1 , where
.
b 0)
and
: ~ E Z )
SECTION 6
104
Let
ro
: GR(L) +
M
denote the evaluation mapping at 0 :=
Then every
b
E
2
0
(E)
.
satisfies
It follows that the chart satisfies
p := PE, F
of
M
about
o
.
for all X E g-l = 2 By differentiation of 3.14.1, it follows that the differential p = r* : gR(L) + aut(M) of the collineation action r satisfies
where the Lie algebra homomorphism defined by 4.13.1.
gR(L) 3 X
+
X#
E
T(2)
is
6.21 EXAMPLE. Suppose G and H are Ranach Lie groups with Lie algebras g and h , respectively. Let r : G + H be an analytic group homomorphism. Then r induces an analytic action of G on M := H by r(g,h) := r(g)h for all (g,h) E GxH The differential r* : g + aut(M) induces right-invariant vector fields on H , i.e., r* : g + h is a continuous Lie algebra homomorphism inducing the commutative diagram
.
6.22
LEMMA.
Suppose
r1
and
r2
are analytic actions of a
BANACH LIE GROUPS
connected Banach Lie group 1 = r,2 implies rl = r 2 r*
G
.
105
on a Ranach manifold
M
.
Then
2 For any m e M , the analytic mappings r1 m and rm from G into M coincide on the neighborhood exp(g) of e E G Since G is connected, 3.1 implies the assertion.
PROOF.
.
O.E.D.
As a consequence, suppose r1 and r2 are analytic homomorphisms from a connected Aanach Lie group G into a 1 2 Ranach Lie group H Then Te(r ) = Te(r ) implies 1 2 r = r
.
.
LEMMA. Suppose g and h are Ranach Lie algebras and p is an action of h on a Aanach manifold M Let 4 : g + be a continuous homomorphism. Then P O $ : g + aut(M) is an action of g on M which is locally uniform if p is locally uniform and which is analytic if p is analytic. Now suppose G and H are Ranach Lie groups with Lie algebras g and h , respectively. Let r be an action of H on M and let f : G + H be an analytic homomorphism. Then rof : G + Aut(M) is an action of G on M which is continuous if r is continuous. If r is an analytic action, then rof is an analytic action and its differential satisfies (rof), = r*of, 6.23
.
.
6.24 EXAMPLE. Suppose g is a Ranach Lie algebra and Z is a Banach space over K Then every continuous homomorphism 4 : g + g k ( 2 ) defines an analytic action of g on 2 Here we have identified
.
.
.
Now suppose G is a Ranach Lie group with Lie algebra g Then every analytic homomorphism f : G + G!t(Z) defines an analytic action of G on 2 whose differential is the continuous homomorphism 4 = f, : g + g k ( Z )
.
6.25
EXAMPLE.
Suppose
g
is a Ranach Lie algebra.
Then
SECTION 6
106
defines a continuous homomorphism ad : g + aut(g) and hence an analytic action of g on g (by derivations). This on g action is called the adjoint action of g Now suppose G is a Ranach Lie group with Lie algebra g , Then Lg induces an automorphism
.
since left and right translations commute. fields in g are right-invariant,
Since the vector
can also be regarded as the differential of the automorphism Int(g) = L R-l of G By 4.5, the mapping 9 9
.
Ad : G
+
Aut( g)
is a homomorphism and hence defines an action of G on (by automorphisms). This action, called the adjoint action of G on g , is analytic since the mappings
and
are analytic by 6.1 and (Ad(g)X),
= Te(Int g)Xe = T
-1
9
( L 1 Te(Rg-1 )Xe g
.
Applying 5.31, to the canonical action of g on G , which is analytic by 6.4 and hence locally uniform by 5.30, it follows that
Hence the diagram
BANACH LIE GROUPS
107
Ad G -Aut(g) exp
exp
’
’
aut(g)
ad commutes and the differential Ad* of the adjoint action Ad of G on g can be identified with the adjoint action ad of g on g
.
6.26
COROLLARY.
For every
g
E
,
G
the diagram
(Lg)* pe
4 Te(G)
commutes.
In particular,
Te(Int g ) (Lg)*
Gg(g)
E
’1 ’
pe
Te(G)
.
6.27 EXAMPLE. Suppose 2 is an associative Ranach algebra over K and let L z := xz and R z := zy denote the left X Y and right multiplication operators on Z , respectively. By 6.23, the continuous homomorphism g(z)Xg(Z) 3 (x,Y)
+
Lx-R
Y
E
gg(2)
defines an analytic action of g(2) x g(Z) particular, the continuous homomorphism g(2) 3 x
+
ad(x) := Lx
-
Rx
E
6.27.1 on
aut(2)
2
.
In
6.27.2
defines an analytic action of g ( 2 ) on Z (by derivations) which is called the adjoint action of g(2) on Z (a special case of 6.25). Now assume that Z is unital and consider the Ranach Lie group G(Z) Then the homomorphism
.
G(Z)xG(Z) 3 (g,h)
+
LgRE1
E
Gg(Z)
defines an analytic action of G(Z) on Z whose differential is given by 6.27.1. In particular, the homomorphism G(2) 3 g + Ad(g) := L R-l 9 9 defines an analytic action of
G(2)
E
Aut(2) on
Z
(by automorphisms)
SECTION 6
108
which is called the adjoint action of G(Z) on 2 and whose differential is given by 6 . 2 7 . 2 . It follows that there is a commuting diagram Ad G(Z) A A u t ( 2 ) exp q (Z 1
1
adaut ( z )
exp
Since Int(g) = Ad(g) I G(2) is linear for all follows that the adjoint action of G(Z) on 2 case of 6.25.
,
it is a special
g
E G(Z)
EXAMPLE. Let E and F be Ranach spaces over D E {R,C,A} and consider the Ranach space Z := L(E,F) Then the mapping the center K of D 6.28
.
GL(F)xGL(E) 3 (a,d)
+
ga,d
E
G&(Z)
over
r
defined by ga,dz := azd-' , is an analytic group homomorphism (over K ) whose differential can be identified with the continuous Lie algebra homomorphism
defined by 6.29
over
6
a,d
z
:=
az-zd
.
EXAMPLE. Let A be an associative Ranach algebra K and consider the Ranach space
F := $ ' r n ( ~ x ~ ,c~ )P ~ + ~ ( A ~ A , w ) of all continuous polynomials f : AxA + W into a Ranach space W which are m-homogeneous in the first variable and n-homogeneous in the second variable. Since the second derivative f"( z ,w) E. L'(AXA,~) is symmetric [ 2 8 , 8 . 1 2 . 2 1 , the formula
defines a continuous Lie algebra homomorphism P : g(A) + g&(F) Since the canonical action of g&(F) on F is analytic, p defines an analytic action of g(A)
.
BANACH L I E GROUPS
on
F
.
109
AS a special case, the formula 6.29.1
.
defines an analytic action of g(A) on P"(A,W) Now assume that A is unital and consider the Aanach Lie group G(A) Then the formula
.
(r(g)f)(z,w) := f(zg,g-lw)
6.29.2
defines a homomorphism r : G(A) + GR(F) Since the mapping of G(A) on F
.
L(A)xFxL(A) + F
and hence an action
,
defined by (X,f,Y) + ((z,w) + f(Xz,Yw)) is analytic by 1.24, it follows that r is an analytic action. By differentiation, it follows that the differential r* of r A s a special case, the formula can be identified with p
.
defines an analytic action of
G(A)
on
67(A,W)
,
with
differential given by 6.29.1.
NOTES.
complete account of the theory of Banach Lie groups, including Lie groups over ultrametric base fields, can be found in C211. Standard references for the finite dimensional case are C26,63,1211. Most authors identify the elements of the Lie algebra of a Lie group G with left-invariant analytic vector fields on G (cf. the remarks at the end of Section 4). It should be noted that the Lie group actions considered in this book are (locally) uniformly continuous and therefore give rise to continuous Lie algebra actions. On the other hand, the actions studied in the theory of group representations are usually only pointwise continuous and their differential is given by densely defined unbounded operators. Also, A
in the theory of operator algebras one is mainly interested in strongly (but not uniformly) continuous groups of automorphisms since these arise naturally in applications to quantum mechanics.
SECTION 7
110
7.
INTEGRATION OF LIE ALGEBRA ACTIONS
The global analytic flow rx(t,m) = exp(tX)(m) associated with a vector field X E aut(M) is obtained by "integration" of an ordinary differential equation ( 5 . 1 ) which is related to the action pX(t) := tX of the abelian Lie algebra R on
.
M In this section, we will generalize this "integration process'' to analytic actions of Ranach Lie algebras.
DEFINITION. A locally uniform action p of a Ranach Lie algebra g on a Banach manifold M is called topologically faithful if for every local representation 7.1
p#
:
g
U,(C?Z
.+
of p with respect to a chart the assignment
x
.+
C
1
(P,p,Z) of
( c f . 5.271,
M
7.1.1
IP#XIC
is a compatible norm on g (as a real Banach space). equivalent condition is that the sets T~ := {
for O
a E
g
x
E
g
: Ip#xIc c a
An
]
> 0 form a fundamental system of neighborhoods of .
Every topologically faithful, locally uniform action p is faithful. Conversely, every faithful action p of a finite-dimensional Lie algebra g on a connected Banach manifold M is topologically faithful since 7 . 1 . 1 defines a norm on g (as a real vector space). 7.2 EXAMPLE. Let G be a Banach Lie group with Lie g Then the canonical action p of 4 on algebra M := G is analytic and hence locally uniform. N o w suppose Then g E G and let I * I be a compatible norm on T ( G ) 9 6.3 implies that the assignment
.
.
INTEGRATION OF LIE ALGEBRA ACTIONS
g
defines a compatible norm on topologically faithful.
.
111
Hence the action
is
p
7.3 LEMMA. Suppose p is a topologically faithful, locally uniform action of a Ranach Lie algebra g on a Ranach Then the mapping manifold M
.
is injective on a suitable neighborhood
of
T
0
E
.
g
--
C Choose a local representation g + O,(C,Z ) of p# p with respect to a chart (P,p,Z) of M about o Let R be a connected open neighborhood of 0 E C such that dist(B,aC) > 0 By 5.6, there exists a convex open neighborhood T of 0 E g such that PROOF.
.
.
defines a real-analytic mapping
r# : T
+
Ooo(R,Z
C
)
satisfying
r#(X,pm) = p(exp(pX)m) for all
m
E
P-'(B)
x
and
E
T
.
since
7.3.1 r ( 0 ) = idg #
and
r'(0)X = p#X , it follows that the real-analytic mapping # F : T -+ O , ( R , Z C ) , defined by F(X) := r#(X) - idR - p#X , satisfies F ( 0 ) = 0 and F ' ( 0 ) = 0 (as a real-linear mapping). Since p-l(R) # fl , it follows that 1x1 := lp#XIR defines a compatible norm on the real Ranach space g Hence
.
we can choose T such that IF'(X)l < 1/2 for all X E T Now assume X1,X2 E T satisfy exp(pX1) = exp(pX2) Then 7.3.1 and 3.1 imply r (X ) = r#(X2) on A and hence # 1 Since T is convex, the mean F(X1)-F(X2) = p#X2-p#X1 value theorem [28; 8 . 5 . 4 1 implies
.
.
.
= I P # x 1- P # x21R = I F ( x ~ ) - F ( x ~ ) ~ ~ < IX1-X21 SUP 1F'CX)l < IX1-X21/2 and therefore X1 = X;?
Ix1-x21
7.4
THEOREM.
Suppose
p
is a topologically faithful, 4 on a Ranach
analytic action of a Banach Lie algebra
0.E.D.
112
SECTION 7
.
manifold M Let G be a subgroup of Aut(M) containing exp(p( g ) ) such that for every g E G there is a commuting diagram
g*
aut (M) p
T
aut ( M I
T P
7.4.1
.4
Ad(g)
I
.
Ad(g) E G t ( g ) Then there exist a unique (Hausdorff) topology T on G and an analytic structure on (G,T) such that ( G , T ) is a Banach Lie group with Lie algebra g Further, the canonical action of ( G , T ) on M is analytic and has the differential
where
.
0 .
PROOF. By [21; 11.71, there exists an open neighborhood TO of 0 E g such that the Campbell-Hausdorff series for the Banach Lie algebra g defines an analytic mapping c : T0 x T 0 + g satisfying
.
By 7.3, we may assume that the mapping for all X,Y E To X + exp(pX) is injective on TO Since c(0,O) = 0 , there exist symmetric open neighborhoods T1 C T C To of 0 E g such that c(TlxT1) C T and c(TxT) c To Put S o := exp(pTo) and S := exp(pT) Define a mapping p : S o + To by p(exp(pX)) := X for all X E TO For any : gS + T by g E G , define a mapping pg Now assume g,h E G satisfy pg(h) := p(g-'h) R := gS n hS # Then h-'g = exp( pX) for some X E TO and
.
.
.
.
.
.
are bianalytic and there exist a topology
T
on
G
and an
113
INTEGRATION OF LIE ALGEBRA ACTIONS analytic structure on
t
(gs,pg,g) : g
E
]
G
defined by the atlas Since
(G,T)
.
= exp(p(c(X,-Y)))
~ X P ( ~ Xexp(pY)-' )
for all
X,Y
E
T1
,
the mapping
SlxSl 3 (g,h) is analytic, where
S1
gh
+
-1
E
S
.
exp(pT1) Since the diagram L hS +ghS
:=
ph LT/'gh commutes for all g,h E G , the left translations L~ are analytic on G By 7.4.1, T := T A Ad(g)-lT is an open g neighborhood of 0 E 4 for every g E C, , and the diagram
.
exp ( pTg 1 p
.
Int(g)
, 1P
1
T T Ad(g) ' g commutes by 5.16, where Int(g)h := ghg-l Hence Int(g) is analytic on an open neighborhood of e := idM in (G,T) By [21;111.1.1.(1)], (G,T) is a Banach Lie group whose Lie algebra can be identified with g Since the intersection of all neighborhoods of e in (G,T) reduces to {e) , is a Hausdorff topology. By 5.32, the canonical action of (G,T) on M is analytic and has the differential p Q.E.D.
.
.
.
.
7.5 COROLLARY. Suppose p is a topologically faithful, analytic action of a Banach Lie algebra g on a Banach manifold M Then the subgroup
.
of Aut(M) generated by p is a connected Banach Lie group with Lie algebra g , acting analytically on M with differential p
.
PROOF.
In order to verify 7.4.1, we may assume
g = exp(pX)
SECTION 7
114 for some
X e g
.
Define
ad(X)
E
L(g)
by 5.34.1
and put
Then the diagram 7.4.1 commutes by 5.3b. By 7 . 4 , G is a Banach Lie group with respect to a Hausdorff topology T and the Lie algebra of ( G , T ) can be identified with g As a union of the connected sets exp(g ) * exp(g) , G is connected with respect to T O.E.D.
.
...*
.
COROLLARY. Suppose p is a topologically faithful, analytic action of a Banach Lie algebra g on a Banach manifold M Then there exist a simply-connected Banach Lie group G with Lie algebra g and an analytic action r of G on M with differential p 7.6
.
.
PROOF. By 7.5, the subgroup G o := < exp(p(g)) > of Aut(M) is a connected Banach Lie group with Lie algebra g , acting analytically on M with differential p Now let n : G + G o he the universal covering group of Go Then the analytic homomorphism n induces an analytic action of G on M with differential p O.E.D.
.
.
.
COROLLARY, Every Lie algebra g of finite-dimension is the Lie algebra of a simply-connected Lie group G (of finite dimension ) 7.7
.
PROOF. By Ado's Theorem [21; Ch.11, there exist a finitedimensional vector space E and an injective homomorphism P : g + gi(E) The corresponding action of g on M := E is analytic and topologically faithful. Now apply 7.6.
.
Q.E.D.
7.8 COROLLARY. Let K be a closed subgroup of a Banach Lie group G with Lie algebra g Then there exist a (Hausdorff) topology T on K and an analytic structure on (K,T) such that ( K , T ) becomes a real Banach Lie group with Lie algebra k , given by 6.8.1. The inclusion homomorphism i : (K,T) + G is analytic.
.
INTEGRATION OF LIE ALGEBRA ACTIONS
115
By 6.8, k is a closed real subalgebra of g and hence a real Banach Lie algebra. By 7.2, the analytic action of k c 9 on M : = G is topologically faithful. Identify K with a subgroup of Aut(M) via left translations. Then PROOF.
.
.
exp( k ) c K By 6.26, we have g, E G & ( g ) for all g E G In case g E K , 5 . 1 6 implies that g* leaves k invariant. Put Ad(g) : = g*lk E G ( l ( k ) It follows from 7 . 4 that K is a real Banach Lie group with respect to a Hausdorff topology T uniquely determined by the property that the Lie algebra of ( K I T ) can be identified with k Since the diagram i (KIT) G
.
T
exp
-
TexP
k
commutes, 6 . 5 implies that As a homomorphism,
i
.
c
i
9
is analytic near
e
is analytic.
E
(KIT)
.
Q.E.D.
EXAMPLE. Suppose L is a Banach space over D E {R,C,H} , endowed with the norm 1 - 1 A Dlinear operator g E GL(L) is called an isometry (or isometric) if (gz( = IzI for all z E L An equivalent condition is The group lgl = 1g-'1 < 1 7.9
.
.
.
U&(L) : = { g
E
GR(L) : lgl = [g-ll < 1 }
of all D-linear isometries of
.
L
is a closed subgroup of
G&(L) By 7.8, U(1(L) is a real Banach Lie group (not necessarily in the operator norm topology) whose Lie algebra can be identified with the closed real subalgebra
of all "infinitesimal" D-linear isometries of 7.10
EXAMPLE.
Suppose
2
L
.
is an associative Banach algebra
.
over K , endowed with the norm I I Let Lxz := xz and R z := zy denote the left and right translation operators Y and R R = R on 2 , respectively. Since L L = L x Y XY X Y Y X , 7.9 implies that
SECTION 7
116
is a closed real subalgebra of u(Z)xu(Z)
3 (x,Y)
Lx-Ry
+
and
g(2) E
7.10.1
ua(2)
is a continuous Lie algebra homomorphism. Now suppose 2 is unital. Then the group U(Z) : = { g = { g = { g
lgl = 1g-ll < 1 } L E U!L(Z) } 4 : R E Ua(2) }
E
G(2)
:
E
G(Z)
:
E
G(2)
.
9
is a closed subgroup of G ( 2 ) By 7.8, U ( 2 ) is a real Banach Lie group (not necessarily in the norm topology) whose Lie algebra can be identified with
The homomorphism U(Z)xU(Z)
3
(g,h)
+
-1 LgRh
E
UX(2)
is analytic and its differential is given by 7.10.1. For any Banach space L over D E { R , C , E } , A := L(L) is an associative unital Banach algebra over the center K of D , and we have UI1(L) = U ( A ) and ua(L) = u ( A )
.
7.11 EXAMPLE, For Banach spaces E and F over D E { R , C , E } , consider the Banach space 2 := L(E,F) over the center K of D , endowed with the operator norm. By 7.9, the groups UL(E) , UI1(F) and U!&(Z) are real Banach Lie groups with Lie algebras ua(E) , uk(F) and u a ( 2 ) , respectively. The mapping Ua(F)xUa(E) 3 (a,d)
+
ga,d
E
ua(Z)
I
defined by ga,dz := azd -1 for all z E 2 , is an analytic group homomorphism whose differential is the continuous Lie algebra homomorphism
INTEGRATION OF LIE ALGEBRA ACTIONS
defined by
6
a,d
z
.
az-zd
:=
117
For infinite-dimensional Ranach Lie groups G , the "analytic" topology 7 defined on a closed subgroup K C G may be strictly finer than the topology induced from G
.
7.12 EXAMPLE. For 1 < p < +m , consider the Ranach space G := L ( 1 , R ) of all (equivalence classes of p-integrable P Then G i s a K-valued functions on the unit interval I Lie group under addition and
.
K :=
{ g
G : g(t)
E
z
6
for almost all
t
6
I ]
is a closed subgroup of G , since every convergent sequence in G has a subsequence which converges almost everywhere. Further, every homomorphism $ : R + K is trivial. In fact, let t E R and put f := $(t) Then f/2n = $(t/2n) E K for all integers n # 0 Therefore { s E I : f ( s ) = n } is a set of measure 0 , showing that f = 0 It follows that the "analytic" topology 7 on K is discrete.
.
.
On the other hand, topology induced from consider the curve
.
K G
I 3 t
. +
is arcwise connected in the To see this, let g E K and
gt := g'l[o,tl
E K
which is continuous by the dominated convergence theorem and Here l[o,tl denotes the satisfies go = 0 and g1 = 1
.
.
characteristic function of [O,tl In the following, an important class of closed subgroups of Banach Lie groups is introduced, for which the "analytic" topology T coincides with the induced topology. 7.13 DEFINITION. Let G(A) denote the group of all invertible elements of an associative unital Ranach algebra A over K A subgroup G of G(A) is called alqebraic of degree < d if there exists a family F of (Ranach space
.
SECTION 7
118
valued) continuous polynomials
f
on
< d
of degree
AxA
such t h a t
[
G =
g
: f(g,g
G(A)
E
-1
)
o
=
for all
G C G(A)
Since every a l g e b r a i c subgroup
f
.
j
F
E
is c l o s e d , 6 . 8
implies t h a t
{ x
g :=
: exp(tx)
g(A)
E
is a c l o s e d real s u b a l g e b r a . a r e K - a n a l y t i c on :=
{ 5
in his
( c f . 2.10.5).
:=
{
7.14
E
cX :
THEOREM.
larg(5))
Suppose
.
c
f
g(A)
E
.
F
Put
and l e t
= C\R-
As i n 2 . 1 1 ,
a l g e b r a i c of degree CA(g)
}
TI
R }
E
d e n o t e t h e ( p r i n c i p a l b r a n c h of t h e ) l o g a r i t h m
+ A
A
5
is a s u b a l g e b r a o f
LgXLgRk
GxG
commutes for all g E G and k E K , F is an immersion. Since K is a closed subgroup, F is a homeomorphism onto a closed subset R C GxG which is an equiva ence relation such By 8.1, R is a submanifold of G x G and that M = G/R F : GxK + R is hianalytic. Let n 1 : GxG + G denote the projection onto the first factor. Then the commuting diagram F GxK R-
t
.
nl
\ J
"1
G
shows that n l : R + G is an analytic submersion. By 9.14, M is a Hausdorff space and carries a unique manifold structure such that n : G + M is an analytic submersion. For the product mapping rG : GxG + G , we have a commuting diagram GxG G+----idGxn
I
GxM
rG
1.
AM
r Since idG x n is a surjective analytic submersion, 8 . 4 implies that the left translation action r of G on analytic. Now consider a splitting g = k @ m of the Banach by space g and define an analytic mapping + : g + G Then there exists a commuting diagram
8.13.1.
4 M-m
where space
M
a
k
,G
IT J,
is the continuous projection onto and
m
with null-
is
SECTION 8
138
= ro(exp X) = (exp X)K
8.19.4
.
for all X E m Taking differentials, we obtain the commuting diagram
is an analytic submersion and Te(n) is surjective and
Since TI : G + M 8.9 implies that
K =
TI
-1
( 0 )
,
.
Ker T,(TI) = Te(K)
.
By 8 . 1 2 , we have Te(K) = pe(k) It follows that T o ( $ ) is an isomorphism. By 4.1, there exist open neighborhoods T of 0 E m and P of o E M such that J, : T + P is -1 bianalytic. Then p := + : P + m satisfies 8.19.2. Since r = TI , 6 . 1 3 implies that 0
has the null-space Let r : G Banach manifold
+
k
Aut(M) be an action of a group M The set
.
is called the orbit of :=
Q.E.D.
ro(G) = { r(g,o) : g
G * o :=
G~
.
{
o
g E G :
E
M
under
E
G
]
G
,
and
G
on a
r(g,o) = o J
is called the isotropy subqroup of
G
.
at
o
.
Then
defines a bijection + : G / G o + G * o The action r is called transitive if M = G - o for some point o E M The transitivity of an action of M is a "global" property. It is often convenient to use a "localized" version of transitivity.
.
139
SUBMANIFOLDS AND QUOTIENT MANIFOLDS
8.20
DEFINITION.
analytic action
An
r
of a Ranach Lie
group G on a Ranach manifold M is called locally transitive at o E M if the evaluation mapping r0
: G + M
.
is an analytic submersion at e E G The action r is called locally transitive on M if r is locally transitive In terms of the differential at every point rn E M p : 4 + aut(M) of r , local transitivity is equivalent to the condition that
.
is surjective and has a split null-space
8.21
PROPOSITION.
Let
r
Ker(po)
.
be an analytic action of a Ranach
Lie group G on a Ranach manifold M which is locally Then the orbit G o o is an open subset transitive at o E M
.
of M , the isotropy subgroup K : = Go is a Ranach Lie subgroup of G , and the canonical bijection
is bianalytic. In terms of the differential p : g + aut(M) of r , the Lie algebra k of K is given by k = Ker(po) PROOF.
Since
ro
is a submersion at r0
e
E
G
and the diagram
G
commutes for all g E G , ro : G + M is a submersion. Ry -1 G * o = ro(G) is open. By 8.9, K = r0 ( 0 ) is a 8.4.iv, submanifold and hence a Banach Lie subgroup of G with tangent space pe(k) = Te(K) = Ker Te(ro) = pe(Ker p o l Hence
h = Ker(po)
.
.
For the canonical projection
.
SECTION 8
140
n : G
+
G/K
the diagram
commutes. Since ro bianalytic by 8.4.
and
'II
are analytic submersions,
4
is
O.E.D.
8.22 COROLLARY. Every locally transitive analytic action of a Banach Lie group G on a connected Ranach manifold M is transitive.
.
PROOF. By 8.4.iv, every orbit of G is open in M Since M is the disjoint union of all orbits, it follows that every orbit is also closed in M Since M is connected, the assertion follows. Q.E.D.
.
8.23 EXAMPLE. Suppose G is a Ranach Lie group. Then the direct product group GxG acts on M := G via the analytic action r(g,h)(m) := gmh -1 for all g,h,m E G This action is clearly transitive and the isotropy subgroup of GxG at e E G can be identified with G (embedded as the diagonal). The differential p of r satisfies
.
.
for all x,Y E g Since 2(X,Y) = (X+Y,X+Y) + (X-Y,Y-X) I it follows that G x G acts locally transitively on G By 8.20, there exists a hianalytic mapping
In particular, for every Banach space L over there is a bianalytic mapping (over the center
In case
L = Dn
is finite-dimensional, we get
.
D E {R,C,H} K of D )
,
SUBMANIFOLDS AND QUOTIENT MANIFOLDS
141
8.24 LEMMA. Suppose r is a locally transitive analytic action of a connected Ranach Lie group G on a connected Then the "lifted" analytic action rBanach manifold M of G on the universal of the universal covering group Gcovering manifold M- of M is locally transitive.
.
-
PROOF. Let n G : G- + G and n M- + M denote the M' covering projections. Then the diagram 6.14.1 commutes. For o E M- , put m : = nM(o) E M Then there is a commuting diagram
.
Since
r,
is a submersion and
bianalytic, it follows that
:r
and n M are locally is a submersion. Q.E.D.
nG
NOTES. For more information about immersions, submersions, submanifolds and quotient manifolds, we refer to C97; Ch. 11, §2],[121; LG, Ch. 111, § § 10-121 and C20; § 5, n07-113. Godement's Theorem is proved in C121; L G , Ch. 111, 5 121(in the finite dimensional case) and is stated for Banach manifolds in [20; 5.9.51. Immersions and submersions are special cases of the so-called subimmersions which admit, about any point, linear local representations with split range and nullspace C20; 5.10.31. Subimmersions between finite dimensional manifolds are also characterized by the property that the rank of the derivatives is (locally) constant C20; 5.10.61. In the infinite dimensional setting, there exist two natural notions of submanifold (and Lie subgroup) which coincide for finite dimensional manifolds and Lie groups (cf. C20; 5.8.3 and 5.12.31). The definitions 8.5 and 8.11 correspond to the stronger version often referred to as "direct" submanifold and "direct" Lie subgroup.
142
9.
SECTION 9
BINARY BANACH LIE ALGEBRAS
Binary Banach Lie algebras are certain graded Lie algebras of polynomial vector fields on a Banach space Z which give rise to a quotient manifold of fundamental importance for the description of symmetric Ranach manifolds. Suppose in the following that Z is a Ranach space over K E {R,C} and let P(Z) =
@ Tn(Z) n >-1 denote the graded Lie algebra of all polynomial vector fields on Z We will often identify the Lie algebras To(Z) and
.
ga(z)
.
9.1 DEFINITION. A binary Lie algebra on Z is a subalgebra h of P ( Z ) contained in T,l(Z) @ To(Z) @ T1(Z) , such that T-l(Z) C h
9.1.1
I : = z -aa zE h .
9.1.2
and
For any
b
E
,
Z
define the "constant" vector field
a Yb : = b az
E
rl(Z)
.
9.1.3
9.2 PROPOSITION. A closed binary Lie algebra h Banach Lie algebra and has an additive gradation
on
is a
Z
where
hn If
g
{ X
=
E
h : [X,Il = nX } =
is a subspace of
{
x
E
h : [X,gl
h
containing =
In particular, the center of
to) } h
=
lo}
hnTn(Z). h-1
and
I
,
then
.
is trivial.
0.E.D.
BINARY BANACH LIE ALGEBRAS PROOF. Let and 1.7, h by the norm
143
.
denote the open unit ball of Z By 4.6.2 is a Ranach Lie algebra for the topology induced X + IX(id)l, For X E h , write
R
.
x = x -1
,
+xo+xl
.
where Xn E T n ( Z ) By 9.1.1, h contains X-l and hence X+ : = Xo + X1 By 9.1.2, h contains X1 = [X+,I] and hence Xo Now suppose [X,g ] = { O } Then 0 = [X,Il = X 1-X-l I showing that X E T o ( Z ) Since
.
.
.
.
YXb = [X,Yb1 = 0 for every
b
E
, we
2
X = 0
have
.
O.E.D.
K and consider the Banach algebra B associated with Z via 3.5.1. Let g be a subspace of R which is closed under the commutator product [X,Y] = XY-YX and satisfies the associativity condition. By 4.12, g is a Lie algebra and 9.3
EXAMPLE.
Let
Z
be a Banach space over
(s
:= (az + b d)# b
-
zcz
-
z d )a E
defines a Lie algebra homomorphism from g into p(Z) in addition, g contains all matrices of the form
for
b
E
Z
.
If,
and a matrix 0
for different scalars Lie algebra on
Z
0 s,t
E
1
t*idz K , then
h := g #
is a binary
.
EXAMPLE. Let L be a Banach space over D E {R,C,E} and consider a splitting 3.8.1 of L Then 2 := L(E,F) is a Banach space over the center K of D and, by 3.11, L(L) is a closed unital subalgebra of the Banach algebra R which satisfies the associativity condition. Hence 9.4
.
SECTION 9
144
(z
:= (az+b-zcz-zd
b,
d #
1-aa z
defines a continuous Lie algebra homomorphism from g := g!t(L) onto the binary Ranach Lie algebra h := g # 2 having the components
-- { b a
h-,
: b
ho = { (az-zd)
on
L(E,F) }
E
&:a
E
,
L(F)
d
E
L(E)
}
and hl = { zcz aaz : c 9.5 let
.
L(F,E) }
E
LEMMA. Suppose h is a binary Lie algebra on 6 : h + P ( 2 ) be a derivation. Then
PROOF.
61h-1 = 0
,
hn
,
For
X
E
*6
61 = 0
61 = 0
2
and
= 0 .
implies 9.5.1
Hence 6(hn) c Tn(Z) by 4.11.2. b E 2 , 61h-l = 0 implies Y
For
ho
E
and
(6X)b - t6X,Yb1 = 6[X,Yb1 = 6(YXb) = 0 61ho = 0
It follows that
6X = f(z)
a E
.
For
X
[X,Yb]
E
ho
h,
put
implies
2f"(z,b)
a = aZ
[6X,Ybl = 6[X,Ybl = 0
6(h, = 0
It follows that
E
,
i.e.,
6 = 0
.
.
LEMMA. Supose h is a binary Lie algebra on $ : h + P(2) be a homomorphism. Then
9.6
PROOF.
For
x
E
.
.
T1(Z)
az
Then
X
hn
,
$1 = I
implies
Q.E.D. 2
and let
145
BINARY BANACH LIE ALGEBRAS
$(hn) c ~ ~ ( 2by) 4.11.2. = id implies
Hence
X = h(z)
a
For
E
X
E
h,
and
b
E
Z
hl
h
is a binary Lie algebra on 2 and let 6 : h + P ( Z ) be a continuous derivation. Then the following conditions are equivalent:
9.7
Suppose
LEMMA.
,
(ii)
6 1 = 0
(iii)
6(hn)
(iv)
6 = ad(X)
c
Tn(Z)
,
for all
x
where
E
n T,(Z)
, is uniquely
determined. PROOF.
For any :=
where
Yn
E
b
E
Z
,
1
6(yb) =
Tn(Z)
.
put yn
(finite sum)
t
n >-1 Then
61
E
T,(Z)
implies
,
146
SECTION 9
= Y(61)b
-[I,6(Yb)I
E
h-1
@
.
Hence Yn = 0 for all n > 0 and 6 1 = 0 Therefore (i) implies (ii). By 9.5.1, (ii) implies (iii) and, trivially, (iv) implies (i). Now assume (iii) and define X E T o ( Z ) by 6(Yb) = YXb
9.7.1
for all b E 2 , Then q : = 6 - ad(X) : h + P ( Z ) is a derivation with nlhql = 0 and n I E T o ( Z ) Hence n I = 0 by the first part of the proof. Now 9.5 implies n = 0 It is clear that X is uniquely determined by 9.7.1.
.
.
Q.E.D.
9.8
DEFINITION.
For a binary Lie algebra
h
on
put
Z
and
Then %; is a binary Lie algebra on Z and (%: )" = A binary Lie algebra h on Z is called full if h = h 9.9
LEMMA.
6 : h +
Suppose
h
is a binary Lie algebra on Then
A
X
.
and
2
h is a continuous derivation.
,
6 = ad(X)
where
9.
E
%:
9.9.1
is uniquely determined. A
+ X1 E h , Then Put 1 = x- + xo A n : = 6 + ad(X-l-X1) : h + h is a continuous derivation satisfying PROOF.
nI = 61
-
X-l
-
X1 = Xo
E
A
h,
.
By 9.7, it follows that n = ad(Y) , where Therefore 6 = ad(-X-l+Y+X1) Since
.
Y
E
To(Z)
.
A
BINARY BANACH LIE ALGEBRAS
[Yrhll = n(hl)C it follows that X the centralizer of
A
ho
E
h
h1
147
r
and hence -X-l + Y + X1 in ? vanishes by 9.2, X
E
$
.
Since is uniquely
determined by 9.9.1.
O.E.D.
9.10 COROLLARY. Suppose h is a full binary Banach Lie algebra and let aut(h) denote the Ranach Lie algebra of all continuous derivations of h Then
.
ad : h
+
aut(h)
9.10.1
is an isomorphism of Ranach Lie algebras. It will now be shown that one can associate, in a natural way, a quotient Ranach manifold N with every full binary Banach Lie algebra h on 2 This quotient manifold is closely related to the symmetric Banach manifolds which are "modelled" over Z Suppose h is a binary Banach Lie algebra on Z Then the group Aut(h) of all continuous automorphisms of h is an algebraic subgroup of Gk(h) of degree 6 2 By 7.15, Aut(h) is a Banach Lie group in the operator norm topology
.
.
.
. .
whose Lie algebra can be identified with aut(h) Let H denote the identity component of Aut(h) Then H is an open subgroup of Aut(h) and hence a Banach Lie group with Consider the closed subalgebra Lie algebra aut(h)
.
.
h, of
h
:=
h,
@
hl
9.11.1
and define a closed subgroup
H,
of
H
by 9.11.2
Let n -1
*
P(Z)
+
rl(Z)
denote the canonical projection. 9.11
LEMMA.
Let h
be a full binary Banach Lie algebra on
148
Z
SECTION 9
.
Then H+ is a Banach Lie subgroup of H whose Lie via the isomorphism algebra can be identified with 12, 9.10.1. PROOF. Since h, is a closed suhalgehra of h , every satisfies exp(ad X ) E H+ Conversely, suppose X E h, By X E h satisfies exp(t*ad X) E H+ for all t E R differentiation, it follows that (ad X ) h + C h+ Write X = X- 1 + Xo + X1 Since I E h, , it follows that
.
.
.
X1 - X-l = [X,I] Theref ore
{ X
E
E
h,
h : exp(t*ad X )
.
H+
E
h
is a split subspace of
Hence
.
X-l = 0
for all
t
and
E
X
E
.
h+
.
R } = h,
Further,
n-l(exp(ad X ) I ) = -X -1
+ f(X)X-l ,
where f : h + I ( h - l ) is an analytic mapping satisfying f(0) = 0 , It follows that there exists a neighborhood of 0 E h such that exp(ad X) for all
X
E
T
.
E
H,
x
E
T
h+
Now the assertion follows from 8.13.
9.12 COROLLARY. For a full binary Aanach Lie algebra on Z , the quotient space
Q.E.D.
h
N : = H/H+
is a connected Banach manifold and r(g,hH+) : = ghH, defines There exists a chart an analytic action r of H on N (P,p,Z) of N about o : = H+ such that p ( P ) = Z and every b E 2 satisfies
.
p(exp(ad Y,)*o) PROOF.
=
Consider the analytic mapping $(u) := ro(exp(ad Y,))
.
h
. J, :
Z
+
N
defined by
BINARY BANACH LIE ALGEBRAS
149
Since h = h-l 8 h+ , $ is locally bianalytic at 0 Since h-l is abelian, it follows that for every v E there is a commuting diagram
z
J,
,
$r(g) J,
.
.
g : = exp(ad Yu-v) g(1) = I
u = v
.
Let p translation express the "canonical"
.
Hence
g
E
is injective on
J,
.
It follows P := $ ( Z ) is
H
E
+ Yv-u
Therefore
Z
?M
where Lvu : = u + v and g : = exp(ad Yv) E H that J, is locally bianalytic on Z Hence an open subset of N For u,v E 2 I
satisfies
.
> M
LV.l
z
2
E
: h + aut(N)
H+ 2
.
if and only if 9.E.D.
denote the differential of the left action r of H on N Our next goal is to vector fields p X for X E h in terms of the chart (P,p,Z) of N about o For this we
.
.
need some results about automorphisms of binary Lie algebras. For any binary Banach Lie algebra h = h-l
h,
@
8
hl
on
the sets
2
Ha : = { exp(ad X ) are abelian subgroups of Ho : = { g
E
:
H
X
for
E
ha } a =
+_
1
,
whereas
H : g(1) = I }
is a closed subgroup of
.
H
LEMMA. Suppose h and k are binary Banach Lie algebras on Z and let g : h + k he a surjective continuous ~ homomorphism such that g(1) E h + and ~ - ~ o g l h=- id Then h = k and g E H1 9.13
.
.
PROOF.
Put
g(11 =
x
+ Y
E
ko
8
kl
Then
SECTION 9
150
for all
h
E
.
2
Hence
X = I
and therefore
and
$ : = exp(ad Y )
I t follows that g(Yb)
= +(Yb)
f o r all
b E 2
$I = I
+
E Aut(k) and
.
= I+Y = gI
[Y,Il
Y = -gX
Since
g
is surjective, we have
Since
g
is continuous, the diagram
and
g
h
.
+
h
,
9 $
-1
is injective, i.e.,
og :
.
idh = $-log = goexp(ad X ) h = h
E
.t $-I
h-k
commutes. Applying 9.6 to the homomorphism we get for all x E h
Hence
x
f o r some
9
h-h
exp(ad X )
satisfies
g =
E
H1
. 0.E.D.
9.14 LEMMA. Suppose h 2 and let g : h + P ( 2 )
glh-, 4
E
= id
and
g(1)
E
is a binary Banach Lie algebra on be a homomorphism such that T-l(Z)
@
To(Z)
.
Then
g ( h ) = h
and
H-1
PROOF.
g ( 1 ) = -Y
Put
U
+ X
E
T-l(Z)
@ To(Z)
.
Then
‘b
for all
b
satisfies
E
2
0 k-l
.
Hence X = I and $ := exp(ad Y u ) E H-l = id and g($-lI) = g(I+Yu) = I Applying
.
BINARY BANACH LIE ALGEBRAS
go@-1 : it + P ( 2 ) is injective, i.e., g =
9.6 to the homomorphism g h =
9.15
g
E
h
and
LEMMA.
Aut(h)
g
.
151
,
it follows that
@
E
H-l
.
O.E.D.
Suppose h is a binary Banach Lie algebra and Then the following conditions are equivalent:
(ii)
g(hn) = hn
(iii)
g =
,
y*
for all
where
y
E
n
,
Gk(Z)
is uniquely determined.
PROOF. It is clear that (iii)* (i)+ (ii). Now assume (ii). Put X : = g(1) Then we have for all b E 2
.
Yxb = [x,ybl = [gl,ybl = g[I,g g(g
-1
'b)
X
It follows that Y
Yb
:=
=
b'
=
I
-1
'b]
=
. .
Now define
y
E
GI1(2)
by
9(Yb) = Y*(Yb)
.
for all h Applying 9.6 to the homomorphism -1 y * og : h + P ( 2 ) , the assertion (iii) follows. 9.16 Then
LEMMA.
Let
h
b e a binary Banach Lie algebra on
.
It is clear that HOHl C H+ Now let g E H + -1 E H+ Let 'TI+ : h + h+ denote the canonical projection. Then (n-lo$)(~+og) = 0 implies PROOF. @
:= g
.
Yh - $ ( g Y h ) = (r-lo@)(n-log)Yh and
Q.E.D. 2
.
.
Put
152
SECTION 9
I
og h -1
Hence gI
E
h+
and
.
Ga(h-l)
E
n-lO9lh-l
Now s u p p o s e
.
Define
G E ( ~ - ~ )
E
g
E
H
y
E
satisfies by
G!2(Z)
Y Yb : = n -1 o g ( y b ) = Y*('b) for a l l
b
.
2
E
Then
is a b i n a r y B a n a c h L i e
: = y,'(h)
h_
a l g e b r a on 2 and t h e c o n t i n u o u s isornorphism -1 @ := y * o g : h + k s a t i s f i e s $ 1 E h_+ and
-1
T-l($yb) = "-1(Y* By 9 . 1 3 ,
i t follows t h a t
we have
y*
and
Ho
E
h = k
g = y*$
B i n a r y L i e a l g e b r a s on o p e r a t o r s on
,
2
-1
gyb) = Y* and
2
@
E
.
HOHl
E
= b '
("-1g'b) H1
.
Since
g
E
H
,
O.E.D.
g i v e r i s e t o a f a m i l y of
c a l l e d "Hergmann o p e r a t o r s " .
T h i s name
stems f r o m t h e f a c t t h a t f o r s y m m e t r i c m a n i f o l d s o f f i n i t e d i m e n s i o n , t h e s e o p e r a t o r s a r e c l o s e l y r e l a t e d t o t h e Rergmann kernel functions. 9.17 z
DEFINITION.
E
and
2
h
Let
2
defines a linear operator
.
Let
2hA(z,h-(z,v))
-
B(z,h)
c a l l e d t h e Bergmann
z
operator associated with
K
Then
+
v - 2hA(z,v)
B ( z , ~ ) v :=
.
P2(Z,Z)
E
b e a Banach s p a c e o v e r
L(2)
E
and
.
h
h*(v,h(z))
9.17.1
The mapping
is a n a l y t i c and s a t i s f i e s
I n case
R(z,h)
E
Ga(Z)
,
define the quasi-inverse
z h := B ( z , h ) -1 ( z - h ( z ) ) 9.18
D
E
EXAMPLE. {R, C,El }
center
K
of
.
Let Then
D
.
E
and 2 :=
Given
F
E
2
.
9.17.2
be Ranach s p a c e s o v e r
L(E,F) is a Ranach s p a c e o v e r t h e c E L(F,E) , d e f i n e h E P2 ( Z , Z )
BINARY BANACH LIE ALGEBRAS
by
.
h(z) := zcz
Then
153
h-(z,v) = (zcv+vcz)/2
and therefore
R(z,h)v = (idF-zc)v(idE-cz) for all z,v then R(z,h)
.
