Symmetric Banach Manifolds and Jordan C*-Algebras (North Holland Mathematics Studies, Vol 104)

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

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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

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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 <

E

Z

.

R

I*(

,

i.e.,

if there exist

such t h a t

Any norm e q u i v a l e n t t o a B a n a c h s p a c e norm

is c a l l e d c o m p a t i b l e .

Z

s p a c e norm

2

T h e c h o i c e of a s p e c i f i c R a n a c h

is o f t e n i r r e l e v a n t a n d o n e c a n r e p l a c e

1.1

by a n y o t h e r c o m p a t i b l e norm. Suppose

2

and

t h e norms o f

Z

and o f

danger of confusion).

W

are Ranach s p a c e s o v e r W

The v e c t o r s p a c e

c o n t i n u o u s l i n e a r mappings over

K

by t h e same s y m b o l f

:

Z

+ W

with respect to t h e so-called

K

and d e n o t e

I*I

(without

L(Z,W) of a l l is a R a n a c h space

o p e r a t o r norm

ANALYTIC MAPPINGS ON BANACH SPACES

If

:= sup Ifzl/lzl

.

sup lfzl

=

3

I Z I

Z#O

In particu ar, for

,

W = K

we obtain the Ranach dual space

.

For L(Z,K) of all continuous linear functionals on Z every n E N , the vector space Ln(Z,W) of all continuous n-linear mappings f : Zx...xZ + W (n copies of Z ) is a Banach space over K with respect to the norm

.

Note that L 0 (Z,W) = W and L1(Z,W) = L(2,W) The elements of L 2 ( 2 , Z ) are continuous (non-associative) algebra

.

Continuous multi-linear mappings lead to structures on Z the basic "constituents" of analytic mappings. 1.1

DEFINITION.

A

mapping

f : Z + W

between Ranach spaces

over K is called a continuous n-homogeneous polynomial if there exists f"E Ln(Z,W) such that

.

for all z E 2 Endow the vector space Pn(Z,W) of all continuous n-homogeneous polynomials from Z to W with the norm

.

If1 := sup (fzl/(z(" = Z#O

The multi-linear mapping

sup lfzl
f"

and the condition that f" is symmetric, i.e., invariant under permutation of the argument vectors. More precisely,

is given by the so-called polarization formula [ 4 3 ; 1.1.1.1 f"

f 3 z l,".rzn)

:=

n n!2

Hence

P2(Z,Z)

al...a a =+1

n

f(a z +...+a z ) 1 1 n n

1.1.2

1 an =+1 can be identified with the space of all

commutative (non-associative) continuous algebra structures on

Z

,

whereas

P0 (2,W) = W

and

P1(Z,W) = L(Z,W)

.

SECTION 1

4

1.2 LEMMA. Endow the vector space C(Z,W) of all continuous mappings from Z to W with the topology of pointwise convergence. Then Pn(Z,W) is a closed suhspace of c(Z,W)

.

Let (fj) be a net in Pn(Z,W) , converging pointwise to f E C(Z,W) Let f? E Ln(Z,W) denote the symmetric 3 n-linear mapping corresponding to fj , and define f-: zx...xZ -t w by 1.1.2. Then f- is continuous and the net (f:) converges pointwise on Zx...xZ to fHence fA E L ( Z , W ) and f%(z z) = f(z) for all z E Z O.E.D. PROOF.

.

A

.

,...,

1.3 DEFINITION. Let ( f n-homogeneous polynomials

) nEN

fn

E

.

he a sequence of continuous pn(Z,W) Then

.

m

1.3.1

n=Ofn

.

is called a power series from Z to W The radius of convergence of 1.3.1 is defined as the largest number R < such that the series

+m

1.3.2 converges uniformly for power series 1.3.1

121

< r

,

is called convergent if

radius of restricted convergence of 1.3.1 largest number

R-

< +-

r < R

whenever 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 <

whenever PROOF.

-

. R- .

To prove 1.4.1, put

.

-

R := R

, -fn := fn , Tn := fi ,

1.4.2 r , in r ,

and z :=

For 1.4.2, put R := R(Zm)m>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

<

: p-(f,r)

1.5.1

}

+m

1.5.2

and R"

T h e power s e r i e s sequence

(

,

r <

-

<

p(f ,r)

< r Since

-

p(f,r)

then 1.4 +m

.

,

1.5.3

is c o n v e r g e n t i E and o n l y i f t h e

or t h e sequence denote e i t h e r

-

implies

r

.

6

1fil

.

+m

is bounded.

or

p"(f,r)

Conversely,

.

If

if

converges uniformly f o r

T h i s p r o v e s 1 . 5 . 1 and 1.5.2.

< nnlfn\

Ifn(olzl+..+anzn)(

(

p(f,r)

<

p(f,r)

1; T n ( y )

then

Hence

R < eR"

Li f n

I f n ( 'In)

Let

PROOF.

6

if

supmlzml

, 1.1.2

1

implies

1.5.4

1;

By t h e r a t i o t e s t , t h e power s e r i e s r a d i u s of convergence

1;

nnxn/n!

.

e

n!n

-n n x

has the

t h e power s e r i e s

Similarly,

l/e

h a s t h e r a d i u s of convergence

.

Hence 1.4

implies 1.5.5

w e g e t 1.5.3.

Combining t h i s w i t h 1 . 5 . 4 ,

The l a s t a s s e r t i o n

f o l l o w s from 1 . 4 . 1.6

DEFINITION.

and

K

O.E.D.

Suppose

and

Z

is an open s u b s e t of

D

.

Z

L;

fn

A mapping

o

called a n a l y t i c i f , f o r every point c o n v e r g e n t power s e r i e s

a r e Ranach s p a c e s o v e r

W

D

E

f

: D + W

is

there exists a

such t h a t

m

1.6.1 for all f

z

: D + W

i n a neighborhood o f

o

E

D

.

is c a l l e d d i f f e r e n t i a b l e ( o v e r

A mapping

K

)

at

o

E

D

if

ANALYTIC MAPPINGS ON BANACH SPACES

7

lim Jf(z)-f(o)-f'(o)(z-o)(/lz-ol = 0

1.6.2

z +o

for a (uniquely determined) linear mapping f'(o) E L(ZrW) r called the derivative of f at o If f is differentiable at every point of D , the derivative of f is the mapping

.

.

defined by z + f'(z) for all z E D The mapping called twice differentiable at o E D if f' is differentiable at o E D In this case, the second derivative

is

f

.

.

is defined by f"(o) := (f')'(o) Higher differentiability and the higher derivatives f(")(o) E L ~ ( z , w ) are defined in

a similar way. 1.7 LEMMA. Every polynomial f E pn(Z,W) is differentiable and the derivative f' E pn-l(Z,L(Z,W)) satisfies

Further,

af(z) := f'(z)

a

:

defines a continuous linear mapping

P"(z,w)

+

P

n-1

(zrL(zrw))

.

PROOF. The continuous (n-1)-homogeneous polynomial f' : z + L(Z,W) , defined by 1.7.1

f(z)-f(o)-f'(o)(~-o)=f n =

1 (1'

m=2

f"(z-or.. m

.

h

(2,.

..,~)-f"(o,..

-

,o)-nf ( z - o r o r . .

ro)

ro)

rZ-0r0r.

n -m

It follows that f is differentiable and has the derivative The symmetric (n-1)-linear mapping f' n-1 (Z,L(Z,W)) associated with f' is given by (f')" E L

.

SECTION 1

8

1.8 COROLLARY. Every analytic mapping f : D + W is infinitely often differentiable and the derivatives

are again analytic for all m > 0 , series expansion 1.6.1 about o E D power series expansion m

I:

If f , then

has the power f ( m ) has the

.

1.8.1 fy(z-0) n =m The power series 1.6.1 and 1.8.1 have the same radius of restricted convergence. For each m > 0 I we have f(m)(z) =

1.8.2

In particular, the power series 1.6.1 is uniquely determined by f and o

.

By the ratio test, the power series 1 ; nxn and xn/n have the radius of convergence 1 Hence 1.4 By 1.7, it follows that the power series implies n l ' n + 1 1; fn and

PROOF.

1;

.

I:

.

1.8.3 f; n=l have the same radius of restricted convergence and are therefore uniformly convergent in a neighborhood R of o E D By [28; 8.6.3II f is differentiable on R and the derivative f' has the power series expansion 1.8.3 about o Repeating this argument, it follows that f is infinitely often differentiable and that the derivatives f(m) for m > 0 are analytic on D and have the power series expansion 1.8.1 about o E D , with the same radius of restricted convergence. In particular, f o r z = o , we get

.

.

ANALYTIC MAPPINGS ON BANACH SPACES (m)

= f(m)(o) = m! fm

( 0 )

E

9

.

L~(z,w)

O.E.D.

If f : D + W is analytic and o E D , the numbers p(f,r) and p"(f,r) associated with the power series expansion 1.6.1 of f about o are also denoted by po(f,r) and pi(f,r) , respectively. The order of f at o is defined as o r d o ( f ) := inf

PROPOSITION.

1.9

{ n e

Suppose

.

N :

D

fn #

o }

E

NU{+-}

is an open subset of

. Z

and

o E D For k = 1,2 , let $k : Z + Wk by analytic mappings into Banach spaces wk Let F : W1xW2 + W be a

.

continuous bilinear mapping into another Ranach space. F( $11 4 , ) : D + W is analytic and

,

PROOF. For n k := Ordo($k) $k about o has the form $k(o+z) =

1

the power series expansion of

k $,(Z)

n ank where

$n

F($11$2)

E

Pn(Z,Wk)

.

Since

is analytic on

D

Then

F

, is bilinear and continuous,

and

Q.E.D.

1.10 PROPOSITION. Suppose D C Z and B C W are open subsets of Banach spaces, and let f : D + B and g : B + V be analytic mappings, where V is another Banach space. Then the composite mapping gof : D + V is analytic, and for every

o

If

6

D

we have

f(o) = 0

E

W

,

then

SECTION 1

10

PROOF.

Consider the convergent power series expansions m

f(o+z) = f ( 0 )

+

1

fn(z)

n=l and

about o E Z and f(o) E W , respectively. F o r k > 1 k homogeneous polynomial hk E p ( 2 , V ) , defined by

L

hk(z) :=

nm

l<m
the

1.10.3

...+nm =k

nl+ nl>l,

,

...,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 <

.

+m

1.14.1 defines a continuous mapping series expansions m

of

fJ

about

o

.

fn :

Z +

.

W

Consider the power

.

n=O Since 1.12.2

implies 1.14.2

and { o + Xv/lvl : XI = r } is a compact subset of D , it follows that the net (fA)j in Pn(Z,W) converges pointwise to fn for every n > 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 <

Z :

E

.

}

R

S i n c e t h e power series c o n v e r g e s u n i f o r m l y f o r

PROOF.

lzl < r

,

r < R

whenever

continuous.

Since

fn

,

f

t h e mapping

O(D,W)

E

,

1.14

: D + W

implies

f

is O(D,W)

E

.

O.E.D.

1.16

The v e c t o r s p a c e

COROLLARY.

holomorphic mappings

f

o f a l l bounded

O,(D,W)

is a c o m p l e x R a n a c h s p a c e

: D + W

w i t h r e s p e c t t o t h e supremum n o r m

.

l f l D := s u p ( € ( z ) l ZED PROOF.

By

[28; 7.2.11,

1.16.1

the vector space

bounded c o n t i n u o u s mappings

f

s p a c e u n d e r t h e supremum norm.

is a c o m p l e x R a n a c h

: D + W

s u b s p a c e by 1 . 1 4 ,

t h e assertion follows.

1.17

Let

COROLLARY.

Banach s p a c e

Z

a € ( z ) := f ' ( z )

If

C

C C D

such t h a t

is a c l o s e d

O,(D,W)

Since

of a l l

C,(D,W)

O.E.D.

be o p e n s u b s e t s o f a c o m p l e x R := d i s t ( C , a D )

>

0

.

Then

d e f i n e s a c o n t i n u o u s l i n e a r mapping

is c o n v e x , t h e n 1.17.1

for all PROOF.

f

E

O,(D,W)

By 1 . 1 3 . 2

and a l l

we have

z,w

E

C

.

15

ANALYTIC MAPPINGS ON BANACH SPACES

1.17.2 This proves that a is continuous. Now suppose C is convex and let [z,w] := { tz + (1-t)w : 0 < t < 1 } denote the line

.

segment in C spanned by z,w E C theorem [28; 8.5.41 and 1.17.2 imply

Then the mean value

O.E.D. As a consequence of 1.17, we obtain Liouville's Theorem. 1.18

COROLLARY.

Every bounded holomorphic mapping between

complex Ranach spaces

and

Z

PROOF. For any fixed pair holds for R = +m

.

is constant.

W

z,w

E

Z

,

the estimate 1.17.1 O.E.D.

The next result is the so-called open mapping theorem (a version of the maximum modulus principle). 1.19 space

THEOREM.

Suppose

and f E O(D,C) is an open mapping. Z

D

is a domain in a complex Ranach

is not constant.

Then

f : D

+

C

PROOF. By 1.11, f is not constant on every non-empty open Hence it suffices to show that f(D) is a subset of D neighborhood of f(o) E C for every o E D Put Then there exists a unit vector v E 2 R := dist(o,aD) such that the holomorphic function g ( h ) := f(o+Xv) , defined on A := { X E C : 1x1 < R } , is not constant. Hence there exists 0 < r < R such that

.

.

.

s :=

inf Jg(X)-g(O)l > 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<-g(O)l < s/2 } and is therefore a neighborhood of g ( 0 ) E C Since f ( D ) D g ( b ) and 0.E.D. f(o) = g ( 0 ) , the assertion follows. \<-g(O)l-' = Ih(O)l <

IXI

.

.

1.20 PROPOSITION. Suppose Z and W are complex Ranach be a sequence of holomorphic spaces. Let fj E O m ( D , W ) mappings on an open subset D of Z such that

c := sup IfJID < +a j and for some point o E D the sequences (fj)(")(o) E Ln(Z,W) converge for all n sequence (fJ1 is uniformly convergent for whenever r < dist(o,aD)

.

.

Then the I z-01 < r

,

.

We may assume 0 = 0 E Z Put !f := (fJ)(")(O)/n! By 1.12, we have for every z ~ ~ : = { z ~ ~ : l z l < r }

PROOF.

.

m

fhz) = and 1.13 implies

1

.

fA(z)

n=O lfAl < c/Rn

, where

R := dist(o,aD)

Hence

+ 2c

1

(zln/Rn

n >N

.

It follows that Ifi-fJ I R

Given

s

> 0 , there exists N >

0

such that

.

A N A L Y T I C M A P P I N G S ON BANACH S P A C E S

17

By assumption, there exists an index k such that n lfA-fil*r 6 s/N for all n < N and all i,j > k Therefore lfi-fJIB < 2s for all i,j > k , and the assertion follows from 1 . 1 6 .

.

Q.E.D.

Many results about real-analytic mappings can be deduced from the corresponding results for holomorphic mappings by using the method of "complexification". For a real Banach space X , the complexification

xC

x oR c

:=

=

x

B ix

is a complex vector space containing X as a real subspace. If 1 * I is a compatible norm on X , then Xc can be endowed with a complex Banach space norm 1 - 1 satisfying

for all

x,y

.

X

E

For example define

If Y is another real Banach space, every continuous polynomial f E Pn(X,Y) has a unique extension c c satisfying fC I X = f fc E pn(x ,Y

.

1.21

For

LEMMA.

f

E

Pn(X,Y)

,

we have

If( < l f C l < 2"lf-l

.

PROOF. Since fC \ X = f and the embeddings X C Xc and Y C Y c are isometric, it follows that If1 < lfCl Now let be the symmetric complex n-linear mapping (fc)- E Ln(XC,YC) Then associated with fc

.

.

Ifc(x+iy)l

=

I

n =

I

I:

m=O

c -

( f ) (x+iy,...,x+iy)

(ijimf-(x

,... ,x,y ,...,y ) n-m m

I

1

SECTION 1

18

1.22 COROLLARY. Suppose X and Y are real Ranach spaces and fn is a power series from X to Y with radius of convergence R > 0 Then the complexified power series

1;

.

00

1.22.1 n=O from

Xc

to

is convergent and

Yc

m

f(x) =

is analytic for

1x1 < R/2e

.

1 fn(x) n=O

PROOF. By 1.21 and 1.4, the radius of convergence Rc of 1.22.1 satisfies R"/2 < Rc < R , where R and R" denote the radius of (restricted) convergence of En By 1.12, the power series 1.22.1 defines a holomorphic mapping for ( z ( < R%/2 Therefore f is analytic for

1;

.

1x1 < R/2e

6

R"/2

.

.

O.E.D.

The following inverse mapping theorem will be frequently used in the sequel. Recall that a bijective linear mapping 4 : 2 + W between Ranach spaces is called an isomorphism if -1 are continuous. By Banach's open mapping $ and $ theorem [15; 48.11, it suffices to require that $ is continuous. 1.23 THEOREM. Suppose Z and W are Ranach spaces over K and let f : D + W be an analytic mapping defined on an open subset D of 2 such that the derivative f'(o) E L ( Z , W ) is an isomorphism. Then there exist open neighborhoods C of o E D analytic mapping h : B + C foh = idB I h o f = idc *

and €3 of f(o) E W and an such that f(C) = B and

PROOF. By considering the complexification of Z and W in case R = R I we may assume K = C We may further assume

.