E
2
E
Gk(2)
If idF-zc and
zh = z(idE-cz) -1
Gk(F)
E
and
(idF-zc)-1 z
=
idE-cz
E
Gk(E)
I
,
i.e.,
is given by a Moebius transformation.
9.19 2
LEMMA.
h
Suppose
is a binary Banach Lie algebra on
and let
a X = h(z) az
Then every
b
h1 '
satisfies
2
E
E
a exp(ad X)Yb = R(z,-h)b az PROOF. 9.17.1. 9.20 Z
[hl,hl]
Since
= 0
,
.
the assertion follows from O.E.D.
LEMMA.
h
Suppose
is a binary Banach Lie algebra on
and let
exp(ad X) exp(ad Yb)H+ = exp(ad(bh &))H+ if
b
PROOF.
E
and
2
For
b,v
is small.
Ibl E
2
,
define
g : = exp(-ad Yb)exp(-ad X) exp(ad Yv) Since 9.19 implies
E
H
.
SECTION 9
154
exp(-ad X) exp(ad Y v ) I
a = -Y B(z,h)v -
I
-
-
+
(h-(v,h(z))
X
= exp(-ad
az
-
V
X)(I-Y
V
)
=
+ (z+2hrr(zIv))a a
2hrr(z,hrr(z,v))- h ( z ) ) jy
I
it f o l l o w s t h a t
Now s u p p o s e
u
E
.
2
Then 9.19
implies
exp(-ad X)exp(ad Yv)Yu
= exp(-ad
X)Yu = R(z,h)u
a az
and hence
Hence 9 . 1 6 9.21 on
THEOREM. 2
g
implies
E
h
Let
.
H+
O.E.D.
b e a f u l l b i n a r y Banach L i e a l g e b r a
and consider t h e c a n o n i c a l c h a r t
N := H/H+
o := H+ ,
about
:
p
h
+ aut(N)
P*(PX)
PROOF.
x
E
h
Suppose
=
x
. X
E
h
of
Then t h e d i f f e r e n t i a l
of t h e l e f t t r a n s l a t i o n a c t i o n
for all
(P,p,Z)
and put
r
of
H
on
N
satisfies 9.21.1
BINARY BANACH LIE ALGEBRAS
P*(PX) = f(z) Let
y
be a smooth curve in
2
.
a
such that
y(0) = b
exp(t*ad X) exp(ad Y b ) I i + = exp(ad Yy(t))H+ if It1 is small. 5.0.4 implies
y(t) = p(exp(t*adX)(p-'b))
Then
2 (0) .
f(b) = In order to show n E {O,l,-l} y(t) : = b + tv f(h) = v For
.
.
155
and 9.21.2
and
9.21.3
9.21.1, we may assume X E hn for For X = Y E h-l , the curve V satisfies 9.21.2. Hence 9.21.3 shows X E ho , 5.33 implies
exp(t*ad X)exp(ad Yb)H+ = exp(t*ad X) exp(ad Yb) exp(-toad X)H+ = exp(ad(exp(t0ad where
X)Yb))H+
=
exp(ad Y y(t)lH+
.
y(t) = exp(tX)b Hence 9.21.3 Now suppose
f ( b ) = X(h)
.
X = h(z) a
E
implies
.
h,
By 9.20, the curve y(t) = bth = B(b,th)-l(b-th(b)) satisfies 9.21.2.
Since
B(b,th)z = z
-
differentiation of 9.21.4 f(h) =
B ( b , O ) = idZ
2tL(b,~
9.21.4
and
+ t 2 (2c(b,h^(b,~))-hn(z,h(b)
shows
d -(= B(b,th)t=O)b
-
h
NOTES.
The concept of binary Lie algebras and the basic results about their derivations and automorphisms are due to
M. Koecher [95]. The construction of the homogeneous Banach manifold N associated with h appears in C841. A shorter proof of 9.21 can be based on a modified version of 9.6.
,
SECTION 10
156
10.
LOCALLY UNIFORM TRANSFORMATION GROUPS
In classical Lie group theory, i t is often the case that analytic properties follow from algebraic-topological conditions. For example, every closed subgroup of a finitedimensional real Lie group is a Lie group in the relative topology and every continuous homomorphism between real Lie groups is analytic. Lie groups can also be characterized in algebraic-topological terms as locally compact groups having "no small subgroups" [ 7 7 ] . In the following two sections it is shown that similar results hold also in the infinitedimensional case of Ranach Lie groups. The basic idea is to consider actions on Banach manifolds which satisfy a strong continuity condition. 10.1 DEFINITION. Suppose r is a continuous action of a topological group G on a Banach manifold M over K A local representation of r with respect to a chart (P,p,Z) of M about o is a continuous mapping
.
10.1.1 where S i s a neighborhood of e E G and D domain in Zc : = ZOKC such that every g E S
is a bounded satisfies
10.1.2 and
whenever
m
E
p-l(D)
and
r(g,m)
E
P
.
10.2 LEMMA. Let r# : S + Om(D,Z C be a local representation of r with respect to the chart (P,p,Z) M about o Then r#(e) = idD and
.
for all z in a neighborhood of gh belong to S
.
0 E D
whenever
g,h
of
and
LOCALLY UNIFORM TRANSFORMATION GROUPS
157
The first assertion follows from 1.11. Now suppose Then 10.1.2 implies n : = r(h,m) E p-'(D) and g,h,gh E S r(g,n) = r(gh,m) E p-I(D) for all m in a neighborhood of o E P. Hence 10.1.3 implies
PROOF.
.
Now the assertion follows from 1.11, 10.3
DEFINITION.
group G on if for every M about o with respect
A
0.E.D.
continuous action
r
of a topological
a Ranach manifold M is called locally uniform point o E M there exists a chart (P,p,Z) of such that r has a local representation r# to this chart.
10.4 PROPOSITION. Every analytic action r of a Ranach Lie group G on a Banach manifold M is locally uniform.
PROOF.
Choose a chart
(P,p,Z) of
M
about
o
,
an open
neighborhood S of e E G and a bounded domain D in Z c -1 containing 0 , such that r(S,o) c p (D) and there exists an analytic mapping F : S ~ +D zC satisfying F(g,pm) = p(r(g,rn)) whenever (g,m) E Sxp-l(D) and r(g,m) E P We may assume that there exists a convex open neighborhood T of 0 E g such that exp : T + S is bianalytic and
.
ZED where to X
D1 E T
mapping of r
.
denotes the first partial derivative with respect Then r#(g)(z) := F(g,z) defines a continuous S + O,(D,Z C ) which is a local representation r# :
.
10.5 LEMMA. Let representation of
Q.E.D.
.
C
r# * S + O,(D,Z ) be a local r with respect to a chart (P,p,Z) of M about o and let C be a convex open neighborhood of 0 E D such that R : = dist(C,aD) > 0 Suppose g E S
.
158
SECTION 1 0
satisfies
j < k
for
1
gj
,
1 < j < k
for
S
E
where
I
r#(gj)-id
d
1 Hence r (g) = idC , showing that r(g,m) = m for all m in a # neighborhood of o E P Since M is connected, 3.1 implies for all
g
r(g) = idM
and a l l integers
H
E
.
.
Since
r
is faithful, we get
H = {e}
.
O.E.D.
10.7
COROLLARY.
A Ranach Lie group
G
has no small
subgroups. PROOF. of
Since
G
is locally connected, the identity component
is an open subgroup.
connected. M :=
G
G
Hence we may assume that
The faithful left translation action of
is
G
G
is analytic and hence locally uniform by 10.4.
apply 10.6.
on Now O.E.D.
A s the main result of this section, it will be shown that
for real Ranach Lie groups, the converse of 10.4 is also true, i.e., every locally uniform action of a real Ranach Lie group case
G G =
is in fact analytic. R
.
r
We consider first the special
is a locally uniform action of Consider a local representation r # : S + O O D ( D , Z L ) of r with respect to a chart (P,p,Z) of M about o and let C be a convex open neighborhood of 0 E D such that R := dist(C,aD) > 0 Then the limit 10.8 THEOREM. Suppose R on a Ranach manifold
M
.
.
h := lim (r (t)-id)/t # t+O exists and there exists IaxC 3 (t,z)
0
+
C
> 0 such that
r#(t,z)
defines an analytic mapping.
10.8.1
Om(C,Z )
E
E
2
C
IaC S
and
SECTION 10
160
.
PROOF'. Put d := R/3 and define Ck : = Ukd(C) By 1.17, af(z) : = f ' ( z ) defines a continuous linear mapping
a
C : om(D,2 ) -+ A :=
Hence there exists
T
>
C
om(C2, L(2
such that
0
.
))
and
IT C S
\r#(t)-idI,, < d
10.8.2
and
1
(r#(t)'-e A
< 1
10.8.3
c2
for all t E IT , where e ( z ) = id denotes the unit element A Define of the associative Ranach algebra A
.
where
0
< T
0 Then for any h E Om(C,Zc) , the vector field
.
.
X := h(z)
a
E
T(C)
generates a local analytic flow rh : IxR t E I has the following property:
+ C
such that every
Whenever gn E G and kn E R are sequences satisfying knt + +- , gk E S for all k < qn : = [knt] and such that the limit
LOCALLY UNIFORM TRANSFORMATION GROUPS
h = lim kn(r#(gn)-id) n+exists, it follows that
Here
E
Om(C,ZC)
and define R1 : = U d ( B ) By 5.7, u , >~ 0 such that X generates a r h : ITxR1
+
C
and the mappings
ht : = (rh(t)-id)/t
Since
IhtIBl
.
.
Put d : = R/2 there exist constants PROOF.
satisfy
10.13.1
< q
[q] denotes the greatest integer
local analytic flow
165
0/2
rh(t) = t*ht+id
Ih;IB,
I
0 Put d : = R / 2 and C := Ud(13) Given X,Y E g , define may assume r(S,o) C p-'(C)
.
9,
exp(tX)
:=
, qt
:=
.
exp(tY)
and put
X1 : = X+Y
E
.
We
T(M)
X2 : = [X,Y] E T ( M ) , h : = ( X# +Y # )id E om(C,Z C ) and IC ) h2 := [X ,Y ]id E Om(C,Z For t > 0 and s := t1j2 #
.
#
, ,
define
and 2
n'
: = 's/n
+s/n +-s/n '-s/n
By 10.13, the vector fields
generate local analytic flows r : I T x R + C having the j property stated in 10.13. Using the Campbell-Hausdorff series [21; 11.71 we may assume that for k < n j , whenever that
since
is closed.
S
r#
contains all powers It1 < T We may also assume
S
.
Then 6 . 7 implies
is continuous,
exp(tX.1 3
E
S
and
By 10.12, we have
Therefore 10.13 implies
.
By Hence r.(t) = r (exp(tX.)) on B whenever 0 C t < T 3 # 3 differentiation, it follows that (X.p)(m) = h.(pm) for all 7
3
SECTION 10
168
.
m E p-l(B) showing that r* is an action of 9 on M * g M + O_(C,Zc) as in 10.11. Then r,(g)CgM Now define p# * and the mapping p#or, : g is real-linear since
holds on p-l(C) all g E S Let
.
4
such that gk : = exp(X/k) 0
E
.
-
.
r*
+
C Om(C,z )
10.14.2
is real-linear and the identity
We may assume that (r#(g)-idlD < d/4 for T be a star-like open neighborhood of exp(T) C S For X E T and k > 1 , put
.
Then 10.5 implies klr # (gk )-idlC c 2 r#(exp X)-idlC
6
d/2
.
.
Hence For k + , we get Ip#(r+X)lc c d/2 for all X E T 10.14.2 is a continuous mapping, show ng that the action r* O.E.D. of g on M is locally uniform. 10.15 THEOREM. An action r of a Banach Lie group G on a Banach manifold M is analytic if and only if r is locally uniform and its differential r* : g + aut(M) C J(M) is linear. By 10.4 and 6.12, every analytic action r is locally uniform and has a linear differential. Conversely, suppose that r is a locally uniform action whose differential r* is linear. Since the action r* of g on M is locally uniform by 10.14, 5 . 3 0 implies that r* is an analytic Lie algebra action. Now apply 5.32. O.E.D. PROOF.
10.16 COROLLARY. An action r of a real Ranach Lie group G on a Banach manifold M is analytic if and only if it is locally uniform. 10.17 COROLLARY. Suppose r is a locally uniform action of a topological group G on a Banach manifold M Let H be a real Lie group and let 4 : H + G be a continuous homomorphism, Then roe : H + Aut(M) defines an analytic
.
169
LOCALLY UNIFORM TRANSFORMATION GROUPS
action of
H
on
M
.
PROOF. Since r is locally uniform and 4 is continuous, the action ro$ is locally uniform and hence analytic. O.E.D.
10.18
COROLLARY.
Every continuous homomorphism
4 :
H + G
between real Lie groups is analytic. PROOF. Apply 10.17 to the (analytic) left translat on act on of G on G Q.E.D.
.
10.19
COROLLARY.
A continuous homomorphism
4 :
G + H
between complex Lie groups is holomorphic if and only if its differential $ * : 4 + h is complex-linear. PROOF. Since 4 is real-analytic by 10.18, 4 is holomorphic if and only if Te(4) : Te(G) + Te(H) is complexlinear. O.E.D. It will now be shown that, for locally compact groups and manifolds, every continuous action is already locally uniform.
In general, the set C(M,N)
:=
{ f
: M + N
: f
continuous ]
of all continuous mappings between locally compact spaces M and N will be endowed with the compact-open topology [921 having a subbasis of open sets (K;Q) := { f
C(M,N)
E
: f(K) C Q
]
for all compact subsets K of M and all open subsets 0 of N In case N is a uniform space, the compact-open topology coincides with the topology of uniform convergence on
.
all compact subsets of
M
.
10.20 PROPOSITION. Every continuous action r of a topological group G on a locally compact complex manifold M is locally uniform.
170
SECTION 1 0
PROOF.
Since
r : GxM + M
is continuous and
M
is locally
compact, the homomorphism
is continuous [92]. Let (Q,p,Z) be a chart about o E. M and let P be an open connected neighborhood of o E Q such that P is compact and contained in Q Then D := p ( P ) is a bounded domain in 2 and S : = { g E G : r(g) E ( 7 ; Qn ) ( o ; P ) } is an open neighborhood of e E G For (g,m) E SxP , define O.E.D. r#(g)(pm) : = p(r(g,m))
.
.
.
Note that a connected Ranach manifold
M
over
K
is
locally compact if and only if M is finite-dimensional. For locally compact groups, we have the following somewhat deeper result. 10.21 THEOREM. Every continuous action r of a locally compact group G on a locally compact manifold M is locally uniform. PROOF. Choose a chart (O,p,Z) of M about o , a compact neighborhood T of e E G and a fundamental system { Bn : n E N } of connected open neighborhoods €3, of -1 0 E Zc such that On : = p (R,) satisfy r(T,Qn) C Q Then pn(f) := fop defines an injective continuous linear
.
mapping
We may assume that there exists a compact neighborhood K of p(Q) in Zc Ry Montel's Theorem [log; 1.61, the set
.
Kn
:=
{ f
E
C)(Rn,ZC
: f(Rn)
C K }
is compact. Since 4,(g) := pOr(g))Qn defines a continuous -1 (pnKn) mapping +n : T + C(Qn,Z C ) , it follows that Tn : = $n is a compact subset of T and an := p n-1 0 4 ~: Tn is a + Kn continuous mapping. For every g E T , K is a neighborhood
LOCALLY UNIFORM TRANSFORMATION GROUPS
.
of p(r(g,o)) E Zc Hence there exist such that the diagram
commutes.
Ry definition,
g
E
Tn
and
un
n
E
N
171
and
.
f = an(g)
f
.
there exist open neighborhoods p(m)
Zc
E
N
of
m E
Q
and
C
Kn
Since
is a Raire space [ 9 2 ] and T = T, , Tn has an Since m : = r(h,o) E Q interior point h for some n
G
E
,
of
such that there is a commuting diagram
is holomorphic. N o w choose a closed neighborhood S of e E T with hS C Tn and a connected open neighborhood B of 0 E Bn such that r(hg,o) E N and an(hg)(R) C C for all g E S Then r(g,o) E Q and r#(g) : = foan(hg) defines a continuous mapping
where
f
.
r# :
S + O(R,Z
C
)
satisfying
r#(g)(pm) = f(p(r
.
-1 for all m E p ( B ) N o w et D be a relatively compact connected open neighborhood of 0 E R , define P : = p -1 ( D ) and replace S by { g E S : r(g,o) E P } Q.E.D.
.
NOTES.
The concept of locally uniform (or "strongly continuo u s " ) action as well as the proof of Theorem 10.8 are due to W. Kaup. The main results 10.14-10.19 appear in C137,1381. For a direct proof of 10.18, see C 2 1 ; Ch. 111, 5 8, nol, Thgor6me 1 1 . The technical results 10.5 and 10.13, of basic importance in this section and in Sections 11 and 13, are due to H. Cartan (cf. [log; Ch. 91 and C106; Ch. V,
5
2, Lemma 11).
In the more general setting of analytic spaces of finite dimension, Theorem 10.21 is proved in C801.
172
11,
SECTION 11
ANALYTIC TRANSFORMATION GROUPS
Whereas in Section 10 we considered actions of Ranach Lie groups, it is the aim of this section to endow certain topological groups acting on Ranach manifolds with the analytic structure of a Ranach Lie group. Suppose G is a (Hausdorff) topological group with unit Consider a fundamental system of neighborhoods element e
.
S
of
e
E
G
.
Then the sets Ns
:=
{ (g,h)
E
G : g-lh
E
S
}
.
form a basis of the so-called left-uniform structure on G In the following we will assume that G is complete with respect to this uniform structure, since this topological
condition is necessary for the existence of a Ranach Lie group structure on G (cf. [ 2 1 ; 3.1.11 and [15; 5.191 1. Now suppose G acts on a Ranach manifold M over K via the homomorphism r : G + Aut(M) Since an analytic action of a
.
Ranach Lie group on M is necessarily locally uniform (cf. 10.4), we assume in the following that r is a locally uniform action. As a crucial additional assumption, we consider act ions which are '~topological.lyfaithful".
11.1 DEFINITION. A faithful locally uniform action r of topological group G on a Ranach manifold M is called topologically faithful if for every local representation
of r with respect to a chart closed neighborhood S o of e
for 6 > 0 e E G .
E
a
(P,p,Z) of M there exists a S such that the sets
form a fundamental system of neighborhoods of
11.2 LEMMA. Every faithful continuous action r of a locally compact group G on a connected Ranach manifold
M
ANALYTIC TRANSFORMATION GROUPS
173
is topologically faithful.
PROOF. By 10.20, r is a locally uniform action. Let C r S + O,(R,Z ) be a local representation of r and let # . S O be a compact neighborhood of e E S Since r is faithful and M is connected, the continuous mapping r# Is0 is injective and hence a homeomorphism onto its image. O . E . D .
-
.
In the following, suppose G is a topological group which is complete with respect to the left-uniform structure and let r be a topologically faithful, locally uniform action of G on a For any point representation of M about o Let 0 E D such that dist(R,aC) > 0 the sets
.
.
be a local with respect to a chart (P,p,Z) of B C C be convex open neighborhoods of R : = dist(C,aD) > 0 and We may assume that S is closed and that r
> 0
for
e
.
connected Ranach manifold M o E M , let r++ : S + O,(D,Z C )
E
G
.
form a fundamental system of neighborhoods of Now choose > 0 such that B < dist(R,aC)
11.3.1
and
sB sB u s-ls c s B B BY continuity of satisfies
,
r#
.
we may assume that every
Ir#(g)-idlD c R / 4
6 .
PROOF. For all m,n E N , 11.3.2 imp1 11.3.1 implies r ( S x R ) C C Since B
.
g
.
11.3 LEMMA. Suppose (9,) is a sequence in (r#(gn)) is a Cauchy sequence in 0, converges in S
#
11.3.2 E
SB
11.3.3 S,
such that
174
SECTION 11
and
-1 i d g = r # ( g m ) ( r# ( g n 1
-
r#(gilgn)
-
r # ( gm 1 )
is a Cauchy s e q u e n c e i n
(r#(gn))
E
Om(B,Z
,
c,'_(B,Zc)
C
1
it follows
from 1.17 t h a t
for
m,n +
.
m
+ e E G , showing t h a t (g,) gm g n w i t h r e s p e c t to t h e l e f t - u n i f o r m
Hence
i s a Cauchy s e q u e n c e i n structure.
11.4
Since
gk
applied to
,
SB
E
gj
S
E
t h e r e e x i s t s an i n t e g e r for
8
0 6 j
E
gj
,
d := R / 2
S
for all
B
for all
k M
For 11.5
k := k ( g )
and
.
,
define
Suppose
LEMMA.
Then 10.5,
fn(0)
c 28
and t h e r e f o r e
n
E
g = e
is f a i t h f u l .
.
+m
Om(D,Z
+ 0 E
,
Q.E.D.
is a s e q u e n c e i n
(g,) #
= idg
k ( g ) :=
f n := r ( g )
the functions
.
implies
> 0 Hence r , ( g ) is c o n n e c t e d and r g = e
> 0
j
k Ir#(g)-idlg c 21r#(gk)-idlg
since
2
S
C
S
5
such t h a t
satisfy
)
11.5.1
D
and
11.5.2 Suppose i n a d d i t i o n t h a t f o r e v e r y
nk
E
N
such t h a t
k(gn)
>
k
k
for a l l
E
there exists
N n
> nk
.
Then
g n + e E G . PROOF.
For
Then 11.3.3
n
> nk
a n d 10.2
put imply
+n := r
#
( gk 1
n
and
Q ~ : = r # ( g kn+ l )
.
ANALYTIC TRANSFORMATION GROUPS
175
JI, -- fnVn on
C
.
11.5.3
Consider the power series expansions
and
about f;
E
0
E
C
and
.
p'(Zc,Zc)
$n(0) E. D , respectively, where Then 11.5.3 implies
'
$n
where FA denotes the symmetric X-linear mapping We first show by induction on corresponding to fn $ p )
k
This is trivial for from the estimate
together with 11.5.1, induction on
X
0
0.
$;
k
that
I
11.5.5
and the induction step follows
11.5.5 and 1.17.
We next show by
that
€A for all X > 2 for 2 < X < m
=
+
I
. .
+
11.5.6
0
Suppose m > 2 and assume that 11.5.6 We show by induction on k that 4;
-
holds
11.5.7
k f;
c c
-
where, for sequences a,,@ n E Pm(Z ,Z ) , an 8, means a -8,+O. For k = 0 , 11.5.7 is trivial. For the n induction step, observe that 11.5.4 and 11.5.6 imply
SECTION 11
176
since 11.3.3 and 1.13 imply for every
-.
By 11.5.5, 11.3.3, 11.5.7, we have
p
sup < + 11.5.9 n 1.17, 11.5.2 and the induction hypothesis
...
k-2 ) o 1 k-1 ofA(0) , where Further, 0, = Cg'(0) = y z n )ofA(zn n Therefore, 11.5.2 implies zj := r ( g 7 ) ( 0 ) + 0 by 11.5.5. # n 0; + id Together with 11.5.8, we get (k+l)f: , showing that 11.5.7 is true for all k Applying 11.5.9 to p = m , we get fm + o , i.e., 11.5.6 is true for all x n + o for every Together with 11.5.5 and 1.17, we get f(")(O) n m > 2 By 11.3.3 and 1.20, this implies (fn-id(B + 0 Since r is a topologically faithful action and gn E S B , it follows that gn + e . O.E.D.
.
.
:J$
.
.
11.6 g E
LEMMA. SB
and
where
and
.
and
There exist constants k < k(g) implies
a,b
> 0 such that
177
ANALYTIC TRANSFORMATION GROUPS
for some
a > 0
.
Hence j
6
a/lf-idlC
11.6.2
and therefore
.
Izj+l I < 2jlf(0)1 < 2ac
11.6.3
j = k , this implies 11.6.1. Since B < dist(0,aC) 1.17 implies that there exists b > 0 such that
For
,
< blzl *If-idIC
.
Since for all z E UB(0) 1.6.3 that follows from
I
f
zj
E
Llg(0)
j < k(g)
for
,
it
< (f'(z.)-f'(O) ' z.)-id( 7 3
t b z.l*lf-idlC + If'(O)-id 3
where
T : = 2abc + d
.
For
A1,
...,Ak
E
L(Zc)
,
we have
k
< (l+lf-idlCT)k < (l+aT/k)k < exp(aT)
.
O.E.D.
c
aut(M) denote the set of all complete analytic vector fields X on M such that there exists a of G (unique) continuous 1-parameter subgroup (g,) satisfying 10.10.1. By 10.11, there exists a mapping Let
g := g M
P# : g
+
O,(C,Z
C
)
satisfying 10.11.1. 11.7
that
LEMMA. gn + e
Suppose gn E S and E R , k n + +m and
kn
E
R
are sequences such
S E C T I O N 11
178
11.7.1
Then t h e r e exists a v e c t o r f i e l d
exists.
.
h = p#X PROOF.
gn + e
Since
e # gn
SB *
E
Define
a n := k n / k ( g n )
.
c l u s t e r p o i n t of
2
k(g,)
w e may a s s u m e
is a b o u n d e d s e q u e n c e .
(a,,)
By 1 1 . 7 . 1 ,
,
r,(e) = idD
and
9 such t h a t
X E
a
,
implies
a > 0
be a
< 1
By
.
i f n e c e s s a r y , w e may a s s u m e Since
h E om(C,Z
C
,
)
10.13 i m p l i e s
that the vector f i e l d
r h : I xR + C
g e n e r a t e s a l o c a l a n a l y t i c flow
rh(t) t
for
+ 0
.
+
11.7.2
i d B E Om(B,Zc)
W e may a s s u m e
[ k n t l < kn
q n :=
we have
By 5 . 7 ,
p r o p e r t y s t a t e d i n 10.13.
having t h e
T
0 : gt
E
h := p#X
sa
is connected,
.
.
O.E.D.
Put
for It1 c s }
.
SECTION 11
180
.
Since S a is closed, it suffices to show that T > 1 Since dist(R,aC) , it follows that r#(gt)(R) C C for a < f3 Therefore the mean value theorem [ 2 8 ; 8 . 5 . 4 1 (tl < T implies
.
.
Then there exists u > 0 such that Now assume T < 1 gs E S and ( r (g )-idIC < a(1-r) whenever ( s I < a a # s assume It\ < T Then gt+s E S aS a C S and
.
LEMMA. g
11.10
.
Now
is a real Ranach space with respect to the
X + lxlp,c : = I P # X I C
norm PROOF.
is a sequence in g such that is a Cauchy sequence in Om(C,Zc) Put
Suppose
hn : = p#Xn
(X,)
.
.
h := lim hn E Om(C,Zc) n+After multiplying by a constant, we may assume lhnlC c B for t t E S Put exp(tXn) = r(gn) Then 11.9 implies gn all n and
.
.
Ir#(gn)-idlg t < Bltl
.
Put d : = dist(R,aC) whenever (tl < 1 Then 1.13 implies R1 := UdI2(B)
.
It( c f,(t,z)
T
z
E
.
:=
:t fn(t,z) f o r all
and
sup (h'l < +n B1 t := min(l,d/26) , 1 1 . 1 0 . 1 implies r#(gn)(R) C B1 t := r#(gn,z) satisfies the differential equation IJ
For and
11.10.1
R
, with
=
hn(fn(t,z))
initial condition
fn(O,z) = z
.
By the
181
ANALYTIC TRANSFORMATION GROUPS
well-known theorem about comparison of solutions of ordinary differential equations [ 2 R ;
10.5.1.11,
it follows that 11.10.2
.
.-
where
c Ih -hnlR + 0 for m,n + m Hence m,n C" 1 r#(gn) E Om(R,Z 1 is a Cauchy sequence for all It\
0 such that
for all n E N and all Y hn := exp(-X/kn) Since
.
E
T
.
kn
+
Now define +m , it follows that
.
0 = lim kn (r# ( g nh n )-id) E Om(R,Z L 1 n+m On the other hand, -X/kn E T for (almost) all
Since
X # 0
implies
u kn Ir#(gn)-idlR
knl r#(gnhn )-idlB by 11.17.2.
n
,
this is a contradiction.
O.E.D.
11.18 COROLLARY. Suppose G is a locally compact group and r is a faithful continuous action on a connected finitedimensional manifold M Then G is a real Lie group of finite dimension.
.
PROOF. By 10.20 and 11.2, the action and topologically faithful. Since G
r is locally uniform is complete with
respect to the left-uniform structure, 11.17 implies the assertion. O.E.D.
S E C T I O N 11
186
11.19 COROLLARY. Suppose G is a closed suhgroup of a finite-dimensional Lie group H Then G is a real Lie
.
group. Since the identity component of H is an open subgroup we may assume that H is connected. The left translation action of G on M := H is faithful and continuous. Since G is locally compact, the assertion PROOF.
follows from 11.18.
-
NOTES.
O.E.D.
Theorem 11.14 is one of the central results of this
book. For locally compact groups acting on finite dimensional analytic spaces, the theorem and the related results 11.1511.18 are due to W. Kaup 1.801. Our proof of 11.14 and of the
preliminary results 11.7-11.13 follows arguments have been used by J.P. Vigui! case of automorphism groups of bounded Banach spaces (cf. Section 13). Lemmas
C137,1381. Similar C1481 for the special domains in complex 11.5 and 11.6 are also
due to Vigug. Lie group structures on transformation groups play a fundamental role in modern differential geometry C931. For example, a theorem of S. Bochner and D. Montgomery shows that every locally compact group of diffeomorphisms on a finite dimensional connected differentiable manifold is a Lie transformation group (cf. C106; Ch. V, 9 2, Th. 21 and C93; Ch. I, Th. 3.31). Similarly, the group of isometries of a connected Riemannian manifold (of finite dimension) is a Lie transformation group according to a theorem of S.B. Myers and N. Steenrod (cf. 1. 93; Ch. 11, Th. 1.21 ) . These results are closely related to the solution of Hilbert's fifth problem characterizing Lie groups among all locally compact groups in topological terms, for instance, by being locally euclidean or by admitting no small subgroups (cf. 10.6 and 10.7). The first result along these lines, the well known theorem of H. Cartan about the automorphism groups of bounded domains in Cn , will be generalized in Section 13. For a self contained and concise account of the topological characterization of Lie groups, see [77; Ch. 111.
METRIC AND NORMED BANACH MANIFOLDS
12.
187
METRIC AND NORMED HANACH MANIFOLDS
For analytic manifolds useful to endow
M
M
of finite dimension, it is often
with a Riemannian (or hermitian) metric
and to study automorphisms of
preserving this metric.
M
For
example, the group of all hianalytic isometries of
M is a finite-dimensional Lie group [931, whereas, in general, there is no such structure for the group of all hianalytic
.
Metric structures, not necessarily automorphisms of M induced by tensor fields on the tangent bundle, are also important for manifolds of infinite dimension. The corresponding Lie theory for groups of isometries will he developed in Section 13. Suppose
M
is a Ranach manifold over
K
.
A
(continuous) pseudo-metric on M is a (continuous) mapping satisfying d(m,n) = d(n,m) , d(m,m) = 0 and d : M x M + R, the triangle inequality m,n,o
E
M
.
d(m,o)
If, in addition,
d(m,n) + d(n,o)
6
d(m,n) = 0 implies
.
for all m = n
,
is called a metric on M An analytic mapping g between Banach manifolds M and N , endowed with pseudometrics d M and dN , respectively, is called a contraction if dN(gm,gn) 6 dM(m,n) for all m,n E M A bianalytic mapping g satisfying dN(gm,gn) = dM(m,n) for all m,n E M is called an isometry. The group of all hianalytic isometries g : M + M of a Ranach manifold M , with respect to a pseudo-metric d , is denoted by Aut(M,d) For any continuous pseudo-metric d on M , let then
d
.
.
denote the open
d-ball with center
and radius r > 0 Two metrics d l and d-l on M are called uniformly equivalent [ 2 8 ; 3.141 if, for k = 2 1 and for
.
every
‘I
> 0 , there exists
.
a
>
0
o
E
M
such that
whenever m,n E M A metric d on M is called locally compatible if for every o E M there exists a chart
188
SECTION 12
(P,p,Z) of M about o such that, on P , d is uniformly equivalent to the metric (m,n) + Ipm-pnl Here 1 - 1 denotes a compatible norm on 2 Since Id(m,n)-d(h,k)( < d(m,h) + d(n,k) by the triangle inequality, it follows that every locally compatible metric d on M is continuous. I € , in addition, d generates the topology of M , then d is called compatihle. A metric Ranach manifold (M,d) is a Ranach manifold M endowed with a compatible metric d
.
.
.
12.1 LEMMA. Suppose 6 is a locally compatihle metric on a connected Ranach manifold M Then there exists a compatible metric d on M invariant under Aut(M,6)
.
PROOF.
Since
M
.
is connected, the formula
d(m,n) := inf { u > 0 : there exists a smooth curve from m to n such that 6(y(s),y(t))
6
u
for all
y
s,t }
defines a metric on M which is invariant under Aut(M,6) It follows that Rd(o;r) is and satisfies 6(m,n) 6 d(m,n) contained in the connected o-component C(o;r) of On the other hand, C(o;r/2) C Rd(o;r) Since R6(o;r) 6 is locally compatible, the sets C(o;r) form a basis of Hence d is a compatihle metric. open subsets of M Q.E.D.
.
.
.
.
12.2 P R O P O S I T I O N . Suppose M is a connected complex Ranach manifold and let 6 be a continuous metric on the open unit disc A := { z E C : I z ( < 1 } which is either invariant under Aut(A or coincides with the euclidean metric. Then m,n) :=
sup b(f(m),f(n)) 12.2.1 fEl)(M,A) defines a continuous pseudo-metric on M invariant under Aut(M) Every holomorphic mapping f : M + N between connected complex Ranach manifolds is a contraction with In particular, every biholomorphic respect to 6,,, and 6 N mapping is an isometry. &M
.
.
METRIC AND NORMED BANACH MANIFOLDS PROOF.
For a chart
(P,p,Z)
subset of D := p ( P ) 1.13 implies If(rn)-f(n)l
,
M
of
such that
let
C
189
be a convex open
R := dist(C,aD) > 0
.
< Ipm-pnl/R
Then
12.2.2
.
This proves the for all m,n E p-l(C) and f E O(M,A) assertion for the euclidean metric 6(z,w) := Iz-wl on A Since Aut(A) acts transitively on A by 3 . 1 8 , every invariant continuous metric 6 on A satisfies
.
6#(m,n) -
SUP 6(0,f(n)) f & O( M,A) ,f(m)=O 6 , for every 7 > 0 there exists
By continuity of
.
with o A C all m,n E Pm-Pn
I
By 12.2.2 and 12.2.3,
+ 6#(m,n)
< Ro
0
this implies for
.
7
.
12.2.4
Therefore, the no -empty set N := { n E M : gM(m,n) < +O } is open and closed, and hence coincides with M Further, bM
is a continuous pseudo-metric on
M
.
.
O.E.D.
The continuous pseudo-metric 6 M is called the Carathgodory pseudo-metric on M associated with 6 It is clear that 6 M is a metric on M if and only if the Banach space O,(M,C) of all bounded holomorphic functions on M separates the points of M , i.e., m # n implies f(m) # f(n) for some f E O,(M,C) On the other hand, 6 M = 0 if and only if every bounded holomorphic function on M is constant. This is true for every compact connected complex manifold M (as a consequence of the open mapping theorem) and, by 1.18, for every complex Ranach space M = Z
.
.
LEMMA. Suppose M is a complex Ranach manifold such 6 M is a metric. Then every holomorphic mapping f : C + M is constant.
12.3 that
PROOF.
Since
6c = 0
,
the assertion follows from 12.2. O.E.D.
.
190
SECTION 1 2
12.4
Let
PROPOSITION,
.
Z
Ranach space
Then
be a bounded domain in a complex 6 D is a compatible metric.
D
The Hahn-Ranach Theorem [ 1 5 ; 34.81
PROOF.
implies
I z l = sup I f ( z ) l
12.4. I
f ES
,
z
E
Z
where
-
{ f
.
If1
< 1 } On the compact set K : = A/2 , every continuous metric 6 on A is uniformly equivalent to the euclidean metric [28; 3.16.5 and 3.17.21. Put R := I i d 1 Then, for every f E S , the Hence 12.4.1 implies that mapping f/2R maps D into K for every T > 0 there exists u > 0 such that for all
S :=
.
whenever
z,w
D
E
.
E
L(2,C)
:
.
Combining this with 12.2.2
respectively, it follows that
or 12.2.4,
is a compatible metric.
bD
O.E.D.
LEMMA.
12.5
Suppose
D =
{
z
E
Z
: ( z (
.
unit ball of a complex Ranach space 2 continuous metric on A invariant under satisfying
, whenever
Let
is the open be a
6
Aut(A) 0 < u
6 ( D , J z J ) NOW suppose f E I)(M,A) Applying Schwarz’ Lemma [27; VI.2.11 to satisfies f ( 0 = 0 the functions fZ(X) : = f(Xz/Jzl) on A (for z f 0 ) , we get
.
(f(z)( = (fz(lz()I < I z (
By 12.2.3,
the assertion follows.
Note that 12.5 on
A
and therefore
.
For domains
D
Q.E.D.
is not true for the euclidean metric
in a complex Ranach space, the
Carathgodory (pseudo-) metric
6,,
can be used to study
6
METRIC AND NORMED BANACH MANIFOLDS geometric properties of
D
.
6D
Since
191
is invariant under
holomorphic transformations, the basic argument is often given
by an extension theorem for holomorphic mappings. Suppose M is a non-empty open subset of a connected complex Ranach manifold N For any complex Ranach manifold Q , let
.
denote the restriction mapping
f
+
f(M
.
In case
W
is a
complex Ranach space, define the restriction mapping
By 3 . 1 ,
these mappings are injective.
12.6 LEMMA. Suppose M is a non-empty open subset of a connected complex Ranach manifold N. Then the following statements hold: C
is surjective if and only if where A is the open unit disc. p,
(i)
(ii)
pc
is surjective if and only if r of C
for all open subsets
(iii)
Suppose
pw
.
.
pA
is surjective,
pr
is surjective
is surjective for all complex Ranach
Then the mappings p! and spaces W surjective for all convex open subsets
pc
C
are of W
.
Suppose first that p A is surjective. If O,(M,C) is not constant, then g : = f/(flM E O ( M , A ) by 3.2. Let G E O ( N , A ) satisfy G ( M = g Then C F := IflM*G E O , ( N , C ) satisfies F I M = f Hence p, is C surjective. Conversely, suppose that p, is surjective. Then f E O(M,A) has an extension F E o,(N,C) Suppose F(N) & A Then F is not constant and, by 3 . 2 , there exists b E F ( N ) \ x Hence g(m) := (f(rn)-b)-l defines a function g E O,(M,C) which has no holomorphic extension to N This contradiction s h o w s F E O(N,A) This proves (i), and ( i i ) follows from a similar argument. Now assume p w is PROOF.
f
E
.
.
.
.
.
.
.
192
SECTION 12
Surjective for all complex Ranach spaces W Then f E O(M,C) convex open subset of W F
.
()(N,W) 3 0 . 7 1 implies E
( i i ) to $of follows that unit ball C
.
.
Let C be a has an extension
Suppose there exists b E F(N)\C $(b) 6 + ( C ) f o r some $ E L ( W , C ) and pc
of
. .
Then [ 1 5 ; Applying
+(C) , we obtain a contradiction. It is surjective. Applying this to the open W , it follows that p y is surjective.
r
:=
O.E.D.
12.7 LEMMA. Suppose (M,d) is a metric Ranach manifold such that the group Aut(M,d) of all hianalytic isometries acts transitively on M , Then d is a complete metric. PROOF. Since d is a compatible metric and M is a Ranach manifold, there exists a closed ball Rd(o;r) which is
.
complete with respect to d Let (mj) be a Cauchy sequence Then there exists an index i such that in M d(mi,m ) < r for all j > i Choose g E Aut(M,d) such j Then mj belongs to the complete space that g ( o ) = mi B d (mi ;r) = g(Rd(o;r)) for all j > i Hence the sequence (mj) is d-convergent. O.E.D.
.
.
.
12.8
COROLLARY.
.
Suppose
is a homogeneous domain in a
D
complex Ranach space 2 such that the Carathgodory metric 6 D is compatible. Then tiD is a complete metric. 12.9 LEMMA. Let M be a homogeneous non-empty open subset of a connected complex Ranach manifold N such that the Carath6odory metric tiM is compatible and the restriction mapping p: is surjective. Then M = N
.
PROOF. Applying 12.2 to the inclusion mapping i : M + N , we get tiN(rn,n) < 6M(m,n) for all m,n E M Since p A is surjective by 12.6, it follows that 6# and 6 N coincide Now suppose M # N Since M is non-empty and N on M is connected, we have aM # a’ Let ( m . ) be a sequence in 7 M converging to a point n E aM Since 6N is continuous Since by 12.2, we have 6 (m.,n) + 0 N 7
.
.
.
.
.
.
METRIC AND NORMED BANACH MANIFOLDS 6,(mi,m.) = GN(mi,m.) < 3
it follows that respect to 6 M 6
m
(mj)
j
+
Let
,
m
is a Cauchy sequence in there exists m E M
12.7,
fjM
,
GN(rni,n) + 6 (m ,n) N j
3
. Ry (m ,m) + 0 . Since M j
193
M with with
is compatible, this implies
a contradiction.
O.E.D.
We will now apply 1 2 . 9 in two important special cases. co(Q) denote the convex hull of a subset Q of a vector
space. Suppose U and V are complex Ranach spaces and M is a domain in Z := UxV such that (e,n) E M for some e E U and (u,eitv) E M whenever (u,v) E M and t E R Then 12.10
THEOREM.
.
N := { (U,SV) : (u,v)
M
E
, n
0 , x 1 + x 2 < 1 and j 2 2 2 2 12.14.1 x1+x2 - c(x1+x2) t c(y 1+y 2 ) = 1 - c ,
.
.
.
.
.
.
.
Hence x : + xg < 1 and yi + y i < l/c It follows that S A T A is compact and is contained in S1 Since x 1 + x2 < 1 on SnTA except at the points al and a2 have a(n(SnTA))C n ( S n T r ) Since A c B , the maximum
.
.
, we
METRIC AND NORMED BANACH MANIFOLDS
,
g o n -1
principle 1.19, applied to
I I SnTA
< lgl
Tr
197
gives 12.14.2
*
.
0 < x and x + x2 < r Choose c < 1/2 such 1 2 2 j Then there exist E R that c(l-xl-x2) < 1 - x1 - x2 j' such that y1(1-2cxl) + y2(1-2cx ) = 0 and 12.14.1 is 2 Applying 12.14.2 to satisfied. It follows that rA c TS the functions z + g(z+ia) , for all a E X , the assertion
Now suppose
.
.
follows
.
O.E.D.
12.15 LEMMA. For j = 1,2 , let in X Put neighborhood of r j exists a convex open neighborhood every f E I),(TC,C) has a bounded
.
Cj be a convex open C := C 1 u C 2 Then there D of A in X such that holomorphic extension to
.
TD *
PROOF. Let E be the set of all s E [0,1] for which there is a convex open neighborhood D of sA in X such that Then E f has a bounded holomorphic extension F to TD is open in [0,1] and contains 0 Now let t E i? Since r = rlu r2 is compact, we have R := dist(r,aTC) > 0 Let s E E satisfy (s-t(- 1 id1 < R/2 Choose F E O,(TD,C) as A above. Since C and D are 0-starlike, the open sets B := C u D and C A D are connected. It follows that f has an extension F E Om(TB,C) Choose r < s such that Ir-tl - 1 id[ < R/2 and apply 12.14 to sr and the j derivativesA F(n) E O(TB,C) Since dist(Tr,aTB) > R , 1.13 imp1 ies
.
. .
.
.
for all
about Fa
E
a
N
n
E
E
rT
A
.
It follows that the power series
has a radius of convergence
O(R(a;R),C)
satisfies
> R
and
.
.
19 8
SECTION 12
Iz-a( < R/2
whenever
.
It follows that
f
has a bounded
holomorphic extension to the convex open neighborhood
.
Hence of tTA that E = [0,11
.
t
E
.
E
.
O,(TC,C)
+
E
is closed, showing O.E.D.
12.16 THEOREM. Suppose X with convex hull C p :
Therefore
O,(TR,C)
B is a connected open subset of Then the restriction mapping is surjective.
PROOF. Let E be the set of all pairs (a,b) E R x R such that there exists a convex open neighborhood D of the segment [a,b] in X with the following property: For every f E Om(TB,C) there exists F E O,(TD,C) such that f and F coincide on a neighborhood of T Then 12.15 implies Iathl
.
E
E
,
(a,c)
E
E 3 (b,c)
E
E
.
12.16.1
Let o E R be fixed. For j = 1,2 and a, E R , there exist polygons in B with vertices a 0 = o , 1 J an = a j j * 1 1 ) E aj'-*' Then [o,a,l C R implies that j(o,a. E By 12.15, we get J 1 2 3 1 2 (ai,ai) E E Since (al,al) E E , we get (a2,al) E E 1 2 2 2 Since (a2,a2) E E , we get (al,a2) E E Cont nu ing this way, we get (a ,a = (ay,a;) E E , Hence E = BxB Now 1 2 Choose suppose x E [al,bll A [a2,b21 , where a , h . E R 1 1 convex open neighborhoods Dj of [aj,bj] such that for every f E O,(TB,C) there exists F E Om(TD ,C coinciding j with f on a neighborhood of T {a.,b . } ' Silice (al,a2) E E , there exist a convex'opdn neighborhood D of [al,a2] and a function F E OO(TD,C) such that f and F Applying 12.15, it coincide in a neighborhood of T t a 1 4 follows that there existsaconvex open neighborhood C of A in X such that F has a bounded holomorphic extension to TC It follows that F 1 , F and F2 coincide on TIXI Therefore f has a holomorphic extension fl to TQ , where
.
.
.
.
.
.
.
:=
u
O 0 such that
PROOF.
.
Rd(o;r) C G * o F o r every n E G o o n Rd(m;r) Hence
for some
.
m
E
,
there exists
m
E
Bd(n;r) = Bd(g*o;r) = g*Rd(o;r) c
g
E
G
.
Hence
G
o
d
Goo
is also closed in
M
. O.E.D.
standard way of constructing a metric on a Ranach manifold M , generalizing the concept of Riemannian or hermitian metrics, is based on "differential metrics" on the tangent bundle T(M) Unlike the finite-dimensional case, the differential metrics of importance for Ranach manifolds are not always based on tensor fields, i.e., scalar products on the tangent spaces, but are given by more general "tangent A
.
200
SECTION 12
norms" satisfying certain continuity conditions. Let M be a Banach manifold over K with tangent bundle T(M) A (continuous) tangent semi-norm on M is a (continuous) mapping b : T(M) + R+ such that for every m E M , the restriction bm := b(Tm(M) is a semi-norm on Tm(M) [15; 28.71. If, in addition, h,(v) = 0 implies v = 0 , then b is called a tangent norm. An analytic mapping g between Ranach manifolds M and N , endowed with tangent semi-norms b M and bN , respectively, is called a contraction if bN(T(g)v) < bM(v) for all v E T(M) A bianalytic mapping g satisfying bN(T(g)v) = bM(v) for all v E T(M) is called an isometry. The group of all bianalytic isometrics g : M + M of a Ranach manifold M , with respect to a tangent semi-norm b , is denoted by Aut(M,b)
.
.
.
12.19 DEFINITION. A continuous tangent norm h on a Ranach manifold M is called compatible if for every o E M there exist a chart (P,p,Z) of M about o and constants 0 < r < R such that every v E T(P) satisfies
normed Ranach manifold (M,b) is a Ranach manifold endowed with a compatible tangent norm b
A
.
M
For every smooth curve y : I + M , identify Tt(I) = R by 4.0.3. Then Tt(y) E Ty(t)(M) and the mapping t + b(Tt(y)) is continuous. The integral
is called the arc length of 12.20 LEMMA. f : y(1) + R,
y
with respect to
h
.
Let y : I + R be a smooth curve and let be an integrable function. Then
PROOF. For a = 2 1 , the set { t countable union of intervals :1 ,
E
I : a y'(t) > 0 } is a The substitution rule
METRIC AND NORMED BANACH MANIFOLDS
201
implies
For I o : = { t E I : y'(t) = 0 } , y ( I o ) has Lebesgue measure 0 by Sard's Theorem [105]. Hence the measure dp(s) := f(s)ds on y(1) satisfies p(y(1 0 1 ) = 0 Hence
.
O.E.D.
12.21
COROLLARY.
Banach space
a
1.1
(Y)
Z
,
Let
y
: [0,1] + 2
endowed with the norm
> Iy(l)-y(O)l
.
be a smooth curve in a
1-1
.
Then
.
PROOF. Put v : = y ( 1 ) - y ( 0 ) According to the Hahn-Ranach Theorem (15; 44.21, choose $ E L(2,R) such that I $ \ < 1 and l$vI = IvI Applying 12.20 to Q o y : [0,1] + R and f := 1 , we get
.
PROPOSITION. Let b be a compatible tangent norm on a connected Banach manifold M Then 12.22
.
d(m,n) := inf { kb(y) :
defines a compatible metric on Aut(M,b)
.
PROOF. It is clear that d let (P,p,Z) be a chart of
M
piecewise smooth curve in M from m to n }
y
which is invariant under
.
is a pseudo-metric on M Now M satisfying 12.19.1 such that
202 D := p(P) w := p(n)
SECTION 12 is convex. F o r m,n E P put z := p(m) and Then p(y(t)) = z + t(w-z) defines a smooth
.
curve y : I I := [ 0 , 1 1
+
.
with Ty(t)(~)Tt(y) = w - z Therefore 12.19.1 implies P
Rb(y) =
11
by(t)(Tt(y))dt
,
where
< R Iw-21
0
Hence d d(m,n) < piecewise connected imply
.
is a continuous pseudo-metric on M satisfying Let y : I + M be a Rlpm-pnl for all m,n E Let J he the smooth curve with y ( 0 ) = m E P -1 Then 12.21 and 12.19.1 0-component of y (P)
.
.
.
12.22.1 Therefore (I < r dist(pm,aD) implies that d generates the topology of M open subset of D satisfying
.
Iz-wJ
sup
Bd(m;a) C P
Now let
< dist(R,aD)
Suppose m,n E p ( R ) satisfy rlpm-pnl > d(m,n) there exist p < dist(R,aD) and a smooth curve y from m to n such that R (y)/r < min(p,lpm-pn( b
.
showing be an
.
Z,WEB
-1
R
,
.
Then in M
.
By
12.22.1, we have y ( I ) A aP # pI Hence p(y(1)) meets the boundary of R (pm;p) Then 12.22.1 implies p < Rh(y)/r , 1.1 a contradiction. It follows that rlpm-pnl < d(m,n) for all m,n E P -'(I31 , showing that d is a compatible metric.
.
O.E.D.
The Carathsodory pseudo-metric 6 M on complex Ranach manifolds M has an "infinitesimal" analogue. 12.23 PROPOSITION. Suppose M is a complex Ranach manifold and let v E Tm(M) be a tangent vector. Then
METRIC AND NORMED BANACH MANIFOLDS
203
defines a continuous tangent semi-norm on M invariant Every holomorphic mapping f : M + N between under Aut(M)
.
complex Ranach manifolds is a contraction with respect to 8, and B~ In particular, every biholomorphic mapping is an
.
isometry.
.
PROOF.
For f E O(M,A) -1 g ( z ) : = (2-w)(l-w*z)
g(w) = 0
and
put w : = f(m) Since defines an automorphism of
g’(w) = (l-Iwl2)-’
(cf. 3 . 1 8 ) ,
Tm(gof)v = (l-lf(m)l Hence the suprema defining a chart of D := p(P)
with
A
it follows that
.
2 -1 1 Tm(f)v
Bm(v)
coincide. Let (P,p,Z) be a convex pen subset of Then 1.13 and dist(R,aD) > 0
and let R such that R : = M
he
.
1.17 imply
for all
u
E
,
T,(M)
the mapping
v
8, : T(M)
E
+
Tn(M) R,
and
m,n
E
p
is continuous.
The continuous tangent semi-norm Carathgodory tangent semi-norm on
M
.
BM
-1
(R)
.
Hence Q.E.D.
is called the
12.24 LEMMA. Suppose M is a complex Banach manifold such that B M is a tangent norm. Then every holomorphic mapping f
: C +
PROOF.
M
is constant. BC = 0
Since
,
the assertion follows from 12.23. Q.E.D.
12.25
PROPOSITION.
Banach space PROOF.
r
Let
Z
R
:= dist(B,aD)
.
Let Then
D 6,
be a bounded domain in a complex is a compatible tangent norm.
be an open subset of
> 0
.
D
with
Then 1.13 implies 12.25.1
204
for all implies
SECTION 12 (z,v)
E
BxZ = T(B)
On the other hand, 12.4.1
Iv( < R(BZ(v)l for all
(z,v)
E
DxZ = T(D)
,
where
R :=
lidlD
.
9.E.D.
12.26 P R O P O S I T I O N . Let D he a circular domain in a complex Banach space Z , with Carathgodory tangent semi-norm Then the convex hull is given by DxZ -+ R+ 8, : co(D) = { z E 2 : BD(0,z) < 1 }
.
.
Since co(D) is a convex circular domain, there exists a continuous semi-norm h on Z such that co(D) = { z E 2 : b ( z ) < 1 } Ry [15; 4.4.21, there exists for every z E Z a linear form $ : Z + C with $(cO(D)) C d and I $ ( z ) ) = b(z) Then f := $ID E O(D,A) by 1.19. Therefore PROOF.
.
.
BD(O,z) > If'(O)z( = l $ ( z ) l = b ( z )
. .
Conversely, suppose f E O(D,A) satisfies f(0) = 0 Let D" denote the balanced hull of D By 12.12 and 12.6, there exists an extension F E O(D",A) of f For z E D , f Z ( X ) := F(Xz) defines a holomorphic function : A + A satisfying F Z ( 0 ) = f ( 0 ) = 0 Applying the FZ Schwarz Lemma [27; VI.2.11, we get
.
.
.
If'(0)zl = lF'(O)zl = lFi(O)I < 1
.
Therefore, BD(O,z) < 1 for all z E D , The continuity and the semi-norm property imply 8 , ( 0 , z ) < 1 €or all z E co(D) Hence BD(O,z) < b(z) for all z E 2 O.E.D.
.
.
12.27 EXAMPLE. Let A he the open unit disc in C , with Carathgodory tangent norm B , By 12.26, we have B(0,v) = lvl for all v E C By 3.18, g ( z ) := (z+a)(l+a"z)-l defines an automorphism of A whenever a E A Since g ( 0 ) = a and g ' ( 0 ) = 1 - Ial2 ,
.
.
12.23 implies
METRIC AND NORMED BANACH MANIFOLDS
.
f o r all
A
205
(z,v) E T(A) = A x C The "integrated" metric w on associated with B i n 12.22 is called the Poincar'e metric
Let y : I + A be a piecewise smooth curve satisfying y(0) = 0 and 0 < r := y ( 1 ) < 1 Since Re(v)/(l-Re(z) 2 ) < lvl/(1-lz12) [ 4 3 1 and can be computed as follows.
.
v E c and z E A , 12.27.1 implies that the curve Hence we may Re(y) : I + A satisfies R (Re(y)) < g B ( y ) B assume y(1) C R Applying 12.20 to the measure P = (l-s2)-lds on [0,1] , we get f o r all
.
.
-21 On
0
1+r = tanh -1 ( r ) log 1-r
the other hand, y(t) : = tr to r with arc length
j1
=
Rg(Y)
0
defines a smooth curve from
r d t - 1' 1-t r
0
* 1-s 2
It follows that w ( 0 , r ) = tanh-l(r) under Aut(A) , it follows that
for all
x,y
E
A
,
.
.
=
tanh-l(r) Since
w
. is invariant
where
is another compatible metric on
A
invariant under
Aut(A)
.
12.28 PROPOSITION. Suppose D is a bounded domain in a complex Banach space 2 , with Carathgodory tangent norm 8, Let d be the associated "integrated" compatible metric on D Then d(x,y) > wD(x,y) for all x,y E D , where w D denotes the locally compatible Carathgodory metric on D associated with the Poinear6 metric w on A
.
.
.
PROOF. For f E ()(D,A) and (z,v) E DxZ , we have ~~(f(z),f'(z)v)6 bD(z,v) by 12.23. Hence
206
SECTION 1 2
for every piecewise smooth curve w(f(x),f(y)) < d(x,y) for all follows from the definition of
in
y
x,y wT)
.
E
D
D
.
.
Ry 12.27, we get
Now the assertion O.E.D.
12.29 COROLLARY. Let D be the open unit ball of for all z E D d(0,z) = w D ( 0 , z ) = w ( 0 , l z l )
.
2
.
Then
11, there exists f E L(Z,C) with = (21 By 12.28, we get
PROOF. By [15 (fl < 1 and
.
W(0,
.
Then g(X) := Xz/lzl defines a Now assume z , 0 holomorphic mapping g : A + D By 12.23 and 12.27, we get d(O,z) = d(g(O),g((zl)) w(o,lz() O.E.D.
.
12.30 COROLLARY. Suppose the open unit ball homogeneous. Then d = w D ' PROOF. Aut(D)
Ry 12.22 and 12.2 Now apply 12.29.
.
,
d
and
wD
D
of
Z
is
are invariant under O.E.D.
The following method of constructing invariant tangent norms on homogeneous Ranach manifolds will be frequently used in the sequel. 12.31 PROPOSITION. Let r be an analytic action of a Ranach Lie group G on a Banach manifold M which is transitive and local y transitive. Suppose there exists a point o E. M such that To(M) carries a compatible norm 1 * ( invariant under To(r 9 ) ) whenever r(g,o) = o Define
.
for all v E Tm(M) r(h,m) = o Then tangent norm on M
.
PROOF.
Since
r
and m E M , where h E G satisfies b : T(M) + R+ is a G-invariant compatible
.
is transitive and the norm
1.1
on
To(M)
METRIC AND NORMED BANACH MANIFOLDS is invariant under the isotropy subgroup norm b is well-defined. Since r : G 0
Go
207
,
the tangent is an analytic
+ M
submersion, 8.3 implies that there exist a chart (P,p,Z) of M about o and a real-analytic mapping P 3 m + hm E G such that ho = e and ro(hil) = m for a l l follows that the G-invariant tangent norm b
.
P It is continuous.
m
E
Since the mapping
is real-analytic, we may assume P , where imp1 ies m
E
Z
ITo(p) Tm(r(hm))v and therefore v
E
.
T,(M)
ITm-idZI
1/2
carries the norm induced by
-
T,(P)V~
for all To(p)
.
This
0 Then C C N Now let u E N n D and r < d st(u,aD) F o r every
.
.
v E R(u;r/4) , there exists B(v;r/4)C R(w;r/2) , 13.1.1 (C,A(w;r/2)) R(u;r/4) C N
and
.
.
w E A(v;r/4)n N Since holds for the pairs
.
(B(w;r/2),R(v;r/4))
Since
D
is connected,
F o r any bounded subset F of O , ( D , W ) induced by the norms I I B for open balls
-
satisfying
.
Hence N
=
, A
D
.
O.E.D.
the topology in D
> 0 is called the topology of locally
disttF3,aD)
uniform convergence on
D
.
Suppose in the following that
M
is a connected complex
.
Let Ranach manifold endowed with a compatible metric d Aut(M,d) denote the group of all biholomorphic isometries of
.
Our first aim is to define a topological group structure on Aut(M,d) Since M is connected , every nonempty open subset P of M induces a metric (with values
0
2 10
SECTION 13 13.2.1
is an open h a l l a b o u t
R := p ( P )
and
where
.
13.2.2
d e n o t e s t h e c o n n e c t e d 0-component
'D
13.3 M
,
> o
dist(R,aDa)
satisfying
0
Suppose
LEMMA.
P
Then t h e m e t r i c s
and
Q
and
dp
of
.
p(R(o;u))
are admissihle h a l l s in
on
dQ
Aut(M,d)
are
uniformly equivalent. PROOF.
Since
dp(g,h) = dp(h
g,h
E
G := A u t ( M , d )
>
0
there exists
a
,
-1
g,id)
> 0
B
such t h a t
I
dQ ( g , i d ) 6 B
whenever
g
.
G
E
Let
P
with respect to a c h a r t s a t i s f y 13.2.1
a point
.
m
for all
i t s u f f i c e s t o show t h a t f o r e v e r y
dp(g,id) c a
be a n a d m i s s i b l e p - h a l l
of
(P1,p,ZI
M
and l e t
Assume f i r s t t h a t
and 13.2.2.
In f a c t , f o r
3 g ( B ( o ; a ) I C: P1
n
B(o;a)
E
P C B(o;a)
implies
d(n,m) c d(n,o)
for a l l Do
,
m,n
+
d(m,o)
that IP09-Plw Further,
o
w e have
d (g,id) d a
and s u p p o s e
>
u >
.
d ( g ( n ) , o ) c d(g(n),g(m))+d(g(m),m)+d(m,o)
T
about
Then
dQ(g,id) 6 a
exists
13.3.1
0
with
0 satisfy 13.2.1 and k B N such that It follows dp(gjrgi) = dp(gi g j ,id) < u for all i,j > k
admissible p-ball about o 13.2.2. Then there exists
.
-’
that g-’g.(P) C Rd(o;20) C P1 k 7 d(g-’g.(m),o) < d(g-’g.(m),m) k 1 k 7 Hence the limit
since + d(m,o)
for all
m
E
.
P
exists and satisfies h(P) C p(Bd(o;2u)) C p(Bd(0;4a)) cp(P1) Therefore g := g k op-loh E O(P,M) satisfies
.
lim dp(gj,g) = j+m
.
0
13.5.1
Put m := g(o) E M and choose T > 0 such that Bd(o;2.r) C P , Let 0 be an admissible ball contained in By 13.5.1, there exists i0 E N such that Rd(m;r) d(gi(o),m) < T for all i > i0 It follows that -1 -1 -1 -1 gi ( Q ) C P , since d(gi (n),o) c d(gi (n),gi (m)) -1 + d(gi (m),o) = d(n,m) + d(m,gi(o)) < 2.r for all n E Q
.
.
.
.
Therefore d ( g g-l,id) < dp(gj,gi) for i > i o By 13.3, 0 ji-, it follows that (gj is also a left-uniform Cauchy sequence. Hence there exist mappings g’ E O(M,M) , for u = +1 - , such that lim dp(gp,gu) = 0 j+a j for every admissible ball
for all m E M g’ E Aut(M)
.
,
P
in
M
.
Since
.
it follows that g-’ogp = idM Hence Since d is a continuous metric, the mappings
GROUPS OF HOLOMORPHIC ISOMETRIES g”
213
are isometric.
O.E.D.
13.6 PROPOSITION. The canonical action r of Aut(M,d) on M is locally uniform and topologically faithful. If is a topological group and 4 : H homomorphism, the action ro+ of 4
uniform if and only if
+
is continuous. o
,
T
> 0 satisfy
PROOF. Let P be an admissible p-ball about respect to a chart (P1,p,Z) of M , and let
C P
Bd(o;-r)
.
For
,
D := p ( P )
S T := { g
E G
H
Aut(M,d) is a H on M is locally
G : = Aut(M,d)
: d(g(o),o)
0
dp(g,h) whenever
g,h
> 0 such that
8
there exists
8 =3 Ir#(g)-r#(h)lD
6
.
a
is uniform Y continuous and is therefore a local representation of r . Now let R : = bD , where 0 < b < 1 Then Q := p -1 R ) is E
ST
It follows that
6
o
an admissible p-ball about
r#
.
.
By 13.3, the sets for a > 0 form a
: = { g E G : dq(g,id) < a } ‘a fundamental system of neighborhoods of
a
>
0
Ipm-pnl < 8 whenever g
E
S7
m,n
,
E
P1
.
Since
it follows that S 8 :=
Hence the action
{
g
E
r
G
E
.
For every
> 0 such that
8
there exists
idM
3
g(Q)
c
d(m,n) < a for a l l
g(P) C P1
contains the set
Ta
S T : Ir#(g)-idlg
< 8 }
.
is topologically faithful.
that every continuous homomorphism
$I
: H + G
.
13.6.2 It is clear induces a
Conversely, locally uniform action roe of H on M suppose the action ro$ is locally uniform. Then there exists a neighborhood
T
of
e
E
H
such that
214
SECTION 13
d($(h)(o),o)
6 T
for all
h
T
E
.
We may assume that the
mapping
is continuous. Then for every 6 neighborhood T B of e E H such ( sB ) B>0 is a fundamental system idM E G I it follows that $ is
> 0
there exists a that +(TB) C S B of neighborhoods of continuous.
.
Since O.E.D.
13.7 PROPOSITION. Suppose (g,) is a sequence in Aut(M,d) such that there exists a chart (P1,p,Z) of about o with
gn(o)
+
o
M
13.7.1
M
E
and a(pogn)
aP Then gn + idM convergence. PROOF.
( 0 )
+
idz
E
Ga(2)
.
13.7.2
in the topology of locally uniform
We may assume that there exists an admissible p-ball
.
P about o , with respect to (P,,p,Z) Define r# : S T + o m ( D , Z ) as in 13.6. Then 13.7.1 implies for almost all n Put
.
gn
E
S
T
For any k E N , choose nk E N such that Then d(gn(o),o) < r/(k+l) for all n > nk
.
d(gA(o),o) < j d(gn(o),o) < jT/(k+ll
.
whenever j 6 k+l and n > nk assertion follows from 11.5. 13.8
and 13.9
COROLLARY. To(g) = id COROLLARY.
Suppose
g
D
g;
T E
S5
,
and the O.E.D.
Aut(M,d)
E
for some point Suppose
Therefore
6
o
E
M
.
satisfies Then
g(o) = o
g = idM ,
is a domain in a complex Ranach
GROUPS OF HOLOMORPHIC ISOMETRIES
space
such that the Carathgodory metric
2
6D
215
is locally
compatible. Let (gn) be a sequence in Aut(D) such that gn(o) + o and gA(o) + idz for some point o E D Then
.
gn
in the topology of locally uniform convergence.
idD
+
13.10 space
COROLLARY. Suppose D is a domain in a complex Ranach Z such that the Carathgodory metric 6 D is locally
compatible. Let g E Aut(D) g'(o) = idz for some point
satisfy o E D
.
g(o) = o
and g = idD
Then
.
13.10 is known as Cartan's uniqueness theorem. For any point
o
,
E M
the local representation
of r on D := p(P) , defined by 13.6.1, is said to be Note that associated with the open p-ball P about o
.
since d(g(m ,o) < d ( m , o ) + d(g(o),o) < O + T < 20 for all m E Rd(o;u) and g E S T Since p(P1) is bounded by
.
t follows that
assumption, Om(DU,Z)
.
r#(ST) is a bounded subset of 0 < b < c < 1 Then R := bD and
Now choose
.
>
C := cD are open balls about 0 satisfying dist(R,aC) and R : = dist(C,aD) > 0 By 13.5, there exists
.
B < dist(R,aC)
11.3.2
for
every
g
E
S SB
S , defined by 13.6.2, satisfies B By 13.1, we may further assume that
such that
:=
ST
.
satisfies
> 0 such that every
c
sup ( (r#(g)(O)l,(r#(g)'(O)-idl
Ir#(g)-idlC c c PROOF.
.
Ir#(g)-idlD < R / 4
13.11 LEMMA. There exists a constant g E S' satisfies
sequence
}
.
Arguing by contradiction, assume there exists a (9,)
0
in
ST
such that every
n
E
N
satisfies
216
SECTION 1 3
Since r#(S ) is bounded on C , it follows that gn(o) and a(pogn)/ap(o) + idz Hence 13.7 implies gn + idM Define Therefore we may assume gn E S B for all n k n := k(gn) (cf. 1 1 . 4 ) and 'I
.
kn hn := gn
.
E
S'
+
.
o
,
.
On the other hand, 13.11.1 and 11.5 Then Ir#(hn)-idlB > 6 imply hn(o) + o and a(pohn)/ap(o) + i d z Applying 13.7, we get a contradiction. 0.E.T). 13.12 LEMMA. For p := ?/3 Then there exists a constant imp 1ie s
.
,
-1 suppose p (c) c R d ( o ; p ) X > 0 such that g,h E S p
Since z : = r#(g)(O) and w := r#(h)(O) and 1.17 imply for d : = dist(D,aDa)
and
belong to
D
,
.
1.13
GROUPS OF HOLOMORPHIC ISOMETRIES
13.13 space
217
COROLLARY. Suppose D is a domain in a complex Ranach Z , endowed with a compatible metric d Let G he a
closed subgroup of mapp i ng
Aut(D,d)
.
.
Then, for every
is a homeomorphism onto a closed subset of
o
D
E
,
the
.
DxL(2)
Suppose g,h E G satisfy g(o) = h(o) and h'(o) Then f := h-'g E G satisfies f(o) = o and Hence = (h-l)'(go)g'(o) = (h-')'(ho)h'(o) = id, f = idD by 13.10, showing that the continuous mapping defined be a sequence in G by 13.13.1 is injective. Now let (g,) Then there with gn(o) + z E D and g;(o) + T E L ( 2 )
.
=
.
.
exists k E N such that d(gn(o),gk(o)) < p n > k Replacing (4,) by the sequence we may therefore assume gn E S p for almost (gn(o)) and (gA(o)) are Cauchy sequences respectively, 13.12 implies that (g,) is a in G which is convergent by 13.5.
for all -1 gk g n ) in C, all n Since n Z and L ( Z ) Cauchy sequence
.
.
,
O.E.D.
13.14 THEOREM. Suppose (M,d) is a connected complex metric Banach manifold and let G be a subgroup of Aut(M,d) which is closed in the topology of locally uniform convergence. Then G can be endowed with a Hausdorff topology T such that ( G , T ) becomes a real Ranach Lie group whose Lie algebra can be identified with the real subalgebra g :=
of
{ X
T(M)
E
.
aut(M) : exp(tX)
E
G
The canonical action
for all t r
of
E
@
: H + (G,T)
13.14.1
on
(G,T)
analytic and, for every real Ranach Lie group homomorphism @ : H + G , the action ro@ of analytic if and only if
R }
H H
M
is
and every on M is
is analytic.
O.E.D.
218
SECTION 13
PROOF. By 13.4, 13.5 and 13.6, G satisfies the conditions of 11.14. Now suppose X E aut(M) satisfies gt := exp(tX) E G for all t E R Since the global flow is on M generated by X is analytic, it follows that (g,) a continuous 1-parameter subgroup of G By 11.8, g is a
.
.
real subalgebra of T ( M ) and the Lie algebra of ( G , T ) can be identified with g It is clear that every analytic induces an analytic action r o e homomorphism $ : H + ( G , T ) of H on M Conversely, suppose the action r o e is analytic. Then r o e is locally uniform by 10.4 and 13.6 implies that 4 : H + G is continuous. It follows that every Y E h induces a continuous 1-parameter subgroup The gt : = g(exp(tY)) of G , hence an element $*Y e g infinitesimal generator (roe), : h + aut(M) of r o e satisfies (roe)* = r,o+* Since the action r* of g on M is topologically faithful by 11.13, it follows that $* : h + g is a continuous homomorphism. Hence cp is O.E.D. analytic.
.
.
.
.
13.15 COROLLARY. Let M be a connected complex Ranach manifold endowed with a locally compatible metric 6 Then Aut(M,G) can be endowed with the structure of a real Ranach Lie group whose Lie algebra can be identified with the real subalgebra
.
aut(M,G) :=
of
T(M)
{ Xeaut(M)
: exp(tX)eAut(M,G)
for all
teR }
.
By 12.1, there exists a compatible metric d on such that Aut(M,G) is a closed subgroup of Aut(M,d) PROOF.
.
M
O.E.D.
13.16 COROLLARY. Let D be a domain in a complex Banach space 2 such that the Carathgodory metric dD is locally compatible. Then aut(D) is a real subalgebra of T ( D ) and Aut(D) can be endowed with the structure of a real Banach Lie group whose Lie algebra can be identified with aut(D)
.
GROUPS OF HOLOMORPHIC ISOMETRIES
219
13.17 COROLLARY. Let (M,b) be a connected complex normed Ranach manifold. Then the group Aut(M,b) can be endowed with the structure of a real Ranach Lie group whose Lie algebra can be identified with the real subalgebra aut(M,b)
{ Xeaut(M)
:=
:
exp(tX)eAut(M,b)
for all teR ]
PROOF. By 12.22, there exists a compatible metric d on M such that Aut(M,h) is a subgroup of Aut(M,d) which is closed by 1.13. O.E.D. In the finite-dimensional case, the "analytic" topology T can he specified more precisely. 13.18 PROPOSITION. Suppose M is a connected complex manifold of finite dimension and 6 is a continuous metric
.
on M Then Aut(M,G) is a real Lie group of finite dimension in the compact-open topology.
M is locally compact, the metric 6 is locally compatible. The topology of locally uniform convergence on Aut(M,G) coincides with the compact-open topology. N o w apply 11.17. O.E.D. PROOF.
Since
13.19 COROLLARY. For every bounded domain D in Cn Aut(D) is a real Lie group of finite dimension in the compact-open topology. 13.20
PROPOSITION.
There exists a constant
every
X
satisfies
aut(M,d)
E
IP#XIC < c PROOF.
n > 1
For p
#
Since
gn
E
,
c > 0
sup { ((P#X)(O)I,I(P#X)'(0)I put
gn : = exp(X/n)
.
Ir#(gn)-idlC < c
SUP {
n
,
such that
I
.
Then 10.8 implies
X = lim n(r (g # n n+m
s T for (almost all
r
13.11 implies
220
SECTION 13
Now multiply by
n
and let
n +
-.
O.E.D.
13.21 COROLLARY. For every closed subalgebra g aut(M,d) , the real-linear mapping
of
is a homeomorphism onto a closed real subspace. 13.22 space
COROLLARY. Suppose D is a domain in a complex Ranach 2 , endowed with a locally compatible metric 6
.
Then, for every
o
E
D
,
the real-linear mapping
a
aut(D,6) 3 h ( z ) E
+
(h(o),h'(o))
E ZxL(2)
is a homeomorphism onto a closed real subspace. 13.23 COROLLARY. Suppose D is a domain in a complex Ranach space 2 such that the Carathhodory metric 6 D is locally compatible. Then, for every o E D , the real-linear mapping aut(D)
3 h(z)&
+
(h(o),h'(o))
E
ZxL(2)
is a homeomorphism onto a closed real subspace. 13.24 PROPOSITION. Suppose (M,d) is a connected complex metric Ranach manifold and let G be a subgroup of Aut(M,d) which is closed in the topology of locally uniform convergence, Put K := { g E G : g ( o ) = o } and Then there exists a compatible k := { X E g : Xo = 0 } norm on g such that exp(adX) E Ua(g) for all X E k Further,
.
.
and
1 K + G a ( 2 ) and define topological isomorphisms r# : 'pfl : k + g a ( 2 ) onto a closed subgroup of GR(2) and a closed real subalgebra of g & ( Z ) , respectively, such that
GROUPS OF HOLOMORPHIC ISOMETRIES
221
there is a commuting diagram
PROOF. Let P be an admissible p-ball about o with respect to a chart (P,,p,Z) of M about o Put D := p(P) and Consider the local C := cD , where 0 < c < 1 representation r# : S T + o,(D,Z) and p # : + ooD(C,Z) There exists a K-invariant open neighborhood Q of o E p-l(C) , endowed with the tangent norm induced by p , such that
.
.
IT,(^)(
sup {
:
g
E
.
K
,
m
E Q
} < +-
,
Hence
1x1
:=
sup IP#(g*X)Ip(Q) 9EK
.
defines a compatible norm on g invariant under K Now the It is clear that 'r is first assertion follows from 5 . 3 4 . # a continuous group homomorphism and 'p# is a continuous homomorphism of real Lie algebras. Now let (9,) be a sequence in K such that lr ( g ) + idz E G L ( Z ) Since # n r ( g ) ( O ) = 0 , it follows from 13.7 that gn + idM E K in # n the topology of locally uniform convergence. Hence 'r# is a topological isomorphism from K onto a subgroup of G!?,(Z) Since K is closed and hence complete in the left-uniform structure, 'r#(K) is also complete in the left-uniform structure and hence closed. Similarly, let (X,) be a 1 sequence in k such that p#(Xn) + 0 E g k ( z ) Since p ( g ) ( O ) = 0 , it follows from 13.20 that Xn + 0 E k # " 1 Hence is a topological isomorphism onto a real p# subalgebra of g L ( Z ) which is closed since k is closed in 9 and therefore complete. For X E h , r#(exp(tX)) defines a local analytic flow on C with infinitesimal By 5.2, it follows that genera tor p#X
.
.
.
.
.
SECTION 13
222
aat Hence
= 1p
'r#(exp(tx))
#
(x)
'r#(exp(tX)) = exp(t*lp#X)
.
'r#(exp(tx)) for all
t
E
R
13.24.3
. O.E.D.
13.25
PROPOSITION.
Let (M,b) he a connected complex normed Then h := { X E aut(M,b) : Xo = 0 }
Ranach manifold. satisfies PROOF. that
k
ik
A
.
= {O}
Z : = To(M)
There exists a compatible norm on
.
'p#(k) C u l l ( 2 )
c
For
X
h n ik
E
3 s+it + lr (exp(sX+tY)) #
E
put
Y := iX
such
.
Then
U ~ ( Z )
defines a holomorphic mapping which is constant by Liouville's theorem 1.18. p#(X)(O) = 0
By differentiation, we get
,
13.21 implies
.
X = 0
'p#(X)
=
0
.
Since
O.E.D.
The next result is a refinement of 11.7 and is crucial in the theory of symmetric complex Ranach manifolds. 13.26
THEOREM.
Suppose
is a connected complex metric
(M,d)
G be a subgroup of Aut(M,d) which is closed in the topology of locally uniform convergence. Let (gn) be a sequence in G with gn + idM such that, for
Banach manifold and let
the mappings hn
:=
the sequences in Z field
2"(r
#
(g )-id) n
(hn(0))
(hA(0))
and L ( Z ) , respectively. X E g such that p
#
PROOF.
and
O,(D,Z)
E
X = lim hn n+m
I
are Cauchy sequences
Then there exists a vector
E
Om(C,Z)
By assumption, there exist constants
. KO,Kl > 0
.
and Ir#(gn)'(0)-idl < 2-"K1 that Ir#(gn)(0)l < 2-"K0 d := R/3 and define Ck : = U k d ( C ) Then 13.11 implies
.
(r#(gn)-id( where
K
> 0
.
c2
< 2-"K
,
Then 10.5 implies for almost all
such Put
13.26.1 m
GROUPS OF HOLOMORPHIC ISOMETRIES
9,"
E
223
s p
13.26.2
and k Ir#(gn)-idl
< d/4 c2
.
and n > rn Put whenever k < kn := 2"-" 2-m(h n -hm) = $n + (r#(fn)-r#(grn)) , where $n := kn(r ( g )-id) - (r (f )-id) Hence #
n
#
2-rnlhn-hml, By 10.5 and 13.26.1,
f n := g ,k n
.
Then
.
n
hnIc+
lr#(fn)-r#(grn)Ic
.
13.26.3
we have
d n := 2 s u p ( r ( g k )-id[ < 2-" 3K < 3 kn Ir#(gn)-idl k < k n i+ c1 c2 and
l$nlC
< 7 k n d n Ir ( g )-idlC #
6
3K2
9
.
2-2m
d
n
13.26.4
On the other hand, we have
and
for all 13.26.2,
n > m , where 13.12 implies
arn
+
0
.
Since
fn,grn
E
Sp
by
SECTION 13
224
where
L1,L2 > 0
.
Combining these estimates with 13.26.3,
we
get
.
for all n > m Hence (h,) is a Cauchy sequence in Denote the limit by h Ry 11.7, there exists Oo3(C,Z) X E g with p#X = h Q.E.D.
.
.
.
13.27 PROPOSITION. Suppose M is a connected complex Ranach manifold such that every holomorphic mapping f : C + M is constant. Then a u t ( M ) A i aut(M) = { O }
.
PROOF.
For
X
E
aut(M)n i
aut(M)
and
g =
s + it
E
C
,
defines a homomorphism C 3 5 exp(gX) E Aut(M) such that the mapping C x M 3 ( g , m ) + f m ( c ) := exp(gX)(m) E M is holomorphic. By assumption, the mapping fm : C + M is constant for every m E M By differentiation, we get
.
xm=o.
Q.E.D.
13.28 COROLLARY. Let M be a connected complex Ranach manifold and assume that the Carathgodory pseudo-metric & M is a metric or that the Carathgodory tangent semi-norm 8, is a tangent norm. Then aut(M) n i aut(M) = { O }
.
By 12.3 and 12.24, every holomorphic mapping
PROOF.
f : C
+
M
is constant.
Q.E.D.
13.29 COROLLARY. Suppose L is a complex Ranach space and Z is an associative unital complex Ranach algebra. Then UQ(L) A i
and
UQ(L) = {0}
13.29.1
GROUPS OF HOLOMORPHIC ISOMETRIES
225
.
PROOF. Let D be the open unit ball of L Then the restriction mapping X + XID is an isomorphism from u&(L) Now 13.29.1 follows onto a closed subalgebra of aut(D) is an from 13.28. Since the left translation mapping z + L
.
isomorphism from u ( Z ) onto a closed subalgebra of 13.29.2 follows from 13.29.1. Let
be a Ranach manifold.
M
analytic vector field
X
E
T(M)
uQ(2)
,
O.E.D.
Define the order of an
at
o
Ordo(X) := Ordo(Xp)
E M
by
I
13.30.1
is a chart of M about o , and the order of the analytic mapping Xp : P + Z at o E P is defined as in 3.0.1. If (O,q,W) is another chart of M about o , we
where
(P,p,Z)
have Xq(m) = s ( m ) Xp(m) aP for all
m
P
E
A
Q
Ordo(Xq)
.
By 1.9, it follows that
> Ord 0
(9)+ aP
Ordo(Xp) = Ord,
.
since Ordo(aq/ap) = 0 It follows that 13.30 1 is independent of the choice of the chart (P,p,Z) of M about o By definition, we have Ordo(X) > 1 if and only
.
if
Xo = 0
To(M)
E
.
As a
consequence of 13.20, we get
13.30 PROPOSITION. Suppose (M,d) is a connected complex metric Banach manifold and let X E aut(M,d) have order > 2 at some point o E M Then X = 0
.
.
13.31 LEMMA. Let r : il + N be a local analytic flow on an open subset M of a Banach manifold N , with infinitesimal Let o E M Then Ordo(X) > 1 if and generator X E T ( M ) only if r(t,o) = o for all t E Qo
.
.
.
PROOF. Choose an open interval I about 0 such that By 5.1, ro : I + M is the unique solution of ro(I) C M the differential equation Tt(ro) = Xr(t satisfying lo) r(0,o) = o Hence Xo = 0 if and only if r(t,o) = o for
.
.
SECTION 1 3
226
all
t
E
I
.
no
Since
is an interval, the assertion follows
from 3 . 1 . 13.32
O.E.D.
LEMMA.
Let
r
IxM + N
:
be a local analytic flow on
an open subset M of a Ranach manifold N , with infinitesimal generator X E T(M) For t E I , put Let o E M Then Ordo(X) > 2 if and gt(m) := r(t,m) only if gt(o) = o and T (g,) = id for all t E I
.
.
.
.
0
By 1 3 . 3 1 , Ordo(X) > 1 if and only if gt(o) = o for Assuming this property, let (P,p,Z) be a chart all t E I of M about o and put PROOF.
.
Then $t : = pOgtop-l satisfies $t(0) = 0 and, by 5.2, the derivatives $ ; ( O ) E L ( 2 ) solve the differential equation
with initial condition
for all $;(O)
t
E
= idz
I
.
$;)(O)
= idz
It follows that
for all
t
E
I
.
.
Hence
h'(0) = 0
if and only if O.E.D.
The infinitesimal version 1 3 . 3 0 of Cartan's uniqueness theorem shows that, for every connected complex metric Ranach manifold (M,d) , a vector field X E aut(M,d) vanishes identically provided it has order > 2 at some point o E M Now assume in addition that every holomorphic mapping f : C + M is constant. Then g := aut(M,d) satisfies g n i g = { O } by 1 3 . 2 7 . It follows that the complexification
.
g
C := gaRC
.
of 4 can be identified with a subalgebra of T(M) general, the holomorphic vector fields in g c are not uniquely determined by their derivatives of order < 1
In (the
GROUPS OF HOLOMORPHIC ISOMETRIES
t'l-jet") at some point field
o
E
M
.
227
For example, the vector
on the open unit disc A belongs to aut(A)' but vanishes of order 2 at 0 E A This example is somewhat typical, for it will now be shown that, in general, the vector fields in g c are uniquely determined by their derivatives of order < 2 at some point o E M
.
.
13.33 THEOREM. Let (M,b) be a connected complex normed Banach manifold such that every holomorphic mapping f : C + M have order
is constant. Put g : = aut(M,b) and let > 3 at some point o E M Then X = 0
.
.
.
X
PROOF. Put X = Y + iY2 , where Y E g By 4.6.2, 1 j Hence [X,Y1] = i[Y2,Y1] E ig has order > 2 at o [X,Y1] = 0 by 13.30. For j = 1,2 , put gi : = exp(tY.) 3 By 13.32, we have g-t(o) 1 = g-t(exp(tX)(o)) 1 for all
t
E
.
R
=
exp(t(X-Yl))(o)
.
=
2 1 f(s+it) := g,(g-,(o))
Hence
gC
E
.
exp(itY2)(o) defines a
holomorphic mapping f : C + M which is constant by assumption. Hence gi(o) = o for all t , showing that Ordo(Y.) > 1 Endow 2 : = To(M) with the Ranach space 3 Since gtj E Aut(M,b) , the assignments norm bo t + To(g:) are continuous 1-parameter subgroups of U k ( Z ) with infinitesimal generator
.
D
j
.=
a -
j at To(gt)t=o
E
Uk(Z)
.
.
By 13.32, we have D1 + iD2 = 0 Hence D = 0 by 13.29. j Therefore Ordo(Y.) > 2 , and 13.30 implies Y = 0 O.E.D. 3 j
.
13.34 COROLLARY. Let M be a connected complex Ranach is manifold such that the Carathsodory tangent norm 8 , compatible. Put 4 : = aut(M) and let X E g have order
> 3
at some point
o
E
M
.
Then
X = 0
.
SECTION 13
228
PROOF.
Apply 12.24.
13.35 COROLLARY, L e t D be a domain in a complex Ranach space Z such that the Carathgodory tangent norm BD is Then the linear mapping compatible. Put g : = aut(D)
.
is injective for every
o
E
D
.
NOTES. The main result 13.14 appears in [137,1381. Independently, the special case stated in Theorem 13.16 has been proved by J.P. Vigui! L1481. For the proof of H. Cartan's original result (13.19), see [log; Ch. 9 1 or [ 93; Ch. 111, Th. 1.23. A somewhat related result is a theorem of S. Bochner and D. Montgomery showing that the group of all holomorphic automorphisms of a compact complex manifold is a complex Lie transformation group with respect to the compact-open topology C93; Ch. 111, Th. 1.11. The basic Lemma 13.1, using Hadamard's three circles theorem, is due to J . P . Vigug [145,148 I. A s shown by counterexamples U45,1481, the "analytic" topology on G := Aut(D) , for a bounded domain D in a complex Banach space, can be strictly finer than the topology of locally uniform convergence. However, it can be shown that for bounded symmetric domains (cf. Section 20), both topologies coincide since in this case G can be realized as an open subgroup of a linear algebraic group (acting on aut(D)' ) and hence Theorem 7.14 applies. The topological and uniform versions 13.7-13.13 of the classical Cartan uniqueness theorem 13.10 are due to Vigu6 U45-148,1541. For a direct proof of Theorem 13.10, see [log; Ch. V, Proposition 11. The complexified version of Cartan's uniqueness theorem [901 plays a central role in the study of complete holomorphic vector fields on Siege1 domains (cf. [90,911 and Section 16).