ANALYTIC MAPPINGS ON 3ANACH SPACES

19

o = 0 f(o) = 0 , Z = W and f’(0) = idz (otherwise, consider f’(O)-lof : D + Z 1. Since g := idz - f satisfies Ordo(g) > 2 , there exists s > 0 such that z E D whenever I z ( < s and %

PO(9,S)

S

< 3

1.23.1

.

Put ho := 0 and define inductively hj+’ := idZ+gohj By 1.10, the mappings hj are holomorphic in a neighborhood of 0

E

andsatisfy

Z

This is clear for

hj(0) = 0

j = 0

.

Further,

, and the induction step follows from

1.10.2 and 1.23.1 :

Now consider the power series expansions m

and

ml+

...+m n =k

an induction argument shows

.

OrdO(hjfl-hj) > j the mappings hj are analytic on

By 1.23.2, B :=

{

z

E

Z : I z ( < s/2 }

and satisfy

s u p lhjl, < s

By 1.23.3,

j for any fixed

m

1.23.3

,

. the sequence

((hj)(m))j

SECTION 1

20

.

converges to (hm)(m) Hence 1.20 implies that the sequence (h') converges to the mapping m

h :=

1

:h

m= 1

.

which is analytic on R and satisfies Ihl, < s It follows that the mappings fohj and foh are well-defined and analytic on B Further,

.

> 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
X#O

.

particular, for every Ranach space L over D E {R,C,A} , the set A = L(L) := L(L,L) of all continuous D-linear "operators" on L is an associative Ranach algebra over K In case under the operator norm, with unit element idL

.

L = Dn is finite-dimensional, we obtain the algebra Ln(D) := L(Dn) of all (nxn)-matrices with coefficients in D .

2.5 EXAMPLE. Suppose D is an open subset of a complex Banach space 2 and W is a complex Banach algebra. Then of all bounded holomorphic mappings the Banach space O,(D,W) f : D + W , endowed with the supremum norm 1.16.1, is a complex Banach algebra under pointwise multiplication. If W is unital, then O,(D,W) is unital. An element x E A of an associative unital Ranach algebra A is called invertible if xy = yx = e for some (uniquely determined) element y E A , called the inverse of x and denoted by y = x -1

.

2.6

PROPOSITION.

Suppose

A

is an associative unital Ranach

BANACH ALGEBRAS algebra and x invertible and

E

27

.

A satisfies le-xl < 1 Then x x-l is given by the Neumann series m

x-l =

1

.

n (e-x)

2.6.1

n=O PROOF. The series 2.6.1 converges in complete and

A

,

since

m

0

1

I(e-x)"l

1

G

n=O

is

le-xIn < +-

n=O

A

is

.

Further, OD

(e-x)n

y := n=O satisfies m

.

xy = (e-(e-x))y = y The identity

(e-x)' = (e-x)' = e n=l is shown similarly. Hence y = x -1

yx = e

O.E.D.

COROLLARY. Let A be an associative unital Ranach algebra. Then the group G(A) of all invertible elements in A is open, and the mapping x + x -1 is analytic on G(A). More precisely, if y E G(A) and x E A satisfy IY-XI < IY I -l , then x E G(A) and 2.1

m

2.7. 1 1 (Y-5Y-x)) ny -1 n=O PROOF. By assumption, le-y -1 X I = ly-1 (y-x)l < [y-'I*[y-x\ < 1 Hence y 'x E G(A) by 2.6 and the inverse (y-1 x)-1 = x -1y is given by the convergent series

x-l =

.

m

.

1 (y-l(y-x~)~ n=O n=O This proves 2.7.1. By 1.4, the power series 2 . 7 . 1 about y E G(A) has the radius of convergence R > ly-ll-' since x -1 y =

I(Y

.

-1

1

(e-y-lx)" =

n -1 v) Y

I

m

-1 n+l < IY lvln

I

for all v E A It follows that the inversion mapping -1 x + x is analytic on G ( A ) O.E.D.

.

SECTION 2

28

Another important example of an analytic mapping on associative unital Ranach algebras is the exponential mapping. 2.8 PROPOSITION. Let A be an associative unital Ranach algebra. Then the convergent power series m n 2.8.1 exp(x) := 1 n=O

5

defines an analytic mapping exp: A + G(A) exp(0) = e If x,y E A commute, then

satisfying

.

.

exp(x+y) = exp(x) exp(y) In particular, every

x

E

A

2.8.2

satisfies

exp(x)-l = exp(-x)

.

2.8.3

Ixn/n! I < Ixln/n! for all x E A and by 1.5.5, 1.4 implies that the power series (n!) 2.8.1 has the radius of convergence R = +Applying 1.15 (in case K = c ) or 1.22 (in case K = R ) , it follows that exp: A + A is analytic. If x,y E A commute, the binomial theorem implies

PROOF.

Since

-'In

+ 0

.

n

Since 2.8.1

r:

n-m m Y m=O converges in norm, we get (x+y)" =

(;b

Since x and y := -x commute, it follows that exp(A) C G(A) and exp(x1-l = exp(-x)

.

2.9 EXAMPLE. Let By 2.4, A := L(L) over the center K

L be a Banach space over D E {R,C,H} is an associative unital Ranach algebra of D Hence the group

.

.

GI~(L):= G(A) = { g

of all invertible

Q.E.D.

E

L(L) : g invertible ]

D-linear operators on

L

is an open subset

BANACH ALGEBRAS

29

of L ( L ) and the exponential mapping exp : L(L) + GQ(L) is analytic. In case L = Dn is finite-dimensional, we write GRn(D) := GQ(D") and have the exponential mapping : Ln(D)

exp

.

GRn(D) The inversion mapping and the exponential mapping for associative unital Ranach algebras are special cases of .+

analytic mappings constructed via the so-called (analytic) functional calculus. For any associative unital complex Banach algebra A with unit element e , the spectrum c,(x)

:=

{ x

E

c

:

Xe-x d G(A) }

of x E A is a compact subset of C [17; 5.81 which is nonempty if A # { O } [17; 5.71. If A is an associative unital real Ranach algebra, the complexification R := A R R C is a complex Ranach algebra and we put

for all

x

E

A

.

Then

CA(he+x) = X + CA(x) if A

is an associative unital Ranach algebra over K If L is a Ranach space over D E {R,C,H} ,

A E

.

K

2.10.1 and

consider the Ranach algebra A := L(L) over the center K of D and, for any x E L ( L ) , define the "operator spectrum" CRL(x) := CA(x)

.

2.10.2

Recall that a subset il of a vector space is called starlike 1 imply about o E il if x E il and 0 < t o + t(x-o) E $2 Every starlike set is connected and simplyconnected.

.

2.10 over

LEMMA. Let A be an associative unital Ranach algebra Then the set

K

.

30

SECTION 2

is open in about

a

E

A

K

,

whenever

R C C

then

is starlike about

RA

is open.

If

R

ae

.

is starlike

PROOF. By considering the complexification of A in case K = R , we may assume K = C Let y E R A Since is continuous on the closed subset C \ R 1 + (Xe-y)-’ E A and satisfies

.

lim

.

I(he-y)-’\ =

o

,

\XI+-

it follows that

inf I(Ae-y)-’\-’ >

r :=

hEC\R

Now suppose

for all x E RA 2.10.1.

let

x

E

A

satisfies

Ix-yl < r

o

.

. Then

.

.

X e C \ R By 2.7, we have Xe-x E G ( A ) Hence and R A is open. The last assertion follows from O.E.D.

In order to describe the holomorphic functional calculus, A be an associative unital complex Banach algebra. For

any open subset R of C , consider the associative unital complex algebras O( R,C) and O(RA,A) of holomorphic mappings, endowed with pointwise multiplication. If f E O(Q,C) and x E QA , define f(x) = fA(x) by the ” C au c hy i n teg ra 1I’

is any compact subset of with piecewise smooth a K and non-empty interior i? containing zA(x) Applying 2.7, 2.10 and 1.14 to the open set i? C C , it follows that fA : Q A + A is holomorphic. By [17; 7.4 and 7.61, the mapping where K boundary

is a unital homomorphism satisfying

.

(idQ)A = id

and nA

BANACH ALGEBRAS

for all

x

have for

ilA

E

f

E

.

If

(,l(Q,A)

$2

and

and

g

( 4 W A= fA : QA

In particular,

are open subsets of

A

.

,

we

2.10.4

is biholomorphic if

AA

C

(,l(A,C)

E

gAOfA

+

31

f

: Q + A

is biholomorphic. As special cases, we obtain the exponential mapping expA : A

G(A)

+

induced by the exponential function

exp : C

and the

+ Cx

logar i thm logA : ilA

2.10.5

+ A

induced by the principal branch of

log :

61 + C

on

The holomorphic mapping expA has the power series expansion 2.8.1 about 0 E A , whereas logA has the power series expans ion logA(x) about

e

E

QA

2.11

LEMMA.

over

R

,

=

-

m

1

-(e-x) n=l n

n

2.10.6

Let

A

.

1

with radius of convergence

be an associative unital Ranach algebra

and define for

0 < s <

TI

and A’

:=

{ 5

E

c

: IIrn(<)1

Then the analytic mappings expA : A:

+

nAS

<

s

1

,

32

SECTION 2

and logA :

Qi

S

AA

+

are inverse to each other and hence bianalytic. PROOF. Since exp: AS + Q s is a biholomorphic function with inverse function log : as + A s , 2.10.3 and 2.10.4 imply the assertion in case K = C Now assume K = R and consider the complexified Banach algebra B := ARRC Since the power series 2.8.1 and 2.10.6 have "real coefficients", it is clear

.

S

.

S

S

.

S

that expB(AA) = expB(AB n A) c SZB n A = LiA Since $2; is starlike about e E A , a similar argument using 1.11 shows S S S S Now define logB(QA) = logB(QB n A) C AB n A = A A expA := expBIA

and

l o g A :=

S

logglnA

.

.

O.E.D.

COROLLARY. Let W he a closed suhspace of a Ranach space 2 over K and suppose x E L ( 2 ) satisfies suplIm zaZ(x)I < P and exp(x)W = W Then x(W) C W 2.12

.

PROOF.

By considering

assume

K =

.

c

2'

:=

and

ZBKC

Wc := WR&

x

, we may

,

where A := L ( 2 ) Hence 2.10.3 implies g := exp(x) E and x = logA(g) Let K he a compact neighborhood of EEZ(g) such that C \ K is connected. By Runge's Theorem in Q" [27; VIII.1.191, the restriction loglK is a uniform limit of a sequence (pj) of polynomials. The continuity of the functional calculus [17; 7.4.iiil implies

.

By assumption, we have

.

=

lim p.(g) j+m

pj(g)W C W

for all

Suppose W 2.13 LEMMA. space 2 over K Let x(W) C W Then

.

PROOF.

A:

.

x = logA(g Since

E

.

j

,

Ql

.

J

the assertion follows.

O.E.D.

s a closed subspace of a Ranach x E L ( 2 ) satisfy Ci2(x) C R and

Consider the closed unital subalgehra

BANACH ALGEBRAS

33

c

A := { x E L ( 2 ) : x(W) W } of L ( 2 ) and the continuous Then [17; 5.4 unital restriction homomorphism p : A + L(W) and 5.141

imply

C

ZaW(x)

ZA(x) = caZ(x)

.

.

0.E.T).

An algebra g over K , with product denoted by brackets (x,y) + [x,yl , is called a Lie algebra if [x,x] = 0 and “X,YI,Zl

[y,zl,xl + “z,xl,yl

+

2.14.1

= 0

.

The identity 2.14.1 is called the Jacobi for all xly,z E g identity. If g is a Banach algebra with respect to the

g

(bracket) product, t en Let 2.14 E X A M P L E . product (x,y) + xy

A

is called a Banach Lie algebra.

be an associative Ranach algebra, with Then the commutator product

.

[XIYI := xy - yx

.

makes A into a Ranach Lie algebra, denoted by g(A) In particular, for every Banach space L over D E {R,C,A} and A := L(L)

,

.

is a Banach Lie algebra over the center K of D In case L = Dn is finite-dimensional, we write gan(D) = ga(Dn) 2.15 PROPOSITION. Let Banach algebra over K

2

.

.

be a (not necessarily associative) Then the set

of all continuous derivations of Z is a closed subalgebra of ga(2) and hence a Banach Lie algebra. The set Aut(2) := { g

E

GR(2) : g(zw) = (gz)(gw) }

of all continuous automorphisms of 2 is a closed subgroup of Ga(2) and there is a commuting diagram ga(2) U

aut(2)

-

-e

exP

Ga(2)

v

Aut(2)

SECTION 2

34

PROOF.

For

6

E

aut(2)

and

g := exp(6)

,

2.1 implies

O.E.D.

2.16 PROPOSITION. algebra. Then

Let

A

be an associative Ranach

defines a continuous Lie algebra homomorphism ad

:

g(A)

+

aut(A)

.

For $ E Aut(A) whose kernel is the center of A $ ad(x) $I-' = ad(gx) and 6 E aut(A) , we have [6,ad(x)l = ad(6x) In particular

and

.

.

is an ideal in aut(A) The elements of inner derivations of A More generally, we have 2.17

PROPOSITION.

.

g

Let

ad(x)y :=

int(A)

are called

be a Ranach Lie algebra.

Then

[x,yI

defines a continuous Lie algebra homomorphism ad : g

+

aut(g)

.

For $ E Aut( g) whose kernel is called the center of g and 6 E. aut(g) , we have $ ad(x) 4 - l = ad($x) and In particular, [6,ad(x)] = ad(6x)

.

int(g) := ad( g) is an ideal in

aut(g)

.

The elements of

int(g)

are called

BANACH ALGEBRAS

4

inner derivations of PROPOSITION.

2.18

K

algebra over

.

.

Let

he an associative unital Ranach

Z

Then there exists a commuting diagram

AL Aut(2)

G(Z)

where

35

is the continuous Lie algebra homomorphism defined

ad

by ad(z)w := and

Ad

ZW-WZ

is the continuous group homomorphism defined by Ad(g)z := gzg -1

.

R z := zy denote the left and Y right multiplication operators on 2 , respectively. Then PROOF. Lx

and

Let

R

Lxz := xz

Y

commute on

and 2

.

By 2 . 8 . 2

and 2 . 8 . 3 ,

exp(ad(z)) = exp(LZ-Rz) = exp(LZ)exp(RZ)

we get

-1

Q.E.D.

NOTES.

Our reference for the spectral theory of Banach alge-

_e_

bras is C171, cf. also C1131. The general theory of Banach Lie algebras, with special emphasis on the Baker-CampbellHausdorff formula, can be found in C211. For the basic facts of Banach space theory (e.g., the Hahn-Banach Theorem, the separation theorems and Banach's open mapping theorem), we refer to U 5 1 . There exists an extensive literature studying the geometry of special classes of Banach spaces (see, e.g.,

C 96,99,1001 ) . Corollary 2.12 will be used in Section 7 and appears in C21; Ch. 111,

5

6, n09, Proposition 191.

SECTION 3

36

3.

BANACH MANIFOLDS

In the fol owing, analytic mapp ngs are not only studied on open subsets of Ranach spaces but on more general "global" objects called (Ranach) manifolds. In 1-dimensional complex analysis, this approach leads to the concept of Riemann surfaces. It will be shown that also in the case of infinite dimension many important geometric and algebraic objects are Ranach manifolds in a natural way. We have made no attempt to cover more general Banach analytic spaces with singularities (cf. [ 3 1 ] ) since the analytic objects considered here are often homogeneous under holomorphic transformations. Suppose M is a topological (Hausdorff) space. A triple (P,p,Z) is called a chart of M if P is an open subset of M , Z is a Ranach space over is a homeomorphism onto an open subset of

K Z

and p : P + Z If o E P is

.

a point satisfying p(o) = 0 , (P,p,Z) is called a chart about o The charts (P,p,Z) and (Q,q,W) of M are said to be compatible if the homeomorphism

.

q0p-l : p(PnQ)

+

q(PnQ)

between open subsets of Z and W , respectively, is bianalytic. In this case, Z and W are isomorphic provided PnQ # An atlas of M is a collection of pairwise compatible charts covering M A maximal atlas (under inclusion) endows M with the structure of an (analytic) Banach manifold over K For example, every open subset D of a Banach space Z is a Banach manifold, with maximal atlas generated by (D,id,,,Z) More generally, every open subset of a Banach manifold is again a Ranach manifold in a natural way. In the special case that Z = Kn is finite-dimensional, a chart (P,p,Z) of M is given by the "coordinate : P + K satisfying functions" j'

a

.

.

. .

A mapping

g : M

+

N

between Ranach manifolds is called

BANACH MANIFOLDS

analytic if, for every point

m

E

M

,

37

there exist charts

(P,p,Z) of M and (Q,q,W) of N such that g(P) C Q and there is a commutative diagram P4-*

P where

g+,

m

E

,

P

Q

i

is an analytic mapping, called the local

.

representation of g with respect to p and q Every analytic mapping is continuous and, by 1.10, the composition of analytic mappings is analytic. A bijective mapping g : M + N is called bianalytic if g and g -1 are analytic mappings. The group of all bianalytic automorphisms of M will he denoted by Aut(M) If M is a Ranach manifold and W = Kk is finite-dimensional, a mapping g : M + W is analytic if and only if its "component functions" : M + K , defined by j '

.

are analytic. If M is a Banach manifold and f : M + A is an analytic mapping into a Banach space, define the order of at o E M by Ordo( f) := Ordo( fop-') where (P,p,Z) is a chart of another chart of M about o Ordo(foq -1 )

M

,

,

3.0.1

.

about o If 1.10.1 implies

OrdO(fop-l) OrdO(pOq -1

f

(Q,q,W)

=

is

Ord (fop-') 0

.

since OrdO(poq -1 ) = 1 It follows that 3.0.1 is independent of the choice of chart (P,p,Z) of M about o Many basic results about analytic mappings on Banach spaces can be generalized to Ranach manifolds. An example is the principle of analytic continuation.