PART I 1
SYMMETRIC MANIFOLDS AND JORDAN ALGEBRAIC STRUCTURES
The second part of the book, devoted to the class of symmetric Banach manifolds, is more algebraic in character since symmetric manifolds can be characterized in terms of certain algebraic systems. In the finite dimensional case, the fundamental results of E. Cartan show that symmetric manifolds can be described in terms of semi-simple Lie groups. For hermitian symmetric spaces, M. Koecher has proposed an alternative, somewhat more elementary approach using Jordan algebras instead of Lie algebras [95,1031. In recent years it has been shown (cf. [58,84,148,24,91,871)that the Jordan algebraic approach carries over in a remarkably smooth way to the infinite dimensional situation of symmetric complex Banach manifolds. Since the relevant Banach Jordan algebras are by now well-understood, thanks to the deep result of E. Alfsen, F. Shultz and E. Stgrmer [ 5 1 ,
the algebraic
description of symmetric manifolds will be given in terms of Jordan algebras and the slightly more general Jordan triple systems. The necessary algebraic background on Jordan structures is provided in Sections 18 and 19, whereas Sections 14 and 15 contain the basic facts on ordered Banach spaces and
*
C -algebras. Symmetric manifolds and, as special cases,
bounded symmetric domains and (symmetric) Siege1 domains are systematically studied in Sections17,20,16 and 21. Various automorphisrn groups associated with Jordan algebras are considered in Section 22, and Section 23 studies the structure and automorphisms of the "classical" Banach manifolds generalizing the Grassmann manifolds and their collineations.
229
SECTION 14
230
14.
ORDER UNIT RANACH SPACES
In this section, we introduce a class of real Ranach spaces which are endowed with an ordering induced by a "positive" cone. These ordered Ranach spaces are closely related to the algebraic structures associated with symmetric Banach manifolds. A s the prototype of these algebraic structures, the so-called C*-algebras will be studied in Section 1 5 . Let X be a real Ranach space. A subset C of X is A cone C is called a cone if tC C C for all t > 0 convex if and only if C + C C C In the following, let C
.
.
-
be an open convex cone. Then the closure closed convex cone in X satisfying
c +x+cc Now assume
C
X+ := C
.
is a
14.1.1
contains a point
e
.
Then
0
E
X+
and
x < y : e y - x EX+ defines a semi-order (i.e., a transitive and reflexive relation) on X Since X, is a convex cone, we have
.
x < y , u < v + x + u < y + v . 14.1 LEMMA. Let R := dist(e,aC)
lxle 2 RX > 0
.
.
1-1
be a compatible norm on X Then every x E X satisfies
PROOF. Since Iy-el < R implies e RX/~X~ > 0 if x # 0
.
14.2
PROOF.
COROLLARY. For
x
E
X = X+ X
14.1.2
,
-
X+
we have
y > 0
,
and put
it follows that O.E.D.
. x = (x+(x(e/R) - (x(e/R
.
O.E.D.
14.3 PROPOSITION. lxle : = inf{ t > 0 : te 2 x > 0 } defines a continuous semi-norm on X with null-space X + n -X+
.
PROOF.
By 14.1,
1x1, < (x(/R
.
By 14.1.2,
the relations
ORDER UNIT BANACH SPACES
te
2 x >
and
0
_+
se
y > 0
imply
(s+t)e
Hence I l e is a continuous semi-norm on closed, we have
14.4
X + n -X+ =
{ x
PROPOSITION.
(i)
C
(ii)
X+ A
PROOF.
.
Since
.
X+
is
.
O.E.D.
The following conditions are equivalent:
contains no affine real line.
-x+
=
.
{o}
0
14.3.1
X : lxle = 0 }
E
_+
.
lxlee 2 x > 0 Hence
231
on
1*Ie
The semi-norm
Assume (ii) and let
.
is an order.
X
on
is a norm.
X
x,y
E
X
x _+ ty
satisfy
.
E
C
for
all t > 0 Then x/t + y E C if t > 0 For t + +- , we By (ii), we get y = 0 Hence (ii) implies get +y E X+ (i). Conversely, assume (i) and let +y > 0 Then e + Ry c C + X + C C By (i), we get y = 0 Hence (i) and (ii) are equivalent. By definition, the semi-order < is anti-symmetric if and only if (ii) holds. By 14.3, (ii) and
.
.
.
.
.
(iv) are equivalent. 14.5
DEFINITION.
O.E.D.
An open convex cone
C
in
X
containing
e is called regular if the conditions of 14.4 are satisfied. In this case, the norm I l e on X is called the order unit norm associated with C and the "order unit" e c C . The order unit norm can be used to introduce a topological version of regularity which is more appropriate in the infinite-dimensional setting. 14.6
DEFINITION.
A
regular open convex cone
C
in
X
containing e is called topologically regular if the order on X is compatible. In this case X , unit norm 1 endowed with the norm 1 . 1 = and the order induced by
*Ie
[*Ie
SECTION 14
232
X,
-
:= C
is called an order unit Banach space.
Every regular open convex cone C in X is topologically regular if X is finite-dimensional. For example, X := R is an order unit space with c : = { x ~ R : x > O } , C=R, and e = l . The corresponding order unit norm coincides with the absolute value. In Banach space theory, the continuous linear functionals play a fundamental role, For ordered Ranach spaces, one is interested in linear functionals preserving the order properties. These functionals are called "states" since they arise naturally in the quantum mechanical formalism. 14.7 DEFINITION. Suppose X is an order unit Aanach space with positive cone X, and order unit e A linear A functional f : X + R is called positive if f(X,) C R, positive linear functional f satisfying f(e) = 1 is called a state of X
.
.
.
14.8 LEMMA. norm 1
.
PROOF.
Apply
Every state
f
f
of
X
is continuous with
to the inequalities
(xle 2 x > 0
. Q.E.D.
By 14.8, the set Sx of all states of an order unit Banach space X is a closed convex subset of the closed unit ball B := { f E L ( X , R ) : If1 6 1 } Since R is compact in the weak * topology [15; 44.121, it follows that Sx is a compact convex set called the state space of X The following result, an ordered version of the Hahn-Ranach Theorem, shows that order unit Ranach spaces have "sufficiently many" states.
.
.
14.9 THEOREM. Let Sx be the state space of an order unit Banach space X Then we have for every x E X
.
x > 0 and
f(x) > 0
for all
f
E
Sx
14.9.1
ORDER UNIT BANACH SPACES
233
14.9.2
PROOF.
x ,d X+
Suppose
subset of
X
,
.
Since
115; 34.11
X+
is a closed convex
implies that there exists
.
f(x) < inf f(X+) < 0
f is an open mapping by [15; 48.11, f(C) is a non-empty cone in R not containing f(x) Hence f ( C ) = R+\ { a } It follows Dividing by f(e) , we that f ( X + ) = R, and f(e) > 0 obtain a state $ on X with $(x) < 0 This proves f
E
L(X,R)
with
.
14.9.1.
.
.
.
sup (f(x)l < ( X I fESX te + Xx k, X+ for some X E {:l} :=
.
Ry 14.9.1,
.
f E Sx such that t + X f(x) < 0 f(x) < If(x)l , a contradiction.
exists t < -A 14.10
.
Now assume t
Then
Since
COROLLARY.
and let
x,y
E
X+
Suppose
.
X
Hence O.E.D.
is an order unit Ranach space
Ix+yl > max( 1x1
Then
there
,IY(
.
)
PROOF. For every f E Sx , we have f(x+y) = f(x) + f(y) > max(f(x),f(y)) > 0 Now apply 14.9.2. O.E.D.
.
14.11 EXAMPLE. Let S be a locally compact Hausdorff space and consider the real Banach space X = { f + t*lS : t of all continuous functions infinity. Here lS(s) := 1
c
:=
,
f
E
f : S
+
R
s
R
for all
[ f
E
x
]
C,(S,R)
s
converging at
.
Then
: inf f(s) sss
>
o ]
is a topologically regular open convex cone in X , and the order unit norm with respect to the constant function e := lS coincides with the supremum norm. Under the pairing
u(f)
:=
f(s) d v ( s )
I
S
the state space
Sx
consists of all positive measures on
S
234
SECTION 14
which are bounded with total mass 14.12
EXAMPLE.
Let
E
1
.
be a Hilbert space over
D
{R,C,E}
E
and consider the real Ranach space
x
=
{ x + t*idg
:
t
E
,x
R
= X* E L ~ ( E )}
.
Here Lu(E) denotes the Ranach space (over the center K of D ) of all compact D l i n e a r operators on E Then
.
is a topologically regular open convex cone in X , and the order unit norm with respect to the identity operator e := idE coincides with the operator norm. Under the pairing +(x) := trace(@x) the state space S x operators $I E L ( E )
,
consists of all positive trace-class of trace 1 [118; 1.19.11.
Ranach order unit spaces are closely related to Ranach algebras with involution. 14.13 DEFINITION. Let algebra over K E { R , C }
be a (not necessarily associative) A K-antilinear mapping z + z* on Z is called an involution if ( z * ) * = z and (zw)* = w*z* for all z,w E Z 2
.
.
The unitization
Z'
:=
of a non-unital involutive
ZfBK
algebra 2 over K has a canonical involution. Similarly, the complexification Xc = X 0R C! of a real involutive algebra X has a canonical involution. For a Aanach algebra Z endowed with a continuous involution, the selfadjoint part
x
:=
{ x
E
2 :
.
is a closed real subspace of 2 closed under the algebra product.
x* = x ] In general,
X
is not
Recall that the spectrum for a non-unital associative
ORDER U N I T BANACH S P A C E S
Ranach algebra
where
over
Z
and
K
z
is defined by
Z
E
is the unitization of
Z' : = ZCBK
235
Z
.
14.14 DEFINITION. Let Z he an associative Ranach algebra over K endowed with the norm I * I A closed real suhspace X of 2 is called 'hermitian if every x E X satisfies
.
14.14. I
Cz(x) C R and
.
1x1 = sup ICz(x)l A
continuous involution
if the self-adjoint part 'hermitian every z
of
z + z*
of
X
is called 'hermitian
Z
is 'hermitian.
Z
continuous involution of Z satisfies Cz(z*z) > 0
E
14.14.2
Z
.
A
is called positive if
14.15 LEMMA. Let Z he an associative unital Ranach algebra over K and let X be a 'hermitian closed real subspace of Z containing the unit element e of Z Put
.
x,
:=
{ x
E
x
: Cz(x)
> 0 }
.
Then le-xl < 1 jx 1x1 < 1
I
x
E
E
X,
X+
+
,
14.15.1
le-x( < 1
14.15.2
and x PROOF.
If
E
I(xle-x( < 1x1
X+
le-xl < 1
-1 < Zz(x-e) = Cz(x)-l
1x1 < 1
and x 0 < ~,(e-x) < 1
E
.
.
14.15.3
, [17; 5.81 implies
.
< 1
.
Hence
zZ(x
>
0
.
Now assume
X+ Then 0 < C,(x) < 1 and hence Since e-x E X , 14.14 2 implies
SECTION 1 4
236
le-x( < 1 In order to prove 14.15.3, we may assume 1x1 = 1 In this case the assertion follows from 14.15.1 and 14.15.2. O.E.D.
.
14.16 COROLLARY. interior
X,
is a closed convex cone in
C = X+A G(Z)
=
{ x
E
X : c,(x)
*Ie
X
with
.
> 0 }
The order unit norm I on X with respect to e coincides with the given norm 1 - 1 In particular, topologically regular open convex cone.
.
C
E
C
By definition, X+ is a closed subset of X Now assume satisfying t X + C X+ for all t > 0 x,y E X+ In order to show x+y E X+ , we may assume 1x1 < 1 and Iy( < 1 Then le-xl < 1 and le-y( < 1 14.15.2 and hence
PROOF.
is a
.
.
.
by
.
By 14.15.1, x+y E X+ Hence X+ is a closed convex cone in X Since CZ(lxle 2 x) = 1x1 _+ Z z ( x ) > 0 , we have Hence \ x l e < 1x1 Since lxle _+ C,(x) > 0 lxle 5 x > 0 by 14 3.1, we have IxIe > sup IZ,(x)( = 1x1 Hence = 1x1 By 14.14.1 and 2.10, every x E X satisfying lXle > 0 belongs to the interior C of X, Conversely, Cz(X if x E C then x + te E X, for -T < t < T It follows that 0 # C,(X) O.E.D.
.
.
.
.
.
.
.
.
14.17 LEMMA. Suppose 2 is an algebra over K , endowed with a involution. Then for every x E with y 2 = x Here W denotes generated by e and x
.
.
associative unital Ranach ‘hermitian continuous X+ there exists y E X + n W the closed suhalgehra of 2
.
We may assume 0 < X,(x) < 1 Then satisfies 0 < E,(a) < 1 The power series PROOF.
.
a :=
e - x
ORDER UNIT BANACH SPACES
237
about 0 has coefficients cn > 0 and converges absolutely for X = 1 By Abel's Theorem [27; p. 741, we have
.
m
lim f(r) = 1 cn r+l n=O Since la1 < 1 the limit
.
by 14.14.2, a completeness argument shows that
y := lim f(ra) r+l
E
W
.
exists and satisfies y 2 = e-a = x Since X, is a closed convex cone (14.16) invariant under taking powers (by the spectral mapping theorem), f(ra) E X, for all r < 1 and therefore y E X, O.E.D.
.
14.18 a,b
COROLLARY. E
Every
satisfy
X,
x
E
X
ab = ba = 0
has the form
x = a-b
.
,
where
Let W denote the closed subalgebra of Z generated Let Ew denote the spectrum space of all by x and e 2 continuous unital homomorphisms f : W + C Since x E X+ by the spectral mapping theorem, 14.17 implies that x2 = y2 PROOF.
.
for some we have
y
E
W n X+
C z ( YiX) =
.
.
By 14.14.1
E,(ytx)
=
and [17; 5.14 and 17.131,
{ f(y)kf(x)
: f
E
EIJ }
2 2 Since f ( ~ =) f(x ~ ) = f(y = f(y)2 and f(y) > 0 for all f E cw , it follows that f(y) _+ f(x) > 0 Hence Now put a := (y+x)/2 and b := (y-x)/2 y 2 x E X,
.
.
.
O.E.D.
14.19 LEMMA. Let Z be an associative Ranach algebra over K and denote by Lxz := xz the left multiplication in 2 Then every x E Z satisfies
.
F
PROOF. If [17; 5.41. 2'
:=
Z8K
Z is unital, the assertion follows from Now assume that Z is non-unital and let denote the unitization of 2 Then the left
.
SECTION 14
238
multiplication operator satisfies
-
A idzl
L; = (
L;
E
x
L(Z1)
associated with
idZ - Lx
-!2
x
0
"1
x
E
Z
'
where R x : K + Z is def ned by t X s := sx for all s E K It follows that 0 E Ciz1(L;) and
.
Ekz1(L;)\
{O} =
since
for all x E C ~ , ( L ~ ) \ { O } Cgzl(Li) = Czl(x) = C , ( x )
.
Since the assertion follows.
,
O.E.D.
14.20 LEMMA. Suppose Z is an associative Ranach algebra over K Then every z E u ( Z ) satisfies Z z ( z ) C iR
.
.
.
PROOF. For t E R , put Ct := CE2(exp(tLZ)) Then -1 lexp(tLZ)I < 1 implies that C t and C-t = Ct are contained in the closed unit disc A [17; 5.81. Hence C t C a A for all t E R The spectral mapping theorem [17; 7.41 implies CR2(LZ) C iR Now apply 14.19. O.E.D.
.
.
14.21 DEFINITION. Let 2 be an associative Ranach algebra Let Lxz := xz and over K , endowed with the norm 1 . 1 R z := zy denote the left and right multiplication operators Y on 2 , respectively. A continuous involution z + z* on Z is called -hermitian if Lz and Rz belong to u ! 2 ( 2 ) whenever z = - z * E Z A continuous involution on 2 is called hermitian if it is 'hermitian and -hermitian.
.
.
14.22
COROLLARY.
Suppose
2
is an associative Ranach
algebra over K , endowed with a -hermitian continuous involution. Then C , ( z ) C iR whenever z = -z* E 2
.
14.23 THEOREM. Suppose 2 is an associative unital Ranach algebra over K Then every hermitian continuous involution z + z* of Z is positive.
.
239
ORDER U N I T BANACH S P A C E S
PROOF. Suppose first that Then [17; 5.31 implies
b
satisfies
E Z
-b*b
.
E X+
.
Hence -bb* E X+ Write h = x + u with x* = x and u* = -u Then b*b = (x-u)(x+u) = x2 - u2 - u x + xu and bb* = (x+u)(x-u) = x2 - u2 + ux - xu Since x 2 E X+ and -u 2 E X+ by 14.14.1 and 14.22, 14.16 implies
.
.
-
b*b = -bb* + 2x2
2u 2
E
.
X,
.
Since X + A -X+ = { O } by 14.16 and 14.4, we get b*b = 0 Now let z E Z By 14.18, we have z*z = x-y , where x,y E X + satisfy xy = yx = 0 Put b := zy Then Hence b*b = 0 and y = 0 by -b*b = -YZ*ZY = y 3 E X+
.
.
.
14.14.2. 14.24
z*z = x
It follows that
COROLLARY.
X+ =
{ x2
:
x
E
.
E
.
X+
X }
Q.E.D.
{ z*z
=
: z
E
Z
}
.
Suppose in the following that Z is a unital associative Ranach algebra over K , endowed with a hermitian continuous We will study the so-called Gelfandinvolution z + z* Neumark-Segal construction which yields representations of 2 by Hilbert space operators. By 14.16, the self-adjoint part X of Z is an order unit Ranach space. Identify a state f E Sx with its canonical extension f E L ( Z , K ) satisfying f(z*) = (fz)* for all z E Z
.
.
14.25
LEMMA.
Suppose
f
E
Sx
and
z,w
E
Z
.
Then
If(Z*W)( < f(z*z)l/2 f(w*w) 1/2 and
.
If(z)l < Iz*z11'2 PROOF.
By 14.23, the K-sesqu linear form
14.25.1
(zlw), := f(Z*W) on
Z
satisfies
(zlw); = ( w z),
and
(z z)f
0
.
Now the
SECTION 14
240
Cauchy-Schwarz inequality follows from [28; 6.2.11. f(X has norm 1 , we get If(z)l2 < f(e*e) f ( z * z ) = f(z*z)
lz*zl
6
Since
.
Q.E.D.
14.26 THEOREM. Suppose z + z * is a hermitian continuous involution of an associative unital Ranach algebra 2 over K with self-adjoint part X Then for every state f of X , there exist a Hilhert space Ef over K with scalar product ( 1 I f , a continuous unital *-homomorphism nf : Z + L(Ef) and a unit vector ef E Ef such that every z E 2 satisfies
.
PROOF.
f ( z ) = (eflnf(z)ef)f
.
(z(z)f = 0 } = { z
2 :
14.26.1
By 14.25,
If := { z
2 :
E
6
(ZIUf = 0 }
.
is a closed subspace of Z The quotient vector space z/If carries the strictly positive scalar product ( z + I f l w + y f := (zlw)f
.
Hence the completion Ef of Z/If is a Hilbert space over K Put ef : = e + If , Then (eflefIf = f(e*e) = f(e) = 1 If is a left ideal in Z since
.
for all
x,y,z
E
2
.
Therefore (VfZ)(Y+If) := zy
i .If
.
defines a linear mapping v f : 2/If + 2 / I f Now define $Z := f(y*zy) Then $(X+) C R+ by 14.24. Hence
.
f(y*z*zy) = $(z*z) < Iz*z( +(el = Iz*zI Since the involution is continuous, it follows that continuous and
f(y*y) nfz
. is
.
241
ORDER UNIT BANACH SPACES
nfz
Hence
has an extension to a bounded operator on
Ef
.
,
again denoted by nfz By definition, n f : Z + L ( E f ) is a nf is unital *-homomorphism satisfying 14.26.1. By 14.26.2, 1 < Inf
continuous with norm satisfying
I
< N112 , where
is the norm of the involution. 14.27
COROLLARY.
There exist
and a unital *-homomorphism
x
E
X
and all
z
E
N
Q.E.D.
a Hilbert space
n : 2
+ L(E)
E
over
K
such that all
satisfy
2
Inxl = 1x1
14.27.1
and 14.27.2 PROOF.
Consider the Hilbert sum
l2
E :=
fES
Ef X
and the continuous unital *-homomorphism given by the direct sum
1
n :=
nf : 2 + L ( E )
f ESX
Then 14.26.2 and 14.9.2 In212
=
.
imply
sup lnfzI2 = sup f(z*z) = Iz*z( fESX fESX
In particular, 14.14.2
implies
1nx12 = (x21
= 1x1
2
. O.E.D.
We now consider the special case 14.28
DEFINITION.
Let
respect to the norm 1 - 1 hermitian if lexp(itx)l
E
.
K = C
.
be a complex Ranach space with
An operator x E L ( E ) is called < 1 for all t E R Let
.
242
SECTION 14
denote the set of all hermitian operators on 2 be an associative complex Ranach algebra.
.
E Now let An element
x E 2 is called hermitian if the left and right multiplication operators Lx and R, are hermitian operators on
Z
.
Let H(2) =
{ x
E
Z : Lx,Rx
HP,(Z)
E
}
.
denote the set of all hermitian elements in 2 HP,(E) = H( L ( E ) ) for every complex Ranach space
Then E
.
14.29 PROPOSITION. Let 2 b e an associative complex Ranach algebra. Then H ( 2 ) is a closed real subspace of 2 and XIY E H ( 2 )
If
2
=j
has a unit element
i(xy-yx) e
,
E
then
.
H(2)
e
E
14.29.1
H(2)
,
and
H(z) PROOF,
Since
=
f x
uk(2)
R }
.
is a closed real subalgebra of g S ( Z ) for all x,y E 2 I [Rx,R I = R
,
E Z
: lexp(itx)) c 1
for all
t
E
and [Lx,L I = L IYIXI [XPYl Y Y the first assertion and 14.29.1 are clear. Now suppose 2 has a unit element e Then x + L X is injective. Hence 14.29.2 follows from 13.29. It is clear that every x E 2 satisfying (exp(itx)l c 1 for all t E R is hermitian. Conversely, we have
.
14.30 LEMMA. Let 2 be an associative unital complex Ranach algebra. Then X := H ( 2 ) is a 'hermitian real subspace of 2 , i.e., every x E X satisfies C,(x) C R and s u p IC,(X)l
PROOF.
=
1x1
.
The first assertion follows from 14.20.
By [17; 5.81,
ORDER UNIT BANACH S P A C E S
243
.
we have sup (C,(x)l < 1x1 To show equality, we may assume ICz(x)l < n/2 The principal branch of arc sin(X) has a
.
power series expansion
convergent for if 1 5 1 < n/2 implies
.
IA(
< 1 , and satisfying 5 = arc sin(sin 5 ) Therefore the holomorphic functional calculus m
1
x = arc sin(sin x) =
cn(sin x) n
n=l Since x E H ( 2 ) , it follows that [sin = I(exp(ix)-exp(-ix))/2i( all n , it follows that
XI
1
m
1x1
0 ]
: C,(X)
is a closed convex cone in
X : C,(x)
> 0 }
X+ is the topologically regular open convex cone consisting of a l l strictly positive elements in X Further, on X with respect to e E C the order unit norm coincides with the given norm.
of
.
I*le
PROOF.
B y 14.30, we can apply 14.16.
O.E.D.
14.32 EXAMPLE. Suppose E is a complex Ranach space with Let X = H E ( E ) denote the real Ranach space of norm I * I The elements of all hermitian operators on E
.
X,
.
= HE+(E)
:=
{ x
E
HE(E)
:
CEE(x) > 0 }
244
SECTION 14
are called eositive operators on C = X + A G!?,(E) =
{
X
E
E
,
and the interior
H ! ? , ( E ) : CLE(X)
consists of all strictly positive operators on Let algebra.
>
0
E
.
}
2 be an associative unital complex Ranach By 14.29.2, the complexification
of the "purely real" Ranach space H ( 2 ) can be identified with a complex subspace of Z , It will now be shown that H(Z)' is in fact a closed subspace of 2 For the proof, we need a concept of "states" on Z related to 14.7.
.
14.33 DEFINITION. Let 2 be an associative unital complex Banach algebra. Then f E L ( 2 , C ) is called a state on Z if If( = 1 = f(e)
.
of all states on Z is a weak* compact L(2,C) called the state space of 2
The set S z convex subset of
.
14.34 LEMMA. For every state f of 2 , the restriction f l ~is a state of the order unit Ranach space X := H ( 2 ) PROOF.
For
;1
t > 0
and
z
E
2
,
.
we have
+ f(z) Re f Z) < Iz+e/t( - l/t
and therefore
.
Further,
I lexp(tz)l-le+tzl I Now suppose
x
E
X
.
Re(i f(x))
,
lexp(itx)l = 1
< t 1 $(t) c(le+itxl-1) 1
and hence
. .
it follows that Im f(x) > 0 Replacing f(x) E R Now suppose x E X, Then 0 < C,(x) < 1x1 and hence < 1x1 By 14.30, this implies 0 < CZ((x(e-x) = 1x1 - cZ(x For
x
Then
by
-x
t
+
0
, we get
.
.
.
245
ORDER U N I T B A N A C H S P A C E S
1
Ixle-xl < 1x1 and hence Ilxl-f(x)l = If(lxle-x)l < 1x1 Since f(x) is real by the first part of the proof, it follows that
.
f(x) > 0
.
Q.E.D.
14.35 LEMMA. For every z E 2 , S 2 ( z ) := { f(z) : f E S 2 } is a compact convex subset of C containing C , ( z ) In particular, we have for z E 2 and x E H ( 2 )
.
Sup
IC,(Z)I
< Sup
max(lx(,lyI) Hence H ( 2 ) ' is a closed complex subspace of 2 and (x+iy)* := x-iy defines a continuous mapping on ~ ( 2 ) '
.
PROOF. For every f E (f(x)( < If(x)+if(y)l
S2
.
, 14.34 implies Now apply 14.35.1
and 14.35.2. O.E.D.
14.37 COROLLARY. S (x) = co E,(x) 2
For every
.
x
E
H(2)
,
we have
PROOF. By 14.35 and 14.34, co Cz(x)C S2(x) C R 14.35.2 implies for all t E. R
.
Further,
246
SECTION 1 4 sup [Cz(x)-tl = sup IS2(x)-tl
. Q.E.D.
14.38 y
E
.
H(Z)
Put
PROOF.
S2(x) > a where CZ(Z)
NOTES.
X+
For
SZ(Z)
f
E
and
c c \to}
H+(Z)n G(Z)
and
is i n v e r t i b l e .
>
0
. Then 1 4 . 3 7 i m p l i e s f(z) f(x) + if Y) . Hence 14.37 i m p l i e s
we h a v e
Sz
f(y)
E
y
u := i n f C z ( x
f(x) > u
c
z :=
Then
.
x
Suppose
COROLLARY.
E
.
R
=
I
Q.E.D.
F o r a s y s t e m a t i c a c c o u n t o f t h e t h e o r y o f Banach o r d e r
u n i t s p a c e s a n d t h e i r d u a l i t y w i t h c o m p a c t c o n v e x s e t s , see
C11. The p r o o f o f 1 4 . 2 3 - 1 4 . 2 6
f o l l o w s C17; Ch. V 1 . H e r m i t i a n
e l e m e n t s a n d o p e r a t o r s c a n a l s o b e d e f i n e d i n terms o f c h e n u m e r i c a l r a n g e , s t u d i e d i n [ 1 8 , 1 9 1 . The o r d e r s t r u c t u r e o f
*
C -algebras R.V.
( c f . S e c t i o n 1 5 ) h a s been s t u d i e d i n d e p t h by
Kadison [761.
The most i m p o r t a n t r e g u l a r c o n v e x c o n e s i n R
n
are t h e
" s e l f - d u a l " c o n e s w i t h r e s p e c t t o a b i l i n e a r f o r m . The homog e n e o u s s e l f - d u a l c o n e s h a v e b e e n c l a s s i f i e d a n d a r e i n 1-1 correspondence w i t h t h e so-called f o r m a l l y real Jordan a l g e b r a s ( c f . S e c t i o n 19 and [ 7 5 , 2 5 1 ) . Recently, t h e concept o f s e l f d u a l homogeneous c o n e h a s b e e n g e n e r a l i z e d t o t h e c a s e o f
( r e a l ) H i l b e r t s p a c e s . I t h a s b e e n shown t h a t t h e s e c o n e s a r e c l o s e l y r e l a t e d t o von Neurnann a l g e b r a s ( A . C o n n e s ) a n d more g e n e r a l l y , t o B a n a c h J o r d a n a l g e b r a s w i t h p r e d u a l
[8-13,
661.
S t a t e s p a c e s o f order u n i t s p a c e s a n d Banach a l g e b r a s p l a y
a f u n d a m e n t a l role i n quantum m e c h a n i c s , b e i n g t h e n a t u r a l "dual" o b j e c t s of t h e system of observables ( c f . t h e i n t r o duction of
[221).
I t i s therefore of i n t e r e s t t o characterize
*
t h e s t a t e s p a c e s o f v a r i o u s algebraic o b j e c t s ( C -algebras)
*
Jordan C -algebras
a n d g e n e r a l i z a t i o n s ) among a l l c o m p a c t
c o n v e x s e t s i n terms o f p h y s i c a l l y r e l e v a n t a x i o m s . R e c e n t l y , t h e s e problems have been s t u d i e d w i t h c o n s i d e r a b l e s u c c e s s C2-4,67,123,1281.
C -ALGEBRAS
24 7
a
15.
c*-ALGEBRAS
The theory of order-unit Ranach spaces, their states and representations, will now be applied to study an important class of involutive Ranach algebras, the so-called C*-algebras. C*-algebras are the prototype of algehras connected with symmetric Ranach manifolds and reveal an interesting relationship between infinite-dimensional holomorphy and functional analysis.
Since C*-algebras are
also fundamental for the algebraic formulation of quantum mechanics [120] they provide a link for possible applications of infinite-dimensional holomorphy to problems in mathematical physics [144]. 15.1 DEFINITION. algebra over K
.
Suppose 2 is an associative Ranach An involution z + z * of 2 is called a
C*-involution if Iz*zI =
.
2
15.1.1
z E 2 It follows that ( z I L < Iz*I-IzI and hence < (z*I , i.e., Iz1 = I z * I for all z E 2 Therefore
for all 121
IZI
.
every C*-involution is isometric and hence continuous. 2 Combining this fact with 15.1.1, we get I z z * ( = I z * I = for all 15.2
z
E
2
EXAMPLE.
.
.
Let
E
and
F
121
2
be Hilbert spaces over
D E {R,C,H} For any operator z E L ( E , P ) , define the by (zhlk) = (hlz*k) for all adjoint operator z * E L ( F , E )
.
E ExF Then (z*)* = z and (zw)* = w*z* whenever F,L) and L is another Hilbert space over D Since W E 1 (z*zh k ) = (zhlzk) for all h,k E E , the Cauchy-Schwarz i nequa ity [28; 6.2.11 implies
(hik)
.
SECTION 15
248
Hence z + z* is a C*-involution of the associative unital Ranach algebra Z := L ( E ) over the center K of D , endowed with the operator norm. 15.3 EXAMPLE. Let 2 := g ( S , D ) denote the associative unital Ranach algebra (over the center K of D ) of all bounded D-valued functions on a set S , endowed with the supremum norm. For z E 2 define the adjoint function z* E 2 by z * ( s ) := z ( s ) * , where * denotes the canonical involution of D Since I X * X I = 1 x 1 ~for all x c D , it follows that z + z * is a C*-involution of 2 The closed subalgebras Lm(S,D) , Cm(S,D) and Cu(S,D) of B ( S , D ) , associated with a measure space S , a topological space S and a locally compact space S , respectively, are invariant under the C*-involution.
.
.
15.4 LEMMA. Let 2 be an associative Ranach algebra over K , endowed with a C*-involution. Then z*z = zz* implies n I l/n
I z ( = lim Iz
PROOF.
=
For
I (z*z)
n
N
E
2n-1 2
I
=
,
n+m 15.1.1 implies
2n-2 4 ((z*z) I =
... =
.
15.4.1
2" Iz*zI
2n+1 =
121
Hence 1 2 2"
I2-" = ( z (
.
Since the limit 15.4.1 exists [17; 2.81, it is equal to
.
IzI
O.E.D.
15.5 COROLLARY. Let W be an abelian Ranach algebra over K , endowed with a C*-involution. Let C w be the (locally compact) spectrum space of W Then the Gelfand mapping
.
is an isometric homomorphism onto a closed subalgebra over
K
*
C -ALGEBRAS
separating the points of
PROOF. 15.6
Cw
249
.
Apply 15.4 and [17, $171. LEMMA.
Suppose
Z
O.E.D.
is a non-unital associative Ranach
algebra over
K I endowed with a C*-involution. Then the canonical involution on the unitization 2 ' := 28K of Z
is
a C*-involution with respect to the Ranach algebra norm I z + s I = sup
.
Izw+swI
IWI
PROOF. F o r x E 2 ' , let Liy : = xy denote left multiplication on 2' By definition, 1x1 = ILi12
.
L;
operator norm of
x,y
for all
E
2'
I
on the ideal
2
of
.
2'
and the unit element
e
.
2'
E
is the
Hence
satisfies
Now suppose x = z + s satisfies lel = ILE,I = lid12 = 1 1x1 = 0 Then zw = -sw for all w E 2 If s # 0 , it follows that c : = -z/s is a left unit of 2 Since wc* = (CW*)* = (w*)* = w and c* = cc* = c , c is a unit
.
of
2
.
lL;lz
0
}
.
In particular, €or a Hilbert space E over D E {R,C,B} , the unital C*-algebra U := L(E) over the center K of D gives rise to the "operator tube domain"
Similarly, the unital C*-algebras U = Ls(S,K) or U = Cm(S,K) associated with a measure space S or a topological space S , respectively, give rise to the 'I function tube doma i n II
D~
=
{ u
E
u
: inf (u(s)+u(s)*) s ES
> o }
.
.
Let C be an open convex cone in X A continuous sesqui-linear mapping 8 : V x V + U satisfying 16.0.1 is called C-positive if 8(v,v) E c for all v E V
.
16.6 8
Suppose C is an open convex cone in is C-positive. Then the Siege1 domain
X
LEMMA.
DC,@ : = { (u,v)
is convex.
E
U x V : 2 Re(u)
-
O(V,V)
E
C }
and
265
TUBE DOMAINS AND SIEGEL DOMAINS
PROOF. have
For
(uo,vo)
,
(ul,vl) E DC
and
O < t < l , w e
I @
16.7 LEMMA. Let C be a topologically regular open convex cone in X , with order unit norm 1 - 1 , and suppose that 0 is C-positive. Then
defines a continuous semi-norm on
.
V
PROOF. For f E Sx , let F E L ( U , K ) be the unique extension of f satisfying F(u*) = (Fu)* for all u E IJ Then Foo : VxV + K is a positive self-adjoint form. Hence the Cauchy-Schwarz inequality 128; 6.2.11 implies that
.
lvlf := F ( O ( V , V ) ) ~ / ~= f(O(v,v)) 1/2 defines a semi-norm on IVI
V
=
.
Since 14.9.2
SUP IVlf
implies
I
f ESX
it follows that 16.7.1 defines a semi-norm on continuous since 8 is continuous. It is clear that 16.7.1
defines a norm on
.
V
V
which is O.E.D. if and only
In this case, o is called if @(v,v) = 0 implies v = 0 reqular. As for cones, a topological version of regularity is more appropriate in the infinite-dimensional setting: 8 is called topologically reqular if 16.7.1 defines a compatible norm on V
.
16.8 EXAMPLE. Suppose E and D E {R,C,E} with Hilbert sum F =
(Fi ) .
H
are Hilbert spaces over
S E C T I O N 16
266
Then U := L ( E ) is a unital C*-algebra over the center K of D , with self-adjoint part X : = H i ( E ) and strictly positive cone C : = H i + ( E ) A G R ( E ) , By 15.16, the order unit norm on X coincides with the operator norm. Now put V := L ( E , H ) and define 6 : VxV + U by @(b,v) := b*v for Then 16.0.1 is satisfied, and 15.15 implies all b,v E V Since the operator norm 1 . 1 @(v,v) = v*v E H i + ( E ) =
.
.
~@(v,v)l = \ v 1 2 ,
satisfies Let
TI =
{ (:)
E
is topologically regular.
8
L(E,F)
-
: u+u*
be the Siegel domain associated with E = D , we have
D
=
{
(t)
E
v*v > 0 } C
and
F : u+u* - (vlv) > 0 }
@
.
For
.
16.9 PROPOSITION. Suppose U = Xc and V are complex Banach spaces, C is a topologically regular open convex cone in X and @ is C-positive and topologically regular. Then the Siegel domain DC,@ in U x V is a normed Ranach manifold with respect to the Carath6odory tangent norm. and let b E V , Then PROOF. Put D := D C,@ Lb(u,v) : = u + @(b,v+b/2) defines a holomorphic mapping Lb : D + DC
since
2 Re Lb(u,v) = 2 Re(u) Hence for every
f
E
-
@(v,v)
Sx C L(U,C)
+ @(v+b,v+b)
.
16.9.1
,
Fb(U,v) := exp(-f(Lb(u,v))) defines a holomorphic mapping disc b satisfying
Fb : D
.
+ A
into the open unit
Now let R be a for all (u,v;h,k) E T ( D ) = Dx(UxV) bounded subset of D Then there exists a positive constant R < + m such that f(Re Lb(u,v)) < R whenever (u,v) E I3
.
TUBE DOMAINS AND SI EG EL DOMAINS
and
f(@(b,b))
C
1
.
For
b = 0
-
> exp(-R)
BD(u,v;h,k)
,
267
16.9.2 and 14.9.2
imply
sup If(h)l = exp(-R)lhl fES
.
X
In order to show the inequality bD(u,v;h,k)
> exp(-R)
sup f(@(k,k))1'2 f ESX
=
exp(-R)
Ikl
,
. .
we may assume that f E S X satisfies f(@(k,k)) > 0 Then -1/2 the assertion follows by taking b = _+k f(@(k,k)) I since 2 If(@(b,k))l < If(h+@(b,k))l + If(h-@(b,k))l Together with 12.23, it follows that the Carathsodory tangent norm
BD
is compatible.
Q.E.D.
16.10 LEMMA, Suppose C is an open convex cone in X , 0 is C-positive and IY(w) I < 1 for all w E B Then the Siege1 domain DC,@,R can be realized as an open subset of D x R via the mapping (u,v,w) + (2u,v,w)
.
.
C,@
PROOF.
For
(v,w)
E
VxB
.
since b + Y(w)b = v ( Y ( w )
L(F1,F2) : IwI
0
}
v2 is a Siegel domain of the third kind associated with C , Q and Y The affine transformations g a l b and associated vector fields Ya+b , introduced in 16.1, have the form
.
*
v1 'a,b(" 2
*
u + a + b $ v 2 + v l b T + ( b $ b 2 + b l b ~ ) / 2+b2wbl
I=(
vl+bl+b;w
v +b +wbp 2
w
W
2
1
and
'a+b
= (a+b*v +v b*)-a 2 2 i i a u
+ ( b 2 + w b ta ) T + (b1+b*w)2 aavl
.
For Siegel domains D in complex Ranach spaces, the Lie algebra h := aut(D) of all complete holomorphic vector fields has an interesting structure closely related to the binary Lie algebras studied in Section 9. Many deeper results about Siegel domains are based on a detailed study of h Let us first study a somewhat more general situation.
.
16.13 LEMMA. Let h he a real subalgebra of a complex Lie algebra and suppose h C := h + i h has an additive gradation
hC
=
@
C
h s ,
SES
where
S
is a finite subset of
hsC for some
(i)
E
If
{
= E
h
5
x
. = S
E
hC
: [E,X]
C = SX
and
}
Then the following statements hold:
,
then there exists a splitting
SECTION 16
270
h =
hs
fB
I
SER
where R : = { s E h s := h n ( h E + h zC)
hnih
Im(s) > 0 }
S :
.
,
s
3
.
(ii)
If
(iii)
If M is a connected complex Ranach manifold with compatible Carathgodory tangent norm 6, and h C aut(M) , then the evaluation mapping po : h s + To(M) at o E M is injective whenever s E R satisfies Re(s) # 0
= {O}
we may assume
and
=
.
PROOF.
For
X
E
h , write
r:
x =
xs,
s ES
.
where Xs E h C s Put A := ad(E) and let with real coefficients. Then F(A) E L ( h )
F be a polynomial and hence 16.13.1
Applying 1 6 . 1 3 . 1 for suitable polynomials F , we get Xs + X- E h and i(Xs - Xz) E h This proves (i), Now S assume hnih = { O } Then Xs = 0 implies X- = 0 Hence S we may assume S = Now suppose h c aut(M) , where M is a connected complex Ranach manifold with compatible Carathgodory tangent norm 6, By 1 3 . 2 8 , h is a purely real Ranach Lie algebra. Let s E R satisfy Re(s) # 0 Then for every X E h s , ad(X) E L ( h ) is nilpotent. By [ 1 7 : 5.81, we get
.
. .
.
.
.
.
Now assume p o ( X ) = 0 By 13.24, h carries a compatible norm such that ad(X) E d ( h ) By 1 6 . 1 3 . 2 and 1 4 . 3 0 , it follows that ad(X) = { O } In particular, [X,El = 0 and hence X E h s A h: = {O} , O.E.D.
.
16.14
put
THEOREM.
z :=
zlx
".XZ
Let
n '
Zl,
.
...,Zn
Let
cl,
be complex Banach spaces and E C be constants such
...,cn
TUBE DOMAINS AND SIEGEL DOMAINS
271
that
...
f(Cl) = for some real-linear form set
f(c n ) = 1
=
f : C
+
R
.
16.14.1
Consider the finite
n
n c v :\)EN", 1 u < 2 , l < k < n } . j=l j j j=1 j
1
T:={ck-
Suppose M is a connected complex Ranach manifold and let (P,p,Z) be a chart of M about o such that 16.14.2 for some
E
E
T(M)
.
In addition, assume one of the following
conditions
(i)
The Carathgodory tangent norm B M is compatible and hC is a complex subalgebra of aut(M)' containing E ;
(ii)
There is a compatible metric d and h is a real subalgebra of E Put hC := h + ih c T ( M )
.
.
on M , T = aut(M,d) containing
Then
and there exists an additive gradation
hC
=
hC s
@
,
16.14.4
s ES
where the spectrum S of ad(E) on and h: : = { X E hc : [E,X] = SX }
.
PROOF. For every expansion
X
E
,
T(M)
?
1
p*x =
hC
is contained in
there is a power series
k a 1 fw(Z) VEN azk + Zk is a continuous polynomial v = (ul, wn Put
k=l about 0 E Z , where f kV : homogeneous of multi-degree
Z
T
...,
.
212
SECTION 1 6
c = (clI...,c n )
and
(clv) : = E Y cjvj
.
By 16.14.2, we have
where yS
Ck--(CIV)'S More generally, for A := ad(E) F : C + C , we have
and every polynomial
P*
.
C Since T is Now assume (i) ho ds and let X E h finite, there exists a polynomial F satisfying F-'(O) = T By 16.14.5, F(A)X L h C C aut(M)' has order > 3 at o E M Hence F(A)X = 0 by 13.34. Now assume (ii) holds and let X E h Since T = T , there exists a polynomial F with real coefficients satisfying -1 F (0) = T By 16.14.5, F(A)X E h caut(M,d) has order > 3 at o E M Hence F(A)X = 0 by 13.30. In both cases, it follows that Ys = 0 for all s E C\T Put l v l := En u and suppose p , v E N" satisfy 1 j c - (clv) = ck (clv) for some j,k E {l,...,n} Then j 16.14.1 implies ( p l = l v l In particular, I v I < 2 implies lpl < 2 This proves 16.14.3. Applying 16.14.5 for suitable polynomials F , we get Y, = p*Xs , where Xs L hC This proves 16.14.4. O.E.D.
.
.
.
.
.
.
-
.
.
.
.
such 16.15 COROLLARY. Let D be a domain in Z = 2 1 x... "n is compatible. Let that the Carathsodory tangent norm c hC be a complex subalgebra of aut(D) containing n a E:= 1 c jz j az j=1 j
and the vector field Ye := e -a
az
for some point
e E D
.
Then
hC
consists of polynomial
TUBE DOMAINS AND S I E G E L DOMAINS
vector fields of degree 16.14.4.
2
G
273
and has an additive gradation
.
PROOF. For $ ( z ) := z-e , put M := $(D) Since 0 E M E = $*(E-[E,Y I ) E aut(M)'
.
Then 16.14 implies
,
.
The that aut(M)' contained in T - I ( Z ) tB T O ( Z ) @ T1(Z) and 5.36 implies $ * ( h C = hC since same is true of aut(D)' Ye E hC , $ = exp(-Y e ) and ad(Ye) is nilpotent on h' Now the assertion follows from 16.14, applied to hC C aut(MIC 9.E.D.
.
.
LEMMA. Let = UxVxW Then
16.16 Z
If
K
D = D
.
=
be a Siege1 domain in
C,O , R
c and B
is circular, then
a +
2iF := iv av
a
2iw aw
E
aut(D)
.
PROOF. For t E R , put gt : = exp(2tE) E G & ( Z ) , gt(u,v,w) = (e2tu,etv,w) since e2t > o and
.
2e2tu
-
~
~t v,e ( t v) e = e2t(2u-~w(v,v))
Then
,
.
it follows that gt(D) = D for all t In case K = c Then ht := exp(2itF) E G & ( Z ) Let w E B Then e2itw ht(u,v,w) = (u,e itv ,e2itw) since B is circular. Since Y(W) is anti-linear,
.
(
=
Hence
.
.
,
put
E
F\
idv+y(e2itw))-1eitv = [e2it(e-2itidV+Y(w) ) ] -leitv e -itv = e it (idv+Y(w))-1 v
(e-2it idV+Y(w))
ht(D) = D
for all
t
.
16.17 THEOREM. Suppose U , V spaces and D is a domain in 2 Carathgodory tangent norm 8, of the f o r m e = (e,O,O) and h vector fields
.
. Q.E.D.
and
are complex Ranach := UxVxW with compatible Suppose D contains a point : = aut(D) contains the W
,
SECTION 16
274
= ie
'ie
a au
a 2E = 2u au
+
a v av
and
a 2iF = iv av
a + 2iw aw
.
Then h consists of polynomial vector fields of degree and has an additive gradation
hr = { Y
h : [Y,El = r Y }
E
.
< 2
16.17.2
The evaluation mapping pe : h + Z at e E D is injective on h, for r # 0 There are commuting diagrams
.
16.17.3
and
16.17.4
where
v1/2
= v-1/2
are complex subspaces of
V
,
defined by
Vr : = pe(hr) and X1 C X-l are purely real subspaces of U defined by Xr := pe(hr) X-l is a real Ranach space and C := D T \ X is an open cone in X := iX-l with
.
h - l = { aa - : a ~ i ~ } au
.
There exist sesqui-linear mappings (anti-linear In the first + V such that variable) 8 : V-1,2xV + U and Y : V-1,2xW h
a +
-1/2 = { @(b,v)=
For all
b,b'
E
V-1,2
,
a
av : b
(b+Y(b,w))-
we have
E
V-1,2
}
.
16.17.5
TUBE DOMAINS AND SIEGEL D O M A I N S
275
16.17.6
and
h,
The subalgehra
has a multiplicative gradation = 1 h,
h,
-1 h,
,
where l h
= h A To(Z)
0
satisfies pe( 1 h,) c x , p e is injective on -'h, and -1 pe( h,) is a complex subspace of W Let n w : Z + W denote the canonical projection. Then R := r W ( D ) is a circular domain in W
.
.
PROOF. There exists a linear form f : C f(1) = f((l+i)/2) = f(i) = 1 Since
.
a + -l+i vE+iF = u 2 au h
16.14 implies that
a av
R
+
such that
,
+ i w -a c h aw
consists of polynomial vector fields of
degree < 2 and that the spectrum S of ad(E+iF) is invariant under conjugation and contained in the set
T
=
1-3i i-3 } { 0 , + 1 , e2-2 1 + i- ,-+ i ,- + ( 1 - i ) , i - 2 , 1 - 2 i , ~ , ~
It follows that hc := S
S
c [ Y
E
1 i 0,+l,+23,$i ~2'
:
}
.
[Y,E+iFl = s y
Then an elementary calculation shows
Put
1
.
.
276
SECTION 16
a au
C
a
c { av + (b+aw) '-1/22i/2 aw } hCo ~ { a u -a + a v - +a a w a w }a , au av
16.17.7
I
and
Here b denotes vectors in Z , a denotes continuous linear mappings and c denotes bilinear mappings. Now define
hr
hn(
:=
C
hs)
t~
.
Re(s)=r Then 16.17.1 and 16.17.2 are satisfied. Since ad(E) is a derivation of h , it follows that 16.17.1 is an additive gradation of h Since r # 0
.
.
By 16.13,
is injective on
pe
[iF, h+1/21
'
'+l/2
-
hr
if
16.17.8
.
it follows that V+1/2 are complex subspaces of V It is easy to check that-the diagrams 16.17.3 and 16.17.4 commute. Therefore V1/2 C VelI2 and X1 C x-l C U By 13.28, we Now suppose x = (x,0,O) E C Then have X-l n iX-l = { O}
.
.
(et x,O,O) = exp(tE)(x)
.
for all t E R It follows that Since pe real Banach space X
.
E
D A
x
.
= C
C is an open cone in the is injective on h-l,2 ,
16.17.7 implies that hqlI2 has the form 16.17.5 for uniquely By 16.17.8, 0 and Y are determined mappings 8 and Y conjugate-linear in the first variable. Further, [h-l~,,h-,,21 C h-l implies 8(b,b') - b(b',b) E i X and f o r all b , h ' E V-1,2 Putting b(h,Y(b',w)) = @(b',Y(b,w)) b ' = ib , we get b(b,b) E X The vector field
.
.
.
TUBE DOMAINS AND SIEGEL DOMAINS
for
b e V-
1/2
277
satisfies
exp(Yb)(u,v,w) = (u+@(b,v+ b+y(brw)) ,v+b+Y(b,w) ,w) 2 Hence Q ( b , b ) = 0 implies (e,tb,O) E D for all t E c C Hence b = 0 by 12.24. Now define 'h0 := h n h, and -1 c c 1 -1 ho is a h o := hn(hi+h-i) Then h, = h, B c c C hs+t C Since multiplicative gradation since [hs,ht]
.
[
it follows that
1
.
.
.
hO,Yiel
1 pe( h,) C X
c
h-,
.
Since
I
.
it follows that pe(-'h0) is a complex subspace of W Now suppose Y E -lh0 satisfies pe(Y) = 0 Then Since TI W is [Y,iF] = iY e h and 13.28 implies Y = 0 continuous and open, R is a domain which is circular since (e,O,O) E D and iF E h O.E.D.
.
.
.
16.18
COROLLARY.
Suppose in addition that
(locally) transitively on D W = pe( -1h,) , the mappings
.
@
Then and
for all b,v E V , where u + u* on U with self-adjoint part X
acts
, V = V- 1/2 ' are continuous and
U = X Y
Aut(D) C
is a continuous involution
.
: h + Z
is surjective. It follows that V-1,2 = V , pe(-'h0) = W and U = Xc By [ 1 5 ; 48.11, the continuous X-l B X = U , i.e., R-linear bijection p, : h -1/2 + V is a homeomorphism. By 16.17.5, it follows that 0 and Y are continuous. Since X is a closed real subspace of U , the involution u + u * is continuous. Now 16.18.1 follows from 16.17.6. Q.E.D. PROOF.
By assumption, the evaluation mapping
.
pe
278
S E C T I O N 16
transitively on w E R and
D
,
idv + Y(w)
is invertible on
D A (UxW) = DCxR
.
DC : = C @ iX
where
V
for all
,
16.19.1
Then
is a Siege1 domain. Here Re(u) := (u+u*)/2 and aw(b,v) : = 0((idv+y(w))-lb,v) Further, C is convex and homogeneous under linear transformations, 0 is C-positive, = D A (UxV) are convex and homogeneous and DC and D
.
CI@
under affine transformations. PROOF.
Since
idv
+
Y(w)
R
,
the
UxVxR : 2 Re(u) - Re aW(v,v)
E
C }
is invertible for all
w
E
set i? :=
{ (u,v,w)
E
is well-defined and satisfies i? n ( U x W ) = D C x R , where DC = D A U = C @ iX By 16.1, i? is invariant under the group H< : = exp(h-l @ h-l,2) of affine transformations of
.
Z
.
Since g,,(u,v,w)
=
(u+Q(b,v)/2,0,~)
w E R and b := -(idv+Y(w)) -1 v , it follows that i? = H 1
.
.
for all t > 1 @ is C-positive.
It follows
For t + By 16.6,
is convex. NOTES.
H' we
,
+m
r
D'
Q.E.D.
Siegel domains in general were introduced by
I. Pjateckij-Shapiro 11121 for the study of automorphic functions on domains in
Cn
.
The importance of tube domains
(multivariable "half-planes") in number theory is well known. The basic example is Siegel's upper half-plane generalizing the classical upper half-plane. Note that we consider (generalized) "right" half-planes which make also sense in real Banach spaces. Siegel domains in
dn
play a fundamental role
in complex analysis since every homogeneous bounded domain is biholomorphically equivalent to a Siegel domain of the second kind, homogeneous under affine transformations. This deep result is due to E.B. Vinberg, S.G. Gindikin and
.
I. I. P jatecki j-Shapiro (cf Cll2 ; Appendix 1 ) . The structure of aut(D) for Siegel domains second kind in
D
of the
Cn was analyzed by W. Kaup, Y. Matsushima
and T. Ochiai. The generalization to the infinite dimensional case C901 is based on a new and simpler proof using Theorem 16.14 (and hence the complexified version 13.33 of Cartan's uniqueness theorem).
2 80
17.
SECTION
17
SYMMETRIC BANACH MANIFOLDS
The Riemannian symmetric spaces are the most important generalizations of the classical non-euclidean geometries to In this section we the higher-dimensional case [ 6 2 ] . introduce the class of Ranach manifolds of primary importance in the following: the so-called symmetric Ranach manifolds. Unlike the finite-dimensional case, these manifolds d o not necessarily carry a metric of Riemannian type. Instead, we consider metric and normed Ranach manifolds, as introduced in Section 12. 17.1 PROPOSITION. Suppose r is a locally uniform action of a compact group K on a Ranach manifold M and let o E M satisfy r(g,o) = o for all g E K Then there exist a
.
chart (P,p,Z) of M about o and a continuous homomorphism * K + G 1 1 ( Z ) such that for every g E K there is a 'r # * commuting diagram
.
Since K PROOF. Let (Q,q,Z) be a chart of M about o is compact and r is a continuous action leaving o fixed, there exists an open neighborhood N of o E Q such that r(KxN) C Q The homomorphism 4 : K + GI1(To(M)) , defined by $(k) := To(r(k)) , is continuous at e E K since the action r is locally uniform. Hence 4 is continuous. Let dk denote the normalized Haar measure on K Then
.
.
defines an analytic mapping and To(p) =
K
p : N
+
Z
satisfying
p(o) = 0
.
T o ( q ) $(k)-l To(q) -1 To(q) $(k) dk = To(q)
By 4.1, there exists an open neighborhood that (P,p,Z) is a chart of M about o
P
.
of For
o g
E E
N K
such
,
SYMMETRIC BANACH MANIFOLDS
281
define
A chart
(P,p,Z)
called K-linearizing.
having the properties stated in 17.1 is Important special cases are the finite
group K = U ( R ) = {?I}
= Z2 =
and the compact group K = U ( C ) = { X us first consider actions of U ( R )
.
E
C :
{o,l}
1x1
=
1 }
.
Let
.
17.2 DEFINITION. Let M be a Ranach manifold.over K A symmetry of M about o E M is an automorphism j E Aut(M) of period 2 having o as an isolated fixed point. 17.3 LEMMA. Suppose M is a Ranach manifold and j is a Then To(j) = -id and there exists a symmetry about o E M U(R)-linearizing chart (P,p,Z) of M about o such that j(P) = P and there is a commuting diagram
.
P
J
> P
17.3.1
PROOF. Applying 17.1 to the action r : Z 2 + Aut(M) defined by r(n) := j " , it follows that there exists a chart (P,p,Z) of M about o such that j(P) = P and there is a commuting diagram
2 82
SECTION 1 7
where '2
:=
g
E
{
z
point of
GB(Z)
z
E
j
T0 ( 1 ) = -id
r
.
has period
: g ( z ) = az
.
2
.
1
it follows that
o
Since = {O]
'2
-1
,
where is an isolated fixed
2 = '2
Hence
@
. Hence
Z
g = -id Z
and
O.E.D.
17.4 DEFINITION. Let r be an analytic action of a real Ranach Lie group G on a Ranach manifold M which is transitive and locally transitive. Suppose there exist a symmetry j of M about some point o E M and an analytic group automorphism J of G having period 2 such that the diagram
17.4.1
.
commutes, where Int(j)g := jgj-l Then (M,A) is called a symmetric metric G-manifold if (M,d) is a metric Ranach manifold, r(G) C Aut(M,d) and j E Aut(M,d) Similarly, (M,b) is called a symmetric normed G-manifold i f (M,h) is a normed Banach manifold and r(G) C Aut(M,b)
.
.
For an arbitrary point m = r(g,o) M about m
m
.
E
M
,
choose
Then jm : = r(g) j r(g) and the diagram
-1
g
Aut(M)
G
JInt ( jm) Aut ( M )
A
r
.
C,
with
is a symmetry of
G
Int(g) J Int(g)-'J
E
commutes, where Int(g)h := ghg-' It follows that the In symmetry condition in 17.4.1 is independent of o E M case (M,b) is a symmetric normed G-manifold, we have j E Aut(M,b) , since for every m E M there is a commuting diagram
.
SYMMETRIC BANACH MANIFOLDS
It follows that 17.5
LEMMA.
jm
Let
AUt(Mrb)
E
for all
m
283
E
M
.
be an analytic group automorphism of a
J
.
Then the real Ranach Lie group G having period 2 associated semi-direct product G XI Z2 is a real Ranach Lie group containing PROOF.
By
.2,
g
of
algebra
G
as an open subgroup.
the left translation action of the Lie G on M : = G is analytic and topologically
dentify faithful. Since of Aut ( M )
G XI Z2 with the subgroup H : = L G W L G J J, E Ga(g) , 7.4 implies that H is a
real Ranach Lie group with Lie algebra open subgroup of 17.6 Then of
.
H
on
symmetries PROOF.
G
.
G
is an
is a symmetric G-manifold. G XI Z2-manifold (for the action
)
and
r(G#Z2) = r ( G ) U r(G)j
j,
for
m
By 17.5,
Hence
O.E.D.
COROLLARY. Suppose M M is also a symmetric J
g
E
M
H : = G#Z2
.
contains all the
is a real Ranach Lie group
containing G as an open subgroup. The action r of G on M has an extension to an analytic, transitive and locally transitive action
r
r ( H ) = r ( G ) u r(G)j
of
.
H on M such that For every m E M , we have
O.E.D.
In view of 17.6, we may always assume that, for a M , all the symmetries jm are induced symmetric G-manifold by elements of G However, for many examples (cf. Section 22) it is more natural to assume that the symmetries "normalize" the action of G An analytic action p of a Banach Lie algebra g on a
.
.
connected Ranach manifold M is called locally transitive if, for every m E M , the evaluation mapping pm : g + Tm(M) is surjective and has a split null-space. 17.7 PROPOSITION. Suppose p : g + aut(M,b) is an analytic action of a real Banach Lie algebra g on a connected normed
284
SECTION 17
Banach manifold (M,b) which is topologically faithful and locally transitive on M Let j E Aut(M) be a symmetry about o E M such that there is a commuting diagram
.
.
where Ad( j ) E GR( g ) Let G be a subgroup of Aut( M,b) Then (M,b) is a symmetric normed containing exp(p(g)) G-manifold if one of the following conditions holds:
.
(i)
For every
(ii)
G = < exp
E
G
,
p ( g )
>
.
g
there is a commuting diagram
PROOF. Since (i) follows from (ii), we may assume that (i) holds. By 7.4, G is a real Banach Lie group with Lie algebra g and the canonical action r of G on M is -1 analytic and (locally) transitive Further, J(g) := jgj defines a group automorphism of G having period 2 such that the diagrams
.
9 Ad(j)i 4
commute.
-IJ>
exP
G
Aut(M)
G
Int( j ) Aut(M)
It follows that
r
J
is analytic.
Q.E.D.
EXAMPLE. Suppose G is a real Banach Lie group whose Lie algebra g can be endowed with a compatible norm I * I such that the adjoint action A d : G + U!i( 9) is isometric. By 12.32, G carries a tangent norm b which is invariant 17.8
under the analytic, transitive, and locally transitive action r(g,h) (m) := gmh-l of the real Banach Lie group
SYMMETRIC BANACH MANIFOLDS
r
:=
GxG
on the Banach manifold
M := G
.
285
Ry 6 . 6 ,
inversion mapping j ( m ) : = rn-l is a symmetry of e and there is a commuting diagram
where of r ( GxG
G
the about
J(g,h) := (h,g) defines an analytic group automorphism having period 2 Hence G is a symmetric normed -manifold
.
.
.
Ry 17.9 EXAMPLE. Let Z be a unital C*-algebra over K 15.22, the unitary group U(Z) is a real Ranach Lie subgroup of G ( Z ) whose Lie algebra can be identified with u ( Z ) Let I I be the norm on u( 2) induced from 2 Then Ad : U(2) + Ua(u(2)) , since Ad(g)x : = gxg* for all g E U(2) and x E u ( 2 ) By 17.8, it follows that U(Z) is a symmetric normed (U(Z)xU(Z))-mani€old.
-
.
.
.
17.10 EXAMPLE. Let L be a Hilbert space over D E {R,C,H} By 8.10, the sphere M : = { m E L : ( m l m ) = 1 } is a real submanifold of
.
L
with
tangent bundle T(M) = { (m,h)
E MxL :
Re(mlh) = 0 }
.
The assignment b(m,h) := (h)h)1’2 defines a compatible tangent norm b on M which is invariant under the canonical transitive action r of UP(L) on M Ry 15.21, I J P ( L ) is a real Ranach Lie group with Lie algebra U ( L ) , The
.
is analytic and its differential p satisfies p,(X) = (m,Xm) for all m E M and X E u P ( L ) Consider an orthogonal splitting action
of
L
r
and put
.
0
m : = (1)
.
Then
=
(:)
for
SECTION 1 7
286
where a E u k ( E ) , b E E and d = -d* E D , Hence rm : U E ( L ) + M is a real-analytic submersion. Hence acts locally transitively on M Now assume D = R -idE 0
.
j
:=
defines a symmetry of normed Uk(L)-manifold.
M
.
(
) 0
1
about
m
Uk(L)
Then
E UR(L)
.
Hence
is a symmetric
M
17.11 LEMMA. Suppose r is an analytic action of a real Banach Lie group G on a Banach manifold M , with differential p Let j he a symmetry of M about o such that the diagram 17.4.1 commutes, where J is an analytic group automorphism of G having period 2 , Let 1 -1 g = g B g he the multiplicative gradation of g induced
.
.
by the differential J, of J Consider a local representation p # of p with respect to a U(R)-linearizing Then chart (P,p,Z) of M about o
.
0) = 0
17.11.1
and ' ( 0 ) = 0
.
17.11.2
PROOF. By 17.3.1, we have p*(j*Y) = (-id)*(p,Y) for all Y E T(M) and 17.4.1 implies j*(pX) = p(J,X) for all x E 1t follows that (J,X)# = (-id)* X # , i.e.,
.
p#(J,X)(Z)
= -(P#X)(-Z)
for all z in a neighborhood of for o = 21 implies ( p # X ) ( O ) = (P#X)'(O) =
O
(P#X)'(O)
.
0 -O
E
Zc
.
Hence J,X and
= OX
(p#x)(O)
0.E.D.
In the following we are mainly interested in symmetric complex Ranach manifolds. Suppose (M,d) is a connected complex metric Ranach manifold and let G he a subgroup of Aut(M,d) which is closed in the topology of locally uniform convergence. By 13.14, G is a real Ranach Lie group with respect to a Hausdorff topology T whose Lie algebra g can
SYMMETRIC BANACH MANIFOLDS be identified with a closed subalgebra of Further, the canonical action and its differential
g
action of
on
M
.
r
of
C,
287
.
aut(M,d) on
is analytic
M
can he identified with the canonical
p
Now assume that for every
m
.
E
there
M
of M about m Then j, is as a consequence of 13.8 and
exists a symmetry jm E G uniquely determined by m 17.3. For any point o E M multiplicative gradation g
, =
the symmetry 1 g fB -1g of
17.12 LEMMA. The evaluation mapping null-space l g
jo
g :
po
g
C,
E
. +
induces a
To(M)
has the
.
By 17.11.1,
PROOF.
X
-
E
'gA
13.30,
17.13
.
17.14
g
+
To(M)
LEMMA.
.
Ker(po)
Then 17.11.2
.
Now suppose implies
.
Ordo(X) > 2
By
O.E.D.
COROLLARY.
:
po
Ker(po)
X = 0
c
lg
M
is a symmetric G-manifold if and only if
is surjective for every
Let
o
E
M
.
--
S T + Om(D,Z) be the local r# r associated with an admissible p-hall For 0 < c < 1 , put C := c D and assume
representation of P about o E M -1 p ( C ) C Bd(o;p) , where p := ~ / 3 Then there exists a constant X > 0 such that every m in a neighborhood of
.
o
E
P
.
satisfies
I r# ( jm )-r# ( jo 1
C < A
In particular, the mapping m + jm from M into G with the topology of locally uniform convergence) is
(endowed
continuous. PROOF.
Since
d(jm(o),o)
= d(jm(o),jm(m))
+ d(m,o)
< d(jm(o),m)
+ d(m,o)
for all m E M follows that there exists 0 < b < c such that jm E whenever m E p-l(B) , where R : = bD Further, = 2 d(m,o)
.
, Sp
it
288
SECTION 17
and
where
A.
A1
and
17.15 LEMMA. from M into T0 (f) = 2 id
are suitable constants.
Now apply 13.12. O.E.D.
For any o E M , the mapping f(m) := j,(o) M is differentiable at o E M and
.
PROOF. Since f is continuous by 17.14 and f(o) = o , we -1 may assume f ( p (C)) C P Put R : = dist(C,aD) and -1 h := pofop : C + D For z E C and m := p -1 ( 2 ) , we have
.
.
Hence Taylor's formula [28; 8.14.31, applied to about z , implies
r#(jm)
17.16 THEOREM. Suppose (M,d) is a connected complex metric Banach manifold and let G be a subgroup of Aut(M,d) which is closed in the topology of locally uniform convergence. Assume that for every rn E M there exists a symmetry jm of M about m Then (M,d) is a symmetric G-manifold.
.
By 17.13, it suffices to show that p0 : g + To(M) surjective f o r every o E M Let (P1,p,Z) be a U(R)-linearizing chart of M about o and consider an admissible p-ball P about o For v E Z , we have Put zn := 2-"v E c for almost all n PROOF.
.
.
.
E
C,
is
SYMMETRIC BANACH MANIFOLDS
.
and gn : = jnojo E G Then 17.14 implies Now define hn : = 2n-1 (r#(gn)-id) Then hn(0) = 2"-'rg(gn)(O)
.
=
289
g,
+
jo 2 = idM
and 17.15.1
2n-1 r#(jn)(0)
.
implies
.
lim Ir#(jn)(0)-2znl/lznl = n n+Multiplying by 2"-l , we get hn(0) + v E 2 Applying Taylor's formula [28; 8.14.31 to r#(jn)' ahout 0 , we get
.
Since r (j )'(zn) = -id and Z # n r # (jn ) ' ( O ) = r # (gn )'(0)r#(jo)'(O) by 2" implies
=
-r#(gn)'(0)
I
multiplying
2
Since r#(j,)"(O) + r#(jo)"(0) = 0 E L (Z,Z) according to + 0 E L(2) R y 13.26, there 17.15 and 1.17, we get h,!(O) such that (p#X)(O) = v and (p#X)'(O) = 0 exists X E -1 Then X E by 17.11 and To(p)(poX) = v O.E.D.
.
.
17.17 COROLLARY. The mapping and the mapping (m,n) + jm(n)
m
+
from
jm M
.
from M into ( G , T ) x M into M are
real-analytic. Since G acts analytically on M by 13.14, it suffices to prove the first assertion. By 17.15, the evaluation mapping ro : G + M is an analytic submersion. Ry 8.3, there exist an open neighborhood 0 of o E M and an analytic mapping h : 0 + G such that h(m)(o) = m for all Since jm = h(m) jo h(m)-' , the assertion follows. m E Q PROOF.
.
O.E.D.
17.18 COROLLARY. The isotropy subgroup K : = { g E G : g ( o ) = o } at o E M is a Banach Lie
2 90
SECTION 17
subgroup of G and the canonical bijection G / K + M is realbianalytic. The Lie algebra of K can he identified with lg and the symmetry j o lies in the center of K For every g E K , the automorphism g, E Aut(g) respects the 1 -1 g multiplicative gradation g = g @
.
.
PROOF. The first assertion follows from 8.21. The Lie = lg For any algebra of K can be identified with Ker(p-) U -1 g E G , we have - g jo g Hence jog = g jo for jg(0) all g E K and, by 4 . 5 , (jo),g, = g*(jo)* It follows that U Q.E.D. g, ( ug = g for u = 21
.
.
.
It will now be shown that K can be realized as a group of linear transformations. Since K is n o t compact in general, this result does not follow from the elementary integration techniques used in the proof of 17.1. 17.19 THEOREM. Suppose (M,d) is a symmetric connected complex metric G-manifold. Then there exists a chart (P,p,Z) of M about o such that P is K-invariant and p(exp(Yu)(o)) = u -1
g for all u in a neighborhood of 0 E Z I where yu E satisfies To(p)(poYU) = u There is a topological isomorphism 'r * K -+ G Q ( 2 ) onto a closed subgroup of
c
.
# *
G!L(Z)
such that the diagram P
g
.P
ip
p1 2-2
17.19.1
r#(d commutes for all * lg + ga(2) p# * such that every X
g
E
K
.
There is a topological isomorphism
o n t o a closed real subalgebra of E
lg
xp = (IP#X)OP PROOF. p#
ga(Z)
satisfies
.
17.19.2
Consider the local representations r# and : g + C)-(C,Z) associated with a U(R)-linearizing chart
SYMMETRIC BANACH MANIFOLDS
291
.
(P,p,Z) of M about o By 17.16, there exists for every such that u E Z a unique vector field Xu E 'g hU : = p ( X ) satisfies hU(0) = u Define # u 1 -1 c Yu : = -(X g C T(M) 2 u -iX iu ) E
.
and put
fU
:=
(hu-ihiu)/2
E
Om(C,Z)
a
P#YU = f u ( z ) z
.
Then
.
The mapping u + fU is complex-linear and continuous, since u + hU is a real-linear continuous mapping by 17.11 and By 5 . 6 ,
13.20.
there exists an open neighborhood
R
of
0 E Z such that F(u) := exp(p,Yu)(0) defines a holomorphic mapping F : B + 2 with F(0) = 0 Further,
.
d F(tu)t,O F'(0)u = dt
- -aa t exp(t.p,Yu)(~)t=O
= f ( 0 ) = (u-i(iu))/2 = U
Hence
F'(0) = idz
u
.
and, by 1.23, we may assume that
D : = F(B) is open in C and F : R + D is biholomorphic. Let Q be a connected K-invariant open neighborhood of o E p-'(D) and define q := F lop : Q + Z Then
.
.
Now define lr # : K + G Q ( Z ) and q(o) = 0 : l g + g Q ( 2 ) as in 13.24. Every g E K satisfies Q-1 -1 g,( g) = g by 17.18, and hence g,Xu = Xv , where
For
u
E
q(Q)
Since 13.24.3
,
5 . 1 6 implies
implies
SECTION 1 7
292
for all
X
E
,
lg
17.19.2
follows from 17.19.1.
O.E.D.
The chart (P,p,Z) satisfying 17.19 is uniquely determined up to domain of definition and linear isomorphism. In the following, (P,p,Z) will be called a canonical chart of M about o As a special case of 17.4, we define
.
17.20 DEFINITION. A connected complex metric Ranach manifold (M,d) is called symmetric if M is a symmetric Aut(M,d)-manifold. Similarly, a connected complex normed Banach manifold (M,b) is called symmetric if M is a symmetric Aut( M,R)-manifold. Every symmetric connected complex normed Ranach manifold (M,b) is clearly a symmetric metric Ranach manifold with respect to the compatible metric d associated with b by 12.22, since Aut(M,b) is a subgroup of Aut(M,d) It will now be shown that, conversely, every symmetric connected complex metric Ranach manifold (M,d) can be endowed with a compatible continuous tangent norm b invariant under Thus the two concepts are essentially equivalent. Aut(M,d)
.
.
17.21 PROPOSITION. Suppose (M,d) is a symmetric connected complex metric Banach manifold. Then there exists a compatible tangent norm b on M such that G := Aut(M,d) is a symmetric is a closed subgroup of Aut(M,b) and ( M , b ) normed Banach manifo d. PROOF. Let (P,p,Z) be a canonical chart of M about o 'r#(g)(D) = D such that D := p(P) is bounded. By 17.19.1, for all g E K Le 1 . 1 be a compatible norm on 2 Then
.
defines a compatible K-invariant norm on 12.31.
.
Z
.
Now apply O.E.D.
293
SYMMETRIC BANACH MANIFOLDS
Note that with respect to the K-invariant norm
, we have
'r : K + Ua(2) and 1 13.25, we have # l g n i * g = { O }
2
.
--
.
I*I
on
l g + ua(2) By Hence there exists an * *
P#
injective homomorphism 'gC + g a ( z ) of complex Lie p# 1 c algebras satisfying 17.19.2 for all X E g
.
17.22 THEOREM. Suppose (M,b) is a symmetric connected complex normed Banach manifold and let (P,p,Z) be a canonical chart of M about o Then there exists a continuous anti-linear mapping 2 3 u + u * E P2(2,2) such
.
that P*?g) PROOF. 17.19.
a
{ ( u - u * ( z ) ) -a z
=
: u
z }
E
.
For u E 2 , define Xu , Yu , hU and f, For u,v Then 17.11 implies hi(0) = 0
[Xu,Xvl
1
E
.
g
and 4.6.2
E
as in 2 we have
implies
-
lp#[Xu,XvI = h:(O)(v,-)
.
h!(O)(u,-)
It follows that the vector field X := [Xiu,Xvl + [Xu,Xivl - i[XurXvl + i[Xiu,Xivl in
'9'
vanishes since +
'p#(X)
0
.
[Xu,Xiv1 = [Xu'Xv1
Yu,Y V I = 0 exp(p,Yu)(0) = u for all u 0 E 2 Hence
It follows that
=
.
By 13.25, we have
-
[Xiu'Xiv1 = 0
By 17.19, we have in an open neighborhood
.
tu+v = exp(p,Ytu+v)(0) whenever
v
. C
of
= exp(t*p,Yu)(v)
and It1 is small. R y differentiation, we get fU (v) = u , i.e., f, is constant. Since [Xu,Yv] E for all v E 2 , it follows that h U ( 2 ) = u - u * ( z ) , where u * E ? ( Z , Z ) The mapping u + u* is continuous by 13.20 and anti-linear since u + fU is complex linear. Q.E.D. E
C
.
294
SECTION 1 7
Let us now consider actions of the circle group U(C) Recall that a domain D in a complex Ranach space Z containing o is called circular if eitD = D for all t E R Then
.
.
a i I := iz az
E
aut(D)
.
17.23 PROPOSITION. Suppose D and R are circular bounded domains in a complex Ranach space 2 and let g : D + R be a Then g is biholomorphic mapping satisfying g(0) = 0 1 inear.
.
.
Put gt(z) : = e itz for all (t,z) E RxZ Then E Aut(D) since R is circular. Since ht : = g-'ogtOg g ( 0 ) = 0 , it follows that ht(0) = 0 = gt(0) Since gt belongs to the center of G L ( Z ) , we have PROOF.
.
Since i.e.,
gt E Aut(D) gogt = gtog
for all t , 13.10 implies ht = gt , By 12.12, the power series expansion
.
m
of g about 0 converges locally uniformly in norm on Since every t E R satisfies m
e it f,(z)
n=l it follows that linear.
m
=
gt(gz) = g(gtz) = fn = 0
whenever
L
e int fn(z)
n=l n > 2
.
Hence
D
.
17.23.1 g
is
Q.E.D.
The notion of circular domain can be "globalized" to the case of manifolds. 17.24 DEFINITION. A connected complex metric Ranach manifold (M,d) is called circular about o E M if there exists an analytic isometric action r : U ( C ) + Aut(M,d) such that r(X,o) = o and To(r(X)) = X id for all X E U ( C )
.
SYMMETRIC BANACH MANIFOLDS
295
It follows that gt : = r(eit) defines a global flow on M , whose infinitesimal generator iI E aut called the circle vector field about o By 13.8, iI are uniquely determined.
.
17.25 LEMMA. Suppose (M,d) is a connected complex metric Then there Banach manifold which is circular about o E M
.
exists a chart (P,p,Z) of M about o such that P invariant under the circle action r , D := p(P) is a bounded circular domain and the diagram r(h) , p
P p1
Amid,,’ commutes for all iI
E
aut(M,d)
is
1p D
.
X E U(C) The circle vector field about o satisfies p,(iI)
=
iz a az
.
PROOF. Since r is an isometric action with fixed point o , the U(C)-linearizing chart (P,p,Z) of M about o can he chosen such that P is invariant, connected and D : = p(P) is bounded. O.E.D. 17.26 PROPOSITION. Suppose (M,d) is a connected complex metric Banach manifold which is circular about o E M Let g = 1 g fB lg be the multiplicative gradation of
.
g : = aut(M,d)
induced by the symmetry
j = exp(ni1) of of
M M
about about
h : = p*( 1g )
o o
.
.
Let (P,p,Z) be a U(C)-linearizing chart Then
is a closed real subalgebra of
exists a continuous anti-linear mapping defined on a closed subspace
W
of
Z
w
To(Z)
3 u +
such that
U*
and there E
P2 (
2 , ~ )
296
SECTION 17
PROOF. Applying 16.15 to n = 1 and the circle vector field iI E g , it follows that gc : = g + ig has an additive C gradation gc = g-i @ !g fB :g , where =
{ X
.
sX } Further, of polynomial vector fields of degree < 2
gcS
E
gc : [X,iI]
=
C P*(gmi) Since
iI
E
g
and
=
P*(4
S = { O,+i} = 9 =
1
gfB
-1
9
C
Tm
5 , 16.
p*(gc) and Z)
consists
.
3 implies
17.2fi.2
1
.
-1 g := g A (g-i C fB gi) C Since , it follows that j , ( n g) = n id for n = 21 Hence the multiplicative gradation of g induced by j is given by 17.26.2. For every X E 'g , we have (p#X)'(O) = 0 by 17.11. Therefore 13.21 implies that To(p)opo : -1 g + 2 is a real-linear homeomorphism onto a It follows that there exists closed real subspace W of 2 a continuous real-linear mapping w 3 u + u* E P ~ ( z , z ) -1 -1 satisfying 17.26.1. Since [iI, gl c g , it follows that W is a subspace of Z and u + u* is an anti-linear mapping O.E.D.
C and 1 where g := gng0 pojop-' = einidD = -idD
.
.
.
NOTES. In the finite dimensional case, the theory of (Riemannian or hermitian) symmetric spaces is due to E. Cartan. This approach was Lie theoretic, using semi-simple Lie groups and Lie algebras, and led to a complete classification of symmetric spaces. For modern accounts of this theory, see C 62,1011. The fundamental Theorem 17.16, allowing a more natural definition of symmetric complex Banach manifolds, as well as the related results 17.14, 17.15 and 17.17, are due to J.P. Vigui C148,1541. The construction of the canonical chart (17.19) and Theorem 17.22 appear in c841. Proposition 17.23 is due to H. Cartan.
JORDAN TRIPLE SYSTEMS
18.
297
JORDAN TRIPLE SYSTEMS
The classical theory of Riemannian symmetric spaces (of finite dimension), developed by E. Cartan, is closely related to the structure theory of semi-simple Lie groups and Lie algebras. In the infinite-dimensional case of symmetric Ranach manifolds, Ranach Lie groups and Lie algebras still play an important role but there is another algebraic structure which seems to be more appropriate for the study of symmetric Ranach manifolds, at least in the complex case: The so-called Jordan algebras and various generalizations. The principal reason €or preferring Jordan algebras over Lie algebras is the fact that the Jordan algebras connected with symmetric Ranach manifolds are well-understood (although not classifiable) whereas there is no completely satisfactory general theory of Banach Lie algebras. Actually, the algebraic objects most directly related to symmetric manifolds are not Jordan algebras but the slightly more general so-called "Jordan triple systems". 18.1 DEFINITION. Let Z be a vector space over K E {R,C} A triple product on 2 is a mapping
.
zxzxz
3 (x,u,y)
which is K-bilinear in (x,y) u and satisfies the identity
,
18.2 Z
.
...,z ...,zn n
F(zl,
zl,
E
PROPOSITION. Suppose Then the two identities
E
z
K-linear or K-antilinear in
.
{xu#y} = {yu# x} Here an equation it holds for all
#
{xu y}
+
= 0
18.1.1
is called an identity if
z . xu # y}
is a triple product on
xv # x}} = {x ux # v} # x} = {{xu # x}v # x}
18.2.1
and U#Y} = {.tux
#
#
ul Yl
18.2.2
SECTION 18
298
are equivalent to the Jordan triple identity #
{xu [YV
# 211
+ “XV
#
# YIU
#
-
21
#
{YV {xu z ] ]
#
#
{x{uy v ] z } 18.2.3
=
and imply the fundamental formula {x(u{xv#x}~u}~x]= {[xu # x}v # {xu # x]}
.
18.2.4
PROOF. Suppose first that the identities 18.2.1 and 18.2.2, expressing some weak form of associativity, hold. Replacing
x by in x
x + y in 18.2.1 and collecting all terms of degree 2 and of degree 1 in y , we get, using 18.1.1, 2 {xu#{xv#y}] + {yu#{xv#x]} = 2 [x{ux # v) # y] + {x{uy # v} # x]
18.2.5
.
The process described above is called polarization and will he frequently used in the sequel. Polarizing 18.2.2 in a similar way, by replacing u by u + v and collecting all terms of degree 1 in u and v , we get, using 18.1.1,
I
xu # x]v # y} + {{xv# x}u # y} = 2 (x{ux# v # y)
Subtract i ng 18.2.6
from 18.2.5,
2 {xu# {xv # y]]
-
#
.
18.2.6
.
18.2.7
we get #
{{xu x}v y} = [x{uy#v
#
x}
Another polarization and division by 2 yields 18.2.3. Conversely, suppose that 18.2.3 holds. Then 18.2.2 follows from {x{ux # u} # y} = { {xu#x}u#y} + { {
Further,
{xu#{xu # y}} = {{xu# x}u # y }
3 {xu # {xv # x}] = 3 {x{ux # v] # x}
XI1 #
y}u # x}
. follows from
JORDAN TRIPLE SYSTEMS
{xu# {xv # x}} = 2 {{xu # x v # x} 2
=
2 {xu# {xv # x}} -
(
{xu# [xv# x}}
= 4
-
-
299
{x{ux # v
x{ux # v} # x}
3 {x{ux # v} # x}
)
-
.
In order to derive the fundamental formula, put y : = [xu# x} Applying 18.2.7, 18.2.1 and 18.2.7,
.
we get
{x{u{xv # x} # u} # x} = 2 {{{xv # x}u # x}u # x}
- {{xv # x}u # y} #
{XIUY v}
=
Applying 18.2.2,
#
XI +
18.2.1
=
2 {{xv # y}u # x} - {{xv # x}u # y} #
#
{YV Y} -
and 18.2.2,
#
x}u Yl
'
18.2.8
we get
{x{uy # v} # x} = {x[u{xu # x} # v} # x} =
[x{{ux # u)x # v} # x} = {x{ux # u } # {xv# x}}
=
{{xu # x}u # {xv# x}}
Combining 18.2.9
.
18.2.9
and 18.2.8, we get 18.2.4.
O.E.D.
In the following we are mainly interested in triple products which are anti-linear in the inner variable u A triple product on Z having this property will be denoted by {xu*y} instead of [xu # y} Given x,u E Z , define a linear operator x u* on Z by putting
.
.
( x n u * ) z := [xu*z} for all z ox = O(x)
E
Z
on
18.3.1
,
and define a K-antilinear operator 2 by putting oxu = O(X)U : = {xu*.}
.
18.3.2
18.3 DEFINITION. A Jordan triple system (or Jordan triple, for short) is a vector space Z over K , endowed with a triple product {xu*y} (anti-linear in u ) which satisfies the Jordan triple identity 18.2.3 or, equivalently, the identities 18.2.1 and 18.2.2. I€, in addition, Z is a
300
SECTION 1 8
B a n a c h s p a c e a n d t h e t r i p l e p r o d u c t is c o n t i n u o u s o n then
ZxZxZ
,
is c a l l e d a R a n a c h J o r d a n t r i p l e .
Z
The f o l l o w i n g e x a m p l e s a r e somewhat t y p i c a l o f t h e R a n a c h Jordan t r i p l e s a s s o c i a t e d w i t h symmetric manifolds. 18.4
EXAMPLE,
over
K
,
Let
b e a n a s s o c i a t i v e Banach a l g e b r a
Z
endowed w i t h a c o n t i n u o u s i n v o l u t i o n
.
z + z*
Then
the continuous t r i p l e product 18.4.1
{xu*y} := (xu*y+yu*x)/2 on
s a t i s f i e s t h e i d e n t i t i e s 18.1.1,
Z
18.2.1
and 1 8 . 2 . 2 .
is a B a n a c h J o r d a n t r i p l e . Note t h a t {xu*.} = o x u = xu*x f o r a l l x , u E 2 , t h u s m o t i v a t i n g o u r notation f o r Jordan t r i p l e products. In particular, every
Hence
Z
C*-algebra 18.4.1.
over
K
Hence f o r
,
2 = Lm(S,D)
is a Ranach J o r d a n t r i p l e w i t h r e s p e c t t o
D
E
c,(S,D)
{R,C,H} and
,
t h e Banach s p a c e s
c,(S,D)
,
associated with a
m e a s u r e s p a c e , a t o p o l o g i c a l s p a c e or a l o c a l l y c o m p a c t space
,
S
center
K
respectively, of
,
D
{XU*Y](S)
DEFINITION.
18.5
triple
2
with t r i p e product = (x(s)u
A closed subspace
o f a Banach J o r d a n
W
is c a l l e d a J o r d a n s u b t r i p l e i f x,u
Since
are Ranach J o r d a n t r i p l e s o v e r t h e
w
6
=3 {xu*.}
2 [xu*y} = { ( x + y ) u * ( x + y ) }
E
-
w
.
{xu*.}
18.5.1
-
{yu*y}
,
a
c l o s e d J o r d a n s u b t r i p l e of a R a n a c h J o r d a n t r i p l e is i n v a r i a n t u n d e r t h e J o r d a n t r i p l e p r o d u c t and is t h e r e f o r e a Ranach J0rda.n t r i p l e .
I n case
K =
c ,
t h e c o n d i t i o n 18.5.1
can be
r e p l a c e d by t h e f o r m a l l y weaker c o n d i t i o n 2 E
18.6
EXAMPLE.
Let
E
w
j {zz*z} E
and
F
w
.
be Hilbert spaces over
18.5.2
301
J O R D A N T R I P L E SYSTEMS
D
E
{R,C,E] and consider the Hilbert sum L :=
(L) .
Then A := L ( L ) is an associative Ranach algebra over the center K of D , endowed with the canonical involution. By 18.4, A is a Banach Jordan triple with respect to 18.4.1. The Banach space Z := L ( E , F ) is a closed Jordan subtriple of A via the embedding
Hence
Z
is a Banach Jordan triple with respect to 18.4.1.
DEFINITION. Every closed Jordan subtriple 2 of , for Hilbert spaces E and F over D , is called a JC*-triple. Then Z is a Banach Jordan triple over the By 15.19, every C*-algebra over K is center K of D (triple isomorphic to) a JC*-triple. Hence the JC*-triples over K can also be characterized as the closed Jordan subtriples of C*-algebras over K 18.7
L(E,F)
.