.

3.1 THEOREM. Suppose f and g are analytic mappings between Banach manifolds M and N , where M is

38

SECTION 3

connected. Suppose further that there exists a non-empty open subset P C M with flP = glP Then f = g

.

PROOF.

Let

I

.

denote the set of all points

o

E

M

such that

.

f l u = glU for a suitable open neighborhood U of o E M Then C contains P and is obviously open. Now suppose and let (Q,q,E) be a chart about m E M such that m E Q is connected. Then Q n C # fl , and 1.11, applied to the domain D := q(Q) C E , implies flQ = g1Q since f and g coincide on a non-empty open subset U C Q Hence C = M

c

.

.

O.E.D.

In case R = C , analytic mappings and hianalytic mappings are also called holomorphic and biholomorphic, denote the set of all holomorphic respectively. Let i ) ( M , N ) mappings g : M + N between complex Ranach manifolds M and N If W is a complex Ranach space, then i ) ( M , W ) is a complex vector space and the subspace O , ( M , W ) of all bounded holomorphic mappings f : M + W is a Ranach space with respect to the supremum norm

.

( f l M := sup I f ( M ) (

.

.

Here 1 . 1 denotes the norm of W If W is a (unital) Banach algebra, then i) M , W ) is a (unital) algebra and i),(M,W) is a (unital) Banach algebra. 3.2 THEOREM. Suppose M is a connected complex Ranach manifold and f : M + C is a non-constant holomorphic function. Then f is an open mapping. PROOF. By 3.1, f is non-constant on any non-empty open Now cover M by coordinate domains given by subset of M charts (P,p,Z) , and apply 1.19 to the non-constant

.

holomorphic functions

fop-’

.

O.E.D.

Every Banach manifold is locally connected and locally simply-connected. Hence every connected Ranach manifold M admits a universal covering n : M- + M , where MA is a connected and simply-connected space and 71 is a covering

39

BANACH MANIFOLDS

projection [ 2 6 ; Ch. 111. PROPOSITION. Let n : M + M be the universal covering of a connected Ranach manifold M Then M" can be endowed with a unique Ranach manifold structure such that n is locally hianalytic. 3.3

.

PROOF. Since n is a covering projection, M can be covered by charts (P,p,Z) such that n-'(P) is the disjoint union of open subsets Pi of M- such that n : Pi + P is a homeomorphism. Then pi := ponlPi determines a chart To see that any two charts of M(Pi,pi,Z) of Me constructed this way are compatible, suppose that (Q,q,W) is

.

another chart of M such that n-'(Q) is the disjoint union of open subsets Qj of M" such that n : Qj + Q is a Then pi(PinQ.) C p(PnQ) homeomorphism. Put qj := qonlOj J and q ,(PinQ.) c q( PnQ) are open subsets and there is a J J commuting diagram

.

p( PnQ)

Since

qop-l

q(PnQ) q0p-l is analytic, the assertion follows. f

0

O.E.D.

o E M such that IT(;) = o Let E be a connected and simply-connected Banach manifold and let e E E Then the following statements hold: 3.4

(i)

COROLLARY.

.

Choose points

M"

and

.

For every analytic mapping f : E + M satisfying f(e) = o there exists a commuting diagram

for an analytic mapping (ii)

E

fh

with

f"(e) =

If 4 : E + M" and J, : E + M" are analytic mappings such that n o $ = no$ and @(e) = $(e)

. ,

40

SECTION 3

then

(I

= $

.

PROOF. Since TI is locally bianalytic, the existence and uniqueness of analytic liftings follows from [26; Ch. 111. O.E.D.

In the theory of Riemann surfaces, the so-called fractional linear transformations or Moebius transformations play a fundamental role. Similarly, the Ranach manifolds studied in the following admit (groups of) fractional linear transformations as bianalytic automorphisms. To describe Moebius transformations in the infinite-dimensional case, we first associate with every Banach space Z over K a certain non-associative Banach algebra B over K As a Ranach space, B is the topological direct sum

.

3.5.1

and the elements of

are written as matrices a b 3.5.2 g = ( 1 ' c d where a E L ( Z ) (acting from the left), b E Z ? c E L2 ( Z , Z ) and d E L ( Z ) (acting from the right For z,w E 2 ,we let az E Z , zd E 2 and zcw = c(z w) E z denote the values of a , d and c , respectively. B

.

Accordingly, the product in L(ZIleft is given by (ala2)z := al(a2z) , whereas the product in L(Z)right is given by z(dld2) := (zdl)d2 The product in F) is defined

.

by [a1

bl) (a2

b2) = (ala2tblc2

al 2+b ld 2

1 -

c2 d 2 1a2+d 1c2 1b 2 1d2 c1 d l Here we have used the conventions introduced above. For example, blc2 E L(Z)left is defined by (blc2)z := b 1 c 2z = c2(bl,z) , and dlc2 E L2(Z,Z) by z(d c )w := (zd )c w := c2(zdl,w) 1 2 1 2 Now let g E B be given by the matrix 3.5.2. implies that

.

is defined

Then 2.7

BANACH MANIFOLDS

is an open subset of

41

and that

Z

defines an analytic mapping

g # : fig + Z called the Moebius transformation (or fractional linear transformation) g#

at

0

E

.

In case is given by

associated with

g

$1

g

d

GE(Z)

E

,

the derivative of

gi(0)v = (av-bd-'cv)d-l 3.6 D E F I N I T I O N . A subset associativity condition i f

(

(alu)c2v (uc1v)c2w

h o l d s for all

r

of

(alu)d2

I = ( (uclv)d2 U,V,W

E

R

al(ud2)

ucl (vc2w)

ucl (vd21

3.7

PROPOSITION.

r

Let

associativity condition. g,h

E

r

and a l l

z

E

Z

)

3.6.1

and all "matrices"

Z

.

r

is said to satisfy the

a 1 (uc2v)

Note that it suffices to check 3.6.1 generators of

3.5.3

for a set of algebra

be a subset of Then

R

satisfying the

g#(h#z) = (gh)#(z)

for all

such that both sides are well-

defined. PROOF.

Put g : = ( al

bl)

,

h := ( a 2

b2)

c1

c2 d2 al(w(c z+d ) ) = (alw)(c z+d ) and 2 2 2 2 for all z,w E Z Hence c1(w(c2z+d2)) = (c1w)(c2z+d 2 -1 al((a2z+b2)(c2z+d2) 1 = (al(a2z+b2))(c2z+d2)-l and whenever cl((a 2 z+b 2 )(c2z+d2) -1 ) = (c1 (a2 z+b2))(c2z+d2)-' c2z+d E GE(2) I t follows that z E fi if and only if 2 gh h#(z) E fi and in this case we have 4'

Then 3 . 6 . 1

imp1 ies

.

.

SECTION 3

42

-1

g # (h# z) = (al(a2z+b2)+bl(c2z+d2)1 (c2z+d2)

-1 -1 ((c, (a2z+b2)+d 1 (c 2 z+d2)) (c2z+d2) ) I

3.8

=

(a, a 2 z+b2)+bl c2z+d2) 1 (cl(a2z+b2)+dl(c2z+d2)1

=

(gh #(Z)

DEFINITION.

.

-1

O.E.D.

A closed subspace

E

of a Ranach space

L

over D E {R,C,E] is called split (or direct) if E has a topological complement in L , i.e., a closed subspace F C L such that L = E 8 F is the topological direct sum. Any such decomposition is called a splitting of L and is often written symbolically as L = 3.9 D

EXAMPLE. E

{R,C,A}

,

(E) .

3.8.1

Suppose L is a (right) vector space over endowed with a real bilinear "scalar product"

satisfying (hlk)* = (klh) and (hlka) = (h(k)a for all Then (hlh) E R and (halk) = a*(h(k) for all a E D a

E

D

. .

Assume

(hlh) > 0

for all

h # 0

.

Then

Ihl := (hlh)1'2 defines a norm on L satisfying L is called a Hilbert (ha1 = ( h ( * ( a l for all a E D space over D if L is complete withrespect to this norm. For every measure space ( S , p ) , the space L2(S,D) of all (equivalence classes of) measurable square-integrable D-valued functions on S is a Hilhert space over D with respect to the scalar product

.

.

h(s)*k(s)dp(s)

( h ( k ) := S

is L2(S,D) for some measure space Dn is a Hilbert space over D ln h?k , where

I t can be shown that every Hilhert space over

isometrically isomorphic to S As a special case, L = with scalar product (hlk) =

.

1

i

j

D

BANACH MANIFOLDS

43

. Every closed subspace

E

of a Hilbert space split, since the orthogonal complement

L

over

D

is

F := EL : = { h c L : (hlE) = 0 }

gives a splitting 3.8.1 of product in L is given by

.

L

In th s case, the scalar

.

It can be shown that a Ranach for all k i E F and hi E E J J space L is isomorphic to a Hilbert space if and only if every closed subspace of

L

is split [ 9 8 ] .

3.10 EXAMPLE. Let L be an infinite-dimensional separable Then the Ranach space L(L) (over Hilbert space over D the center K of D ) contains the set L w ( L ) of all compact D-linear operators on L as a closed subspace which is not

.

split [30; p. 1131. of the Ranach space

Similarly, the closed subspace C u ( N , D ) over D is not split [ 9 9 1 .

Cm(N,D)

Topological splittings give rise to Moebius transformations. 3.11 EXAMPLE. Let L be a Ranach space over D E {R,C,H} and consider a splitting 3.8.1 of L Then 2 := L(E,F) is

.

a Banach space over the center algebra B associated with 2

.

K of D The Ranach defined in 3.5.1, contains

,

as a closed unital subalgebra via the mapping

The corresponding Moebius transformation (c a

3.11.1

d b, # (z) := (az+b)(cz+d)-]

is an analytic mapping on the open set of all

z

E

Z

with

SECTION 3

44

cz+d

E

Gk(E).

Since

satisfies the associativity

L(L)

condition, 3 . 6 implies

for all g,h defined.

z E 2

and all

L(L)

E

such that both sides are

It will now be shown that, for every Aanach space over D , the set M g ( L ) : = { E C L : E split subspace }

L

3.12.1

of all split subspaces of L can be endowed with the structure of a Banach manifold over the center R of

D

.

,

called the Grassmann manifold over L In case L = D" is finite-dimensional, we obtain the classical Grassmann manifold

M ~ ~ ( D:=) MX(D") , In order to define a manifold structure on consider a splitting 3.8.1 of L and define pF := { H For any

z

E

L(E,F)

E

MR(L) : L = F @ H

,

the "graph" of

zh

EZ

:=

{ (h)

: h

E

belongs to PF , and the mapping L(E,F) onto PF Let

.

L

,

LEMMA.

+

z

-t

: D1

pk := ' E k , F k PROOF.

Ez

is a bijection from

L(E,F)

For any two splittings

pE ?

Ez) -

+

D2 for

:=

z

.

L = F @ E l = F2@E2

Dk := p ( P n P F 1 C L(Ek,Fk)

the sets

p20p;'

.

}

z

denote the inverse mapping defined by 3.12

,

}

E

pE,F : PF

Mk(L)

are open and is an F 1 analytic mapping. Here k = 1,2

.

We will use the symbolic notation = [F2)

L = (F1) 1

E2 2

.

Of

BANACH MANIFOLDS

Suppose first that F1 - F2 with b e L(E~,F~)

.

zh

E 2 = { ( h)l

Hence that

dh = h

45

Then there exists

1

:

+ bh defines an isomorphism

*

d : El

+ E2

such

.

is an analytic mapping on D1 = L(E1,F1) Now suppose El = E2 Then there exists c E L(F1,E1) with

.

Hence that

ak = k

+ ck defines an isomorphism a

: F1 + F2

such

is an analytic mapping on the open subset D1 = { z

E

L(E1,F1) : id-cz

To treat the general case, suppose E E PFnPF Then the identity 1

2

.

E

D1

GL(El) }

.

is not empty and let

-1 -1 holds on the set (pEfF10p1 1 (p, ( P F n P F 1 ) which is an tF1 1 2 open neighborhood of pl(E) by the second special case. -1 Since E is arbitrary, D1 is open and p20p1 is analytic. 0.E.D. It follows from 3 . 1 2 that there exists a unique topology

SECTION 3

46

on the set MI1(L) such that the sets PF are open and the mappings PE,F : PF + L(E,F) are homeomorphisms. This topology is metrizable [ 3 1 ; 8 2 . 1 1 and therefore Hausdorff. Further, 3.12 implies that

constitutes an analytic atlas on

.

MI1(L) It follows that is a Banach manifold over K By [15; 3 4 . 1 0 1 , every closed subspace E of L having finite dimension or finite codimension belongs to MB(L) In particular, the so-called projective space

.

MI1(L)

.

P(L) := { E C L : dim(E) = 1 }

is an open and closed subset of MI1(L) and hence a Ranach manifold. In case L = Dn is finite-dimensional, we write 'n-1 (D) := P(Dn)

.

3.13 LEMMA. mapping

Put

Lx := L \

: LX

1T

defined by n(k) := k D connected component of PROOF.

.

{O}

,

P(L)

+

,

Then the surjective

is analytic. ML(L)

Further,

.

For a fixed vector

o

E

P(L)

is a

consider a splitting

Lx

such that 0

Put

P

:=

PF n P(L)

(?)

=

.

,

p := pD,F

and

n

:=

{(t)

LX :

is an open neighborhood of diagram

E

o

E

2 Tl

:= L ( D , F ) = F

.

Then

# 0 }

and there is a commuting

BANACH MANIFOLDS

47

3.13.1 where Since

h

-'

is analytic. hn is connected if L f R

I ) ( ~ )

L~

:=

Hence

,

is analytic. is connected.

'TI

P(L)

O.E.D.

We will now describe an important class of bianalytic automorphisms of the Grassmannmanifold M = ME(L) associated with a Ranach space L For any element H E M , the image

.

.

space gH under g E GI1(L) is a split subspace of L Hence r(g)H := gH defines a bijection of M For any splitting 3.8.1 of L there is a commuting diagram

.

.

for all z E L(E,F) Since Int(g)z := gzg-' ,L(gE,gF)) is a chart of M it follows that ( 'gF E ,gF r(g) is bianalytic. The automorphism r(g) E Aut(M) is called the collineation on M induced by g It is clear that the mapping where

.

is a homomorphism. The projective space P(L) C M is invariant under all collineations. Now consider the chart (P,p,Z) of M defined by P := PF , p := pE,F and 2

:=

L(E,F)

.

For

consider the open subset il

of

2

.

9

:=

{ z

E

2

: cz+d

E

GR(E) }

Then there is a commuting diagram

48

SECTION 3

a - z

I

g# where g # ( z ) :=

(az+b)(cz+d)-'

3.14.1

.

is the Moebius transformation associated with g The connected complex manifolds of dimension 1 are called Riemann surfaces. By the Riemann mapping theorem [27; VII.4.21, every simply-connected Riemann surface is biholomorphically equivalent to the complex projective space P1(C) = P(C2 ) , to the Gauss plane C or to the open unit disc A

:=

{

z

E

IzI

C :

< 1 }

.

We will now show that the biholomorphic automorphisms of these manifolds can be described in terms of collineations and Moebius transformations. Consider the collineation homomorphism

with kerne 1

Let (P,plC) be the canonical chart of complex line

o := Then

P = M

\

{O]

0 (c) E

M

M := P1(C) about the

.

and

.

Via the mapping p -1 : C + M we can for all z E C identify C with an open subset of M , and the "Riemann sphere" C U { m } can be identified with M , the point m corresponding to the line By 3.14.1, the

(t) .

BANACH M A N I F O L D S

collineation

r(g)

49

associated with a matrix

has the local representation g#(z = (az+b respect to the chart (P,p,C) of M . 3.15

PROPOSITION.

Cx : = C \

For

Aut(CX) = { z

+

az : a

have z

E

+

a

- : a

cX 1

E

where the semi-direct product of the additive group and the multiplicative group G(C) is defined by Z2 = 2 / 2 2 the action u*a = j‘(a) Here j a ) = l/a is the inversion mapping.

.

and cons der a sequence (zj) in has no Cx converging to 0 Since the sequence ( g ( z i ) ) cluster point in Cx I it has a subsequence conveGging to 0 Since z + g(z)-l belongs to Aut(CX) I we may or to m Consider the Laurent expansion assume ( g ( z j ) ) + 0 Let

PROOF.

g E Aut(CX)

.

.

of

9

about

127; V 1.211

.

.

0 The Casorati-Weierstrass Theorem implies that g and the automorphism

-1 z + g z ) of C x have no essential singularity at 0 Hence g(z) = zn *p(z) , where p is a polynomial with p(0) # 0 Since (g(zj)) + 0 , it follows that n > 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 <

T

is also

Us(R)

.

}

5.5 PROPOSITION. Let B C C be open subsets of a complex Ranach space 2 such that R := dist(f3,aC) > 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

<

t

Since t h e polynomials

g

Cx : larg(5)I

E

logA : SiA

,

AxA

for all

G

E

Then

.

s

is a subgroup of

G

.

d

6

<

define

g

Let

logA(g)

E

.

g

E

w h i c h is

G(A)

satisfy

G

In particular,

is a

G

Banach L i e g r o u p i n t h e norm t o p o l o g y whose L i e a l g e b r a c a n b e

PROOF.