.
In terms of the operators x o u* and Q, on Z defined by 18.3.1 and 18.3.2, the basic Jordan triple identities can be expressed as follows.
,
18.8 PROPOSITION. A vector space Z over K , endowed with a triple product {xu*y} , is a Jordan tr ple if and only if it satisfies the operator dent ities
( x 0 u*
18.8.1
Q, = Qx(uo x*
and (Qxu)o u* = x
18.8.2
(Qux)*
or, equivalently, the operator Jordan triple identity [ x o u * , y o v*l = xn{uy*v}* In this case,
2
-
{yv*x}o u*
.
18.8.3
satisfies the operator fundamental formula
S E C T I O N 18
302
QxQuQx = Q(Qxu) Note that 18.8.3
-
18.8.4
is equivalent to the operator identity
[xa u*,ycI v * ] = {xu*y} a v*
-
x
0
{uy*v}*
.
18.8.5
18.9 DEFINITION. A linear mapping g : 2 + W between Jordan triples Z and W over R is called a (Jordan triple) homomorphism if it satisfies the identity
or, equivalently, any of the operator identities
or
An invertible homomorphism is called an isomorphism. Let Z be a Jordan triple. A linear mapping 6 : Z + 2 is called a (Jordan triple) derivation if it satisfies the identity 18.10
DEFINITION.
or, equivalently, any of the operator identities 18.10.1 or
Here 18.11
Q(x,y)u := {XU*Y}
.
Let 2 be a Banach Jordan triple over K Then the group Aut(Z) of all continuous automorphisms of Z is a closed subgroup of Gk(2) and a real Banach Lie group in the operator norm topology whose Lie algebra can be identified with the closed real subalgebra aut(2) of g k ( Z )
.
PROPOSITION.
303
JORDAN TRIPLE SYSTEMS
consisting of all continuous derivations of PROOF.
Since
of degree
< 2
Aut(2)
,
2 ,
is a real algebraic subgroup of
the assertion follows from 7.14.
Gk(2)
Q.E.D.
Note that U ( K ) can be identified with a subgroup of the center of Aut(2) via the mapping X + X*idZ By 18.8.3 and 18.10.1, the real-linear span
.
is an ideal of aut(2) whose elements are called inner derivations of 2 In case K = C , the mapping z + iz belongs to (the center of) aut(2) , generating the continuous 1-parameter group (t,z) + e itz in Aut(2) Further,
.
.
int(2) for u u
is spanned by all derivations of the form E 2 , since o
v*-vn u* = i((u+iv) o (u+iv)*-uo u*-vo v*)
i * u n u*
.
18.11.1
In the complex case, it is easily verified that a triple product on 2 satisfies the Jordan triple identity 18.8.3 if and only if the operators i * u a u * are derivations of 2 (in the sense of 18.10.1) or, equivalently, if and only if exp(it-uou*) E Aut(2) for all t E R and u E 2 In the following, for a subspace p of a Lie algebra
.
g , we will denote by [ p , p ] the linear span of all elements [X,Yl with X,Y E Let {xu*y} denote a continuous triple product on a Banach space 2 Ry 18.1.1, there exists a continuous mapping
.
.
2
18.12.1
2 3 u + u* E p (2,Z)
such that all
x,~,u,z E 2
satisfy
u*(z) = {zu*z} and
2 {xu*y} = u*(x+y)-u*(x)-u*(y)
.
18.12.2
SECTION 18
304
We will now characterize Ranach Jordan triples in terms of the polynomials
u*
.
As
the graph of a continuous real-linear
mapping I
p : = { (u-{zu*z})-aa z : u is a closed real subspace of
E
z }
18.12.3
.
T-l(Z) fB T 1 ( Z )
18.12 LEMMA. Suppose Z is a Banach space over K , endowed with a continuous anti-linear mapping 18.12.1. Then the following statements hold: (i)
If Z is a Banach Jordan triple, then the commutator relations
and
are satisfied. Identifying linear operators on Z with the corresponding vector fields, we have further k -
:=
[ p , p 1 = int(z1
h
:=
{
(uniform closure)
and
x
E
T ~ ( z ): [x,pl
c
p } = aut(z)
.
-
Further, g := h fB p is a real Ranach Lie algebra containing 5 := k fB p as a closed ideal. (ii)
PROOF.
For K = c , Z is a Banach Jordan triple if and only if 18.12.4 and 18.12.5 hold. For
u
E
Z
, put xu :=
(u-{zu*z})-a az
Then 4 . 6 . 2
implies
.
18.12.6
JORDAN TRIPLE SYSTEMS
1 2 [Xu,Xv1
305
= ({uv*z}-[vu*z}~-a
az
.
+ ( { { z u * z } v * z } - { z u * { z v * z }a} ~ ~ It follows that 18.12.4
is equivalent to the identity
{{zu*z}v*z} = {zu*{zv*z}}
18.12.7
which follows from 18.2.1. Further, the Jordan triple identity 18.2.7 implies 18.12.5, being equivalent to 2 {{uv*z}w*z} - {uv*{zw*z}} = {z{vu*w}*z}. In case
K = C
polarization.
,
18.12.8
follows from 18.12.4
18.12.8
and 18.12.5
The remaining assertions are clear.
by
Q.E.D.
2 The polynomials u* E p ( 2 , Z ) , associated with a Ranach Jordan triple Z via 18.12.2, give rise to Rergmann operators (cf. 9.17). More precisely, for u,v,z E 2 define B(U,V*)Z = z - 2 {uv*z} + ( 2 {{zv*u v*u} - {zv*{uv*u}} z - 2 {uv*z} + {u{vz*v}*u
=
)
I
i.e., B(u,v*) = idz
-
2 u o v* + Q,Qv
E
L(2)
.
18.12.9
In the special case 2 = L ( E , F ) , endowed with the triple product 18.4.1, we have R(u,v*)z = (idF -uv*)z(id E -v*u) The Banach Jordan triples of interest in the following can be endowed with a distinguished norm related to the Jordan
.
structure. 18.13 DEFINITION. endowed with a norm unv*-vou*
E
A
ui(2)
Banach Jordan triple 2 over 1 * I , is called -hermitian if for all
u,v
E
2
if and only if every
u
E
2
,
.
By 18.11.1, a complex Banach Jordan triple
-hermitian
K
2
is
satisfies 18.13.1
S E C T I O N 18
306
18.14 LEMMA. A closed Jordan subtriple Ranach Jordan triple 2 is -hermitian. PROOF. For I exp(t(uo
u,v v*-VD
E
W and t E R u*)Iw)( < 1
.
,
W
of a -hermitian
we have Q.E.D.
18.15 EXAMPLE. Let Z be an associative Banach algebra over K , endowed with a -hermitian continuous involution z + z* Ry 18.4, 2 is a Banach Jordan triple with respect to 18.4.1. Let L(x)z := xz and R(y)z := zy denote the left and right multiplication operators on 2 , respectively. Then
.
Since x : = uv*-vu* and y : = v*u-u*v are skew-adjoint and the involution is -hermitian, we have L(x),R(y) E u A ( 2 )
.
.
Hence 7.9 implies u a v*-vo u* E u g ( 2 ) It follows that Z is a -hermitian Banach Jordan triple. By 15.7, every C*-algebra over K is a -hermitian Banach Jordan triple with respect to 18.4.1. In particular, the Banach Jordan triples of 0-valued functions on a measure space or topological space S introduced in 18.4 are -hermitian with respect to the supremum norm. Ry 18.14, it follows that every JC*-triple over K is a -hermitian Ranach Jordan triple with respect to the operator norm. In particular, for Hilhert spaces E and F over D E {R,C,E} , the Banach Jordan triple L(E,F) over the center K of D is -hermitian with respect to the operator norm. -Hermitian Ranach Jordan triples can also be characterized in terms of Lie algebras.
,
18.16 PROPOSITION. A Banach Jordan triple Z over K endowed with a norm 1 . 1 , is -hermitian if and only if the closed real-linear subspace g : = h fB p , for
is a real subalgebra of
T(Z)
.
In this case,
3
cg
.
JORDAN TRIPLE SYSTEMS
307
.
PROOF. By 7.9, k is a closed subalgebra of Hence g is a Lie algebra if and only i f [p,p] C k Since 18.12.7 imp1 ies
.
18.16.1 this condition is equivalent to
int(Z) C U ( Z )
.
O.E.D.
As the main result of this section, we will now characterize the class of symmetric complex Ranach manifolds algebraically in terms of Banach Jordan triples. First, we associate a Banach Jordan triple with each symmetric complex Banach manifold.
18.17
THEOREM.
Suppose
(M,b)
is a symmetric connected complex normed Ranach manifold and let ( P , p , Z ) he a Endow Z with the norm canonical chart of M about o induced by To(p) : (To(M),bo) + 2 Let
.
9 =
1
g @
-1
.
1R.17.1
4
g
be the multiplicative gradation of the symmetry
-hermitian
jo
of
M
about
o
.
:=
Then
aut(M,b) Z
induced by
becomes a
Banach Jordan triple such that p*(
a
-1
g ) = { (u-{zu*z}kaz : u E z }
.
18.17.2
PROOF. By 17.22, there exists a triple product on Z 1 satisfying 18.17.2. Since P*( g ) c T o ( Z ) by 17.19, it := p*(-'g) satisfies 18.12.4 and 18.12.5. By follows that 1 18.12, Z is a Banach Jordan triple. Since p*( g ) c u i ( 2 ) , it follows that Z is -hermitian. O.E.D.
The inverse process, associating a symmetric Banach manifold with each -hermitian (complex) Ranach Jordan triple, requires some more preparation. 18.18 Put
LEMMA.
Let
Z
be a Banach Jordan triple over
K
.
S E C T I O N 18
308
hl
{ {zu*z}-aaz
:=
: u
E
c T1(Z)
2 }
(uni€orm closure) and
Then h := h-l on 2
.
PROOF.
@
h0
By 18.12.7,
@
hl
is a full binary Ranach Lie algebra
[hl,hl]
= {O)
.
For
u
E
2
,
put
a Yu := u az
and
a YU := [zu*z} az
.
Then 18.12.8 implies [Yw,[Yu,Yv]] = 2Yb , where b : = {uv*w} Hence [hl,h,l] c h, Since h, is a subalgebra of T o ( Z ) by the Jacobi identity 2.14.1, it follows that h is a Lie algebra containing
.
.
a I := z az
O.E.D.
Let H denote the identity component of Aut(h) and By 7.15, H is a define h , and H, by 9.11.1 and 9.11.2. Banach Lie group over K in the operator norm topology whose By 9.11, Lie algebra can be identified with h (cf. 9.10). H+ is a Banach Lie subgroup of H The quotient space
.
N := H/H+
18.18.1
is a connected Banach manifold and the left translation action r of H on N is analytic. Let p : h + aut(N) denote the di€ferential of r By 9.12 and 9.21, there exists a chart (P,p,Z) of N about o := H, such that all u E 2 and X E h satisfy
.
P(exp(ad Y ~ ) +H
=
18.18.2
JORDAN TRIPLE SYSTEMS
309
and P*(PX) =
x
.
18.18.3
Consider the closed real subalgebra ideal
4.
=
@
p
-
g
of
@ p
=
h
of
and the
defined in 18.12.
18.19 LEMMA. Suppose G is a subgroup of H containing Let u E Z satisfy idZ-u u* E GE(Z) Then the exp(g-) -1 orbit G - m of m := p (u) in N is open.
.
.
PROOF.
For
v
E
Z
define
xv
:=
Xv
xv
by 18.12.6
+ [Xu,Xv1/2
and put E
9_
.
Then the real-analytic mapping $(v) := exp(Xv)(m) from Z into N satisfies T ~ ( + ) v = pm(xV) , since v + xV is reallinear. By 18.18.2, there is a commuting diagram
-
.
-
where $,(v) = v {uv*u} + {uv*u} {vu*u} = (idZ-uo u*)v By 4.1, it follows that + is real-hianalytic on a Hence the orbit G * m 3 $(Z) is a neighborhood of 0 E Z neighborhood of m E N and is therefore open. O.E.D.
.
is a group acting transitively on a connected topological space M Let G be a normal subgroup of I' such that the orbit G * m is open f o r some m E M Then Gem = M 18.20
Suppose
LEMMA.
'l
.
.
.
PROOF.
For
n
E
M
,
there exists
g
E
l'
with
n = g*m
.
Hence
-
G * n = G*(g*m) = g(g 'Gg-m) = g(G*m)
is open in closed.
M
Since
.
Therefore every G-orbit in M is open and M is connected, the assertion follows. O.E.D.
310
SECTION 1 8
18.21
.
g_c g c
that
exp
c o n n e c t e d s u b m a n i f o l d on subset.
s
By 1 8 . 1 2 ,
M := G ( o )
such
of
is a n o p e n c o n n e c t e d s u b m a n i f o l d o f
,
u := 0
Applying 18.19 t o
h
be a closed real subalgebra of
g
Let
LEMMA.
and
5
exp p ( 4 )
it follows t h a t
N
is a n o p e n as a n open
G(o)
G(o)
and 5 . 3 6 i m p l i e s
exp(X) exp(Y) exp(-X) = e x p ( e x p ( a d X ) Y )
for a l l
X
and
E
.
G
subgroup o f By 9.21,
Y
E
3
By 1 8 . 2 0 ,
.
5
It follows t h a t
we have
the restricted action
:
plg
a n a l y t i c and t o p o l o g i c a l l y f a i t h f u l .
9
By 7 . 1 4 ,
18.22
The c a n o n i c a l a c t i o n
LEMMA.
r
of
O.E.D.
is
G
is a
G
.
M
.
+ aut(N)
c o n n e c t e d real Banach L i e g r o u p w i t h L i e a l g e b r a a n a l y t i c a l l y on
is a n o r m a l
M = G(o) = G ( o )
on
9
I
acting
is
M
locally transitive. PROOF.
Since
show t h a t
r
: G + M 0-
c 9c 4 P c Let (PnM,p,Z) Then 1 8 . 1 8 . 2
a c t s t r a n s i t i v e l y on
G
is a submersion.
M
,
it s u f f i c e s to
Since
.
,
w e have 4 = k B P , where k := gn h , denote t h e canonical c h a r t of M about o
.
implies t h a t t h e diagram
commutes , w h e r e
a JI h ( Z ) z ) := h ( 0 ) It follows t h a t
s s u r j e c t i v e with null-space
po
18.22.1
k
. O.E.D.
Let
n : M-
+
M
denote t h e universal covering manifold
JORDAN TRIPLE SYSTEMS
of
M
o
and choose a "base point"
.
satisfying
M"
E
311
h
n(o) = o E M Let p denote the analytic action of g M" obtained by lifting the analytic action p of g on
.
M
on
h
Since
is topologically faithful, the group
p
< exp
G- :=
> C Aut(Mh)
pA(g)
becomes a connected real Banach Lie group with Lie algebra
g
,
acting analytically on
.
M"
PROPOSITION. The action rh of G* on M h is locally transitive and hence transitive. There exists a chart 18.23
(Prr,p",Z) of
M
about
g
.
The isotropy subgroup
for all
X
E
o
K := { g
such that
E
G- : r-(g,o) = o }
is a connected Banach Lie subgroup of G* with Lie The canonical mapping G-/K = M h is realalgebra h bianalytic. There exists an analytic homomorphism lr" K + Gi(Z) such that for every g E K there is a # . commuting diagram
.
-
rA ( g ) -1 ( PA )nPh p"
rh (g
5
> P-
.z
2
1 P--
'r#(g) PROOF.
The first assertion follows from 18.22.
By 8.21,
K
is a Banach Lie subgroup with Lie algebra Ker(pi) = { X
E
g : pi(X) = 0
Further, the canonical bijection
}
.
G-/K = M"
is real-
is simply-connected in the quotient is connected, [26; p. 59, Cor. 11 implies that K is connected. Let (PnM,p,Z) denote the canonical chart of M about o and choose an open
bianalytic. topology,
Hence
Since
G"/K
G-
312
SECTION 18
neighborhood PI of o E Me such that nlP- is a homeomorphism onto an open subset of P A M Then pe := ponlP" defines a chart (P",ph,Z) of Mn about satisfying 18.23.1. Hence there is a commuting diagram
.
where $I is defined by 18.22.1. 18.23.1 and 5.16, the diagram rh (g -1 ( p" )nPh Z
exp(X) E
18.24 THEOREM. over K . Then
Z
Let Mh
,
.
Ry
rh ( q ) > PA
I commutes for every X Since K = < exp(lz) >
Ker(pi) = k
Therefore
o
I'
18.23.2
p"
2
k C T o ( Z ) and g := exp(X) the assertion follows.
E
K
.
O.E.D.
-
be a hermitian Banach Jordan triple is a symmetric normed G--manifold.
) with the compatible norm bo such PROOF. Endow T : ( M that To(pA) becomes an isometry. Since e x p ( X ) ~ U!(Z) for all X t k and K is connected, 18.23.2 implies
.
By 12.31, there To(rh(g)) E Ue(To(M")) for all g E K invariant under exists a compatible tangent norm b on M" Now consider G" and inducing the norm bo on To(M-) the automorphism s of having period 2 , defined by slk := id and s ( p := -id The universal covering group G1 of G h is a simply-connected Banach Lie group with Lie algebra g acting analytically and (locally) transitively
.
.
.
M A with differential P Hence M h 5 G1/K1 , where K1 is a connected Banach Lie subgroup of G1 with Lie algebra k Let J be the analytic automorphism of G1
on
.
having period 2 and satisfying J, = s and J induces a symmetry j E Aut(M%) satisfying
j,o
P
= PO s
on
. Now
.
Then JIK1 = id about o f M-
apply 17.7. Q.E.D.
JORDAN TRIPLE SYSTEMS
18.25
THEOREM.
In case
K
313
, the simply-connected normed -
= C
Banach manifold Mk associated with a hermitian Banach Jordan triple Z is circular and a symmetric normed Banach manifold with Jordan triple Z . PROOF.
Consider the vector field
Then 18.23.1 implies pT(p&(iI)) = iI , showing that t + exp(t pd(iI)) defines an isometric circle action on M- with fixed point o Hence M- is circular about o
.
.
and jo := exp(a p”(i1)) is a symmetry of M* about o Since r N is transiti;re on Mu , the assertion follows. Q.E.D. NOTES.
The fundamental correspondence between (simply-connec-
ted) symmetric complex Banach manifolds and hermitian Banach Jordan triples is the main result of C841. In this paper it is further shown that this 1-1 correspondence is functorial and hence gives a categorical equivalence. The construction of the homogeneous Banach manifold N associated with a binary Banach Lie algebra h (cf. Section 9) also appears in C841. In the finite dimensional case, presented in C1031, the (compact) symmetric manifold associated with a Jordan triple can be constructed using different methods closer to algebraic geometry (cf. C 103; 5 71 1. In this case, the Jordan triple product reflects curvature properties of the underlying manifold (cf. C103; Th. 2.101 1. It should be noted that not all real-analytic symmetric manifolds give rise to a Jordan algebraic structure. The real case is more appropriately described in terms of Lie algebras and Lie triple systems C 62,1011 Unlike the finite dimensional case, it is not possible to classify symmetric Banach manifolds in general. However, the subclass of symmetric Hilbert manifolds has been characterized in terms of hermitian Hilbert Jordan triples and allows a complete classification up to isometric isomorphism [861.
.
SECTION 1 9
314
19.
JORDAN ALGEBRAS
The triple product structure associated with symmetric Ranach manifolds reflects properties of certain Lie algebras of complete analytic vector fields and is thus "Lie-theoretic" in nature. In this section, we will investigate the connection between these triple products and another class of nonassociative algebras, the so-called Jordan algebras. 19.1 DEFINITION. A Jordan alqebra is a commutative (not necessarily associative) algebra 2 over K E {R,C} , with product denoted by zow , satisfying the Jordan identity
z2o(z0w)
.
zo(z 2ow)
=
19.1.1
Note that 19.1.1 expresses a weak form of associativity, since 2 is commutative. The complexification Xc = X BR C of a real Jordan algebra X is a complex Jordan algebra. Similarly, the unitization 2 ' = 2 (B K of a non-unital Jordan algebra 2 over K is a unital Jordan algebra. F o r z E 2 , let MZw := zow define the multiplication operator quadratic representation Pz = P(z) : = 2 M :
-
M(z 2 )
MZ
=
M(z)
,
and define the
.
In terms of the multiplication operators, the Jordan identity 19.1.1 is equivalent to the operator identity [M(z),M(z')I
= 0
.
19.1.2
Associative algebras give rise to Jordan algebras via the so-called "anti-commutator" product. Rased on this type of example, Jordan algebras were originally introduced in order to provide an algebraic foundation for the quantum mechanical formalism. 19.2
EXAMPLE.
Let
2
be an associative algebra with
JORDAN ALGEBRAS
315
.
product zw Then Z becomes a Jordan algebra under the anti-commutator product zow := (zw+wz)/2 In this case, we have
M,
(LZ+RZ)/2
=
.
19.2.1 and
Pzw = zwz
,
where L, and R, denote the left and right multiplication operators, respectively. EXAMPLE. Suppose Z is an associative algebra over K , endowed with an involution z + z * Then the selfadjoint part 19.3
.
x
:=
{ x
2 : x* = x
E
}
is closed under the anti-commutator product 19.2.1 general, not under the given associative product) therefore a real Jordan algebra. This applies in to the involutive algebra L ( E ) over the center where E is a Hilbert space over D E {R,C,A} set
.
:=
H!L(E)
{ x
E
L(E)
: x* =
(but, in and is particular K of D , Hence the
x }
of all self-adjoint (or hermitian) bounded D-linear operators o n E is a unital real Jordan algebra under the antin is finite-dimensional, commutator product. In case E = D The set H R 3 ( 0 ) of all selfwe write H R , ( D ) := H ~ ( D " ) adjoint 3x3-matrices over the octonions 0 is a unital real Jordan algebra (of dimension 27) under the anti-commutator product which is called the exceptional Jordan algebra since it cannot he embedded as a Jordan subalgebra of an associative algebra.
.
19.4 EXAMPLE. Let Y he a real Hilbert space with scalar Put X := R @ Y and define a commutative product (xly) product on the real Banach space X by
.
21'2(a+x) whenever
a,B
E
0
R
and
(B+Y) := (CxB+(XlY)) x,y
E
Y
.
+ (ay+f3x)
19.4.1
It is easy to verify that
SECTION 19
316
X becomes a real Banach Jordan algebra under the product 19.4.1, with unit element e := (21’2,0) This Jordan
.
algebra is called a (real) spin factor since it is closely related to the spin systems of quantum mechanics arising in the study of the canonical anti-commutation relations. As an example, the real Jordan algebra Ha2(D) of all self-adjoint 2x2-matrices over D E { R , C , E , O } is (isomorphic to a) spin factor of dimension 3,4,6 and 10, respectively. Under this isomorphism, we have Y = { x
19.5 LEMMA. identities
E
H R ~ ( D ): trace(xr = 0 }
Every Jordan algebra
2
.
satisfies the operator
and
By 19.1.2, we have the operator identity 2 2 [M(x+y),M((x+y) ) ] = 0 Since ( x + Y ) =~ x2 + 2xoy + y 19.5.1 follows by considering the terms of degree 2 in x of degree 1 in y Another polarization gives 19.5.2. PROOF.
.
.
, and
D.E.D.
.
19.6 LEMMA. Suppose 2 is a Jordan algebra and x,y E 2 Then the commutator [Mx,My] is a derivation of 2 , and for every derivation 6 of 2 we have [6,M X 1 = M 6 x PROOF.
By 19.5.2, we have
since the middle term is symmetric in
y
and
u
.
Therefore
J O R D A N ALGEBRAS
317
yo(uo(zox)) - uo(yo(zox)) = (yo(uoz))ox
- (uo(yoz))ox + zo(yo(uox)) showing that
[My,Mu]
-
zo(uo(yox))
is a derivation of
X
.
,
The last
assertion is clear.
O.E.D.
Jordan algebras and Jordan triples are closely related. More precisely, a Jordan triple can be regarded as a family of Jordan algebra structures. PROPOSITION. Suppose Z is a Jordan triple with triple Then for every e E 2 , Z becomes a Jordan product {xu*y} algebra under the product 19.7
.
xoy : = {xe*y}
.
19.7.1
The multiplication operators and the quadratic representation of this Jordan algebra are given by Mx = x o e* and
.
Px - OxOe PROOF.
The product 19.7.1 is commutative by 18.1.1. z : = {xe*y} , 18.2.7 implies x 2o(xoy) = {{xe*x}e*z} = 2 {{ze*x}e*x}
-
Putting
{x{ez*e}*x}
and 2 {ze*x} = {ye*{xe*x}} + {x{ey*e}*x} Applying 18.2.1
.
twice, we get
{x{ez*e}*x} = {x{e{xe*y}*e}*x} = {x{ex*{ey*e}}*x}
=
{xe*{x{ey*e}*x}}
.
Combining these identities, it follows that x 2o(xoy) = {{ye*{xe*x }e*x} + { {x{ey*e}*x}e*x
Hence
2
{xe*{x{ey*e}*x}}
=
is a Jordan algebra.
{ye*{xe*x}}e*x}
=
Further, 18.2.7
implies
(yox2 ox
.
SECTION 1 9
318
pXy := 2 {xe*{xe*y}} - {{xe*x}e*y] =
{x[ey*e}*x} = QXQey O.E.D.
In order to show that, conversely, Jordan algebras give rise to Jordan triples, consider a Jordan algebra 2 over K with product zow , and define a (trilinear) triple product on
Z
by putting
.
Since 2 is commutative, this triple for all x,y,z E 2 product is symmetric in the outer variables (x,y)
.
19.8 LEMMA. The triple product associated with a Jordan algebra Z via 19.8.1 satisfies the Jordan triple identity {{ZUX}VYI
+ {{ZUY}VX}
-
PROOF. Define a linear operator P(x,y) P(x,y)z := {xzy} Then 19.8.1 implies
on
.
+
P(x,Y) = M M X Y and, in particular, imp1 ies
P(x,x) = 2 M 2 X
M M
Y X
-
+
MyouMx
- M xM uM y
-
My'uMx
implies
-
M(x 2 )
M((x0y)ou) = MxOyMu
Therefore 19.8.2
= tX~~ZVIY1
{ZUIXVY}l
+
Z
by putting
M(x0y)
19.8.2
.
Now 19.6.1
=
Px
MUOXMY
J O R D A N ALGEBRAS
-
MxowMx +
-
-
-
M Mwox x
2 MwMx2
+
MxMxow
2
M ( X2
-
MxowMx
+
) M ~ 2 M
MwM(x 2 ) - 2 M x2M w
+
= 2[Mx,MxowI
Polarizing the identity
319
[MwtM(X
2
)I
~
+
+M
2 ~M
~2
M
~
M(x 2)Mw
= 0 *
2 P ( X O W , X =) MwPx
+ PxMw
,
we g e t
and t h e r e f o r e wo{xvy} Since
[MZ,Mu]
19.8.3
imply
+
{x,wov,y} =
{WOX,V,Y~
is a d e r i v a t i o n o f
Z
19.8.3
+ t X ~ V ~ W O Y 1
by 19.6,
19.8.1
and
{ z u { x v y } } = [MZ,MuI x v y } + M ( z o u )
19.9
COROLLARY.
Every J o r d a n a l g e b r a
Z
over
K
satisfies
t h e fundamental formula
.
19.9.1
by 1 9 . 8 . 1 ,
the assertion follows
PxPuPx = P ( P x u ) PROOF.
Since
Pxz = { x z x }
O.E.D.
f r o m 1 9 . 8 a n d 18.2.4.
Recall t h a t a n i n v o l u t i o n o f a J o r d a n a l g e b r a
K
is a K - a n t i l i n e a r b i j e c t i o n
z + z*
of
Z
Z
over
such t h a t
SECTION 19
320
(z*)*
=
in case
z
and
K = R
(zow)* = w*oz*
,
for all
z,w
E
2
.
Note that,
the identity mapping is an involution. is a Jordan algebra over K z + z* Then 2 becomes a
19.10 COROLLARY. Suppose endowed with an involution
Z
.
,
Jordan triple under the triple product
-
{xu*y} = xo(u*oy) satisfying PROOF.
Q,U
=
{xu*.}
=
u*o(yox) + yo(xou*)
pX(u*)
19.10.1
. Q.E.D.
Apply 19.8 and 18.2.
19.11 EXAMPLE. Let 2 he an associative algebra with an By 19.2, Z is a Jordan algebra under involution z + z* the anti-commutator product and, obviously, the given involution is also an involution of this Jordan algebra. By 19.10, 2 is a Jordan triple under the triple product 19.10.1, which reduces to {xu*y} = (xu*y+yu*x)/2
.
.
19.12 DEFINITION. An element e E Z of a Jordan triple over K is called unitary if e n e* = idZ , i.a., if tee*.} = z for all z E z
Z
.
19.13 PROPOSITION. There is a 1-1 correspondence between unital involutive Jordan algebras and Jordan triples with distinguished unitary element. More precisely, if Z is an involutive Jordan algebra with unit element e , then e is unitary with respect to the Jordan triple product 19.10.1. Conversely, if Z is a Jordan triple and e E Z is unitary, then Z becomes a Jordan algebra with unit element e , product
xoy := {xe*y} z*
and involution :=
{ez*e}
.
19.13.1
These two constructions are inverse to each other. PROOF. The unit element e of an involutive Jordan algebra satisfies e* = e , and is therefore unitary with respect to the triple product 19.10.1. Conversely, suppose Z is a By 19.7, Z is a Jordan triple with unitary element e
.
321
JORDAN A L G E B R A S
Jordan algebra with unit element By 1 8 . 2 . 7 , we have
e
for the product 19.7.1.
{e{ez*e}*e} = 2 {{ze*e}e*e} - {ze*{ee*e}} {ze*e} - {ze*e}
= 2
for all z + z*
z E Z of 2
,
=
z
showing that the K-antilinear endomorphism has period 2 Further, 18.2.3 implies
.
z*ow = {{ez*e}e*w} = {e{ze*e}*w}
-
{{ez*w}e*e} + {ez*{we*e}} = {ez*w}
whereas 1 8 . 2 . 1
,
implies
(zow*)* = {e{ze*w*}*e} = {ez*{e(w*)*e}} = {ez*w}
.
.
Hence z + z* is an involution of the Jordan algebra 2 Further, the Jordan triple product 1 9 . 1 0 . 1 derived from 1 9 . 7 . 1 and 1 9 . 1 3 . 1 coincides with the original Jordan triple product, since 1 8 . 2 . 3 implies {xuy} : = xo(u0y) - uo(y0x) + yo(x0u) =
{xe*{ue*y}} - {ue*{ye*x}}
=
{x{eu*e}*y}
+ {ye*{xe*u}}
. Q.E.D.
19.14
K
DEFINITION. Let with a norm
, endowed
Z
1-1
be a Banach Jordan algebra over A continuous involution
.
on 2 is called -hermitian if Mu u* = -u and [Mx,My] E u L ( 2 ) whenever x adjoint. z + z*
Note that in case from the first.
K = C
,
E
U9.(2)
and
y
whenever are self-
the second condition follows
EXAMPLE. Let 2 be an associative Ranach algebra over K , endowed with a norm 1 - 1 Let z + z* be a -hermitian continuous involution on 2 By 19.11, Z 19.15
.
.
SECTION 1 9
322
becomes an involutive Banach Jordan algebra under the anticommutator product. Now assume u * = -u , x* = x and Then the commutator [x,y] : = xy - yx is skewy* = y adjoint and hence
.
and 4
[MxIM I = L([x,yl) + R([y,xI) Y
U ( Z )
E
.
19.16 P R O P O S I T I O N . Let 2 be a Ranach Jordan algebra over R I endowed with a norm 1 . 1 and a -hermitian continuous involution. Then the associated Banach Jordan triple is - hermitian. Conversely, if Z is a -hermitian Banach Jordan triple with unitary element e E Z , then the associated Jordan algebra involution is -hermitian. PROOF.
By 19.10.1, we have u n v * = [M(u),M(v*)] + M(uov*)
.
Hence
Now write u = x + y and v = h h* = h I Y* = -y and k* = -k
.
[M(u),M(v*)l
-
[M(V),M(U*)I
+ k
I
where
x* = x
,
Then
= 2 ([MxIMhl
-
[MyiMk1)
It follows that Z is a -hermitian Banach Jordan triple. Conversely, suppose Z is a -hermitian Ranach Jordan triple with a unitary element e E Z The associated Jordan algebra has the multiplication operators MZ = z o e * Since 18.2.3 imp1 ies
.
.
{e(z*)*w} = {e{ez*e}*w} = {{ze*e}e*w} {{ze*w}e*e} it follows that
-
{ze*{ee*w}} = {ze*w}
,
+
JORDAN ALGEBRAS
,
Z O ~ =* e o ( z * ) * Hence
u* = -u
implies
u o e* = -ea u*
2 MU = u n e * - e o u *
Now suppose imply
x* = x
323
and
ug(Z)
E
.
y* = y
19.16.1 and hence
.
Then 18.8.3 and 19.16.1
[Mx,M J = [ x o e * , y o e*] = {xe*y}o e*
Y
=
M(xoy)
-
-
y o {ex*.}*
y o x*
and [Mx,M Y 1 = [ e o x * , e o y * J = {ex*e}oy* - eo{ye*x}*
-
= xay*
since
xoy
{xe*y}o e* = x n y *
is self-adjoint.
-
~ ( x o y ),
Combining the above equalities,
we get
19.17
DEFINITION.
.
Suppose
element e An element exists an element v E Z
Z
is a Jordan algebra with unit
u E 2 is called invertible if there 2 such that uov = e and u ov = u
.
19.18 PROPOSITION. An element u E 2 is invertible if and only if Pu is invertible. In this case, the "inverse" v = u-l of u is uniquely determined and given by the -1 -1 formula u-l = Pu ( u ) Further, P(u-l) = P(u)
.
PROOF.
v
:=
Suppose first that Pu is invertible and put -1 (u) Pu Then [Mu,PuJ = 0 by 19.1.2 and hence
.
PU e Since
.
P,
=
u2 =
M U = U
is injective,
Mu Pu v = Pu M u v = PU (uov)
uov = e
2 u 2ov = M(u 2 )V = 2 Mu"
-
.
Therefore
P v = 2 U
.
uO(u0V)
-
U
=
U
.
SECTION 19
324
Now suppose u is invertible and choose v 2 Then M(uov) = idz Hence [MU ,M(v ) ] = 0 19.5.1. Therefore
.
2 PU (v
= 2 :M
= 2 M(v2)u2 = M M(u V
M(V
-
2
)e
-
M(U
2
)v
as in 19.17. = [Mv,M(u2)I by
2
M(u2)v2 = u20v2 = M(u
2 )v = Mvu = e
2
)
MVv
.
Putting z := v2 , the fundamental formula 19.9.1 implies Hence Pu is injective and idZ = Pe = P(Puz) = PuPzPu surjective. Therefore Pu is invertible and u-l = v satisfies
.
2 P v = 2 Mu" U
-
M(u 2 )v = 2 uoe - u = u
and is therefore uniquely determined as pU
= P(Puv) = PuPvPu
whence
P(u-')
= P-' U
.
,
it follows that
.
-1 (u) Since u-l - Pu PUPv = idZ = PvPu , O.E.D.
An algebra 2 is called abelian if it is associative and commutative. It follows that Z is a Jordan algebra and the anti-commutator product 19.2.1 coincides with the original product. For any abelian algebra Z with involution, the self-adjoint part X of Z is an abelian real subalgebra of 2 For K E {R,C} , the Banach algebras L - ( S , K ) , C-(S,K) and C u ( S , K ) , associated with a measure space, a topological space or a locally compact space S , respectively, are abelian.
.
19.19 DEFINITION. the identity
A Jordan triple
2
is called abelian if
{xu*{yv*z}} = {{xu*y}v*z}
is satisfied.
19.19.1
An equivalent condition is that
z o z * := { x n u * : x,u
E
2
}
JORDAN ALGEBRAS
325
z
is a commutative set of linear operators on
.
For R E {R,C} , the Jordan triples Z of K-valued functions on a set S , defined in 18.4 , are abelian. 19.20 Z
For any abelian Jordan triple is closed under the operator product.
LEMMA.
Z*
PROOF.
By 18.2.3 and 19.19.1,
{x{uy*v}*z} = [xu*{yv*z}} = {xu*{yv*.}}
Hence
-
Z
,
the set
we have
{yv* xu*z
.
( x n u * ) ( y a v * ) = xo{uy*v}*
E
ZnZ*
.
O.E.D.
19.21 LEMMA. Suppose 2 is an abelian Jordan triple, and let e E Z Then 2 becomes an abelian Jordan algebra under the product 19.7.1. Conversely, if 2 is an abelian Jordan algebra with involution, then 2 becomes an abelian Jordan triple under the triple product 19.10.1.
.
PROOF,
Putting
u = v = e
in 19.19.1,
we get
xo(yoz) = {xe*{ye*z}} = {{xe*y}e*z}
.
= (xoy)oz
Hence the Jordan algebra (Z,o) is associative and therefore abelian. Conversely, for any abelian Jordan algebra with involution, the Jordan triple product 19.10.1 reduces to {xu*y} = xo(u*oy) and is therefore abelian. O.E.D. For any associative algebra, the subalgebra generated by a single element is abelian. The same is true for Jordan algebras, but requires some more argument. Define powers inductively by putting x1 : = x and xn+l := xoxn for
.
n > 1 element. 19.22
For a unital algebra, let
PROPOSITION. M(X~)= 2
Suppose
2
-
xo
denote the unit
is a Jordan algebra. P ~ M ( ~ ~ - ~ )
Then 19.22.1
326
SECTION 1 9
for a l l integers m > 3
M(xm)
.
In particular,
the operators
b e l o n g t o t h e c o m m u t a t i v e a l g e b r a g e n e r a t e d by
W e p r o v e 19.22.1
PROOF.
x = z
by i n d u c t i o n o n
Now s u p p o s e n > 3 a n d 1 9 . 2 2 . 1 n-2 3 < m < n-1 P u t u := X
.
and
Mx
hypothesis.
.
Putting
we g e t
i n 19.6.1,
since
m
Mx
holds for a l l Then
.
commute o n
Mu
m
satisfying
by t h e i n d u c t i o n
Z
Hence
M( x n ) = M(x 2 o u ) = M(x 2 ) M U
+
2 M(x
= 2 M(x
since
[Mx,Q]
n-1
n-1
= 0
)
.
)Mx
-
2 MxMUMx
Mx
-
(
2 M:
2 M(x )
-
T h e r e f o r e 19.22.1
)
Mu
,
holds for
m = n
.
0.E.r).
19.23
x
E
Z is a J o r d a n a l g e b r a a n d l e t T h e n w e h a v e f o r a l l m,n > 1
COROLLARY.
Z
.
Suppose
.
M(xm) M ( x n ) x = M(xm+n ) x PROOF.
W e p r o v e 19.23.1
19.23.1
by i n d u c t i o n on
n > 1
.
For n = 1
19.22 i m p l i e s M(xm) M
X
Now a s s u m e 1 9 . 2 3 . 1
x = Mx
M(xm)x =
xO(x
is t r u e f o r some
m+l
n > 1
T h e n 1 9 . 2 2 a n d t h e f i r s t p a r t of t h e p r o o f
and e v e r y imply
m
.
,
JORDAN ALGEBRAS
M ( x m ) M(xn+') =
-
-
327
x = M(xm) ( ( x o x n
M(xm) Mx M ( x n ) x = Mx M(xm) M x") x m+n
Mx
M ( X ~ + ~ ) X= M(x
)
m+n+l Mxx = M(x )x
. O.E.D.
19.24
For any J o r d a n a l g e b r a
COROLLARY.
x
s u b a l g e b r a g e n e r a t e d by PROOF.
By 1 9 . 2 3 ,
we have
x
and
2
2
E
,
the
is a b e l i a n . m n x Ox =
X
m+n
for all
> 1
m,n
,
and t h e r e f o r e k
xko(x"ox") for all
k,m,n
> 1
.
m ) o xn
19.24.1
= ( x ox
19.25
Suppose
LEMMA.
2
putting
is a J o r d a n t r i p l e a n d
z
D e f i n e " t r i p l e powers" z ( l ) := z
z
E
2
,
is a b e l i a n . E
i n d u c t i v e l y by
Z
and Z
.
is
O.E.D.
Then t h e J o r d a n t r i p l e g e n e r a t e d by PROOF.
x
Hence t h e s u b a l g e b r a g e n e r a t e d b y
a s s o c i a t i v e and t h e r e f o r e a b e l i a n .
( 2 n + 3 ) :=
for all
n > 0
product
xoy := { x z * y }
By 1 9 . 7 ,
2
QZ(Z
(2n+l)
is a J o r d a n a l g e b r a u n d e r t h e
and an i n d u c t i o n argument u s i n g 18.2.1
shows z
for all
n > 0
abelian,
18.2.3 tz
.
(2n+l) = Zn+l
S i n c e t h e J o r d a n a l g e b r a g e n e r a t e d by
z
is
implies
( 2 m + 1 ) ( 2 ( 2 k + l )) , Z ( 2 n + l ) } = { z m + l { z ( z k ) * z } * z n + l j
= (Zkozm+l)oz"+1
= z
k+m+n+2 =
+
k oz n + l
(2
m+l )oz
-
Zko(zm+lozn+l)
(2(ktm+n+l)+l)
I t f o l l o w s t h a t t h e l i n e a r s p a n o f a l l powers n > 0 is a n a b e l i a n J o r d a n t r i p l e .
z
(2n+l)
for 0.E.D.
328
SECTION 1 9
19.26 LEMMA, Suppose x E Z is invertible.
2 is a unital Jordan algebra and -1 Then M(x-’) = Px Mx commutes with
Mx ’ PROOF,
Putting
y := x - l
and
+
-1 ) Mx2
Mi M(x-l)
M(x
u := x M(x
=
2
in 19.6.1, )
M(x-l)
+
Mx
we get
.
Now M(x-’) and M(x2) commute by 19.22.1. Further, M(x-’) and P, commute as a consequence of 19.18. It and M(x-l) commute. Therefore follows that :M
M,
=
2 2 Mx M(x-l)
-
M(x
2
)
M(x-’) = P, M(X-’)
. Q.E.D.
19.27
COROLLARY.
Let
e
be a unitary element of a Banach
Jordan triple Z and consider the associated unital involutive Jordan algebra. Then u E Z is unitary if and only if u is invertible and u-l = u*
.
Suppose first that u is invertible and satisfies u-l = u* Since M(u) and M(u*) commute by 19.26 and, by 19.10, the Jordan triple product 19.10.1 coincides with the given one, we get PROOF.
.
u o u* = [M(u),M(u*)]
+ M(uou*)
=
idZ
.
Conversely, suppose that u is unitary. Then uou* = {ue*{eu*e}} = 2 {{ue*e)u*e) - {e{eu*u}*e) = 2e - e and hence u 2 ou* = 2 uo(uou*) - {uu*u} = 2 uoe - u = u By definition, u is invertible and u* = u-l Q.E.D.
.
.
NOTES. Jordan algebras were introduced in order to provide an algebraic foundation of the quantum mechanical formalism (cf. [74,75,1201 and the introduction of C221). Although this program was only partially successful, it turned out that Jordan algebras have interesting algebraic properties [25,69,1251 as well as important applications to other parts of mathematics 11041. Jordan algebras of Hilbert space operators were studied in [132-1361 whereas a satisfactory theory of abstract Banach Jordan algebras was developed by E. Alfsen, F. Shultz and E. Stpkmer 15,521.
BOUNDED SYMMETRIC DOMAINS AND JB*-TRIPLES
BOUNDED SYMMETRIC DOMAINS
20.
AND
329
JR*-TRIPLES
A class of symmetric Ranach manifolds of particular importance
are the so-called bounded symmetric domains in complex Ranach spaces.