.

g

i d e n t i f i e d with

For every p a i r

r

the analytic action

of

G(A)

: = i f

E E

m, n

i s a c l o s e d s u b s p a c e of

Emln

( 1 5 ; 44.21

g

if

implies t h a t

f(g,g-l)

Since

f(g,g

= -1

G =

o )

for all

,

m+n < d

consider

on t h e R a n a c h s p a c e s

E

: f(g,g

-1

)

= 0

G(A)

f

E

,

belongs to

G

g

E

1

G

Theorem i f and o n l y

and e v e r y p a i r ,n it follows t h a t

(m,n)

.

for all

(m,n)

}

F~

G ( A ) : r ( g ) F m , n = Fm,n

Now c o n s i d e r t h e Ranach s p a c e E :=

for all

a n d t h e Hahn-Ranach

E

= r(g)f(e,e)

{ g

with

NxN

E

Then

d e f i n e d by 6 . 2 9 . 2 . Fm,n

(m,n)

L~+"(A,K)

.

119

INTEGRATION OF LIE ALGEBRA ACTIONS

of all continuous (m+n)-linear mappings : Ax..,xA .+ K For 1 < p < m and 1 < v n , define continuous unital homomorphisms

O(Zl I

. . .

IZ

U

x I . . . , zmPW18

tYWv?.

.

#Wn)

.

For any fixed x E A , the operators T (x,e) , Tpv(e,x) for UV p < m , v < n commute. By [17; 5.4, 15.4 and S171, the operator m n

satisfies

1 u p - 1 8,

CiE(Tx) C {

U

: aU,Bv

E

CA(x) }

V

.

7.14.1

.

Let P : g ( A ) .+ gi(Em,, ) denote the differential of r The Banach space Em,n can be realized as a closed subspace of E in a natural way and 6.29 implies p(x) = TxlEm Now suppose g E G satisfies sup larg EA(g)l < n/d Put x := logA(g) E g ( A ) Then 2.11 implies B y 7.14.1, it follows that sup IIm CA(x)l < n/d sup IIm EiE(Tx)l < n Since 2.12 implies exp(Tx)Fmln = exp(px)F = r(g)Fm,n = F m,n m, n p(x)FmIn = Tx(Fm,n) C Fm,n Hence x E g O.E.D.

.

. . .

.

7.15 EXAMPLE. Let Banach algebra over

Z K

.

be a (not necessarily associative) Then the group

.

of all continuous automorphisms of Z is algebraic of degree < 2 and is therefore a Ranach Lie group in the operator norm topology. Its Lie algebra can be identified with the closed subalgebra

120

SECTION 7

.

of all continuous derivations of Z Now suppose 2 is associative. Let Lxz := x z and R z := zy denote the left and right multiplication operators Y on 2 , respectively. Then g(Z)

3 z

+

ad(z) := L Z - R Z

E

aut(2)

is a continuous Lie algebra homomorphism. unital, G(Z)

3

g

+

Ad(g) := L R - l 9 9

E

In case

7.15.1 2

is

Aut(Z)

defines an analytic group homomorphism whose differential can be identified with 7 . 1 5 . 1 . NOTES. Theorem 7.14 is due to L. Harris and W. Kaup C611. For a special case, cf. [ 2 1 ; Ch. 111, 9 6, no9, Corollary 11. As shown in [ 6 1 ; Example 2 1 , linear groups defined by polynomial equations of unbounded degree are not necessarily Lie groups. As an application of Theorem 7 . 1 4 , it is shown in [ 6 1 ; Theorem 21 that the group K of all (surjective) linear isometries of a complex Banach space 2 is algebraic of degree $ 2 and hence a real Banach Lie group provided the open unit ball D of Z is homogeneous under biholomorphic transformations. For arbitrary Banach spaces, K need not be a Lie group (cf. C 61; Example 61 ) . The example 7 . 1 2 is due to K.H. Hofmann C 6 5 ; p . 2 - 0 8 1 . Algebraic groups play a prominent role in classical Lie group and symmetric space theory, since Lie groups and transformation groups on symmetric spaces are often given by algebraic equations. Further, the algebraic framework makes it possible to generalize geometric results (for real or complex base fields) to more general fields. For an account of algebraic group theory in the context of Jordan algebras (cf. Sections 19 and 2 2 ) , see 1 1 2 5 1 .

SUBMANIFOLDS AND QUOTIENT MANIFOLDS

8.

121

SUBMANIFOLDS AND OUOTIENT MANIFOLDS

For a given Banach manifold H it is often convenient to consider two classes of Ranach manifolds naturally associated with H , namely submanifolds (defined, e.g., by analytic equations) and quotient manifolds (under equivalence relations). These "derived" manifolds are of particular importance in case H is a Ranach Lie group. It will he shown later that symmetric Ranach manifolds can always be realized as quotient manifolds of Ranach Lie groups. Before introducing submanifolds and quotient manifolds, we study two types of analytic mappings needed for their "local" description. 8.1 DEFINITION. Suppose g : M + N is an analytic mapping between Banach manifolds and let T,(g) : Tm(M) + T (N) be gm Then the differential of g at m E M

.

(i)

g

is called an immersion at

m

injective and the image space (closed) subspace of T (N) : gm (ii)

if Tm(g) is Tm(g)Tm(M) is a split

g is called a submersion at m if T,(g) surjective and the null-space Ker Tm(g) subspace of T,(M)

.

is is a split

The mapping g is called an immersion (submersion) if g is an immersion (submersion) at every point m E M Locally, immersions and submersions can be characterized as follows.

.

THEOREM. Let g : M + N be an analytic mapping between Banach manifolds. Then the following statements hold: 8.2

(i)

g is an immersion at m E M if and only if for every chart (P,p,Z) of M about m there exists a chart (Q,q,W) of N about g(m) such that 2 is a split subspace of W and there is a commuting diagram

SECTION 8

122

>Q

P'

for a suitable neighborhood (ii)

P'

of

m

E

P n g -1 ( 0 )

.

g is a submersion at m E M if and only if for every chart (Q,q,W) of N about g(m) there exists a chart (P,p,Z) of M about m such that W is a split subspace of 2 and there is a commuting diagram

where

g#

is a continuous projection onto

W

.

Suppose first that qogop-I coincides with a linear mapping g # E L(Z,W) on a neighborhood of 0 E 2 , Then the diagram PROOF.

commutes. Hence g is an immersion at m if g # is the inclusion mapping of a split subspace 2 of W , whereas g is a submersion at m if g# is a continuous projection onto a split subspace W of 2 Now suppose g is an immersion at m Let (Q1,ql,W1) be a chart of N about g(m) and put g# := qlogop-1 By assumption, there exist a Ranach space W and a linear isomorphism y : W + W1 such that Z is a split subspace of W and gi(0) = y l Z Define

.

.

.

.

for all ( z , z ' ) in a neighborhood of ( 0 , O ) E W = ZxZ' , Then c is analytic and c'(0,O) = y By 1.23, there exist open neighborhoods D of ( 0 , O ) E W and Q of g(m) E Q,

.

123

SUBMANIFOLDS AND QUOTIENT MANIFOLDS

such that

c : D

+

ql(Q)

q := c-'oql : Q g(m) such that

+

D

is bianalytic.

defines a chart

Then

(Q,q,W) of

.

about

N

in a neighborhood of 0 E z g is a submersion at m Let (P1,pl,Z1) be a chart of M about m and put -1 g# := qogopl By assumption, there exists a projection

for all

z

.

N o w suppose

.

y

: Z1 + Z1

g;(0)ly(Z1)

such that Ker(y) = Ker(gi(0)) and is an isomorphism onto W Define

.

for all z in a neighborhood of analytic, and the derivative gi'0' c'(0) = (

0

0

I:( id

0

.

Z1

E

Y(Z1) Ker(y)

Then

c

is

1 +z

is an isomorphism. By 1.23, there exist open neighborhoods P of m E P1 and €3 of 0 E 2 such that c : pl(P) + R is bianalytic. Then p := copl : P + B defines a chart (P,p,Z) of M about m such that

for all

(" )

(z)

in a neighborhood of

W'

8.3 COROLLARY. Let g : between Aanach manifolds. (i)

N Then

M +

g is an immersion at m exist open neighborhoods

E

Z

.

Q.E.D.

be an analytic mapping

if and only if there of m E M and Q of g(m) E N and an analytic mapping f : Q + M such that g(P) C Q and fog = idp M

E

P

.

(ii)

g is a submersion at m E M if and only if there exist an open neighborhood Q of g(m) E N and an analytic mapping h : Q + M such that h(g(m)) = m

124

SECTION

and

goh = id

a

Q .

PROOF. Suppose first that g has a "left inverse" Then T (f)oTm(g) = id m satisfying fog = idp gm Tm(g) is injective, Tgm(f) is surjective and

.

Tm(g)oTgm(f) : T

gm

(N)

+

f

.

at Hence

Tgm(N)

is a continuous projection with image Tm(g)Tm(M) and nullspace Ker T (f) Therefore g is an immersion at m gm Now suppose that g has a "right inverse" h at m satisfying h(gm) = m and goh = id Q ' Then T (g)oTgm(h) = id Hence Tm(g) is surjective, T (h) is m gm injective and

.

.

.

T (h)oTm(g) : Tm(M) gm

3

Tm(M)

is a continuous projection with null-space Ker Tm(g) and image T (h) Tgm(N) Therefore g is a submersion at m gm Conversely, suppose g is an immersion or a submersion at m By 8.2, there exist charts (Pl,p,Z) of M about m and (Ql,q,W) of N about g ( m ) with g(P1) C Q1 such that there is a commuting diagram

.

.

.

g

P? p(P1)

-

+ilq q(Q1)

I

g#

where

g # is (the restriction of) a linear mapping. In case g is an immersion, we may assume that Z is a split subspace of W and g# is the inclusion mapping. Let C$ : W Z be a continuous projection onto 2 and define +

f := p -1 o+oq : Q

Q

-*

M

,

is an open neighborhood of g ( m ) E Q1 such that Let P be an open neighborhood of m E P such that g(P) C Q Then + o g # = idz implies

where

+ ( q Q )C p(P)

.

.

fog1 p = P-lo+og#oPl p = idp ,

SUBMANIFOLDS AND QUOTIENT MANIFOLDS

125

In case g is a submersion, we may assume that W is a split subspace of Z and g# : Z + W is a continuous projection onto W Let rl : W + Z he the inclusion mapping

.

and define

where

Q

is an open neighborhood of

q(Q) C p(P)

.

Then

g#oq

=

idw

g(m)

E

Q,

such that

implies

O.E.D.

8.4

COROLLARY.

Consider a commuting diagram M

,N K

for Banach manifolds

M,N

and

K

.

Then the following

statements hold: If f is a surjective submersion and h is analytic, then g is analytic. If h is also a

(i)

submersion, then

g

is a submersion.

(ii)

If f is continuous, g is an immersion and h is analytic, then f is analytic. If h is also an immersion, then f is an immersion.

(iii)

The composition of analytic immersions (analytic submersions) is an analytic immersion (analytic submersion).

(iv)

If g is analytic and a submersion at m E M , then g(M) is a neighborhood of g(m) E N In particular, every analytic submersion is an open

.

mapping. PROOF. 8.5

Apply 8 . 3 .

DEFINITION.

Q.E.D. A

subset

N

of a Ranach manifold

M

is

SECTION 8

126

called a submanifold if for every point o E N there exist a chart (P,p,Z) of M about o and a split subspace W of 2 such that p(NnP) =

wA

P(P)

.

8.5.1

For a submanifold N of M , the charts ( N n P , p ( N n P , W ) satisfying 8.5.1 constitute an analytic atlas on N It follows that N is a Ranach manifold in the topology induced from M The inclusion mapping i : N + M has the local representation

.

.

NnP-P

i

Therefore i is an analytic immersion and the tangent space To") can be identified with the split subspace To(i)To(N) of To(M)

.

8.6 LEMMA. Suppose N is a submanifold of a Ranach Then, f o r manifold M with inclusion mapping i : M + N any Banach manifold K , a mapping f : K + N is analytic if and only if iof : K + M is analytic.

.

PROOF. Since N is a Banach manifold in the topology induced from M and i is an analytic immersion, the assertion follows from 8.4.i. Q.E.D. 8.7 PROPOSITION. Suppose g : M + N is an analytic Then immersion and a homeomorphism onto N' : = g(M) a submanifold of N and the mapping g : M + N' is bianalytic.

.

.

N'

PROOF. Let m E M By 8.2, there exist charts (P,p,Z) M about m and (Q,q,W) of N about g(m) such that g(P) 0 , Z is a split subspace of W and the diagram

of

c

is

127

SUBMANIFOLDS AND QUOTIENT MANIFOLDS

commutes.

Since

g

is a homeomorphism onto

.

,

N'

we may

.

Then q(N'r\Q) = q(gP) = p(P) Let assume g(P) = N ' n Q II be a continuous projection from W onto Z Then the open neighborhood Q' := { n E Q : n(q(n)) E p(P) ] of g(m) E Q satisfies q(N'nQ') = p(P) = 2 n q ( Q ' ) It The tangent space follows that N' is a submanifold of N

.

.

.

satisfies T (q)T (N') = 2 = Tm(p)Tm(M) gm gm =

Tm(qOg) Tm(M)

=

Tgm(q) T,(g)

Tm(M)

.

It follows that T,(g) : Tm(M) + T (N') is an isomorphism gm Ry 4.1, the mapping g : M + N' is for every m E M

.

bianalyt ic.

Q.E.D.

8.8 PROPOSITION. Suppose g : M + N is an analytic mapping between Banach manifolds. Let N ' be a submanifold of N and suppose g is a submersion at every point m E M' := g -1 ( N ' ) Then M' is a submanifold of M , g : M' + N' is an analytic submersion and

.

PROOF. By assumption, there exist a chart (Q,q,W) of N about g ( m ) and a split subspace W' of W such that By 8 . 2 , there exist a chart (P,p,Z) q(N'r\Q) = W'nq(Q) of M about m such that g(P) C Q , W is a split subspace of 2 and there is a commuting diagram

.

PI where

g#

is a continuous projection onto

W

.

Then

2 ' := W'@Ker(g#) is a split subspace of 2 satisfying p(M'AP) = Z'np(P) It follows that M' is a submanifold

.

12 8

SECTION 8

of

M

with tangent space given by 8.8.1.

commutes, the restricted mapping submersion.

g : M'

Since the diagram

+

N'

is an analytic O.E.D.

Suppose f : M + N is analytic and is a submersion at every point of the fibre

8.9 COROLLARY. n E N If f

.

M I := f-l(n) =

is a closed submanifold of satisfies T,(M') = Ker Tm(f) M'

M

.

m

E:

M : f(m) = n

and every

m

E

1 ,

M'

8.10 EXAMPLE. Let E be a real Hilbert space with scalar Then f(h) : = (hlh) defines an analytic product (hlk) mapping f : E + R with f'(h)k = 2(hlk) Hence f'(h) E L ( E , R ) is surjective for h # 0 and the null-space Chl = { k L E : (hlk) = 0 } is a split subspace of E

.

.

Hence f i s an analytic submersion on Hilbert sphere S :=

{ h

E

E : (h(h) = 1

is a closed submanifold of

E

1

=

E \ {O}

f

-1

.

.

By 8.9, the

(1)

.

8.11 DEFINITION. A subgroup K of a Ranach Lie group called a Banach Lie subgroup if K is a submanifold of

G G

8.12

is

.

LEMMA, Let G be a Banach Lie group with Lie algebra g Then every Banach Lie subgroup K of G is closed and a The Lie Ranach Lie group in the topology induced from G algebra of K can be identified with the closed subalgebra

.

PROOF. As a submanifold, closed in G [ 6 3 ; 1.2.11.

.

K is locally closed and hence Consider the commuting diagram

SUBMANIFOLDS AND QUOTIENT MANIFOLDS

129

rG

GxG G ixi

KxK

r

Ti K

I

K

where rG and r K denote the product mappings and i is the inclusion mapping. Since ‘G is analytic, 8 . 6 implies that ‘K is analytic. Identify Te(K) with a subspace of Te(G) via Te(i) Then every X E k satisfies pe(X) E Te(K) , since t + exp(tX) is an analytic curve in K Conversely, satisfy pe(X) 6 Te(K) Let y : R + K he the let X E g Then 6.22 analytic homomorphism satisfying TO(y) = pe(X) implies y(t) = exp(tX) for all t E R Hence X E b

.

.

.

.

.

.

Q.E.D.

8.13

PROPOSITION.

A

closed subgroup

K

of a Ranach Lie

group G is a Banach Lie subgroup if and only if the closed subalgebra k:=

{ X

E

g

: exp(tX) E K

f o r all t

is a split subspace of the Lie algebra g every neighborhood T of 0 E b , exp(T) of e c K .

E

R }

of G , and f o r is a neighborhood

PROOF. Suppose first that K is a Ranach Lie subgroup of G Then there is a commuting diagram

.

k

.1

pe Te(K)

C C

Te(G)

where pe i s a Ranach space isomorphism. Since Te(K) is a is a split split subspace of Te(G) , it follows that k The second assertion follows from the fact subspace of g b is the Lie algebra of K Conversely, suppose K that is a closed subgroup of G having the properties stated T above. By 7 . 8 , there exists a unique Hausdorff topology

.

.

on K such that (K,T ) is a Banach Lie group with Lie By assumption, T is the topology induced algebra k from

G

.

.

Now consider a splitting g =

hem

and define an

SECTION 8

130

analytic mapping

@

:

g

+

G

by 8.13.1

$(X+Y) := exp(X) exp(Y)

.

Then To($) = p, is an for all (X,Y) E m x k isomorphism. By 4.1, there exist open neighborhoods N of 0 E k , T of 0 E m and S o of e E G such that @ : NxT + S o is bianalytic. By assumption, there exists an open neighborhood S of e E S o such that exp(N) 3 S n K Then

.

Since K is a subgroup of submanifold of G

G

.

,

8.13.2 implies that

K

is a O.E.D.

In order to introduce "quotient" manifolds which will be of particular importance in later sections, consider a Ranach manifold M and let R C MxM be an equivalence relation on M identified with its graph. Let M/R be the set of all equivalence classes of M with respect to R Consider the canonical projection n R : M + M/R and endow M/R with the quotient topology. Then 1~ is continuous. For k E {1,2} let nk : R + M denote the projection from R C MxM onto the k-th factor. Since R is a symmetric relation, there is a commuting diagram

.

R

j .R M

where j e A u t ( M x M ) is defined by j(ml,m2) = (m2,ml) €or all ml,m2 e M The following result, known as Godement's Theorem, characterizes the equivalence relations R for which the quotient space M/R can be endowed with the structure of a Banach manifold.

.

8.14 THEOREM. Let R be an equivalence relation on a Ranach manifold M Then the following conditions are equivalent:

.

SUBMANIFOLDS AND QUOTIENT MANIFOLDS

(i)

R

is a closed submanifold of

project ions (ii)

nk : R

+

M

131

and the

MxM

are analytic submersions;

is a (Hausdorff) Ranach manifold such that the M/R projection n R : M + M/R is an analytic submersion.

In this case the manifold structure on determined by the condition that submersion.

nR

M/R

is uniquely

is an analytic

The proof, following [121; LG, 111.121, is divided into several steps. Suppose first that (ii) holds. Then the "diagonal" A : = { ( x , x ) : x E M/R } is a closed submanifold of M/R x M/R Since n R x n R * M x M + M/RxM/R is an analytic -1 (A) submersion by assumption, 8.8 implies that R = ( n R x n R ) is a closed submanifold of M x M satisfying

.

-

T(m,n) (R) = (T,( =

{ ('v,w) E T,(M

-1 R lXTn ('R) xTn(M)

(x,x ( A

: Tm(nR)v =

.

Tn nR)W

1

.

For every Here x : = nR(m) = nR(n) for all (m,n) E R v E Tm(M) , there exists w E Tn(M) such that R since Tn(nR) is surjective. Since (v,w) E T (m,n) Ker Tn(nR) is a split subspace of Tn(M) , it follows that Ker T

(m,n)("l) = { (v,w) = {0} x

E

T (m,n 1

(R) : v = 0 1

Ker Tn (n,)

.

is a split subspace of T (R) It follows that (m,n) n l : R + M and, by a similar argument, n 2 : R + M are analytic submersions. The proof of the converse imp icat on is based on the following lemmas. (i) *(ii) 8.15 LEMMA. There is at most one manifold structure on M/R such that n R is an analytic submersion.

is PROOF. Suppose that the quotient space N := M/R Hausdorff and carries two manifold structures, denoted by

SECTION 8

132

N1 and N-l , respectively, such that n R is an analytic submersion. Applying 8.4.i to the commuting diagrams n

it follows that N 1 = N-1

id

.

:

Na

N

is hianalytic, i.e.,

-a

Q.E.D.

8.16 LEMMA. Suppose open mapping and M/R PROOF.

+

Let

P C M

satisfies (i). Then is a Hausdorff space.

R

be open.

Then 8.4.iv

nR

is an

implies that

-1

n R ( n R P ) = nl((MxP)n R)

is an open subset of M since n 1 : R + M is an analytic submersion. Hence n R is an open mapping. Now suppose m,n E M are not equivalent, i.e., (m,n) p! R Since R is closed, there exist open neighborhoods P of m E M and 0 of n E M such that ( P x Q ) n R = $ Since n R is an open mapping, it follows that n R ( P ) and n R ( Q ) are disjoint open

.

.

neighborhoods of nR(m) and nR(n) , respectively. quotient topology on M/R is Hausdorff.

Hence the

O.E.D.

If

P

is an open subset of

M

,

define

R p := R n ( P x P )

.

.

8.17 LEMMA. Suppose R satisfies (i) Then M has a covering by open subsets P such that P / R p is a Ranach manifold and the canonical projection n

: P + P/Rp

RP

is an analytic submersion. PROOF.

Let

o

H

:=

E

M

.

Since

n2

is an analytic submersion,

Ker T (0,o)(n2 1 = { (v,w)

is a split subspace of

T(o,o)(R)

,

E

T ( 0 , o( )R )

: w

= 0)

which in turn is a split

SUBMANIFOLDS AND QUOTIENT MANIFOLDS

subspace of projection

To(M)xTo(M) from

Q

.

133

Hence there exists a continuous

To(M)xTo(M)

onto

H

.

Write

a(v,w) = (al(v,w),O) I where al : To(M)xTo(M) + To(M) linear. Then B(v) := al(v,O) defines a continuous projection from To(M) onto E

:= { v

E

To(M) : (v,O)

E

T( 0 1 0 )( R ) }

is

.

Therefore F := Ker(6) is a split subspace of To(M) , and hence there exist an open neighborhood Q of o E M and a submanifold N of Q containing o such that F = To(N) Since n l is an analytic submersion, 8.8 implies that

s

:=

.

( N x Q ) n R = ( n l I R Q )-l(N)

is a submanifold of

containing

RQ

( 0 ~ 0 )

such that 8.17.1

It follows that we have an injective mapping

T(o,o)(n,lS) : T ( 0 1 0 ) ( S )

+

To(M)

.

8.17.2

.

Since n 2 is an analytic Now suppose w E To(M) submersion, there exists v E To(M) such that Then (Bv,O) E T(o,o)(R) and hence ( v ~ w )E T ( 0 1 0 ) (R)

.

(V-8ViW) = (V,W) belongs to

T

(S)

-

( B v ~ O )E T(o I 0 ) ( R )

by 8.17.1.

A

(FxTo(M))

is

It follows that 8.17.2

(010)

an isomorphism. By the inverse mapping theorem 4.1 there exist open neighborhoods P1 and P2 of o E Q such that n 2 : S A (P1xP1) + P2 is bianalytic. Therefore P2 C P1 and there exists an analytic mapping 0 : P2 + N n P l such that (g(m),m) E R f o r a l l m E P2 For m E N n P 2 I the points (m,m) and ( g ( r n ) , m ) belong both to S A ( P ~ X P ~ )

.

.

134

SECTION 8

$

1

N A P , = id

.

8.17.3

Since N n P2 is open in N n P1 , it follows that P := $-l(Nn P,) n P2 is an open neighborhood of o E M As a consequence of 8.17.3, we have $(P) = N A P Since $(m) = $(n) implies (m,n) E R , there exists a commuting diagram

.

.

where J, i s surjective. Now suppose m,n E N n P satisfy $(m) = $ ( n ) Then the points (m,n) and (m,m) belong both to (NAPxNAP)A R C S Hence m = n and 6 is bijective. Define a manifold structure on P/Rp such that J, is bianalytic. Then n is an analytic submersion, RP since I$ is an analytic submersion on P by 8.17.3.

.

.

Q.E.D.

The proof of 8.14 can now be completed as follows: Suppose R is an equivalence relation on M satisfying (i). By 8.17, there exists an open covering {Pi} of M such that for every i , the quotient space Mi : = Pi/Rp

i carries a manifold structure such that the canonical projection

is an analytic submersion. Ry 8.4.iv, Qi := n open subset of M The commuting diagram

.

-1

(xRPi) is an

8.17.4

SUBMANIFOLDS AND QUOTIENT MANIFOLDS

135

defines a bijective mapping

a i * Define a manifold structure on the open subset nR(Qi) of M/R such that ai is -1 bianalytic. For Bi : = a onRIQi , the diagram

commutes. Since n1,n2 and n i are analytic submersions, B i is an analytic submersion by 8 . 4 . It follows that is an analytic submersion. Applying 8.15 n R : Qi + vR(Qi) to the open subsets nR(Qi)n n,(Q.)

1

= Qi17 Q./R J

QinQ. 3

it follows that the manifold structures on nR(Qi) and n ( Q ) agree on n,(Qi)nvR(Q.) Since M/R is a R j 3 Hausdorff space by 8.16, it follows that M/R carries a manifold structure such that nR(Qi) is an open submanifold Further, n R is an analytic submersion of M/R for all i by 8.17.4. Q.E.D.

.

.

8.18

EXAMPLE.

Let

L

he a Banach space over

D

E

{R,C,B}

and consider the projective space P(L) of all 1-dimensional subspaces of L Then P(L) is a connected component of the (cf. 3.13) and is therefore a Grassmann manifold M P ( L ) Banach manifold over the center K of D Now put L X := L\{O} , DX := D \ { O } and define

.

.

Then R is an equivalence relation on commuting diagram

M := L x and there is a

where n ( k ) := k D is an analytic submersion on P(L) by 3.13, since the mapping J, defined in 3.13.1 has the differential

136

SECTION 8

.

kl = - 1 E Since + is bijective, it -3llows that M/R has a Ranach manifold structure such that 1 1 ~ is an analytic submersion. for a1

THEOREM. Let K be a Ranach Lie subgroup of a Banach Then the quotient space Lie group G with Lie algebra g M := G/K carries the structure of a Ranach manifold such that the canonical projection II : G + M is an analytic 8.19

.

submersion. G acts analytically on M via the left translation action, defined by r(g,hK) := ghK f o r g,h E G Let p : g + aut(M) be the differential of r Then the evaluation mapping p o : g + To(M) at o : = K E M is surjective and has the null-space .

.

.

Ker(po) =

k = { X

E

g : exp(tX)

E

K

for all t

E

.

R }

8.19.1

For any Banach space splitting g = k (3 m , there exists a chart ( P , p , m ) of M about o such that

x

P(ro(exp x)) = for all

X

in a neighborhood of

0

8.19.2 E

m

.

PROOF. The mapping F : GxK + GxG , defined by F(g,k) : = (g,gk) , is analytic and T(e,e)

( F ) : Te(G) x Te(K)

+

is given by ~ ( ~ , ~ ) ( F ) ( v , = u )(v,v+u) injective and the image space

Te(G) x Te(G)

.

Hence

T(e,e)(F)

is

T ( e r e )(F)(Te(G)xTe(K)) = { ( v , w ) E T ~ ( G ) ~ T ~ (:G )w - v E T ~ ( K ) } 8.19.3

is a split subspace of Te(G) x Te(G) , since Te(K) is a Hence F is an immersion at split subspace of Te(G) (e,e) Since the diagram

.

.

137

SUBMANIFOLDS AND QUOTIENT MANIFOLDS

1 1 F

GxK

GxG

g XRk GxK

>

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

<

R

.

and

In

< d/2

Then every

1 -(r (gk,z)-z) - (r#(g,z)-z) k #

I

<

d

10.5.1 z

E

I

r # (g,z)-z

C

satisfies

I

and

PROOF. For (g,z) E SxD define F(g,z) : = r (g,z)-z # Then 10.5.1 implies suppose z E C

.

W

:= r#(g,z)

C1 : = u R / , ( C )

E

.

0 < j < k

,

1.17.1

Now

By 1.11, we have

since this identity holds in a neighborhood of according to 10.2. It follows that

For

.

0

E

C n p(P)

implies

.

This since C1 is convex and satisfies dist(Cl,aD) > R / 2 implies the first assertion, and the second assertion is an O.E.D. immediate consequence. 10.6

is a topological group r on a connected Then G has no small subgroups, i.e., manifold M exists a neighborhood S of e E G such that every of G contained in S is trivial. COROLLARY.

Suppose

G

a faithful locally uniform action

.

Consider a local representation of r with respect to a chart ( P , p , Z ) choose C C D as in 10.5. Then PROOF.

r# : of

M

admitting Ranach there subgroup

S + om(D,ZC)

about

o

and

LOCALLY UNIFORM TRANSFORMATION GROUPS

T := { g

E

:

S

159

(r#(g)-idlD < R/4 }

.

is a closed neighborhood of e E G Now suppose H Then 10.5 implies subgroup of G contained in T

.

is a

.

k > 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 <

T

.

Then af =

IT r#(t)'dt

A

E

0

is invertible, since 10.8.3 Define

h

E.

C

om(C2,2

IT-1 oaf-e

implies

by

)

I A

h(w) : = af(w)

-1

(r#(T,w)-w)

< 1 c2

.

. 10.8.4

IrxC , Then 10.8.2 implies If Is1 is small, we have r#(s,C2) C D and hence 10.2 implies r (t,w ) = r (s+t,w) # s # where w S : = r#(s,w) It follows that Now suppose (t,z) w := r#(t,z) E C 2

E

.

.

=

Is+T r#(t,w)dt - IT

-

Is

r#(t,w)dt =

r#(t,w)dt = IS(r#(T+t,w)

0

Is+T r # (t,w)dt

T

0

S

,

-

r#(t,w))dt

0

.

Therefore 1 lim -(f(ws)-f(w)) s+o

= r#(T,w)

- w ,

10.8.5

161

LOCALLY UNIFORM TRANSFORMATION GROUPS

Define a continuous mapping 1

A(u) : =

A : C2

f'(w+t(u-w))dt

0

+ L(Z

C

)

by

.

A(w) = f'(w) e G & ( Z C )

, and [28; 8.14.31 implies It follows that A(ws) E G L ( Z C ) f(w S )-f(w) = A(ws)(ws-w) if I s 1 is small and 10.8.4 and 10.8.5 imply

Then

.

It follows that the mapping ITxC 3 (t,z)

is differentiable in equa t ion

+

t

r # (t,z)

E

C2

and satisfies the differential

a r (t,z)

h(r (t,z))

=

10.8.6

#

at

.

with initial condition r#(O,z) = z By 5.7, there exists a local analytic flow F : IU xC + C 2 , where 0 < u < T , with infinitesimal generator

The uniqueness theorem for solutions of ordinary differential equations [28; 10.5.21 implies that r# 1 IU xC = F is analytic. Since 10.8.6 implies r (t,z)-z = #

t

E

Ih'lC < +1 exists.

IT

since

that

r#(s,C) C C 1

by 10.8.2.

Since

by 1.13, it follows that the limit 10.8.1

COROLLARY. Banach manifold 10.9

,

,

0

it follows from [28; 8.5.41

for all

It h(r#(s,z))ds

Q.E.D.

Every locally uniform action of M is analytic.

R

on a

SECTION 10

162

DEFINITION. For a locally uniform action r topological group G on a Ranach manifold M , let

10.10

of a

denote the set of all complete analytic vector fields M for which there exists a continuous homomorphism R 3 t + gt E G satisfying

.

r(gt) = exp(tX)

X

on

10.10.1

We say that the continuous 1-parameter subgroup (g,) of G satisfying 10.10.1 is associated with X It is clear that tX E g M whenever (t,X) E R x g M'

.

10.11 COROLLARY. Suppose r is a locally uniform action of Consider a a topological group G on a Ranach manifold M local representation S + Om(D,Z") of r with respect r# to a chart (P,p,Z) of M about o and let C be a convex

.

--

open neighborhood of 0 there exists a mapping

E

D

P # : 9,

such that

.+

dist(C,aD) > 0

.

Then

0JC,ZC)

uniquely determined by the identity (P#X)OP = xp

I

10.11.1

.

holding on p -1 ( C ) For (t,X) E R x g H , we have If (gt) is a continuous 1-parameter p#(tX) = top (X) # subgroup of G associated with X , then (t,z) + r#(gt,z) defines a local analytic flow on C with infinitesimal genera tor

.

X# : = P#(X)

a

.

PROOF. Since C is connected, 1.11 implies that P # is uniquely determined by 10.11.1. Now suppose X E g, and let (gt) be a continuous 1-parameter subgroup of G associated with X Then 10.8 shows that the limit

.

LOCALLY UNIFORM TRANSFORMATION GROUPS

163

p#X : = lirn (r#(gt)-id)/t E c)oD(C,ZC ) t+O Since r#(gt,pm) = p(r(gt,m)) = p(exp(tX)(m))

exists. all m E p-l(C) choice of 10.12

,

(9,)

it follows that

p#X

does not depend on the

and satisfies 10.11.1.

COROLLARY.

For all

X,Y

E

gM

,

for O.E.D.

we have

( X +Y )id = lim (r ( g h )-id)/t # # # t t t+O

E

O-(C,Z C )

and [ X ,Y lid = lim (r#(gth,g_,h-,)-id)/t2

#

#

E

om(C,Z C )

t+ 0

Here ( g t ) and ( h t ) are continuous 1-parameter subgroups of G associated with X and Y , respectively. Put R := dist(C,aD) , d := R/6 and define C k : = ukd(C) We may assume that Ir#(g)-idl,, < d for all Let T > 0 satisfy gshtgslhtl E S whenever g E S For (t,z) E I x C put s,t,s',t' E I T T wt := r#(ht,z) E C 1 and zt : = r#(gt,wt) E C2 Then the mean value theorem [28; 8.6.21 and 5.12.3 imply PROOF.

.

.

.

.

where the limits

and

exist according to 10.8.

and

Since [28; 8.5.41

implies

.

S E C T I O N 10

164

< a 1 (r#(gs)-id( *lhYICl + Ir (h #

c2

the first assertion follows.

Since

[ X ,Y lid(z) = hi(z)hy(z) #

1 ; i lhYIC2

s

#

-

,

h;(z)hX(z)

.