In the finite-dimensional case, bounded symmetric
domains generalize the non-euclidean hyperbolic geometry of the open unit disc and have many of mathematics, e.g., the theory harmonic analysis on semi-simple theory. In infinite dimensions,
applications to other areas of automorphic functions, Lie groups and number bounded symmetric domains are
closely related to (Jordan) operator algebras and may also play a certain role in the quantum mechanical formalism, e.g., arising as "curved phase spaces" in the (second) quantization procedure [ 1 4 4 1 . The study of the hermitian Ranach Jordan triples associated with bounded symmetric domains is based on spectral properties of the Jordan multiplication operators u o u* A key lemma is the following result about Jordan algebras.
.
20.1 R
E
THEOREM. {R,C}
.
Suppose Let
W
is a Banach Jordan algebra over
Z
denote the closed subalgebra of
generated by an element
u
E
Z
.
Put
:=
S
Z
CkW(MU) U { O }
.
Then 20.1.1 In case Ek2(MU) C R , we have CkW(MU) C EkZ(MU) therefore sup (Ea,(MU)( = sup (CR2(MU)(
.
and
PROOF. By considering the complex Ranach Jordan algebra Z 0 C and its closed subalgebra W O K C generated by u , K we may assume that K = C Then Z ' := 2 fB C is a complex
.
Banach Jordan algebra with unit element containing
as an ideal.
2
e := (O,1)
Therefore 20.1.2
.
.
ERZl(MU) Put W' := W fB c By 19.22, M(WI) : = { Mx : x E W' } is a commutative subspace of
since L(2')
0
.
E
Let
A
be a maximal abelian subalgebra of
L(Z')
3 30
SECTION 2 0
Containing
M(W’)
.
Then CIIZl(MU) = CA(MU)
20.1.3
.
Ry 20.1.2 and 20.1.3, by [17; 15.41. Now let A E CkZ(MU) there exists a continuous unital homomorphism f : A + C such that A = f(MU) For any 5 E C , we have
.
and 20.1.4 where
cl+c2
=
2 f (MU) = 2X
20.1.5
.
20.1.6
and
c1c2 ,
For j = 1,2 Hence 0
= f(PJ
E
x
put
j
Z,(P(x.)) 3
: = g.e-u 3
E
W’
.
= EQZ,(P(x.)) 3
Then
f(P(x.)) = 0 3
.
.
20.1.7
.
Suppose now that xj has an inverse y E W ’ Then e = P(x. )y2 and the fundamental formula 19.9.1 implies 3 2 Hence P(x.1 E G R ( Z ’ ) , a idZl - Pe = P(xj) P(y 1 P(x.) 3 3 contradiction to 20.1.7. Therefore [17; 5.43, applied to the abelian unital Ranach algebra W ’ , gives c j E C w , ( u ) = CILWl(MU) C S , since
.
(a-idZI-MU)-’ = (a*idZ-MU for all
a
E
C\S
,
NOW 20.1.1
x
a-1
follows from 20.1.5.
second assertion follows with 2.13.
The O.E.D.
Suppose Z is a Banach Jordan triple over K Given u,v E Z , let W denote the smallest closed subspace of 2 invariant under u a v* and containing u 20.2
.
COROLLARY.
.
* BOUNDED SYMMETRIC DOMAINS AND JB -TRIPLES
Put
.
S : = ELw(uo v*) w {O}
Eaz(QUQv) C E L Z ( u n v*)
and
SS
c R ,
.
E L ( u n v*) C CLz(un v*) W v*)(
.
IEL~(UO
sup IziW(un v * ) J = sup
,
E L z ( u o v*) C (S+S)/2
Then
EL,(B(u,v*)) C (l-S)(l-S)
we have
331
In case and hence
PROOF. By 19.7, 2 becomes a Ranach Jordan algebra under the product xoy := {xv*y} , and W is the closed subalgebra of
u
generated by
Z
.
With respect to this Jordan algebra -
structure, we have U P v* = Mu r OuQv - P,, and We may assume that R(u,v*) = id - 2 MU + Pu (cf. 18.12.9). K = C
and define
20.1. Then Mu Therefore 20.1.6
,
W'
2'
and
A
as in the proof of
,
P, and B(u,v*) = Pe-u and 20.1.4 imply
belong to
A
.
and
20.3
DEFINITION.
called 'hermitian
A
Ranach Jordan triple
if all
u,v
E
2
2
over
K
is
satisfy
E L ( U P v*+vo u * ) C R
20.3.1
Z
and sup
IEL~(uou*)I
= (unu*l
.
20.3.2
Banach Jordan triple Z over K is called hermitian if it is both -hermitian (18.13) and 'hermitian.
A
By 18.13.1 Jordan triple hermitian.
and 14.30, every Z
-hermitian
is automatically 'hermitian
complex Ranach and hence
Suppose in the following that Z is a hermitian Ranach Jordan triple over K with respect to a norm I * I Put is defined by 18.12.6, and -'g : = { Xu : u E Z } , where Xu 1 1g @ -1 g is put g : = aut(z) A u k ( z ) c ~ ~ ( 2 ) Then g :=
.
.
a real Ranach Lie algebra contained in the binary Ranach Lie
332
SECTION 2 0
algebra h = h - , @ h o @ h, over K , associated with 2 in 18.18. There exists an analytic action p of h on the homogeneous Banach manifold N := H/H+ introduced in 1 R . 1 8 . 1 such that the canonical chart (P,p,Z) of N about o : = H+ satisfies 18.18.3 and
a
p(exp(p(u,,))(o))
=
u
.
for all u E Z By 18.19, the orbit M : = G ( o ) C N of o under the group G := < exp p ( g ) > is a domain in N, and the universal covering manifold n : M" + M of M is called the simply-connected symmetric Ranach manifold associated with Z There exists an analytic action pof g on M" such that the canonical chart (P",p",Z) of M" about o satisfies 18.23.1. Now define
.
Since CI1z(uuu*) c R by assumption, 2.10 implies that Q 2 is open and starlike about 0 Hence Q z is a simplyconnected domain. In case K = C , Q z is circular and hence balanced.
.
PROPOSITION. Suppose 2 is a hermitian Banach Jordan triple. Then the canonical chart (P",p",Z) of M" about o can be chosen such that Q 2 cp"(P") 20.4
.
.
PROOF. We first show that p -1 ( a z ) is contained in M Assume that there is a point u E Q z such that p -1 ( u ) M Then tu E nZ for 0 < t < 1 and -1 (tu) E N defines a continuous curve in N with mt : = mo = o E M and m1 E!, M Hence mt E a M for some t Now tu E Q Z implies idZ - (tu)o (tu)* E G E ( Z ) By 18.19, the orbit G(mt) is open in N , in contradiction to the fact that mt E a M Therefore p -1 ( Q , ) C M Since Q z is
.
.
.
.
simply-connected, t h e universal covering canonical chart p"(P") 3 Q 2
.
.
.
(P",p",Z)
The triple product
such that
II
defines a
pn = pon
and Q.E.D.
(x,u,y)
+
-{xu*y}
endows
2
with
*
BOUNDED SYMMETRIC DOMAINS AND JB -TRIPLES
333
the structure of a hermitian Banach Jordan triple, called the dual Jordan triple. Clearly, Z and its dual Jordan triple have the same derivations and automorphisms.
a;
+n
:=
{
z : 1 + K*Ef,z(UUU*) > 0 }
U E
{ u
For
< 1 } , since
K
=
.
is hermitian. The binary Banach Lie algebra h and the homogeneous Banach manifold N are independent of K define l g K : = aut(2)n u f , ( Z ) C T o ( Z ) and -19 K : = { X," : u E 2 } , where
Then
ilz
=
52;
Z
E
xc
: IuOu*l
:=
(U +
a
K{ZU*Z})E
+_ , put
Z
.
.
Now
20.4.1
.
Then g K := 'gK 6i -'gK is a closed real subalgebra of h In case K = C , g - and g + have the same complexification g
c -- g-l c
@
C
go
,
gF
where
and
By 18.19, the orbit
is a domain in N manifold of G K ( o ) (PK,pK,Z) of MK M~ satisfying
GK(o)
.
of
o
X
E
gK
.
.
N
under the group
Let MK denote the universal covering By 1 8 . 2 3 , there exist a chart and an analytic action p K of g K on
.
(PK)*(PKX) = for all
E
By 20.4,
x
20.4.2
we may further assume
c pK(PK) By definition, Mis the simply-connected symmetric Banach manifold associated with the Jordan triple Z , whereas 'M is the "dual" simply-connected symmetric Banach manifold associated with the dual Jordan triple.
334
SECTION 2 0
20.5 K = ‘I
Define Q := C \ { t E R : t C -1 } Then there exist holomorphic functions + C and a : K Q + C such that
and let
LEMMA.
t :
.
-KQ
,
c*rK(c2 ) = tanK(5c) : =
+
K =
and
PROOF. Let $+ : C \ R + + { h E C : Im(h) > 0 } and 0- : C\R- + { h E C : Re(h) > 0 } denote the holomorphic branches of 5 li2 determined by $+(-l) = i and -1 (principal branch) are 1$-(1)= 1 Then tanK and tan-K holomorphic on $ K ( C \ R K ) Hence
.
.
T ~ ( s ):=
o,c)/+,e
tanK($
and
are holomorphic on C \ R K and C\R-K , respectively. For I h l < 1 , there exist convergent power series expansions m
tanK($h)
1
=
K
n bn h 2n+l
n=O
and tan-l(I) = with real coefficients
bn
m
( - K )
n h2n+l
n=O For 1 5 1 < 1
.
,
/(
2n+l)
define
m
and
Since
(C\RK) U A =
-KQ
and
(C\R-K) U A = KG
,
the assertion
BOUNDED SYMMETRIC DOMAINS AND JB*-TRIPLES
follows. 20.6
O.E.D.
Suppose Z is a hermitian Banach Jordan Then there exist real-bianalytic mappings
PROPOSITION.
K
triple over
for
335
K
=
.
-+ , such that 20.6.1
for all
u
E
-K
0,
.
For
Iuo u*l < 1
, we have
tan
'(uo u*)1/2 K 2 20.6.2 (u n u*) 1/2 satisfies whereas the inverse mapping T -1 : G ; + -1 T~ (v) = aK(v) , where for I v u v * ) < 1 we have n/2 'I
(U)
=
tanK -1 ( v n v*) 1/2 u p ) = PROOF.
For
u
E
-K
Gz
( v n v*) 1/2
, we have
C1lz(uou*) C
v . -KQ
20.6.3
.
Therefore
20.5 and the holomorphic functional calculus applied to L(Z C ) , for Zc : = 2 QK C , imply that T ~ ( u ) := T ~ ( U U U * ) U defines a real-analytic mapping 'IK : + zC satisfying Similarly, CLZ(vo v*) 20.6.2. Hence T ~ ( O ; ' ) c Z by 3.1.
c KQ for v E G i implies that aK(v) := U ~ ( V C I V * ) V defines + Zc satisfying 20.6.3. a real-analytic mapping uK : Hehce uK(G;) c Z by 3.1. Now let W denote the closed Jordan subtriple of Z generated by u Then v : = T ~ ( u )E W Hence 20.5 implies
.
.
VOV*(w =
'I
L
(UOU*) (UUU*)
maps I-K : = t E R : 1 - Kt > o 1 into IK := { t E R : 1 + Kt > 0 } , the spectral mapping theorem Therefore 20.2 implies implies CiW(vo v*) C I Similarly, 20.5 implies c g Z ( v n v*) c IK , i.e., v E n KZ for v : = oK(u) E W
Since
5 + c*TK(c)2
.
.
v o v * I w = a ( u o u * )2 ( u o u * ) K
.
SECTION 2 0
336
5 +
Since
v* )
5I
, we get T-~;u) : =
is connected and independent o f commu t i ng d iagram
k
.
Hence there is a
BOUNDED SYMMETRIC DOMAINS AND J B ^-TRIPLES
345
is a real-bianalytic mapping satisfying $ ( o ) = o and To(pZ) To($) = To(p;) Since Q is also G-equivariant, it follows that $ is biholomorphic. By 1 8 . 2 5 , there exists a circle group (gtItER on M; about o satisfying
where
$
.
-
p2(gtm) = e
it
-
p2(m)
..
20.20.1
.
for all m in a neighborhood of o E M 2 A power series argument (cf. 17.23.1) shows that a holomorphic 2-valued mapping f , defined on an open connected neighborhood of o E M; and satisfying f(o) = 0 and f(gtm) = eitf(m) for all m in a neighborhood of o E. is uniquely determined by its differential To(f) Define a holomorphic mapping $ : by
.
Mi+Z
Mi
$(m for all and
m
.
M;
E
Then
$ 1 = To(nl
20.20.2 n
.
for all m E M2 Now choose g E G R ( Z ) such that goTo($) = To(p;) Then ( g o $ ) ( o ) = g ( 0 ) = 0 = p",o) and it 20.20.2 implies (go$)(gtm) = e (go$)(m) on M; Hence 20.20.1 implies that go$ and p; coincide in a neighborhood of o E M; Let p-' : 2 + P c N be the canonical embedding. Then the holomorphic mappings n 2 : M; + M2C N and p-1 ogo$ : M; + P C N coincide in a neighborhood of o E M2 and are therefore equal. Hence
.
.
.
n
and p : M2 doma in
+
Z
defines a hiholomorphic mapping onto the
346
SECTION 2 0
-
-1 Since n l ( $ M2) = R , 20.20.2 implies that JI(M';) is a circular bounded domain. The same is true for D , since is linear. Since M2 is homogeneous, D is homogeneous under biholomorphic transformations. By 12.13, D is balanced and therefore simply-connected. Hence M2 is simply-connected and n 2 is biholomorphic. by 18.12.6. Then For u E Z , define Xu E p
Xu = (p2)*(p2XU) = p*(p2XU) If
{uu*u} = 0
,
then
y(t) : = tu
E
aut(D)
satisfies
. y(0) = 0
.
and
y'(t) = u = u - {y(t)u*y(t)} for all t E R Hence y(t) = exp(tXu)(0) E D Since D is hounded, u = 0 let
.
denote the group of all deck transformations of n l -1 y E r , put m := y ( 0 ) E Mi and u := pi(4m) E D
x and 2 0 . 1 9
:=
i uou* aa z
implies
(pix), = 0 ( p; 0 4
Since
=
$
[XiuiXul/4 = plX
y*(p;X)
0 =
Hence
=
Y*(P;xl0
.
.
Now
.. For Then
k
E
g
20.20.3
Hence
= T,(Y
and therefore
1 * ( P;X 1
=
Tm ( P;O4)
(
PYX m = o .
is G-equivariant, we have (P;O$)*(P;x)
= (P;)*($*(P;x))
=
(P;)*(P;x)
.
=
x
It follows that {uu*u} = 0 and hence u = 0 Since pi is injective on M; , we get m = o and, by 3.4, y = id It are follows that l' is trivial. Since B and M;/r homeomorphic, B is simply-connected. Therefore n l is biholomorphic and pio4on;l
: B + D
.
20.20.4
*
BOUNDED SYMMETRIC DOMAINS AND JB -TRIPLES
is
a biholomorphic mapping.
347
By 17.19, it follows that
h
-
.
for all m in a neighborhood of o E M 2 Since -1 To(pl+ ) = To(pi) , it follows that p; = p;O+ in a Hence 20.20.4 defines an extension neighborhood of o E M; O.E.D. of q = p1
.
.
Suppose in the following that R is a circular hounded symmetric domain in Z By 20.20 and 17.23, we can assume (after applying a linear coordinate transformation) that the canonical chart of €3 about 0 is the inclusion mapping B C Z Then aut(B) = g = k @ p
.
.
PROPOSITION. €3 contains the set { U E 2 : 1 - CLZ(Un u*) > 0 }
20.21
-
Q,
.
.
:=
PROOF.
Ry 20.4, we have
Qi C p(MnP)
= p(M) = B
. O.E.D.
COROLLARY.
20.22
is a positive hermitian Ranach Jordan
Z
triple.
-
PROOF. Ry 20.21, Q z is a bounded domain. If u satisfies C L ~ ( U u*) O < 0 , then tu E for all Hence u = 0 Now apply 20.16.
Qi
.
THEOREM. The spectral norm I - I m is a JB*-triple with respect to
20.23
and ball
2
R
.
on
(*Im
Z
Z
E
t
R
E
.
O.E.D.
is compatible with open unit
i2.i
PROOF. Since { u E Z : luIm < 1 } = is bounded, the continuous semi-norm I * I m on Z is a compatible norm. Since the canonical chart of B about 0 is the inclusion mapping, 20.11 implies B = Since 20.20.4 implies
i2.i .
* I m ) for all u E 2 , it follows that is a hermitian Banach Jordan triple. Since positive by 20.22 and i * u o u*
(Z,l*lm)
E
ua(Z,I
Z
is
SECTION 2 0
348
2
J U O U * l r n= sup Caz(uOu*) = lulm by 14.30,
Z
.
(-Im
is a JB*-triple with respect to
O.E.D.
20.24 COROLLARY. Every circular bounded symmetric domain in a complex Banach space is convex. 20.25 PROPOSITION. A positive hermitian complex Ranach Jordan triple Z is a JB*-triple if and only if every u satisfies 20.8.1.
E
Z
PROOF. By 20.R, every JB*-triple over K satisfies 20.8.1. Conversely, suppose 20.8.1 is satisfied. Then ( u o u * l > luI2 and hence (ul, > lul It follows that lulm is a compatible norm on Z Therefore B := i l Z is a bounded circular domain. By 20.11, B is homogeneous and hence symmetric. By 20.23, 2 is a JR*-triple with respect 3 to I * I m In particular, I[uu*u}(o = (uls for all u E 2 Now assume there exists u E Z such that 1111 < IuI, = 1 Then 20.8.1 implies
.
.
.
.
.
n+= but
for all
n
.
1.1
This contradiction shows
20.26
COROLLARY. Every complex JB*-triple Ua(2) = Aut(2) and u a ( 2 ) = aut(2) = 'gball B of 2 satisfies aut(B) = g -
.
(*Im
=
.
.
O.E.D.
satisfies The open unit
2
PROOF. The inclusion Aut(2) C U k ( 2 ) holds for all JB*-triples over K , Conversely, suppose g E U a ( 2 ) Then g E Aut(B) , where B is the open unit ball of 2 Let g = 1g 0 -1 g be the multiplicative gradation of g := aut(B) with respect to 0 E B Since g is linear -g' = p , it follows that cj,(-lg) = -'g and and
.
.
.
g*xu = xg(u)
for all
u
E
2
.
By 18.12.6,
g
E
Aut(2)
.
O.E.D.
*
BOUNDED SYMMETRIC DOMAINS AND J B -TRIPLES
20.27
DEFINITION.
Suppose
is a u n i t a l complex Ranach
2
J o r d a n a l g e b r a , endowed w i t h a n i n v o l u t i o n {uv*w}
z
.
+ z*
Let
denote the associated Jordan t r i p l e product.
is c a l l e d a J B * - a l g e b r a
2
349
if all
z,w
Then
satisfy
Z
E
and 20.27.2 20.28
EXAMPLE.
A closed u n i t a l Jordan *-subalgebra
u n i t a l complex C*-algebra 2 0 . 9 and 2 0 . 1 0 ,
every JC*-algebra
is a J B * - t r i p l e
T h e c o n v e r s e i m p l i c a t i o n is n o t t r u e :
JB*-algebra.
By
and h e n c e
I t follows t h a t every JC*-algebra
s a t i s f i e s 20.27.2.
of a
Z
is c a l l e d a J C * - a l g e b r a .
A
is a It can b e
shown t h a t t h e c o m p l e x i f i e d e x c e p t i o n a l J o r d a n * - a l g e b r a
is a J B * - a l g e b r a w i t h r e s p e c t t o a s u i t a b l e norm w h i c h c a n n o t b e r e a l i z e d as a JC*-algebra 20.29
EXAMPLE.
[1561.
For a Hilbert space
over
E
D
a l l self-adjoint
D-linear
complexification
o p e r a t o r s on
D =
20.30 and
Here
EC :=
E
in
( c f . 15.11.1).
It follows t h a t
Z
is a
and hence a JR*-algebra. Let
U
b e a Ranach J o r d a n t r i p l e s a t i s f y i n g
Then e v e r y ( n o n - z e r o ) u n i t a r y
1u*1 = ( u I
PROOF.
.
L(EC)
EC := E OR C i n case D = R a n d , i n c a s e is t h e e v e n - d i m e n s i o n a l c o m p l e x H i l h e r t s p a c e
LEMMA.
20.8.1.
Then t h e
c ,
D = H , Ec underlying E JC*-algebra
.
{R,C,H}
c a n be i d e n t i f i e d w i t h a c l o s e d
Z := Xc
u n i t a l Jordan *-subalgebra of
case
E
E
X := H a ( E )
c o n s i d e r t h e u n i t a l real Ranach J o r d a n a l g e b r a
for all
u
E
U
.
e
E
c
>
U
h a s norm
0
such t h a t
The f i r s t a s s e r t i o n f o l l o w s f r o m
1eI3 = I{ee*e}I = lel Iu*I = I { e u * e } I < c l u l
.
There exists for all
u
E
Z
.
Then
1
, of
SECTION 2 0
350
*I
( U * l 3 = I{u*uu*}I = ( { u u * u Hence
c
may b e r e p l a c e d by
w e may a s s u m e 20.31
.
2
IVI
20.32
XI
Then
LEMMA.
z
E
U
Z
Let
x* = x
where
abelian,
and
JvJ < JuI
.
lul
= 2
and O.E.D.
h e a n a b e l i a n Ranach J o r d a n t r i p l e I{uv*w}I < l u l * l v l * l w l
Then
.
,
c > 0 such t h a t < c . 1 ~ 1* I V ) . \ W ( f o r a l l U,V,W
There e x i s t s
1 {uv*w}I
z
E
.
for all
z
since
is
it follows t h a t
c
may b e r e p l a c e d by
LEMMA.
Suppose
self-adjoint. g e n e r a t e d by
.
c = 1
w e may a s s u m e 20.33
< J u J and
E
< (u1 + lu*l < 2 )u(
lu-u*l
=
PROOF.
Hence
u = x+v
Let
s a t i s f y i n g 20.8.1. U,V,W
Q.E.D.
2 1x1 = Iu u * l
PROOF.
Repea ing t h i s a r g u m e n t ,
c = 1 .
COROLLARY.
v* = -v
.
c1I3
< c
c1I3
.
Repeating t h i s argument, Q.E.D.
Z
is a J B * - a l g e b r a
and
x
E
Then t h e c l o s e d u n i t a l s u b a l g e b r a o f
is
Z Z
is a n a b e l i a n C*-algebra.
x
is a n a b e l i a n a l g e b r a i n v a r i a n t u n d e r Hence A is a l s o a n a b e i a n J o r d a n t r i p l e . the involution. Now f o r a l l z,w E ‘2 Hence 20.32 i m p l i e s lzowl < 21 - I w ( By 1 9 . 2 4 ,
PROOF.
A
.
20.30
implies
0 get Re(i f(x)) < 0 Replacing x by -x , it follows that Now let w E Z be a unit vector. By [15; 44.81, f(x) E R there exists $ E L ( 2 , C ) such that 1 $ 1 = 1 = $(w) Then
.
.
.
.
f(z) : = $(zow) If1 = 1 = f(e) satis€ies
.
satisfies Hence $(xow)
E
R
.
(wI
It follows that I (idz+it Mx)wl > Iterating and replacing t by t/n
For n 8.21.
+
+-
I
we get
lexp(itMx)l
Therefore every
for all we get
w
E
Z
,
> 1
for all
t
E
R
t
E
R
. by [17; O.E.D.
20.35 PROPOSITION. Every JB*-algebra Z is a complex JB*-triple under the triple product {uv*w}
.
PROOF.
For
u = x+iy
u n u * = [M(u),M(u*)]
E
Z = Xc
- M(u
o
,
19.10.1
implies
u*) = i[M ,Mxl - M(uou*) Y
.
Since x,y and uou* are self-adjoint, 20.34 and 14.29 imply u m u * E Ha(Z) It follows that Z is a hermitian Banach Jordan triple. Now assume CRZ(uou*) < 0 Let Qez = z* denote the involution of Z Then
.
.
.
u * o (u*)* = Qe(uou*)Qe and hence C9,z(u*o (u*)*) = Cgz(uo u*) < 0 Jordan subtriple W of Z generated by x
.
The closed
is contained in
352
SECTION 2 0
t h e u n i t a l C*-algebra positive.
g e n e r a t e d by
A
.
> 0
Similarly,
u n u * = 2 ( x n x*
it f o l l o w s t h a t
unu*
.
caz(un u*) = { O } u = 0
EXAMPLE.
-
.
u* o ( u* ) *
Since 1
by 1 4 . 3 0 a n d t h e r e f o r e a n d 20.25.
O.E.D.
h e a real H i l b e r t space w i t h
Y
Yc
.
satisfies
Now a p p l y 20.16
Let
complexification
yn y*)
u o u* = 0
Hence
by 20.27.2.
20.36
+
By 2 0 . 2 ,
> 0
CgZ(yo y*)
Hg(Z)
E
.
c g W ( x ox * ) > 0
In particular,
CkZ(xox*)
and is t h e r e f o r e
x
is a u n i t a l r e a l
X := R @ Y
By 1 9 . 4 ,
Banach J o r d a n a l g e b r a c a l l e d a r e a l s p i n f a c t o r . 2 := Xc
Let
d e n o t e t h e c o m p l e x i f i e d Ranach J o r d a n
= C 6l Yc
a l g e b r a , endowed w i t h t h e p r o d u c t
- ( ~ l w )+ ( a w + B z )
2 1 ’ 2 ( a + z ) o ( ~ + w ) = a6
a,B
for of
E
z,w
and
C
w i t h real form
Yc
E
Y
C
i Y
. Here z . Then -
-
+ z
a+z
conjugation o f t h e c o m p l e x H i l b e r t s p a c e 2 R 6l i Y The u n i t e l e m e n t e := (21’2,0) -e = e Further,
. .
-
(a+z)* := a
-
is t h e c o n j u g a t i o n
- + -z
defines a
:= a
w i t h real form of
2
satisfies
z
d e f i n e s an i n v o l u t i o n o f t h e complex J o r d a n a l g e b r a
real form 2
X
.
Z
with
The c o m p l e x i n v o l u t i v e R a n a c h J o r d a n a l g e b r a
is c a l l e d a c o m p l e x s p i n f a c t o r .
A v e r i f i c a t i o n shows t h a t
the corresponding Jordan t r i p l e product s a t i s f i e s 2 {uv*w} = u ( v l w ) + w ( v l u ) for all
u,v,w
E
Z
c a n be shown t h a t Z
norm.
By 2 0 . 3 5 ,
p r o d u c t 20.30.1.
Z
20.30.1
V(W(U)
.
Using t h e C l i f f o r d a l g e b r a o v e r
Y“
2
can be r e a l i z e d a s a JC*-algebra
[581.
becomes a J R * - a l g e b r a
Hence
-
X
under t h e t r i p l e
is a r e a l J R * - t r i p l e u n d e r
20.30.1. NOTES.
The p r i n c i p a l r e s u l t s o f t h i s s e c t i o n are d u e t o W.
Kaup C 8 7 1 , 20.1,
c f . also C58,24,84,1481.
c f . C87,1071.
it
with respect to a s u i t a b l e
is a c o m p l e x J B * - t r i p l e
Therefore
,
F o r t h e b a s i c Theorem
SYMMETRIC SIEGEL DOMAINS
21.
353
SYMMETRIC SIEGEL DOMAINS
Besides the bounded symmetric domains, the most important class of symmetric domains in Banach spaces are the symmetric tube domains and Siegel domains. These domains can be regarded as unbounded realizations of hounded symmetric domains via the so-called Cayley transformations. Algebraically, Siegel domains are closely related to idempotents in Jordan algebras and Jordan triples. 21.1
LEMMA.
Suppose
an idempotent, i.e.,
Z
is a Jordan algebra and
e2 = e
.
2 M i - 3 Me2
e
+
Me = 0 .
2 2 xo(x oz) = x o xoz)
xo(y20z) + 2yo((xoy)oz) = 2(xoy)o(yoz) + y
,
y = z = e Mex
2
,
xoz)
0
.
we get
3 2 2 Mex = 2 Me"
f
is
Then
PROOF. Polarizing the Jordan identity we obtain the identity
For
Z
E
f
.
2 Me"
O.E.D.
21.2 Z
.
THEOREM. Let e be an idempotent of a Jordan algebra Then there exists a splitting Z =
Z (e) @ Z
1
into the eigenspaces
1/2
(e)
@
Zo(e)
Zs(e) := { z
E
21.2.1 Z
: M z = sz
e
}
of
Me
.
PROOF. The polynomials po(e) := (e-l)(2e-l) , pli2(8) := 48(1-8) and pl(e) := e(20-1) have integral The polynomial coefficients and satisfy po + pli2 + p1 = 1 2 p(0) = e(e-1)(20-1) divides the polynomials ps(8) - ps(e) Since and ps(e) pt(f3) in Z [ e l whenever s # t p(Me) = 0 by 21.1, it follows that the operators
. .
P, : = p,(Me) o n Z are idempotent, and satisfy Po + P1,2 + P1 = idz and ysPt = 0 whenever s # t Put Zs(e) := Ps(Z) for s E {OlZ,1} Then 21.2.1 follows, since
.
.
SECTION 2 1
354
the projections
Ps
are pairwise disjoint.
(8-s)ps(e) = p(e) for s (e-1/2)p ( e l = -2 p(e) 1/2 (Me-s-idZ)Ps = 0 ,
E
.
{O,l} Since
and P(M,)
= 0
Further,
,
we get 9.E.D.
The splitting 21.2.1 is called the Peirce splitting of the Jordan algebra 2 with respect to the idempotent e r Z . 21.3 EXAMPLE. Suppose Z is an associative algebra with unit element e Let c E Z he an idempotent. Then c is also an idempotent with respect to the Jordan product 19.2.1, and the corresponding Peirce splitting has the form (c) = (e-c)Zc @ cZ(e-c) and Z,(c) = CZC , Z 1/2 Zo(c) = (e-c)z(e-c) It is convenient to write the elements z E Z as matrices
.
.
U
z = (
vl]
I
v2 where z E CZC , v1 E cZ(e-c) , v2 E (e-c)Ze and w E (e-c)Z(e-c) Then the Peirce spaces have the form
.
u
Z1(C) = Z1/2(C) =
o
o)
: u
E
czc }
,
I(
and
21.4 EXAMPLE. Suppose Z is an associative algebra with unit element e and involution z + z * Let c E Z he a projection, i.e., a self-adjoint idempotent. By 19.3, the self-adjoint part X of Z is a real Jordan algebra with respect to the anti-commutator product and c is an idempotent in X Using the notation of 21.3, the corresponding Peirce splitting is given by
.
.
SYMMETRIC SIEGEL DOMAINS
355
and XO(C) = ZO(C)AX = { ( Oo
'1
w*
:
=
w
E
(e-c)z(e-c)
1
.
In the following we need more general Peirce splittings for Jordan triple systems. 21.5 DEFINITION. An element e of a Jordan triple Z called a tripotent ("triple idempotent") if {ee*e} = e
is
.
21.6 EXAMPLE. Let e be a projection, i.e., a self-adjoint idempotent, of an involutive Jordan algebra 2 Then e is a tripotent for the Jordan triple product 19.10.1.
.
21.7 EXAMPLE. Let Z be an associative *-algebra with Jordan triple product given by 18.4.1. Then e is a tripotent of Z if and only if ee*e = e In the special case 2 = L ( E ) , where E is a Hilbert space over D E {R,C,E} , the tripotents in Z are the so-called partial isometries [47; Problem 98, Cor. 31.
.
21.8 LEMMA. Let operators satisfy
be a Jordan triple. Then the Bergmann Q(B(u,v*)z) = B ( u , v * ) Q, B(v,u*) Z
.
.
Define Q U I Vz := {uz*v} By definition of B ( u , v * ) (cf. 18.12.9), it suffices to show the following identies
PROOF.
Q(QUQvz) = QUQvQ,QvQ, QZ,{UV*Z} Q{uv*Z}
+
= (UOV*)
Q,,Q,Q,~
21.8.1
I
Q, + Q,(VOu*) =
21.8.2
r
4(uoV*)QZ(VD u * ) + QuQvQ, + Q,QvQu 21.8.3
and Q{uv*z} ,QUQVz = ( u 0 v*)QzQVQu + QuQvQ,(vo u*) The fundamental formula 18.8.4 18.2.7,
implies 21.8.1.
.
21.8.4
Further, by
SECTION 2 1
356
2 {zw*{uv*z}}
=
-
Polarizing 18.2.4, Q{uv*z}
+
Hence 21.8.3
.
{uv*{zw*z}} = {z{vu*w}*z} = QZ(vou*)w we get
QQuv,Qzv = QuQvoz
+ ozo~ou
o~,zo~Q~,z
+
follows from
In order to show 21.8.5, observe that polarization of 18.2.2 yields 2 {z{vz*w}*u} = {uv*[zw*z}} + {uw*{zv*z}}
,
i.e., 2 Q Z r U ( v u z * ) = ( u n v*)Q, + Q, oz I
Since
2 ( v o z*)(vo u * ) = QvQz,u
+
.
v
u*
(0,z)o
. by 18.2.7,
21.8.6 we
get 2 (uov*)Qz(vo u * ) = 4 Q
( v o z * ) ( v n u * ) - 2 QuIQZv ( v a u*)
-
Qz,uQvQz,u + 2 QZ,,(Qvz
0
=
QZ,U
ZIU
+
=
QvQ z,u
u*)
-
2 QUrQZV (v 0 u*)
t ( Z D (Qvz)*)QU
Q z , ~ u ~ v-z (Q,VO v*)QU Q ~ , uQv Q ~ , u
+
-
QOzv, Quv
Qz,QuQvz
-
QQZv,OUv
by another application of 21.8.6 and 18.8.2. This proves 21.8.3. In order to show 21.8.4, polarize 18.2.2 to get *QUv, {uv*z} = Q ~ Q v Q ~ , z +
Another polarization yields
Q
~
*vQu , ~ '
21.8.7
357
SYMMETRIC SIEGEL DOMAINS
QQuw, [uv*z} Put
w : = Qvz
.
Q",OvZ
+
QOUv, {uw*z} = Q ~ Q ~ , ~ 'u,zQv,wQu Q ~ , ~ +
S i n c e 21.8.6
and 1 8 . 8 . 1
= Qv(Zn V*) =
.
imply
(VDZ*)Ov
I
i t €01 lows t h a t
--
OuOv,Q z 'u,z V
Qu,z OV,4,Z
+
QU
-
~Quv,Ou,zOvz
= 2 Q Q ( z D v * ) O ~ +, ~2 0,
u v
NOW
18.2.7
*Ouv,Qu,zOvz
(Vu
z*)QvQU
*
implies 2 (zov*)Ou,z
-
QOZV,"
Applying t h i s t o t h e f i r s t t e r m , 18.2.2
,z
+ Qz(vou*) 21.8.6
.
t o t h e s e c o n d t e r m and
t o t h e t h i r d t e r m , we g e t
s i n c e 21.8.6
implies
QQuv,{uv*(QZ~)}= Q u Q v Q Q z ~ , ~ Q ~ , O Z ~ Q v Q ~ +
Q.E.D.
21.9 triple
PROPOSITION.
Z
.
Let
e
b e a t r i p o t e n t of a J o r d a n
Then t h e r e e x i s t s a " P e i r c e " s p l i t t i n g Z =
into the eigenspaces
zl(e)
@
Z
Zs(e) = { z
1/2 E
21.9.1
( e ) @ Zo(e) Z : {ee*z} =
sz }
of
SECTION 21
358
cue* satisfying the "composition rules" {Zr(e)z s (e)*zt (e) }
c z ~ + (e) ~ - ~
21.9.2
and {Zl(e)Zo(e)*Z} = { o ) = {Zo(e)Zl(e)*Z} Here
Zs(e) := { O )
whenever
s
j!
{0,1/2,1}
.
21.9.3
.
PROOF. By 19.7, 2 is a Jordan algebra under the product Since e is clearly an idempotent of this xoy : = {xe*y) Jordan algebra and Me = e o e * , 21.9.1 follows from 21.2. Now consider the Bergmann operators R(u,v*) and define gt := B(e,(l-t)e*) = B((l-t)e,e*) For the polynomials ps introduced in 21.2, 18.12.9 implies for all z E Z s ( e ) and gtz = (po(s)+t*pl/2(s)+t 2p,(s))z
.
.
t
E
R
.
Hence gt = Po
+ t * P1/2
+
2 p1
21.9.4
'
where Ps : 2 + Z s ( e ) denote the Peirce projections, follows that gt is invertible for t # 0 and -1 - gl/t gt By 21.8, we get
It
.
gtIxy*4 =
I (~tx)(~l/tY)*(~tz)l
= {(t2Ux)( l/t)2By)*(t2Yz)}
= t 2 ( a+y -13 1
IXY*ZI
for all t # 0 , if x E Z,(e) , y E Zg(e) and z E Z (el Compar ng with 21.9.4, we get 21.9.2. Y to show 21.9.3, we first polarize 18.8.2 to get
.
2 {xu*y}o u * = x o (QUy)* + y n (Qux)*
In order
21.9.5
and 2 x n {ux*v}* = (Q,U)U
Further, 18.2.7 Q,Z2
V*
+ (Q,V)U
U*
implies
= 2[{ze*e}e*e}
-
{ze*{ee*e)} = z
.
21.9.6
SYMMETRIC SIEGEL DOMAINS
for a l l
z
E
.
Zl(e)
Now l e t
w
Zo(e)
E
359
.
Then 21.9.5
imp1 ies woe* = w
Qew = 0
since
Q
by 21.9.2.
T h e r e f o r e 21.9.6
z 0 w* = Q e ( Q e z ) 0 w* = 2 e
Similarly,
- e
( Q e e ) * = 2 { w e * e } n e*
0
o ( Q ~ w ) *= D
,
implies
{ (Qez)e*w}*
.
( Q e w ) o( Q e z ) * = 0
a n o t h e r a p p l i c a t i o n of 21.9.5
gives
w O z * = w o Q e ( Q e z ) * = 2 { w e * ( Q e z ) } O e*
21.10
EXAMPLE.
over
D
j = 1,2
for Z :=
Suppose
{R,C,B}
E
.
( Q e z ) o (Qew)* =
,
E
w
E
E
and
H1
O.E.D.
H2
are Hilhert spaces
a n d c o n s i d e r t h e H i l b e r t sum
T h e n t h e e l e m e n t s of t h e J B * - t r i p l e
z = ( u
.
c a n be w r i t t e n as m a t r i c e s
L(F F ) 1' 2
where
0
L(E)
.
, v1
L ( H ~ , H ~ )
E
v2 L(H1,E)
V1)
r
, v2
E
and
L(E,H2)
F o r any u n i t a r y o p e r a t o r
e
E
UL(E)
matrix
is a t r i p o t e n t i n
Z
inducing t h e Peirce s p l i t t i n g
I
the
360
SECTION 21
21.11 LEMMA. Let e be a tripotent of a Banach Jordan Then U := Zl(e) is a Ranach Jordan algebra with triple 2 unit element e and involution u* : = {eu*e} Further, every u E U satisfies u * u e * = e o u * E L ( Z )
.
.
.
PROOF. Since U is a Jordan subtriple of 2 with unitary element e , the first assertion follows from 19.13. Now let z E Zs(e) and s E {0,1/2,1} Put v : = u* E U Then 21.9.2 implies {ve*z} E 2 S ( e ) and 18.2.3 gives
.
.
{eu*z) = {e{ev*e}*z} = {{ve*e}e*z} + {{ve*z}e*e}
-
{ve*{ee*z}} = {ve*z}
+ s{ve*z} - s{ve*z}
= {ve*z}
. Q.E.D.
21.12 LEMMA. Let U be a 'hermitian Ranach Jordan triple over K satisfying 20.8.1, with unitary element e Then the closed unital subalgebra IJx of U generated by a selfadjoint element x 8 U is an abelian C*-algebra over K isometrically isomorphic to c ( S , K ) for some compact space
.
s. PROOF. Since x = x* , U, is an abelian Jordan subtriple of 2 containing e Hence 20.32 and 20.30 imply luovl = I {ue*v}I < IuI*Ivl and Iu*l = IuI for all It follows that u,v E U X
.
.
Iu13 = ({uu*u}I = ~ u o ( u * o u 0 and A2n+l Xb Hence 5.36.
Put
AY := ad(Xe)Y : = [Xe,Y]
.
exp(tA)X; = cos(t) Xi + sin(t) AX;
.
For t = n/4 , we get 21.16.1. Since A2XJ = -4X; induction argument shows A2nX; = (-4)"X; and A2n+l Xu = (-4)'AX; for a l l n > 0 Hence
,
an
.
exp(tA)X; = cos(2t)X;
+
sin(2t) AX; 2
.
t = n/4 , we get 21.16.2. Since (ge)* is a Lie algebra homomorphism, the remaining assertions follow. O.E.D.
For
In order to describe the additive gradation 21.15.1 of h in more detail, we introduce the following vector fields:
a az = a-au
Ya : = a-a Yb : =
, (2{eb*z}+b)z a
a +
= 2{eb*v}-
(2{eb*w}+b)=a
,
au ya := -{za*z}-a
az = -{ua*u}-
a -
au
2{ua*v}- a av
-
{va*v}-a , aw
SYMMETRIC SIEGEL DOMAINS
365
Yb : = (2[be*z} - [zb*z})G a - 2 [ ~ b * ~ }a au
=
+ (2{be*u} - {vb*v}
-
2{ub*w})z a
+ (2{be*v} - 2{vb*w} )aw a Here a E iX and b E V and pe(Yb) = p ( Yb 1 = b e
. .
Note that
. pe(Ya) =
21.18 COROLLARY. The additive gradation 21.15.1 following components
h-,
{ ya: a
=
ix)
E
p
e
(Ya) = a
has the
r
h-1/2 = { Y b : b s V } ,
h1
=
hl12
{ Ya: a =
{ Yb
E
: h
iX} , E
v ]
and there exist multiplicative gradations and b- := lh0 = h, A T o ( Z ) = 1 b- 61 -'h-1
-
k O = { X c : C E W } r
h,
= 1 h, @ -1 h,
where
SECTION 2 1
366
21.18.1
By 21.18, we have
Pe(h-l/2) =
Pe(h1l2)
=
v
.
-1 k - ) = X and pe( -1 h ) = W , pe('k-) = {O} pe( Since ge E Aut(M ) , the image D := ge M-) is a symmetric domain We will now show that D is in M+ and h = aut(D) contained in the open subset Z of M+ and is, in fact, a Siege1 domain (of the third kind).
0
.
21.19
Every
PROPOSITION. C :=
{ x
E
x
E
X
satisfies
Mx
.
X
By 21.11 and 18.13, we have 2i Mx = 2i x n e * = i x D e *
Hence
and
X : CRU(Mx) > 0 }
is a topologically regular open convex cone in PROOF.
Ha(U)
E
M,
-
is a hermitian operator on
e n (ix)* U
Since the closed unital subalgebra U x of x is an abelian JB*-triple, 20.32 implies IMxlUxl < IxI-lel = 1x1
.
E
uE(2)
(even on
2)
. .
generated by
IJ
Therefore 14.30 and 20.1 imply
1x1 < (Mxl = suplCRU(Mx)l
G
(MxlUxl
G
1x1
.
21.19.1
Hence the mapping x + Mx is a unital isometric embedding of X into the order-unit Ranach space f f a ( U ) (cf. 14.31). Since C" := H a + ( U ) r\ G I 1 ( U ) is the topologically regular open convex cone associated with Hf.(U) r it follows that c = x n C" is also a topologically regular open convex cone. O.E.D.
SYMMETRIC SIEGEL DOMAINS
367
21.20 LEMMA. If u E U satisfies u+u* > 0 , then u and Mu E L ( U ) are invertible. If le-u( < 1 , then u+u* > 0
.
.
If u + u * > 0 , 21.19 and 14.38 imply MU E G R ( U ) PROOF. -1 Further , v := MU e satisfies [Mu,Mv] = 0 , since this is is small (by 19.26) and u + M-le defines a true if le-ul U holornorphic mapping for u + u * > 0 Therefore u is invertible with inverse v , since u 2ov = M M u = M M u = uoe = u Now suppose u = x+iy v u u v Then 20.31 implies le-xl < 1 Hence satisfies (e-u( < 1
.
.
.
lidU-Mxl = IYe-xl < 1 that CLU(Mx) > 0
by 21.19.1.