1.13 implies

[X ,Y ]id E om(C,Zc) For (t,z) # # ut := r#(hqt,z) E C 1 , v t : = r#(gVt,ut) E C 2 w t := r#(ht,vt) E C3 and zt : = r#(g,,wt) E C4 Taylor's formula [28; 8.14.31, applied to p = 2 imply I

I (r#(g,htg_,h-,,z)-z)/t2

E

. ,

I xC

put

T

Then and 5.12.2

- [X#,Y#lid(z) I

1

t + 0

For on

C

,

,

the last expression converges to

0

as a consequence of 5.12.5.

, uniformly O.E.D.

10.13 LEMMA. Suppose r is a locally uniform action of a topological group G on a Banach manifold M Consider a C local representation r S + Om(C,Z ) of r with respect # . to a chart of M about o and let B be a convex open neighborhood of 0 E C such that R := dist(R,aC) > 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

E

om(Bl,Z C

and

, the mean value theorem implies

Irh(s,z)-rh(s,w)I

Iz-WI

10.13.2

(l+(slo)

.

for I s ( < 'I and z,w E B We may assume U T < d suppose g n E G and k E R are sequences as above. n tn : = t/qn E I T and hn : = (r#(gn)-id)/tn E o , ( C , Z C )

.

Now Put

.

Since

kn/tn

+

1

and therefore

Ih

fn : = r#(gn)

or

, I

I

+ 0 for B1 For n < u for (almost) all := r (t ) , it follows that n h n

10.13.1

implies

< d/qn

Ifn-idl B1

We now show by induction that

Ih-h

.

.

k fn(R) C B1

n +

m

10.13.3

and

whenever 0 < k < qn , Th s is clear for k = 0 , and assuming the assertion for some k < qn , it follows that fk+l is well-defined on R and 10.13.3 implies n

166

S E C T I O N 10

+ (fn-idIB k < d/qn + kd/qn Since k+l < qn hand, we have

,

this implies

fk+'(B)C

R1

.

. On the other

and

O.E.D.

10.14 THEOREM. Suppose r is a locally uniform action of a Hanach Lie group G on a Ranach manifold M Then there exists a locally uniform action r* of the Lie algebra of G on M , called the differential of r , such that the following diagram commutes

.

4

A

r*

aut(M)

PROOF. F o r (t,X) E R x g I rx(t) := r(exp(tX)) defines a locally uniform action of R on M By 10.8, rx defines a global analytic flow on M Let r,(X) E aut(M) be the

.

.

LOCALLY UNIFORM TRANSFORMATION GROUPS

infinitesimal generator of

rX

commutes and

. .

167

Then the diagram 10.14.1

r*(tX) = t r*(X) In order to show that is a homomorphism of real Lie algebras, C consider a local representation r # * S + O o J ( D , Z ) of r Let B with respect to a chart (P,p,Z) of M about o r* : g + T ( M )

-

.

be a convex open neighborhood of 0 E D such that R : = dist(B,aD) > 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

<

*

Assume t h a t

PROOF.

Q.E.D.

such t h a t

s\sB

E

S5

is c o m p l e t e , t h e a s s e r t i o n f o l l o w s .

B

e # g

If

LEMMA.

k(g) > 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

<

.

(a,)

for all

k(gn)

a s i n 1 1 . 4 and p u t

applied to

Then 10.5,

.

n

d := R/2

Let

Assume f i r s t t h a t

passing to a subsequence, kn

> 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

<

T

c 1 , Then 0 < t <

for all

k(gn)

.

T

By 1 0 . 1 3 ,

follows t h a t

r h ( t ) = l i m r # ( gqnn ) E O , ( R , Z n+m qn in S t h e sequence (gn ) B

By 1 1 . 3 ,

gt

:=

qn lim g n n+-

E

S

r # ( g t ) = r h ( t ) on

Since

r#(gt)

+

idR

E

Om(R,Z

1

R

,

converges.

Put

B *

,

it follows t h a t

whenever b o t h s i d e s are d e f i n e d . C

.

C

Further,

11.7.2

gt + e E G

i.e.,

-

gsgt

.

gs+t

implies

Hence t h e r e

exists a unique e x t e n s i o n t o a continuous 1-parameter group of

C,

,

a g a i n d e n o t e d by

r ( g t ) = exp(t)o

for a l l

(gt)

.

t E R

Let

.

X E g

Then

satisfy

h o p = Xp

on

it

ANALYTIC TRANSFORMATION GROUPS

p

-1

(B)

,

h =

i.e.,

p

#

.

X

the above arguments s h o w that h/2a = Then 2aX E g and h = p#(2aX)

.

g

11.8 COROLLARY. mapp i ng g

3 x

.

a > 1

Now assume

p

#

X

179

€or some

X# : = P # ( X )aG

X

E

4

.

O.E.D.

T(M)

is a real subalgehra of

+

Since

and the

T(C)

E

is a homomorphism. PROOF. Given X,Y r(g,) = exp(tX) , imply

E

g

,

define gt,ht E G by r(ht) = exp(tY) Then 10.12 and 11.7

X# + Y

.

#

= p#(E)

a az

and

a

[X# I Y # I = p#(F) az for uniquely determined vector fields that the identities

E,F

9

E

.

It follows

and

.

hold in a neighborhood of 0 E C Since M 3.1 implies X+Y = E E 9 and [X,Y] = F E 9

PROOF.

Put ‘I

exp(tX) = r(g,) :=

and

sup { s > 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\ <

.

'I

By 11.3, the limit gt

lim gt n n+m

:=

E

sB

.

exists and 11.10.1 implies is a topological group and

Ir (gt)-idl Bltl Since G sft - s R t - g n gn for every n , it gn follows that gs+t - gsgt whenever both sides are defined. Hence there exists a unique extension to a continuous 1-parameter subgroup of G , again denoted by (g,) X E g satisfy r(gt) = exp(tX) for all t E R

.

11.10.2

where

cn

+

0

.

For

.

IP#X--hnlB < c 11.11 X

+

Ip#XIc For

t

It1 <

+

0

,

is independent of o

E

M

,

let

Go

Let Then

'I

it follows that

Hence 3.1 implies

The topology of

LEMMA.

PROOF.

o <

implies for

.

g

p#X = h

.

O.E.D.

induced by the norm

(P,p,Z)

and

C

.

denote the set of all quadruples

a = (P,p,Z;C) , where (P,p,Z) is a chart of M about C admitting a local representation * S + Om(D,Z 1 and r# is a convex open neighborhood of 0 E D such that

-

.

o

C

dist(C,aD) > O Let [ X I u = IXlp,C : = IP#XIC be the associated Banach space norm on g Now suppose

.

.

Choose convex open u = (Pj,p.,Z;C.) E Go for j = + 1 j 3 3 0 E C such that there exist analytic neighborhoods B j of j mappings f : R + C satisfying -j j j

.

Then

182

SECTION 11

( f j ) * P +# ( X )

on

f.(R-j) 3

.

= P$X)

It follows that

Since fj(B-j) is a neighborhood of mapping theorem [ 1 5 ; 48.11 implies

the symbol 0

Eo

E

1.1

-

-

0

C

E

,

j

Ranach’s open

.

denoting equivalence of norms on 9 For N be the set of all points m E M such that

, let I * I ‘I

.

for all T E Em part of the proof. Now suppose

o E N by the first and let (P,p,Z;C) E En For every m t p-’(C) , q : = p - p(m) and R := C - p ( m ) defines an element (P,q,Z;R) E Em with -1 [*IqlR Since p ( C ) n N # , it follows that p (C) C N I showing that N = M Q.E.D. ~

.

.

PROOF.

Then n E

.

The automorphism

r(g),

of

leaves

T(M)

g

invariant, since f o r every continuous 1-parameter subgroup (gt) of G t + ggtg -1

,

associated with X E g , the assignment defines a continuous 1-parameter subgroup of

which satisfies = exp(t*r(g),X) with

r(g),X

E

representation

G

r(ggtg -1 ) = r(g)exp(tX)r(g)-l for all t E R and is therefore associated

.

g

Now let

r# : S

(g(P),pog-’,Z;C) E E :r : g ’Sg + O_(D,Zcy

(P,p,Z;C)

+ Om(D,Z

C

.

Put

E

Lo

,

with local

m : = g(o)

,

.

with local representation defined by ri(h) : = r#(ghg-’)

Then

.

Since the diagram

commutes, the assertion follows from 11.11. 11.13

COROLLARY.

4

O.E.D.

is a real Ranach Lie algebra and the

ANALYTIC TRANSFORMATION GROUPS

4

canonical action p of topologically faithful. PROOF.

Ry 11.3.1,

on

183

is analytic and

M

4.6.2 and 1.13, we have

I ~ # [ x r yI HI <

2

Ip#xIC

IP#YI~

I.IPrR

Since the norms I *Ip,c and on g are equivalent, 4 is a Ranach Lie algebra. For +(X,z) := (z,p#(X)z)

, the diagram gxp

-1

(C)

f

T(p-’(C))

commutes. Since + is analytic by 1.24, it follows that acts analytically on M B y definition of the topology of g , this action is topologically faithful.

.

g

O.E.D.

11.14

Suppose

THEOREM.

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 connected Ranach manifold M Then there exists a on G such that ( G , T ) becomes a real Hausdorff topology Ranach Lie group with Lie algebra on M via r

.

PROOF.

Apply

g

,

acting analytically

11.12, 11.13 and 7.4.

O.E.D.

For transformation groups on finite-dimensional manifolds, the conclusion of 11.14 can be made more precise by showing that the topology 1 coincides with the given The proof is based on the following lemmas. topology on G

.

11.15

LEMMA.

dim(g) < +-

.

Suppose

M

is finit -dimensional.

Then

SECTION 11

184

.

a closed neighborhood of 0 E g If (X,) is a sequence in T then hn : = p # X n is a bounded sequence in is

.

om(Cl,ZC )

Ry Montel's Theorem, we may assume that h := lim h n+m

exists.

It Eollows that

n

E

om(C,Z

C

)

is a Cauchy sequence in

(X,)

g

which is convergent by 11.10. Hence T is compact. It follows that g is locally compact and hence finitedimensional. O.E.D. 11.16 LEMMA. Suppose M be a sequence in S B \ {e)

is finite-dimensional. Then the sequence

.

k(g n )(r # (g n )-id)

C

ooo(C,Z

E

has a convergent subsequence with limit PROOF.

For

g

E

SB\{e}

and

k

:= k ( g )

< 21r#(g k )-id\

klr#(g)-idl c1

h # 0

,

Let

(4,)

)

.

10.5 implies

< R/2

c1

and

Here

C 1 := uR,2(C)

.

By Montel's Theorem, the sequence C o(C1,Z ) has a convergent subsequence in

k(gn)(r#(gn)-id) E Since the compact-open topology, with limit # 0 relatively compact in C1 , the assertion follows.

.

C

is Q.E.D.

11.17 THEOREM. 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 o f G on a connected manifold M of finite dimension. Then C, is a real Lie group of finite dimension in the given topology with L i e algebra g

.

PROOF. Since dim(g) < +m by 11.15, it suffices to show that exp(T) is a neighborhood of e E G whenever T is a By 11.15, 11.9 and 5.6, we may neighborhood of 0 E 4

.

ANALYTIC TRANSFORMATION GROUPS

185

assume that T = -T is compact, that exp(T) c S there exists an analytic mapping F : T + O m ( R , Z c ~

.

and that satisfying

F(Y,z) = r (expY)(z) E C for all ( Y , z ) E T x R Now suppose # there is a sequence gn E S \ exp(T) with gn + e Put B k n := k(gn) By 11.16 and 11.7, we may assume that there exists a non-zero vector field X E 9 such that

.

Since

p#X = lirn kn(r (g )-id) # n n+-1 gn & exp(T) and the sets

O,(C,Z

E

T

C

)

.

.

11.17.1

and

{ gn : n E N } u {e} are compact, a continuity argument shows that there exists a > 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

<

12.2.3 o

>

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 <

and ‘I

< 1

.

Then

= 6(0

lSD(O,Z)

Since

PROOF.

< ~(O,T)

6(0,a)

< 1 }

6

is continuous and invariant under rotations,

.

12.4.1 implies 6 D ( ~ , z ) > 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 <

s < 1 }

is a domain in Z and, for every complex Ranach space the restriction mapping pw is surjective. PROOF.

Define a holomorphic mapping 0 : CxZ + Z (u,gv) for all z = (u,v) E Z Then N = @(]O,l]xM) =

u

,

by

.

@ ( s , z ) :=

W

@({s}xM)

O<s
is a domain.

The set

Q :=

{

(s,z)

E

CXxM : O ( 5 , z )

E

M

}

open and, for every z E M , the connected 1-component Q; the open set Qz := { 5 E C x : ( 5 , ~ )E Q } is an open annulus. By 5.8, the set no := { (s,z) E CXxM : 5 E Q i } a domain. For f E O ( M , W ) and z = (u,v) E M f Z : Q; + W , defined by holomorphic mapping fZ(';) := f(u,gv) , has a Laurent expansion

,

is

of

is

the

where

is independent of

r

for

0 < r

E

Qz .

R y 1.14, the mappings

S E C T I O N 12

194

are holomorphic.

fn : M + W

0

E

Choose open neighborhoods R of V such that C is balanced and

e E U and C of BxC C M Since

.

f Z

is holomorphic on A for all z E RxC , 1.11 implies fn = 0 whenever n < 0 Ry 12.10.1, we have for n > 0 and

.

5

0

E

Qz

.

n fn(U?CV) = 5 fn(u,v)

12.10.2

.

w = (x,y) E M and let 1 < R E :Q Put Jx) < R } Since f is continuous, there exist for every X E K open neighborhoods D A of w E M and O X of X E C such that E X : = O(QXxDA)C M and f l E X is bounded. Since K is compact, it follows that there exists an open neighborhood D of w E M such that E : = O(KxD) C M and c := I f I E < f m Then 12.10.1 implies

Now fix a point K := {

x

E

c

.

: 1 c

.

< R-"c

for all

n > 0

and hence

.

1 (fnlD < c 1 R-" = C R - ~ / ( R - ~ ) n >m n >m It follows that the series

12.10.3

m

f(z) =

I:

fn(Z)

n=O

converges locally uniformly in norm on domain, the set

is also a domain.

M

.

Since

no

is a

By 12.10.3, the series

converges locally uniformly in norm on a' and hence defines The domain a holomorphic mapping F : n ' + W

.

is hiholomorphically equivalent to

Q'

and

METRIC AND NORMED BANACH MANIFOLDS

195

g(c,u,v) : = F(c,u,v/c) defines a holomorphic mapping g : A + W Choose a non-empty open subset R of M that there exists r < 1 with for all z E R A := { 5 E C : r < 1 5 1 < l/r } C Clz AxB C A and 12.10.2 implies

.

such

.

Then

.

for all ( ~ , u , v ) E A x 0 Hence ag/ac = 0 on A x R and, by Hence 1.11, a g / a r = 0 D := { z E Z : (<,z) E A for some 5 } is a domain containing N and +(z) := g ( c , z ) defines a holomorphic

.

+

mapping

: D +

w

satisfying

+(M =

f

.

0.E.D.

12.11 DEFINITION. A domain D in a complex Ranach space 2 is called circular if 0 E D and eitD = D whenever t E R D is called balanced if AD c D whenever A E C satisfies 1 x 1 < 1 For every circular domain D , the balanced hull

.

.

D" : =

u

SD

O<s
is the smallest balanced domain in

2

containing

D

.

12.12 COROLLARY, Let D be a circular domain in a complex Ranach space 2 with balanced hull D" Then for every f E O ( D , W ) , the power series

.

W

about 0 E D converges locally uniformly in norm on D and the restriction mapping p w : O ( D % , W ) + o ( D , W ) is surjective. PROOF.

Apply 12.10 and 12.10.3 for

U := { O }

.

0.E.D.

12.13 COROLLARY. Let D be a homogeneous circular domain in a complex Ranach space Z with compatible Carathgodory metric bD Then D is balanced. In particular, every bounded

.

homogeneous circular domain in a complex Banach space is balanced.

196

PROOF.

SECTION 1 2

Apply 12.9 and 12.12.

O.E.D.

Another type of extension theorem concerns the so-called "tube domains". Suppose X is a real Ranach space with complexification 2 = XC For any subset R of X , the set

.

TB := B fB

ix = { z

is called the tube with base

E

Z :

R

Re(z)

E R

.

12.14 LEMMA. For j E {1,2} , let r := [O,a.] he the line j 3 segment spanned by 0 and a + E X Let B be an open J subset of X containing the convex hull A of

.

lgl rTA

< lgl

Tr

PROOF. We may assume that al and a2 are linearly independent and embed C2 as a subspace of 2 by identifying (zl,z2) with the vector z l a l + z2a2 For 0 < c < 1/2 , define $ : C 2 + C by 2 2 $(zl,z2) := z1+z2 - c(zl+z2) and put S : = $-'(l-c) Since TZ($) = (1-2czl,l-2cz2) , it follows that $ l Q x C is a By submersion, where Q := { x1 + iyl E C : lxll < 1/2c } 8.9, S1 : = S A ( i 2 x C ) i s a submanifold of Q x C Further, the projection n(z 1 , z 2 ) := z 2 is injective on S1 and TZ(n) : Ker T Z ( $ ) + C is an isomorphism for every z E S1 By 4.1, it follows that D := n(S1) is open in C and the inverse mapping n-l : D + S1 is holomorphic. N o w assume (z1,z2) E S A TA Put z = x + iy Then j j j ' x > 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
tB

+ (1-t)R

.

.

METRIC AND NORMED BANACH MANIFOLDS

199

Further,

c > 0

since otherwise there exists

such that the hounded

holomorphic function z + (f(z)-c)-l on TR has no extension Repeating this argument, it follows that f has an to Tn extension f- E O(TC,C) with

.

O.E.D.

12.17 COROLLARY. Suppose C is a domain in X such that the tube domain D = C @ iX is homogeneous and has a compatible Carathgodory metric 6,, Then C and D are convex.

.

PROOF.

M

Apply 12.9 to

:=

D = TC

and

N : = Tco(C)

12.18 LEMMA. Suppose r : G + Aut(M,d) is an isometric action of a Ranach Lie group G on a metric Ranach manifold (M,d) which is locally some point o E M Then G acts transitively

.

. O.E.D.

analytic connected transitive at on M

.

.

Ry 8.21, the orbit G o is open in M Since is a compatible metric, there exists r > 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

<

ITm(p)vl/2 < bm(v) < 31Tm p)v1/2 Hence b is compatible.

for all O.E.D.

12.32 EXAMPLE. Let G he a Ranach Lie group with Lie Then the left translation action of G on algebra g M := G is analytic, transitive and locally transitive. Further, Ge = {e} Hence every compatible norm on g

.

.

.

induces a left invariant compatible tangent norm on G Now assume that g carries a compatible norm 1 - 1 invariant

.

The action r of under the adjoint action Ad : G + Aut(g) GxG on M := G defined by r(g,h)(m) : = gmh-' is analytic, transitive and locally transitive. Further, the isotropy subgroup Int(G)

K

(GxG), at e E G corresponds to the group of all inner automorphisms of G Identifying :=

.

on Te(G) is Te(G) with g , we see that the norm 1 . 1 invariant under K It follows that G carries a compatible tangent norm which is invariant under left and right

.

translations. NOTES.

For a deeper study of (invariant) metrics on complex

manifolds, cf. C94,43,114,145-1541. The extension theorems for holomorphic functions on circular domains (12.12) and tube domains (12.16) are due to H. Cartan and S. Bochner, respectively (cf. [16,1091).

208

13.

SECTION 13

GROUPS OF HOLOMORPHIC ISOMETRIES

famous theorem of H. Cartan (cf. [ 1 0 9 ; Ch. 9 I ) shows that the group Aut(D) of all biholomorphic automorphisms of a bounded domain D in Cn is a real Lie group (of finite dimension) with respect to the compact-open topology. In this

A

section, the results of Section 11 will be applied to generalize this theorem to bounded domains in complex Ranach spaces. In this respect, the essential feature of bounded domains is the existence of a compatible metric invariant under Aut(D) In fact, our result holds in the more general setting of metric complex Ranach manifolds. The compact-open topology is of minor importance in the infinite-dimensional setting and will be replaced by the so-called "topology of locally uniform convergence".

.

Let D be a domain in a complex Ranach space Z and consider the Ranach space Om(D,W) of all bounded holomorphic mappings from D into another complex Banach space W Every non-empty open subset R of D induces a norm ) * I R on Om(D,W) defined by

.

A

subset

F of

s called bounded if

Om(D,W) SUP

feF

If

D < + -

.

13.1 LEMMA. Let R and C be open balls in D with Then the norms dist(R,aD) > 0 and dist C,aD) > 0 and I induce the same topology on every bounded subset F of O,(D,W)

.

Ic

PROOF.

every

(*IR

.

Since F-F is bounded, it suffices to show that for a > 0 there exists 6 > 0 such that

whenever u E C

.

dist(B,aD)

.

Put B = B(u;r) and assume first that f E F Then R(U;rl) C c for some r1 < r Put d : = and

r2 : = r+d/2

.

.

Now apply Hadamard's three-

GROUPS OF HOLOMORPHIC ISOMETRIES circles theorem [27; VI.3.131 +(f(u+Xv)) , where and v E 2 has n o r m < 1

x

+

.

209

to the functions

f

,

4 E U W , C ) has norm < 1 Using the Hahn-Ranach theorem E

F

[15; 40.101, we get

where

a = (log r 2 - log r)/(log r 2 - log r l ) and b = (log r - l o g rl)/(log r 2 - log r l ) Since F is bounded, it follows that 13.1.1 holds if C contains the

.

Now consider the set N of all points center u of A u E D such that 13.1.1 holds for some and hence every) open ball R about u satisfying dist(R,aD > 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

<

(M,d)

+-

)

on

.

Aut(M,d)

,

defined by

dp(f,g) := sup d mEP 13.2 DEFINITION. Let (P1,prZ) be a chart of M about o such that p(P1) is a bounded domain in 2 , and , on PI , the metric

d

is uniformly equivalent to the metric An open neighborhood P of o E P1

(m,n) + Ipm-pnl

.

called an admissible such that

p-ball about

o

if there exists

is (J

> 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)

<

d ( o ,m)

.

Q

0

contains

P A Q

13.3.2

<

< a

40

.

By

assumption

since

d(g(n),g(m)) =

and h e n c e

2a

,

a

> 0

on

Pl

Now l e t

he given

,

there

such t h a t

0

E

PI

.

W := P n Q

Define

it f o l l o w s from 13.3.2

.

Applying 13.1 t o

that there exists

p

>

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

<

t h e r e exists

P

=+ I P 0 9 - P l p B

>

0

with

<

'I

B < u

such t h a t

0

such

GROUPS OF HOLOMORPHIC ISOMETRIES

211

for all m,n E P1 ' Hence the assertion 13.3.1 follows in case PAQ # fl Now consider the set N of all points o E M such that 13.3.1 holds f o r all admissible halls P about o Then Q C N by the first part of the proof. Now let P be an admissihle ball containing a point m E N Let R be an admissible ball about m Then 13.3.1 holds

.

.

.

.

for the pair (Q,R) and, since PnR # , the metrics d p and d R are uniformly equivalent. It follows that P C N and

N

is open and closed.

The topology on

.

Q.E.D.

induced by the metrics

dp

N

Hence

Aut(M,d)

=

M

for admissible balls P in M is called the topology of This topology is locally uniform convergence on M metrizahle, Hausdorff and independent of P In case r) is a hounded domain in a complex Ranach space 2 , endowed with the compatible Carathgodory metric 6 D , this topology on coincides with the topology of locally Aut(D) = Aut(D,G,,)

.

.

uniform convergence on the hounded subset Aut(D) of O,(D,Z) For a connected complex manifold M of finite dimension, endowed with a compatible metric d , the topology of locally uniform convergence on Aut(M,d) coincides with the compact-open topology, since every compact subset of M can be covered by finitely many admissible balls. The topology of locally uniform convergence has important topological properties.

.

PROOF.

Let

.

be an admissible ball in M Then + d (g,id) = d (h,id) + dp(g,id) and d P (goh,id) < dp(goh,g) P = dp(g,id) for all g,h E G : = Aut(M,A) Hence dp(g -',id) the multiplication and inversion mappings are continuous at -1 idM E G By 13.3, the mappings g + f gf are continuous at idM since we may assume that f(P) is an admissible hall -1 with respect to the chart (f(Pl),pOf 1 2 ) and = df(p) (g,id) Since the metrics d p on G dp(f-'gf,id) P

.

.

.

are left-invariant, it follows from [ 2 1 ; III.1.1.(1)1

that

212 G

SECTION 13 is a topological group.

13.5 PROPOSITION. structure.

O.E.D.

Aut( M , d )

is complete in the left-uniform

PROOF. Since Aut(M,d) is metrizable, it suffices to show that every Cauchy sequence (gj) converges, Let P be an

and let u > 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)

<

and

,

}

T

with

13.6.1

. .

by r#(g,pm) := p(g(m)) Note define r# : S 7 + O,(D,Z) that g(P) C P,- and g(o) E P for all g E S 7 since 7 < 0 Sr is a closed neighborhood of idM E G For

.

every

a

>

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}

<

(iii) The semi-order (iv)

X

(x+y) > 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 <

cn lsin

XI

n=l 14.31

THEOREM.

1

m

n <

n=l Suppose

c 1

c

=

.

.

Since

arc sin(1) =

n

c

> o

n

for

5. O.E.D.

is an associative unital complex

2

Ranach algebra. Let X = H ( 2 ) denote the real Ranach space of all hermitian elements in Z Then the set

.

x,

= H+(2)

:=

{ x

E

x

x

E

X

{ x

E

of all positive elements X , and the interior C = X + A G(2) =

> 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

<

Sz(z)I

14.35.1

IZJ

and sup (C,(X)l

=

.

Sz(x)I = 1x1

sup

14.35.2

PROOF. Since S z is a compact convex set in the weak * topology and the evaluation mapping S 2 3 f + f(z) is affine and continuous, S 2 ( z ) is a compact convex subset of c . Now let W be a maximal abelian subalgebra of 2 containing z Then C ( 2 ) = Cw(z) by [17; 15.41. Hence

.

2

for every s E E,(z) there exists a unital homomorph sm @ : W + C with Q(z) = s Since 1 4 1 = 1 by [17; 16.31 , the Hahn-Ranach theorem [15; 44.21 implies that there exists a state f E S 2 with flW = Q Hence E,(z) C S 2 ( z ) For 14.35.2, use 14.30. O.E.D.

.

.

14.36

COROLLARY.

For

x,y

.

E

H(2)

,

we have

Ix+iyl > 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

<

.

This contradiction shows

s = 0

.

.

Since

and

I Z I

= IL;I

z

,

*I

is a norm on 2' which coincides on 2 with the original norm. For x E 2' and z E 2 , we have xz E 2 and hence

for all

E

2

it follows that

lLiZ12 =

(XZl2 =

I(xz)*(xz)l

=

2

.

< Iz*I*Ix*xI*(zI =

IZI

Ix*xI

Iz*(x*x)zl

2

It follows that every x E 2' satisfies 1x1 c Ix*xI < I x * I * I x I , hence 1x1 < Ix*l and therefore ( x I 2 = Ix*xI , showing that the involution on 2' is a C*-involution. In order to show that 1 - 1 is a complete norm on 2 ' , consider the left translation homomorphism L ' : 2 ' + ~ ( 2 ), defined by

2 50

SECTION 1 5

.

x + L' Since L' is isometric, the image L'(Z) is X complete and hence closed in L ( 2 ) Therefore L'(Z') = L'(Z) @ K*idZ is also closed in L(Z) and is therefore complete. This shows that Z' is a Banach algebra.

.

Q.E.D.

LEMMA. Every C*-involution on an associative Ranach L, and R, belong algebra Z over K is -hermitian, i.e., to ui(Z) whenever z* = -2 15.7

.

s unital. Put PROOF. By 15.6, we may assume that 2 w : = exp(z) E 2 Since the involution is continuous, we get

.

w* = exp(z*) = exp(-z) = exp(z

- L-1w.

.

Therefore lwI2 = (w*w( = le( < 1 Therefore and exp(RZ) = Rw E U k ( 2 exp(LZ) = Lw E U k ( 2 )

Q.E.D.

15.8 COROLLARY. Every C*-involution on a comp ex associative Banach algebra 2 is hermitian and H ( 2 ) = { x E 2 : x* = x }

.

PROOF. We may assume that Z is unital. By 15.7, every self-adjoint element is hermitian, showing that the involution is 'hermitian and hence hermitian. By 14.29 2, every hermitian element is self-adjoint. Q.E.D.

Note that the identity mapping is a C*- nvolution on , considered as a real Ranach algebra, which is not 'hermitian. Hence 15.8 is not true for K = R . 2 = C

DEFINITION. An associative Ranach algebra 2 over endowed with a C*-involution, is called a C*-algebra if

K

15.9

x = x*

E

2

Cz(x) C R

.

15.9.1

By 15.8, the spectral condition 15.9.1 is redundant for K = c By 15.6 and 14.19, the unitization 2 ' = 2 @ K non-unital C*-algebra Z over K is a C*-algebra.

.

15.10

LEMMA.

Suppose

Z

is a C*-algebra over

,

K

and

of a

W

is

251

C*-ALGEBRAS

a closed *-subalgebra of PROOF. For x = x * E W C \ C,(x) is connected. Cw(x) = C,(x)

.

2

.

Then

W

is a C*-algebra.

, we have

C,(x) c R and hence Therefore [ 1 7 ; 5.141 implies O.E.D.

15.11 EXAMPLE. Let E be a Hilbert space over D E {R,C,E} and consider the associative unital Banach algebra Z := L(E) over the center K of D , endowed with the operator norm and In case D = C , 15.8 the C*-involution defined in 15.2. implies that Z is a complex C*-algebra. In case D = R , Ec : = EORC is a complex Hilbert space and Zc can be identified with L (Ec) Hence x = x * E 2 implies

.

showing that Z is a real C*-algebra. In case D = H , E can be regarded as an even-dimensional complex Hilbert space, denoted by 15.11.1

such that

u

H

carries a conjugation u + and a b z = t L -) : a r b E L (H) } -b a is a purely real closed unital *-subalgebra of L(EC)

-

-

.

au : = au for all a E L(H) and u E H The can be identified with L(EC) complexification 2' above, it follows that 2 is a real C*-algebra.

.

.

Here

As

15.12 EXAMPLE. Let S be a set and consider the associative unital Ranach algebra 2 := B ( S , D ) over the center K of D E {R,CIE} I endowed with the supremum norm and the C*-involution defined in 15.3. In case D = C , 15.8 implies that 2 is a complex C*-algebra. In case D = R , 2 is a real C*-algebra since 2' = S ( S , C ) In case D = H , consider the associative unital complex Ranach algebra A := B ( S , L 2 ( C ) ) of all bounded mappings f : S + L 2 ( C ) , endowed with the supremum norm (with respect to the operator

.

SECTION 1 5

252

norm on L z ( C ) ) and the C*-involution f + f* defined by f*(s) := f ( s ) * for all s E S Then A is a complex C*-algebra by 15.8. and s E S Then a

.

z = t L

Put

-

.

a(s) : = a(s)*

for all

a

E

B(S,C)

b

-)

: a,b E B ( S , C ) } -b a is a purely real closed unital *-subalgebra of A whose can be identified with A Hence Z complexification Zc is a real C*-algebra. qpplying 15.10, it follows that the Ranach *-algebras L m ( S , D ) , C m ( S , D ) and C u ( S , D ) I associated with a measure space S , a topological space S and a locally compact space S , respectively, are C*-algebras over K

.

.

15.13 K , z

E

LEMMA. Suppose W is an abelian C*-algebra over In case K = R , assume in addition that z * = z for all W , Then the Gelfand homomorphism

is an isometric *-isomorphism.

.

PROOF. By 15.9.1, Q is a *-homomorphism into C u ( C w , K ) Hence 15.5 implies that Q is an isometric embedding onto a closed *-subalgebra A of c u ( C w , K ) separating the points of Cw The Stone-Weierstrass Theorem [ 2 8 ; 7.31 implies A = cu(CW,K) O.E.D.

.

15.14 K

.

COROLLARY.

The involution of a C*-algebra

Z

over

is hermitian.

PROOF. For x = x* E Z , consider the closed *-subalgebra W of 2 generated by x By 15.10, W is an abelian C*-algebra over K , Applying 15.13 and [ 1 7 ; 5.141, we get

.

15.15 COROLLARY. An associative Ranach algebra Z over K endowed with a C*-involution, is a C*-algebra if and only if

,

*

C -ALGEBRAS

253

> 0

CZ(Z*Z)

15.15.1

.

In particular, in case for all z E Z C*-involution is positive.

K = C!

,

every

PROOF. The spectral mapping theorem and 15.15.1 imply Cz(x) C R whenever x* = x Conversely, suppose 2 is a C*-algebra. To show 15.15.1, we may assume that 2 is unital. Then the assertion follows from 15.14 and 14.23.

.

Q.E.D.

15.16 COROLLARY. adjoint part X

Let 2 be a C*-algebra over Then the set

.

x+

{ x

:=

of all positive elements in X satisfying

x+= { z

If

x

L

: x

has a unit element C := X + n

X

x }

E

e

G(2) =

x

E

is

{ x

a

{ z*z:

=

,

: C,(X)

K

with self-

> 0 ]

closed convex cone in

2

E

2

}

.

15.16.1

the set E

X : C,(x)

> 0 }

of all strictly positive elements in X is a topologically regular open convex cone in X containing e The order unit norm on X coincides with the given norm.

.

I -Ie

PROOF. In case 2 is unital, the assertion follows from 15.15, 14.16 and 14.24. Now suppose 2 is non-unital and let x E X+ Then x = (y+s) = y 2 + 2sy + s2 , where is self-adjoint. Hence s2 = 0 , i.e., y + s E 2' s = 0 Thus 15.16.1 holds also in the non-unital case.

.

.

Q.E.D.

is a Hilbert space over D E {R,C,E} Let X : = H&(E) denote the self-adjoint part of the unital C*-algebra 2 := L(E) over the center K of D The elements of 15.17

.

EXAMPLE.

.

Suppose

E

254

S E C T I O N 15

are called positive operators on

E

,

and the interior

consists of all strictly positive operators on

E

.

15.18 every

LEMMA. Suppose 2 is a complex C*-algebra. Then for x = x* E 2 , the left and right multiplication If operators Lx and Rx are hermitian operators on 2 x is positive, then Lx and Rx are positive. PROOF.

for all

.

We may assume that

t

E

R

.

Hence

Lx

2

E

is unital.

.

~ a ( 2 )

Ry 15.7,

Since [17; 5.41

2 (Lx ) = C,(x) , it follows that Lx E ~ a + ( z ) if x positive. The analogous statements for Rx follow by C!i

considering the "opposite" C*-algebra of

2

.

implies is O.E.D.

By 15.11 and 15.10, every closed *-subalgebra 2 L ( E ) , for a Hilbert space E over K E {R,C} , is a

of

.