.
21.21
LEMMA.
u
If
E.
.
It follows from 14.15.1 Q.E.D.
U
satisfies
(uI
< 1 , %hen
ge(u) = (u+e)o(e-u)-l If
u
E
U
satisfies
U+U*
g-,(u) PROOF.
.
21.21.1
> 0 , then =
(u-e)o(e+u) -1
.
By 21.20, the right-hand side of 21.21.1
21.21.2 and 21.21.2
is well-defined. By 19.24, the closed unital subalgebra of U generated by u is abelian. Consider the Ranach algebra
associated with
U,
via 3.5.1.
U,
Then the matrix
satisfies the associativity condition 3.6.1 + (Xe)# = Xe E T(UU) , 5.22 implies
on
U,
.
Since
SECTION 2 1
368
+
exp(tXe)(u) = exp(tXe 1 # (u) 1/2 + sin t(e n e*) 1/2
(cos t(en e*)'I2u
=
( e n e*)
1/2 (-e* sin t(e0 e*) (e o e*) 1/2
+
cos t(eo e*)1/2)-1
-'. For It1 < n/4 , put ut cos(t)e - sin(t)u . If lul < 1 , 21.20 implies that ut cos(t)(e-tan(t)u) and M(ut) are invertible since cos(t) # 0 and Itan(t)I < 1 . For 0 ( c o s (t ) e-sin (t
(cos( t ) u+s in (t e
=
u
:=
=
t = n / 4 , we get 21.21.1. If u+u* > 0 and t < 0 , 21.20 implies that ut = cos(t)(e+tan(-t)u) and M(ut) are
invertible. 21.22
For
t = -n/4
COROLLARY.
-
we get 21.21.2.
Q.E.D.
We have
c
=
c
= exp(X) :=
x2
,
:=
{ x2
: x
E
x }
and
{ x
E
:
x invertible }
Further, ge maps the bounded symmetric domain biholomorphically onto the symmetric tube domain DC : = C fB iX in U
. R n U
.
PROOF. By 21.21, D := ge(BnU) is a domain in U and Aut(D) acts (locally) transitively on D since R n U is By 21.17, aut(D) the open unit ball of the JB*-triple U contains the vector fields Ya and
.
u
a = au
iee*u}-a au
.
iX Since e = g , ( 0 ) E D , 16.19 implies D = D = Q+iX for some convex open cone Q in X By n 21.21, we have = DnX = ge(BnX) For every x E X , the closed unital subalgebra Ux of U generated by x is an f o r all
a
E
.
.
SYMMETRIC SIEGEL DOMAINS
369
abelian C*-algebra by 21.12. Further, 21.21 implies Since the Cayley transformation in ge(BnUx) = D n U x c(S,C) with respect to lS maps the open unit ball onto the = (XnUxI2 and right half-plane, it follows that QnUx = exp(XnUx) Since x is arbitrary and a + ? i C Q ,
.
it follows that imp1 ies
20.1 implies i.e., C = Q
7
. -
X2
Q =
cc
.
and
-
and
Q
x
Every
E
.
Q = exp(X)
cC
Since [17; 5.41
.
rjy convexity, 52 is invertible, since
C
cc ,
.
exp(y1-l = exp(-y) for all y E x Conversely, let x be invertible. Then Px(Ux) C Ux and, by 20.2, we have
> 0
E.tu(Px) Px(Ux) =
Ux
.
.
invertible in 21.23
A s in the proof of 2.12,
-1 x-l = Px ( x )
Hence
.
U,
Therefore
For u (u+wI = max((ul,lwl)
PROOF.
LEMMA.
By 21.9.3, ( u+w)
[J
E
and
E
E
UxnC
w
E
W
.
,
C
it follows that and x is
UX
x
E
.
O.E.D.
we have
the Jordan triple (odd) powers satisfy (2n+l)
(2n+l) =
+
(2n+l)
Since the Peirce projections are continuous, it follows from 20.8.1 that Iu+wI < 1 if and only if ( u l < 1 and J W ]
1 , the operator a*idZ _+ Oe+w By 21.9, is bijective and hence invertible [ 1 5 ; 48.11. Qe+, leaves the Peirce components Z,(e) invariant. a(idv 2 Y(w/a)) = a*idV f_ Oe+w is invertible on V
.
Hence O.E.D.
21.25 THEOREM. unit ball B
.
Suppose Z is a complex JB*-triple with open Let 2 = UxVxW be the Peirce splitting with
respect to a tripotent transformation
e
E
Z
.
Then the Cayley
4, := exp($(e+{ze*z})az)a maps
R
onto the symmetric Siege1 domain
D : = {(u,v,w)
2 :
E
Re(2u - @,(v,v))E
(w(< 1
C
} ,
C = { x E X : ZEU(Mx) > 0 } is a topologically regular open convex cone, @(b,v) := 2 {eb*v} defines a C-positive mapping and Y(b,w) := 2 {eb*w} . where
PROOF. cone in
By 21.18, X
.
is a topologically regular open convex DC := C @ iX C U denote he associated
C
Let
tube domain. The sesqui-linear mapping satisfies 16.0.1, since 18.2.3 implies {e{eb*v}*e}
=
since 18.2.3 and 21.9.3 4-’@(Y(b,w),v)
Y :
@,(b,v)
h,
:=
b
V
= {e{eb*w}*v}
-
VxW
+
V
satisfies
=
{be*{ew*v}} = 2-’{bw*v}
.
By 21.24, the R-linear operator is invertible whenever ( w ( < 1 Hence
and
v
.
is well-defined via 16.0.3.
h-, fB h-l,2
u
imply that
{ {be*e}w*v} + { {be*v}w*e} is symmetric in idv + Y(w) on
: vxv +
2 {{be*e}v*e} - {be* ev*e}} = {bv*e}
Further, the sesqui-linear mapping 16.0.2,
@
.
By 16.1,
D
Now put
is invariant under the
.
SYMMETRIC SIEGEL DOMAINS
371
H< : = < exp(h of all affine transformations = exp(Ya+b) €or a E iX and b E V Now define
group
.
ga,b
D' := D x(BnW) = Dn(UxW) C
Then
D
=
H OD'
C Aut(2)
is a real Banach Lie group with Lie algebra k+ I acting analytically and (locally) transitively on every connected + component S + of S For every e E S , the group
.
acts transitively on
S+
.
PROOF. The real Ranach Lie algebra k+ acts analytically and topologically faithfully on Z Hence 7.4 implies that K+ is a connected real Ranach Lie group with Lie algebra k+ I acting analytically on Z As a subgroup of Aut(2) , K+ acts also analytically on every connected component S+ of + can be S By 22.18.1, the tangent space Te(S+) at e E S By 22.19, the evaluation mapping identi€ied with Y : k+ + Y of the analytic action p of k+ on S+ pe induced by the action r of K+ on S + has the form p,(6+~ = y for all 6 E '/z+ and y E Y Since Y = exp(2My) by 22.8, it follows from 4.1 that the 'exp ( y 1 orbit K(e) of the group K := < P : Y ~ Y > C K + is exp(y) open in S+ Since the norm of Z induces a compatible metric on S + invariant under K+ C t J k ( Z ) , 12.18 implies + + K(e) = K (el = S Hence K+ acts locally transitively on S+ Q.E.D.
.
.
.
.
.
.
.
.
22.21 THEOREM. Let Z be a JB*-triple over K and consider the closed real submanifold S of all unitary elements in 2 Then every connected component S+ of S is a symmetric normed real Banach manifold under the K+-invariant tangent norm induced by the norm of Z The symmetry j(u) : = {eu*e} of S+ about e induces the multiplicative gradation 22.19.1 of k+
.
.
.
PROOF.' Let b denote the tangent norm o n S+ induced by + 1.1 Then p : k+ + aut(S , b ) is an analytic, topologically faithful and locally transitive action. The differential j* of the real-bianalytic automorphism j of
.
JORDAN AUTOMORPHISM GROUPS
382
.
satisfies j * l ' k + = o-id for u = _+1 Hence + j, E GE(k ) Since j(S) = S and j(e) = e , it follows -Ithat j(S+) = S + Since Y = Te(S ) and Te(j)lY = -id , it follows that j is a symmetry of S+ about e By 17.7, S+ is a symmetric normed K+-manifold. Q.E.D. 2
.
.
.
.
We now specialize to the case K = C Suppose in the following that 2 is a complex JB*-triple containing a unitary element e Let X := { x E 2 : x* = x } be the self-adjoint part of the corresponding involutive Jordan algebra. By 7.15, the automorphism group
.
of the real Ranach Jordan algebra X is a real Banach Lie group in the operator norm topology whose Lie algebra can be identified with the closed subalgebra aut(x) := { 6
ga(X) : ~ ( x o y )= ~ x o y+ XOSY }
E
.
of all continuous derivations of X Since 2 = Xc , we can embed Aut(X) c Ga(2) and aut(X) c ga(2) by complexification. It is clear that X can also be regarded as a real JB*-triple. Let Aut(X,X) and aut(X,X) denote the corresponding automorphism group and derivation algebra, respectively. By 22.15, Aut(X,X) is a closed subgroup of GL(X) and a real Banach Lie group in the operator norm topology. 22.22 EXAMPLE. For a Hilbert space E over D E {R,C,E} put X := H E ( E ) := { x E L ( E ) : x* = x } By 20.29, the
.
,
complexification 2 := Xc is a JC*-triple. Then gax := axa -1 defines an analytic group homomorphism UE(E) 3 a
+
ga
E
Aut(X)
whose differential is the continuous Lie algebra homomorphism
defined by
Sax := ax-xa
.
SECTION 2 2
384
22.23
We have
PROPOSITION.
Aut(x) = { g
E
Aut(X,X)
:
ge = e } = { g
E
Aut(2) : ge = e ]
and
PROOF. Every g E Aut(X) belongs to Aut(X,X) and to Aut(2) (by complexification). Conversely, suppose g E Aut(X,X) satisfies ge = e Then gX = X and, by
.
22.15.2, 22.23.1
.
for all x E X Applying 2 2 . 2 3 . 1 to e E X and polarizing, we get g E Aut(X) Now suppose g E Aut(2) satisfies ge = e Then g(z0w) = g{ze*w} = {(gz)(ge)*(gw)} = gzogw and a similar argument shows g ( z * ) = (gz)* f o r all z,w E 2 Hence g X = X and g E Aut(X) Every 6 E aut(X) belongs to aut(X,X) and to aut(2) (by complexification). Conversely, suppose 6 E aut(X,X) satisfies 6e = 0 Then 6X C X and
.
.
.
.
.
6*e = 6*{ee*e} = 2{(6*e)e*e} Hence
6*e = 0
and therefore
6
E
aut(X)
Now suppose 6 E aut(2) satisfies 6{ze*w} = {(Gz)e*w} + {z(Ge)*w} + and 6(z*) = G{ez*e} = 2{(6e)z*e} + z,w E 2 Hence 6X C X and 6 E
.
- {e(se)*e)
and
@
aut(x)
.
since
6e = 0. Then ~ ( Z O W )= {ze*(Gw)} = GZOW + zogw {e(Gz)*e} = (6z)* for all aut(X) Q.E.D.
22.24 PROPOSITION. For MX := { Mx : x multiplicative gradations aut(X,X) = MX
= 2 g*e
.
E
X }
,
there exist
22.24.1
385
JORDAN AUTOMORPHISM GROUPS
aut(2) = i MX fB aut(X) PROOF. For every x E X 22.6 and 21.11 implies
,
.
22.24.2
we have
2i Mx = (ix)n e* - e
Mx = x n e* 6 aut(X,X)
(ix)*
.
aut(Z)
L
.
Now suppose 6 E aut(X,X) Then x := 6e 11 : = 6 - M E aut(X,X) satisfies r\e = 0 X t-l E aut(X) by 22.23. This shows 22.24.1. 6 E aut(Z) Then
E
.
.
22.24.3
X
and Hence Now suppose
+ {e(de)*e} = 2 6e + (6e)*
6e = 6{ee*et = 2{(6e)e*e}
.
It follows that 6e = ix & iX Since q := 6 satisfies r,e = 0 , 22.23 implies L aut(X)
-
.
.
Mix E aut(Z) This shows
22.24.2. 22.25
by
Q.E.D.
PROPOSITION.
The real-linear span
int(X) : = R
< [Mx,M
of all inner derivations of
Y X
]
>
: x,y & X
is an ideal in
int(X) = { 6 e int(2) : 6e = 0 } = { 6
E
aut(X)
and
int(X,X) : 6e = 0 }
.
Further, there exist multiplicative gradations int(X,X) = MX fB int(X) and int(2) = iMX fB int(X)
.
PROOF. By 19.6, int(X) c aut(X) and int(X) is an ideal in aut(X) It is clear that M X and int(X) are contained in int(X,X) Similarly, 22.24.3 implies that iMX and hence int(X) belong to int(2) Since 19.10.1 implies
.
.
.
zn w* = [M(z),M(w*)] + M(zow*) f o r all
z,w
E 2
,
every inner derivation
SECTION 2 2
386
I:
& =
xju y? E int(x,x) 7 j of the "Jordan pair" (X,X) can be written as
1 [M(X-),M(yj)I 3
6 =
j where
u = 6e
E
.
X
of the JR*-triple
2
Similarly, every inner derivation
can be written as
22.26 COROLLARY. We have exp(int(2)) C Int(2)
.
PROOF,
Let
x,y
E
X
+ MU
.
exp(int(X,X)) C Int(X,X)
Then
exp(x)
and
is invertible and
exp(2Mx) = P exp(x) E Int(X,X) by 22.8.
Similarly,
exp(ix)
exp(2iMX ) = Pexp( ix)
is unitary and 22.8 implies E
Int(2)
.
Since 6.7 and 22.8 imply
It will now be shown that the open convex cone
JORDAN AUTOMORPHISM GROUPS
C := { x
X
E
:
>
CLX(Mx)
387
}
0
associated with X (cf. 21.19) is a symmetric real Ranach manifold which is "dual" to the symmetric real Banach manifold of unitary elements in Z considered in 22.21. The automorphism group
of C is a closed subgroup of GI1(X) , since C is the interior of X, = C By 7.8, Aut(C) is a real Banach Lie group whose Lie algebra can be identified with the closed subalgebra
.
of all continuous derivations of
22.27
C
.
We have
PROPOSITION.
and
.
PROOF. Since C = exp(X) by 21.22, Aut(X)C Aut(C) Conversely, suppose g E Aut(C) satisfies ge = e By complexification, g is a linear automorphism of the right
.
half -plane DC : = C $ i X C Z = X C
.
associated with C Endow h := aut(DC) with its canonical additive gradation h = h-l @ @ (21.18). Since g is linear, the differential g, E Aut(h) leaves
hl
=
a i { {zx*z}-az
4
-
x
E
x
}
invariant. It follows that g{zx*z} = { z y * z } for all -1 z E Z , where y := g{(g e)x*(g- 'e)} = gx Hence g E Aut(X) If 6 E aut(C) satisfies 6e = 0 , then the
.
.
SECTION 22
388
transformations gt : = exp(t6) E Aut(C) satisfy gte = e for all t E R (13.31). Hence gt E Aut(X) for all t , showing O.E.D. that 6 E aut(X)
.
22.28 LEMMA. For every x E X , we have Px E Aut(C) if x is invertible.
PxC C
and
PROOF. For any u E C , there exists v E C such that v2 = u (21.22). Hence 19.9.1 and 21.22 imply
.
Pv Pxu = Pv Px Pve = P(PV x)e = ( ~ , x ) ~E F -1 By 21.22, v = exp(y) for some y E X Since 21.18.1 implies y o e* E aut(DC) it follows from 22.8 that
.
Pi1 = P(exp y)
=
exp(2 y o e * )
E
Aut(DC)
.
leaves C = Dc 17 X invariant. Hence Pxu E PilF = follows that PxC C , If x is invertible, then Px E GI1(X) and hence Px E Aut(C)
.
22.29 COROLLARY. Put Pc : = { Pu : u E C } exists a semi-direct product decomposition
.
It
O.E.D.
Then there
Aut(C) = PC*Aut(X) with respect to the action Aut(X) on Pc
.
g*Pu := g Pu g-l = P gu
of
PROOF. Given g E Aut(C) , write ge = u2 where u E C -1 (u2 ) = e , i.e., Then h := Pilg E Aut(C) satisfies he = Pu h E Aut(X) Hence g = PUh , and this decomposition is unique, since u2 = v2 for u,v E c implies u = v
.
.
Q.E.D.
22.30 COROLLARY. satisfies
The real Banach Lie algebra
aUt(C) = aut(X,X) = MX PROOF.
By 21.18.1,
MX C aut(C)
CB
.
aut(X)
.
Now suppose
aut(C)
JORDAN AUTOMORPHISM GROUPS
.
x := 6e
aut(C) Then satisfies rle = 0 6
E
22.31
.
COROLLARY.
Hence
X
E
rl
E
and aut(X)
:= 6
rl
389
-
Mx
E
aut(C)
by 22.27.
Q.E.D.
The group
is an open (and closed) subgroup of Aut(X,X) and is therefore a real Banach Lie group in the operator norm topology acting analytically on C
.
PROOF.
Suppose first that g E Aut(X,X) satisfies x := ge E F Then 22.16 and 21.22 imply x E C and hence x-l E C For each u E C there exists v E C with u = vL (21.22). Therefore 22.15.2 and 22.28 imply
.
.
gu = gp e = P
YV
V
P-1ge = P (x-l) x 4"
E
c
.
.
Hence g E Aut(C) Since Pc c Aut(X,X) by 22.17, it follows from 22.29 that Aut(C) is an open subgroup of Aut(X,X) Q.E.D.
.
22.32
PROPOSITION. Int(C) :=
The group
< Px
: x
E
invertible > = Int(X,X)
X
of all inner automorphisms of C is a normal subgroup of Aut(C) and Int(X) := Int(C) n Aut(X) is a normal subgroup of Aut(X) such that there is a semi-direct product decomposition Int(C) = Pc
Int(X)
.
PROOF. By 22.28, Int(C) c Aut(C) and by 22.17, Int(C) is a normal subgroup of Aut(X,X) Since Aut(C) c Aut(X,X) by 22.31, the assertion follows. Q.E.D.
.
22.33.
LEMMA.
We have
exp(int(X))c
int(C) := int(X,X) = satisfies
exp(int(C)) c Int(C)
3
.
@
Int(X) aut(x)
and
SECTION 2 2
390
PROOF.
Apply 22.8.
Q.E.D.
Since C = exp(X) and exp(x)-I = inversion mapping j(x) := x-l defines automorphism of the open subset C C X Since 21.12 implies Te(j) = -idx , j about e
.
22.34 PROOF,
LEMMA.
For
y
E
C
, we
jP j = P-1 Y Y '
have
By 19.9.1, we have for all
exp(-x) , the an analytic having period 2 , is a symmetry of C
z
E
C
jP z = P(P z ) -1 (Pyz) = (P P P )-lP z Y Y Y Z Y Y Y
Y
Y
Y
Y
Z
Q.E.D.
22.35 PROPOSITION. The real Banach Lie algebra k- := aut(C) has the multiplicative gradation
k- =
k-
@
-'k-
22.35.1
where
'k-
=
aut(x) = { 6
E
k- : 6e = 0 }
and -1 k- = { M ~ x : PROOF.
E
x }
=
{
U D ~ *+
eou* : u
Apply 22.30.
z }
.
Q.E.D.
22.36 PROPOSITION. The real Banach Lie group analytically and (locally) transitively on C < Pexp(x) : x acts transitively on
E
C
E
x > c
c Aut(C)
.
pe * k- + X of the canonical action p of k- on C at e has the form pe(6+Mx) = x for all 6 E aut(x) and x E X Since P (el = exp(2x) , 21.22 implies that exp(x) PROOF.
By 22.35, the evaluation mapping
.
JORDAN AUTOMORPHISM GROUPS
391
.
It follows < 'exp(x) : x E X > acts transitively on C that < exp( F ) > and Aut(C) act locally transitively on
c.
Q.E.D.
22.37 THEOREM. Suppose Z is a complex JB*-triple with Let C be the open convex cone unitary element e E 2 Then associated with the self-adjoint part X of 2
.
.
C = Aut(C)/Aut(X) is a symmetric normed real Ranach manifold under the Aut(C)-invariant tangent norm induced by the norm on symmetry j(x) : = x -1 of C about e induces the multiplicative gradation 22.35.1 of aut(C)
X
.
The
.
PROOF. Since Aut(C) acts (locally) transitively on C and the isotropy subgroup Aut(X)C Aut(Z) = U k ( 2 ) at e E C leaves the norm I * I on X = Te(C) invariant, there exists a unique tangent norm b on C which is invariant under on Te(C) The Aut(C) and coincides with 1 . 1 real-analytic automorphism j of C satisfies jgj = g for all g E Aut(X) and jPxj = P(x -1 1 by 22.34. Hence 22.29 implies j Aut(C)j = Aut(C) The differential j* of j satisfies j*('(2- = a-id for u = 21 By 17.7, C is a symmetric normed Aut(C)-manifold. 9.E.D.
.
.
.
22.38 EXAMPLE. For a Hilhert space E over D E {R,C,H} let C be the interior of the closed convex cone
,
.
of all positive D-linear operators on E Then gax := axa* defines a real-analytic group homomorphism Gk(E) 3 a
+
ga e Aut(C)
22.38.1
whose differential is the continuous Lie algebra homomorphism g k ( E ) 3 a + €ia
defined by
&ax := ax+xa*
E
aut(C)
.
By 15.16.1,
22.38.2 every element of
SECTION 22
392
.
C = X+n GR(E) has the form aa* , where a E GL(E) It follows that the analytic action of GR(E) on C defined by 22.38.1 is transitive and, by 22.38.2, also locally transitive. By 8.21, there exists a real-analytic isomorphism
since UR(E) is clearly the isotropy subgroup at e := idE E C In case E = Dn is finite-dimensional, we get C GRn(D)/URn(D)
.
.
NOTES. Jordan triple systems are a special case of the so-called "Jordan pairs" which form the most satisfactory category from an algebraic point of view. For a systematic account of the theory of Jordan pairs, cf. C1021. The concept of Jordan pair leads to a natural definition of the group Aut(Z,Z) , which is called the "structure group" by Jordan algebraists (cf. C25,1251 1 . The group Aut(Z,Z) and the associated Lie algebra aut(Z,Z) play a central role in the "Freudenthal-Koecher-Tits" construction of exceptional Lie algebras (cf. C1041). The close connection between exceptional Lie algebras and the exceptional Jordan algebra is another strong motivation for the study of Jordan algebras. There exists a similar relationship between the exceptional Jordan algebra and the exceptional symmetric spaces. Important classes of Jordan algebras and Jordan triple systems have only inner derivations. In the finite dimensional case, this is true for semi-simple algebras and triple systems ([25; Ch. IX, Satz 3.1lIC103; Corollary 8.91). In the infinite dimensional case, a typical condition is the existence of a Banach space predual C139-1421. The prototype of these results is the Kadison-Sakai Theorem stating that von Neumann algebras have only inner (bounded) derivations C 118; Theorem 4.1.6 1.
CLASSICAL BANACH MANIFOLDS
23.
393
CLASSICAL BANACH MANIFOLDS
The Grassmann manifold MR(L) of all split subspaces of a Hilbert space L has been frequently used to illustrate the general theory of Ranach manifolds and their Lie transformation groups. In this section, we consider more general symmetric Ranach manifolds which can be realized as submanifolds of MR(L) and will be called "classical" Banach manifolds. In finite dimensions, every (irreducible, noneuclidean) symmetric complex manifold is either a classical manifold or belongs to two exceptional types of dimension 16 or 27. Although this is not true for infinite dimensional manifolds, the classical Ranach manifolds still provide the principal class of examples of symmetric Ranach manifolds. The classical Banach manifolds can be realized as quotient manifolds under transitive actions of the so-called classical groups. Suppose L is a Hilbert space over D E {R,C,E} Let Gi(L) denote the group of all invertible D-linear operators on L By 6.9, the identity component GK of GL(L) is a Banach Lie group (over the center K of D ) in the operator norm topology whose Lie algebra can be identified with gK : = g i ( L ) By 1 5 . 2 1 , the identity component G+ of the group
.
.
.
of all unitary &linear operators on L is a real Ranach Lie subgroup of GK whose Lie algebra can be identified with the closed real subalgebra g+ :=uA(L) = {
x
E
gi(L) :
x* + x
= 0
}
.
Let r denote the collineation action of Gi(L) on the Grassmann manifold Mi(L) over L (cf. 6.20). Then every connected component M+ of M i ( L ) is invariant under GK + can be written as Any point o E M
where
.
SECTION 2 3
394
L
is an orthogonal splitting of 6il
j
:= { H
.
is an open subset of
For any
j
E
L(L)
,
inf (h hsH h (=1 Now def ne
MI1(L) :
E
.
MI1(L) -idF
j :=
O
idE
)
E
UP.(L)
.
Then 7.14 implies that the group
is a real Banach Lie group in the operator norm topology whose Lie algebra can be identified with the closed real subalgebra
g
-
=
uP.(F,E) := { X
E
gll(L) : X*j+jX
=
0 }
.
Since every x E ga(L) satisfies - is a split X = (X+jX*j)/2 + (X-jX*j)/2 , it follows that g real subspace of gK Hence the identity component G- of UR(F,E) is a real Ranach Lie subgroup of GK with Lie In case F = Dq and E = dp are finitealgebra gdimensional, we write UI1(F,E) = UI1 (D) and 4rP u&(F,E) = uI1 (D) Since qrp
.
.
.
(ghl jgh) = (h(g*jgh) = (hljh) is invariant under the for all g E UL(F,E) , 6ilj collineation action of UL(F,E) It follows that the connected component M- of 6il containing o is invariant j under GPut 2 := L(E,F) Then there exists an additive K K gradation g K = g-l @ g o @ :g , where
.
.
and
.
CLASSICAL BANACH MANIFOLDS
395
Further, there exist multiplicative gradations g K = l g K @ -'gK for "curvature" K = f , where 1
g
K
(E
{
=
d0 l
: a* = -a
,
= -d
A*
} = u!L(F)xu!L(E)
and
and
,
K = C
In case
gK 8 C =
we have
C
g
'gK
.
0 C = go
,
23.1
PROPOSITION. M+ is an open submanifold of M ! L ( L ) GK acts analytically and (locally) transitively on M+ For 2 = L(E,F) , there exists a chart (P,p,Z) of MS about o such that p(ro(exp X I ) = X
X
for all
E
.
K g-l = 2
p*(pX) = X
.
23.1.1
The differential
of
p
r
satisfies
a
#
and
23.1.2
= (az+b-zcz-zd)-
az
for all
x
=
is an open subset of
M-
p(M
-)
=
K b d I E g
(" c P
B := { z
23.1.3
and E
2 :
z*z < idE }
is the open unit ball of the JB*-triple product
2 {uv*w} = uv*w
+ wv*u
o
.
Then
P := PF
with triple
23.1.4
.
Consider the chart
,
.
For K = + - , MK is a symmetric normed symmetry r(j) about o PROOF.
2
(PF,pE,F,2)
is contained in
M'
GK
-manifold with
of and
ME(L)
about
p := PE,F
396
SECTION 2 3
satisfies 23.1.1 and 23.1.2. In particular, the evaluation mapping Po : g + To(M+) satisfies To(p)(poX) = b for all + is b E 2 , where X is given by 23.1.3. Since o E M arbitrary, it follows that Gu and G+ act (locally) transitively on M+ ~y 20.10, 2 is a m*-triple over K under the triple product 23.1.4 and the operator norm 1 . 1 Since 1z12 = 1z*z1 by 15.1.1 and Hll(E) is an order-unit Banach space by 15.17, it follows that B is the open unit ball of 2 , Since p(Q.nP) = B and p ( P ) = 2 , we get 7 M- = Q . n P The homomorphism X + XI maps g K onto the 3 - ,:h fB h: fB hy , where binary Banach Lie algebra hK :=
.
.
.
g#
h-K 1 = { & :
-
~ E Z } ,
K = { (az-zd)-a : a ho az
,
L(F)
E
d
L(E
E
and
h,K
=
{ Zc&
: C
E
The real Banach Lie algebras 1 hK = { (az-zd)-a
az
L(F,E) } = { { z b * z
hK := g#" : a* = -a
=
a
: b
'hK fB -1 h K
, d*
=
}
.
E
2
}
.
satisfy
-d } c aut(2)
and
a : b -1 hK = { (b+K{Zb*Z})G
E
2
By 20.13 and 18.20, the group < exp( h-) > acts (locally) transitively on B Hence G- acts (locally) transitively on MNow assume
.
.
a
4 = (c
E
GK
.
satisfies r(g,o) = o Then g # ( O ) = 0 = bd-' d E GL(E) and b = 0 Since g E G K , we get therefore c = 0 It follows that
.
.
g =
(t
:)
E
UL(F)xU%(E)
.
Hence d*c = 0
.
By 12.31, the operator norm on Z = T0 (p)T0 (MK) induces a GK -invariant compatible tangent norm on MK , Since
and
CLASSICAL BANACH MANIFOLDS
397
j E UQ(F)xUQ(E) satisfies j # ( O ) = 0 and j ; ( O ) = -id , it follows that r ( j ) is a symmetry of M K about o There is a commuting diagram
.
.
where J U gK : = u = i d Applying 17.7 to the analytic action p# of hK on MK defined by p # ( X # ) := p X , it follows that M K is a symmetric normed GK -manifold. Q.E.D. The manifolds M+ (of "compact type") and M(of "non-compact type") are called classical Ranach manifolds of type I. Similarly, Z is called a classical JB*-triple of type I. In order to describe the Cayley transformation for classical manifolds of type I, we may assume that dim(F) > dim(E) (Otherwise, consider the biholomorphic In this case, there are mapping E + El of M L ( L ) ) . orthogonal splittings
.
and the base point of 'M 0
Putting
and
gK
U :=
L(E)
is
0 := ( 0 )
.
and
E V := L(E,H)
z
L(E,F) =
:=
,
we have
E
U
( Uv )
has the components
4-1 =
i(B
: bl
,
b2
E
V} I
398
SECTION 2 3
and 0
: c1
H2
9;
Further,
x#
0
u
6
= (allu+a12v+bl-uclu-uc
2
+
X =
hK-1 hoR
= =
v-udl- a au
la’’ ::) a
gK
a22
21
c2
\c1
hK
.
L(H,E))
E
(a21u+a 2 2 ~ + b 2 - ~ ~ 1 ~ - V C 2a~ - V d ) z
for a l l
and
c2
t
d l
h a s t h e components
{
biz+ a
b 2 = a :
bl
U , b2
E
E
,
V }
{ (allu+a12v-ud)- a + (a21u+a22v-vd)-a : au av all,d
E
U
a21
I
E
V
I
a12
, a22
L(H,E)
E
E
L(H) }
and h,K = {
(UCIU+UC
c1 Let
e
E
2
v ) L au
u ,
c2
+
( v c u+vc v)-a : 1 2 av
E
L(H,E)
be a u n i t a r y o p e r a t o r i n
I:(
U = L(E)
.
The m a t r i x
; 0
xe
.
Then
E
is a t r i p o t e n t w i t h P e i r c e 1-space V
.
}
:= (;e*
induces t h e Cayley v e c t o r f i e l d
U
a n d P e i r c e p1 p a c e
a)
p(xe)
E
E
g
a u t M+)
satisfying 23.1.6
CLASSICAL BANACH MANIFOLDS
399
Since :X = -Xe I an induction argument shows X2n+2 = (-l)n :X and x2*+l = ( - 1 1 xe ~ for all e follows that exp(tXe) = id
+ (l-cos(t))Xi + sin(t)Xe
n >
o
.
It
.
Therefore the Cayley transformation r (ge ) E Aut (M') associated with the tripotent e is induced by the matrix 0
ge = exp(+xe)
=
2 -e *
satisfying
Since
it follows that (ge)# B = { z E Z : u*u+v*v Siegel domain D := { ():
The symmetry about
In the special case D = { )(:
maps the open unit ball < idE } of Z bianalytically onto the
2 : u*e+e*u-v*v
E
e
E
D
2 :
.
is given by
e := idE E
> o }
I
we get the Siegel domain
U*+u-v*V > 0 }
described in 16.8, with symmetry
In order to describe the other types of classical Banach manifolds? fix u = 21 and let Q be a o-conjugation on a
400
SECTION 2 3
Hilbert space L over D , i.e., an anti-linear bijection of L satisfying Q2 = amidL and (QhlQk) = (k(h) for all h,k E L , Then
x
+
xT
23.2.1
QX*Q-~
:=
L (L) having period
defines a K-linear anti-automorphism of 2
.
By 7.14, the
"a-orthogonal" group
Oka(L) := { g
T Gk L ) : g g = i d L }
E
23.2.2
= { g ~ G k L) : g * m = 8 } is a Banach Lie subgroup of Gk(L) whose Lie algebra can be identified with the closed subalgebra
x
E
{ X
E
gK = okU(L) := { =
gk(L) :
xT+x
= 0
}
gk(L) : X*Q+QX = 0 }
23.2.3
.
It follows that the identity component GK of OLa(L) is a Since Banach Lie group over K with Lie algebra gK (XT)* = (X*)T , the identity component G+ of Ok'(L) n Uk(L) is a real Banach Lie subgroup of GK with The closed subset Lie algebra 4' := oka(L) n u k ( L ) N := { H E Mk(L) : H1 = QH } of ML(L) is invariant under the collineation action r of Oka(L) , since gQg* = Q for every g E Ok'(L) , Hence every connected component M+ of N is invariant under GK Any point o E M+ can be written
.
.
.
as
where
is an orthogonal splitting of
and
q(h) =
L
is a conjugation of
such that
E
.
Put
401
CLASSICAL BANACH MANIFOLDS
Since jT = -j implies (jx*j)T = j(xT)*j , it follows that A UL(E,E) is a real the identity component G- of OL'(L) Banach Lie subgroup of GK with Lie algebra g : = uLa(L) n & ( E , E ) The connected component M- of the open subset N n n of N containing o is invariant j under G-1 E L(E) and put For z E L ( E ) , define zt : = q z * q Since z := { z E L ( E ) : Zt + uz = 0 ]
.
.
.
dt XT=(
ubt a t)
uc
for all
x
3
(" C
=
L(L)
E
K
gK = g-l
there exists an additive gradation
9-1 = { ( :
:
gKl = { ( ;
; ) : c " z } .
)
:
~
E
Z
}
, K fB go fB :g
,
and
Further, there exist multiplicative gradations for "curvature" K = _+ , where 4 = 9 f~ - l g K
and
, where
402
SECTION 2 3
23.2 PROPOSITION, M+ is a closed submanifold of M L ( L ) and G K acts analytically and (locally) transitively on M+ For 2 : = { z E L(E) : zt + uz = 0 } , there exists a chart (P,p,Z) of M+ about o such that
.
p(ro(exp X I ) = for all
X
E
g-,K
m
2
.
x
23.2.4
The differential
of
p
r
satisfies
p*(pX) = X # = (az+b-zcz+zat )-aaz
23.2.5
for all a
x=(
-
p(M ) = B :=
t) -a and
c
is an open subset of
M-
b
P
{ z
E
2 :
K
23.2.6
9
z*z < idE }
is the open unit ball of the JB*-triple 2 , with triple product given by 23.1.4. For K = +_ I MK is a symmetric normed G K -manifold with symmetry r(j) about o
.
PROOF.
Put E F := ( o )
and consider the chart o Then
.
(PFIpE,F,L(E))
of
ML(L)
about
b
Since
2
P:=PFnM+={(-)E:ba2}. idE is a split subspace of L(E) , it follows that
M+
.
is a submanifold of ML(L) The chart p := PE,FIP Of M+ about o satisfies 23.2.4 and 23.2.5. This implies To(p)(poX) = h for all b E 2 , where X is given by 23.2.6. Since o E M+ is arbitrary, it follows that GK and G+ act (locally) transitively on M+ By 20.10, 2 is a JB*-triple over K with open unit ball B Since p(P) = 2 and p(Q.nP) = B it follows that M- = ~ . ~Thep 3 J homomorphism X + X maps k onto the binary Banach L i e algebra hK = gK = #h K @ h! @ hy , where # -1
.
.
.
CLASSICAL BANACH MANIFOLDS R
h-l
{
=
&
: b E 2
}
$=
(az+zat )-a : a az
hlK = {
ZCZ-
403
I
1
L(E)
E
and
a az
The real Banach Lie algebras 1
hK
}
: c E 2
hK :=
{ (az+zat 1-aaz : a*
=
a : b { {zb*z}G
=
=
g;
'hK
=
@
E
-'hK
2
}
.
satisfy
-a } c aut(2)
and -1
h
K
=
a
{ (b+K{zb*z})z : b
}
2
E
By 20.13 and 18.20, the group < exp( h-) > acts (locally) transitively on B Hence G- acts (locally) transitively on MEvery g E GK satisfying r(g,o) = o has the form a 0
.
.
By 12.31, the operator norm on 2 = To - p)To(MK) induces a GK -invariant compatible tangent norm cn MK Further, r(j) is a symmetry of MK about o The commuting diagram 23.1.5 and 17.7, applied to the analyt c action p # ( X # ) : = p X of hK on M K , show that MK is a symmetric normed G K -manifold. O.E.D.
.
.
For u = 1 , the manifolds MK are called classical Banach manifolds of type 11. Similarly, 2 is called a classical JB*-triple of type 11. For u = -1 , we obtain the classical Banach manifolds and the classical JB*-triple of type 111. In order to describe the Cayley transformation for classical manifolds of type I1 and 111, we distinguish two cases. Case 1: e
E 2
dim(E)
Type IIeven and Type 111. be any unitary operator. is even, we can write
If If
2 2
has type 111, let has type I1 and
SECTION 2 3
404
for some Hilbert space K invariant under the conjugation q . Let TI E U R ( K ) be any unitary operator. Then
is a unitary tripotent.
In both cases, the matrix
+
induces the Cayley vector field p(Xe) E aut(M ) satisfying 23.1.6. A s in the case of type I manifolds, it follows that the Cayley transformation r(ge) E Aut(M+) associated with the tripotent e E 2 is induced hy the matrix
satisfying ( g ) ( z ) = (z+e)(idE-e*z)-'
e #
(z+e)(e-z)-le
=
.
Since
it follows that (gel# maps the open unit ball bianalytically onto the tube domain D := { z
E
z
: z*e
+ e*z >
0
R
of
2
}
.
with symmetry (je)#(z) = ez-le about e E D If 2 is of type I11 and e := idE , we get the tube domain -1 D : = { z E 2 : z * + z > 0 } with symmetry (je)#(z) = z
.
.
Case 2: Type IIodd Now assume that u = 1 odd. Then there is an orthogonal splitting
and
dim(E)
is
CLASSICAL BANACH MANIFOLDS
405
for some Hilbert space K invariant under the conjugation q Put u : = { u E L(H) : u t fu = 0 } and V := L(H,D) Then every z E 2 has the form
.
.
where u E U and v E V vh = ( E ( v t ) for a l l h unitary operator. Then
. E
Here
.
H
vL E H Now let
q
is defined by E UR(K) be a
is unitary and
is a tripotent with Peirce 1-space V The matrix
.
0
O 0
51
1 and Peirce --space 2
e 0 0
0
0
0
+
induces the Cayley vector field p(Xe) E aut(M satisfying 23.1.6. Since :X = -Xe ' the same argument as €or manifolds of type 1 implies exp(tXe) = id + (l-cos(t))XL e
+ sin(t)xe
.
In particular, the Cayley transformation r(ge) E Aut(Mf) associated with the tripotent e is induced by t h e matrix
satisfying
SECTION 23
406
2lI2e (u-e -'vt v(e-u)
-1
0
e
Since O
e I
it follows that (gel# maps the open unit ball R of bianalytically onto the Siege1 donain t u*e+e*u-v*v -e*vt D = { ( ,u -vO ) ~ Z :
j
1
2
> O I
with symmetry (je)#(:
--v 0
t
I
- e - (-v
O
0)
-1 ('0
e
U
-lvt 1 )
i
-1
e'-lvt0 -vu e In order to describe classical Banach manifolds o € type IV, consider a Hilbert space L over K E {R,C} , endowed with a conjugation Q Define the anti-automorphism X + XT
.
of L(L) as in 23.2.1 and consider the associated "orthogonal" group O&(L) (cf. 23.2.2) with Lie algebra K g : = o&(L) (cf. 23.2.3). Let GK denote the identity component of O&(L) The identity component G+ of O&(L) n U & ( L ) is a real Banach Lie subgroup of GK with Lie + : = &(L) A d ( L ) algebra By 3.13, the projective space
.
.
P(L) := { H
E
ME(L) : dimK(H) = 1 ]
.
is a connected component of the Grassmann manifold M&(L) The closed subset N := { H E P ( L ) : (HIOH) = 0 } of P ( L )
407
CLASSICAL BANACH MANIFOLDS
is invariant under the collineation action
. Hence every connected component invariant under G K . Any point o M+
r
of
Ok(L)
on
of N is can be written as
M+
P(L)
E
0
,
o = ( O ) K
where 23.3.1
is an orthogonal splitting of 0
L 0
;
Q=[,"
such that " I
where q(z) = y is a conjugation of Put involution of K
1;
0
1
j :=
.
b
E
a. J
A
0
-idz
Since jT = j implies the identity component Banach Lie subgroup of - : = &(L) n u k ( 2 , Z ' ) open subset under G-
Z
.
N
of
E
Ok(L)
and
*
denotes the
.
(jX*j)T = j(XT )*j , it follows that G- of Ok(L) n U k ( 2 , Z 1 ) is a real G K with Lie algebra The connected component M- of the
.
N
containing
o
is invariant
.
Identify 2 = L ( K , Z ) and put Zt : = L ( Z , K ) and c E Zt , define bt E Zt and ct E Z
2
For by
b tz = (612) , and c t := qc** , i.e., bt : = *b*q , i.e., t cz = (TIC) For a E L ( 2 ) , put at : = qa*q E L ( Z ) and -a := qaq E L ( Z ) - -Then at = and az = a z Then there = K tB go K tB ;g , where exists an additive gradation
.
and
.
a*
.
SECTION 23
408 I
o\
0
0
Further, there exist mu It ipl icat ive gradat ions 1 K g K = g @ -'gK for "curvature" K = t , where
and
In case
gK
and
K = C 0 C =
, we gC
have
IgK 0 C = g o
C
,
.
23.3 PROPOSITION. M+ is a closed submanifold of P ( L ) and GK acts analytically and (locally) transitively on M+ There exists a chart (P,p,Z) of M+ about o such that
.
p(ro(exp XI) = X for all
X
K g-l
-
23.3.2
.
The differential t p*(pX) = X# := (az+b-zcz+ct 2 E
Z
p
of
a
- z d ) -a z
r
satisfies 23.3.3
for all -d
-bt 23.3.4
M-
is an open subset of
P
and
is the open unit ball of the JB*-triple product
Z
, with
triple
CLASSICAL BANACH MANIFOLDS
409
-
2{uv*w} : = u(vlw) + w(vlu) - V ( W J U ) For K = symmetry
+_ , r(j)
MK
is a symmetric normed about o
PROOF. Consider the chart (PF,pK,F,F) induced by the splitting 23.3.1 of L
of P ( L ) Then
.
[
-btb/2
). F
Since Z is a split subspace of a submanifold of P ( L ) For
.
: h
,
23.3.5
GK -maqifold with
.
P := PFnM+ = {
.
Z
E
about
o
.
}
it follows that
M+
is
we have
1
-b -btb/2 ro(exp X) = 0 idz b (0 0 1 Hence there exists a chart p : P + Z satisfying 23.3.2. With respect to the chart p , a matrix
!)
C'
23.3.6
ok(L)
E
g =
(!I
(with a , B , y , d E K , b , h ' E Z has a local representation
,
c,c'
E
Zt
and
a
E
L(Z)
b'z t2/2 t y~ 2/2 for all z E 2 with cz + d # y z t 2/2 By differentiation, we get 23.3.3. In particular, we have To(p)(poX) = b for + is all b E Z , where X is given by 23.3.4. Since o E M arbitrary, it follows that GK and G+ act (locally) transitively on M' We have p =
g
az+b cz+d
-
.
.
p(njnP) = and z
z
E
z
:
z*z
-
1ztZl2/4
.