C*-algebra over K Conversely, the Gelfand-Neumark embedding theorem asserts that every C*-algebra 2 over K can be realized as an operator algebra, i.e., as a closed *-subalgebra of L ( E ) for some E

.

Let 2 be a C*-algebra over K , Then there exist a Hilbert space E over K and an isometric *-isomorphism n : 2 + L ( E ) onto a closed *-subalgebra of 15.19

THEOREM.

L(E)

By 15.6, we may assume that 2 is unital. and 14.26, there exist a Hilbert space E over K *-homomorphism n : 2 + L ( E ) such that PROOF.

.

By 15.14 and a

(nzl = ~ z * z I ~ ' =~ I z I for all z E 2 Hence n is isometric and n ( 2 ) is a closed *-suhalgebra of L ( E )

. O.E.D.

C * -ALGEBRAS

15.20

COROLLARY.

C*-algebra PROOF. 15.21

255

The complexification

of a real

2'

is a complex C*-algebra.

2

O.E.D.

Apply 15.19, 15.11, and 15.10. LEMMA,

.

Let

L

be a Hilbert space over

D E {R,C,H} Then a linear operator g E GQ(L) is g*g = id, isometric if and only if g is unitary, i.e., Y

.

In particular, UL(L) = { g

GQ L ) : g*g = id L ]

E

15.21.1

is a real Ranach Lie group in the operator norm topology whose Lie algebra can be identif ed with the closed real subalgebra U ( L ) := {

x

x* + x

gQ(L) :

6

of all skew-adjoint operators on L real Ranach Lie subgroup of GQ(L) PROOF.

Every unitary operator

g

E

.

.

= 0

}

Further,

GQ(L)

UQ(L)

satisfies

is a

lgul

2

= (gulgu) = (g*gulu) = ( u ( u ) = Iu

therefore isometric. g

E

UL(L)

for all u E L and is Conversely, every isometric operator

satisfies

(gulgu) =

ulu)

for all

u

E

L

.

Polarizing this identity (in case D # R 1, we get 15.21.1. It follows that Ui(L) is a real algebraic subgroup of degree

.

< 1 in (9,g-l) Now apply 7 . 1 4 . The last assertion follows from the fact that uR(L) is a split real subspace of gn(L)

.

In case

O.E.D.

L = Dn

Uin(D) := UQ(Dn) 15.22

COROLLARY.

is finite-dimensional, we write

and

uQn(D) : = uQ(Dn)

Let

2

.

be a unital C*-algebra over

K

.

Then

.

is the group of all unitary elements in 2 In particular, U(2) is a real Ranach Lie group in the norm topology whose Lie algebra can be identified with the closed real subalgebra

SECTION 15

256

u ( 2 ) :=

{ x

E

of all skew-adjoint elements in 2 real Ranach Lie subgroup of G ( 2 )

g(Z) : x* + x = 0 }

. .

Further,

is a

U(2)

PROOF. By 15.19 and 15.21, an element g E G ( Z ) satisfies lgl = Ig-ll < 1 if and only if g is unitary. Now apply 7.14. The last assertion follows from the fact that u ( 2 ) is O.E.D. a split real subspace of g(Z)

.

By 15.22, exp(u(2)) any unital C*-algebra 2 more is true:

is a neighborhood of

over

K

.

e

for For abelian C*-algebras, E

U(Z)

15.23 LEMMA. L e t W be an abelian unital C*-algebra over K Then the identity component of the Ranach Lie group

.

U(W)

coincides with

exp(u(W))

.

PROOF. The identity component of U ( W ) consists of all exp(xn) with elements of the form exp(xl) exp(x2) n E N and xl, xn E u ( W ) Now the assertion follows O.E.D. from 2.8.2.

...,

...

.

We are now going to prove two fundamental geometric results about unital complex C*-algebras, namely the Russo-Dye Theorem ( 15.24 and 15.25) and the Vidav-Palmer Theorem (15.27). The proof given here makes decisive use of holomorphic mappings and shows how infinite-dimensional holomorphy can be applied to problems in functional analysis. 15.24 THEOREM. Let 2 be a unital complex C*-algebra with Then open unit ball R and unitary group U ( Z ) -B = co U ( Z )

.

.

B C co U ( 2 )

PROOF. It suffices to show that the mapping

is holomorphic for

( c ( < (B)-'

.

.

For

8

E

R

Ry 1.12, it follows that

,

C -ALGEBRAS

6 = g (0) = B

For

b

E

2

,

1

d5 =

257

2n gB(e it )dt

1

.

15.24.1

0

consider the vector field

Then 5.23 implies that

,

$(b) := exp(Xb)(0) = tanh(bb*)1'2 ( b b * )2'1

15.24.2

defines a real-analytic mapping $ : C + 2 for some open neighborhood C of 0 E 2 Since $ ' ( O ) = idz , we may assume that $ : C + D is bianalytic, where D is an open neighborhood of 0 E B For B = $ ( b ) E D , 5.23 implies Now let f : 2 + 2 gB(5) = exp(Xb)(ge) whenever ( 5 1 < 1 be a real-analytic mapping vanishing on the closed real For every z E M , b - z b * z submanifold M := U ( 2 ) of 2 Hence is contained in TZ(M) = { v E 2 : v*z + z * v = 0 }

.

.

.

.

.

xh is tangential to M , and 5.1.1 implies f(gB(5)) = 0 whenever I 5 1 = 1 Applying this to f ( z ) : = e - z z * and f(z) : = e - z * z , we get

.

gs(5)

whenever for all

(51 =

6

E

I3

1

.

E

15.24.3

U(2)

and I s ( is small. By 1.11, this is true Now the assertion follows from 15.24.1. O.E.D.

15.25 PROOF.

COROLLARY.

B

=

co(exp u ( Z ) )

.

By 15.24, it suffices to show that U ( Z ) C co(exp u ( Z ) ) Let E U(2) and 0 < s < 1 Let W be the abelian unital C*-algebra generated by B For ( 5 ) E U(W) Since 151 = 1 , 15.24.3 implies gS B g o ( 3 ) = ge E exp(u(W)) , 15.23 implies ( 5 ) E exp(u(W)) 's8 By 15.24.1, we get sB E co(exp u(W)) For s + 1 , the assertion follows. O.E.D.

.

.

.

.

.

.

The Russo-Dye Theorem, in its stronger form 15.25, will now be applied to obtain a metric characterization of C*-algebras known as the Vidav-Palmer Theorem. Suppose in the

258

SECTION 15

following that Z algebra such that 15.26 x,y

LEMMA. E

H(2)

,

is an associative unital complex Ranach C

~ ( z )

2 =

.

The mapping x + iy + x - iy , for is an algebra involution of Z

.

h 2 E Z has a unique x,y E X Since h and

PROOF. For any h E X := H ( 2 ) , decomposition h2 = x + iy with h2 commute, 14.29.2 implies hx

-

xh = i(yh-hy)

E

.

X A iX = { O }

. .

Therefore x commutes with h2 and hence with y be a maximal abelian subalgebra of 2 containing x y Then

Let and

.

XZ(h

2

) =

CW(h

2

) =

{ f(x) + if(y) : f

E

Cw }

W

.

Now 14.30 implies f(x) E R , f(y) E R and ~ R Hence f(y) = 0 for all f E C w , zZ(h 2 ) = ~ , ( h ) C showing that C,(y) = Cw(y) = { O } By 14.30, y = 0 It follows that h E X implies h 2 E X and, consequently, xy + yx E X whenever x,y E X Further, i(xy-yx) E X by Then 14.29.1. Now let a = x+iy , b = h+ik E A

.

.

.

.

ab =

Similarly,

.

+ b*a*

=

(x+iy)(h+ik) + (h-ik)(x-iy)

(xh+hx)

-

(yk+ky) + i(yh-hy)

(ab-b*a*)/i

E

X

.

+ i(xk-kx)

Now (ab)* = b*a*

ab = (ab+b*a*)/2 + i(ab-b*a*)/2i

E

.

X

follows from

. O.E.D.

15.27 THEOREM. An associative unital complex Ranach algebra 2 is a C*-algebra if and only if Z = H ( ' 2 ) ' In this case the involution is given by (x+iy)* = x-iy for

.

x,y

E

ff(Z)

.

.

PROOF. Put X := H ( 2 ) Then for every unital complex C*-algebra 2 , X is the self-adjoint part of 2 by 15.8.

c - -ALGEBRAS

259

Hence Z = Xc and the involution is given by (x+iy)* = x-iy C Conversely, suppose Z = X By 14.36 for all x,y E X

.

.

and 15.26, the canonical involution of Z is a continuous algebra involution which is hermitian by 14.30. By 14.27, there exist a complex Hilbert space E and a unital *homomorphism II : 2 + L ( E ) satisfying 14.27.1 and 14.27.2. It follows from 14.36 that II is a homeomorphism onto a unital closed *-subalgebra n ( 2 ) of L ( E ) By 15.10, n ( 2 ) is a C*-algebra. Now assume z E 2 satisfies lnzl = 1 By 15.25, there exists a sequence zn E co(iX) such that Therefore z n + z and hence I z I < 1 = 111zl n(zn) + n ( z ) It follows that since lexp(ix)l = 1 for all x E X NOW assume I z ( < 111zl for I z ( < 111z1 for a 1 z E z z*I < III(z*)\ = ( ( n z ) * l , 14.27.2 implies some z Since

.

.

.

.

.

(IT212 =

z*zI

<

.

<

IZ*I.IZI

I(IIz)*l*(IIzl

= lnzl

2 I

O.E.D.

a contradiction.

*

NOTES. The theory of (complex) C -algebras is by now a major branch of functional analysis, with many applications to operator theory, mathematical physics and the theory of group * representations c29,30,118,22,120 1. Real C -algebras have been considered in connection with "continuous geometries" C141 and occur also in the study of "reversible" Jordan operator algebras C133,134 1. The idea of using Moebius transformations for the proof of the Russo-Dye Theorem 15.24 is due to L. Harris C571, cf. also 117; 5 381. A similar proof applies to the Russo-Dye * Theorem for Jordan C -algebras due to J. Wright and M. Youngson C1571 and to the more general version involving (circular) bounded symmetric domains D with non-empty extremal boundary S C911. More precisely, it is shown in C91; Theorem 5.91 that 6 is the closed convex hull of any connected component of S . The Vidav-Palmer Theorem 15.27 can also be generalized to give a metric characterization * of Jordan C -algebras C159,107,1171.

260

16.

S E C T I O N 16

TUBE DOMAINS AND SIEGEL DOMAINS

The open unit disc A in C is biholomorphically equivalent to the upper half-plane via the Cayley transformation 3.18.2. Using this fundamental fact, many geometric and analytic properties of A (e.g., geodesics in hyperbolic geometry and fundamental domains for discrete groups of automorphisms) can 'be conveniently realized on the upper halfplane. Generalized half-planes, the so-called tube domains and the more general Siegel domains play also an important role in complex analysis for several complex variables (cf. [112]). In infinite dimensions these domains are moreover closely related to operator theory [ 5 9 ] . In this section, we introduce tube domains and Siegel domains in Ranach spaces. The corresponding Cayley transformations will be studied in Section 21. In the following, let U , V and W be Ranach spaces over K e {R,C} and put 2 := UxVxW Suppose U is endowed with a continuous involution u + u* (i.e., a K-antilinear homeomorphism of period 2 1 . Then the self-adjoint part

.

x

:=

{ x

E

is a closed real subspace of U called the real part of u e U open cone in

X

DC :=

.

{ u

1

u

: x* = x

.

and Re(u) : = (u+u*)/2 is Now let C he a connected

Then E

U : Re(u)

E

C }

.

is a domain in U , called the tube domain with base C If C is convex, then DC is also convex. Now suppose in addition that 0 : VxV + U and Y : vxw + V are continuous sesqui-linear mappings (antilinear in the first variable) which satisfy the identities

and 16.0.2

TUBE DOMAINS AND SIEGEL DOMAINS

261

.

€or all v,b E V and all w E W Further, let B be a domain in W such that for Y(w)b := Y(b,w) I the real-linear operator idv + Y(w) on V is invertible for all w E R Put

.

16.1

PROPOSITION. D

C,@

-

*=

The open subsets

{ (u,v)

UxV : 2 Re(u)

E

-

@(v,v)

}

C

E

and D

~

,

:=

~

of U x V D :=

{ (~ u I v I w ) E UxVxB

,

and

DCI@,R

: 2 Re(u)

C }

E

Z , respectively, are connected. Further, is invariant under the affine transformations

ga,b(ulvlw) := (u + a + @(b,v €or a = -a* E U pair, we have 'a,b

- Re OW(vrv)

and

b

E

+

V ,

-

g a r l h l - 'a+a'+(@(b,b'

b+Y(w)b) ,v + b + Y(w)b,w)

If

(a',b')

is another such

16.1.1

)-@(b',b) )/2,b+b'

In particular, H < : = { g a I b : a * = - a ~ U , b ~ v i}s a group of bianalytic automorphisms of D Further, @(b,b') = @(b',b) implies

.

'a,b

'a',b'

-

'a+a',b+b'

defines an analytic flow on and 'ta,tb infinitesimal generator

a Ya+b := (a+@(b,v))- a + (b+Y(b,w))K au PROOF.

Since

D

with

.

and B are open and the mappings and Y are continuous, 2 . 7 implies that D C I O and open subsets of U x V and Z I respectively. For put

A := idv

C

+ Y(w)

.

Then

Re

I

D

w

E

@

are B ,

262

SECTION 16

Now 16.0.2

implies

Taking the real part, it follows from 16.0.1 that D is invariant under the transformations ga,b The composition formula 16.1.1 follows from 16.0.1 and 16.0.2. Regarding U , V and W as subspaces of Z , we get canonical embeddings DC C DC,@ C D and DC x B C D Therefore H< (DCxB) c D Conversely, suppose (u,v,w) E D Then w E R and

.

.

.

b := -(idV+Y(w)) -1 v is well-defined.

E

.

V

Since

.

(DCxB) Since C , I3 and H < are it follows that D = H < connected, D is connected. Similarly, DC,@ = H < DC is connected. Q.E.D. 16.2

COROLLARY.

The real Ranach space h-l

{ Ya

:=

: a = -a*

E

U

} C aut(D)

is an abelian Lie algebra, h-1/2 satisfies

[h-,,2,h-l,,l

-

*=

c

h-,

{ Yb : b and

E

V } C aut(D)

TUBE DOMAINS AND SIEGEL DOMAINS

:= h-l

h,

@

h-l,2c

263

aut(D)

is a nilpotent real Ranach Lie algebra such that : h,

' e

+

{ a

E

u

a* = -a }

:

x

v

is a topological isomorphism.

PROOF.

Ry 1 6 . 0 . 2 ,

we have

-

[Yb,Yb'I = (@(b,b')

.

for all b,b' E V Hence [ h , , h , ] C h-, abelian, h, is a nilpotent Lie algebra.

a

@(b',b))x

.

Since

E

hql

1 is

O.E.D.

,

D C U x V and C,@ c UxVxW are called Siegel domains (of the first kind, DC,O,B the second kind, or the third kind, respectively). 16.3

DEFINITION.

The domains

DC C [J

We are mainly interested in Siegel domains in complex Banach spaces for which the Caratheodory tangent norm is compatible. Note that for K = C , U can be identified with the complexification

Xc = X fB iX

of

X

and

.

DC = C fB iX

PROPOSITION. Suppose U = Xc is a complex Banach space and C is a topologically regular open convex cone in X Then the tube domain DC is a normed Ranach manifold with respect to the Carathgodory tangent norm. 16.4

.

PROOF. By assumption, the order unit norm ( - 1 to a point e E C generates the topology of X complexification, the state space realized as a subset of L ( U , C ) :=

.

Sx of (X,C,e) By 14.9.2,

.

JUI

with respect By can be

sup If(u)l fE S X

.

defines a compatible norm on U For every f E Sx , F(u) := exp(-f(u)) defines a holomorphic mapping F from D := DC

into the open unit disc

A

,

since

S E C T I O N 16

264

.

for all u E D Since F'(u)h = - F ( u ) f(h) (u,h) E T(D) = D x U , we get

=

exp(-IRe ul)

(h(

€or all

.

Together with 12.23, it follows that the Carathgodory tangent norm 8, is compatible. Q.E.D. 16.5 EXAMPLE. For a unital C*-algebra U over K , the set C of all strictly positive elements in X : = ff(U) is a topologically regular open convex cone (15.16). The corresponding tube domain is DC := { u

E

U : Cu(U+u*)

>

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 ) <~ ~(Y(w)12 < 1

and

.Since Put

16.0.2, the convexity of

C

b := (idv+Y(w))

w

E

8 :=

R

-1

,

v

, (idV-Y(w) 2 )h

.

and the positivity of

@(b,b) - @(Y(w)b,Y(w)b) = @(b,b) -

@(Y(W)

we have

Using @

2

,

brh)

we get

268

SECTION 16

16.11 COROLLARY. Suppose U = Xc , V and W are complex Ranach spaces, C is a topologically regular open convex cone in X , 0 is C-positive and topologically regular, R is bounded and IY(w)( < 1 for all w E R Then the Siege1 . in UxVxW is a normed Ranach manifold with doma in DC , ,

.

respect to the Carathgodory tangent norm. PROOF.

Apply 16.10, 16.9 and 12.25.

O.E.D.

16.12 EXAMPLE. Suppose E , H1 and H2 are Hilbert spaces over D E {R,C,B} and consider the Hilbert sum

for k = 1,2 matrices

.

Write the operators U

z =

cv2

z E 2 := L(F1,F2)

as

V

wl)

'

where u E U := L ( E ) , (vl,v2) E V := L(H1,E) x L(E,H2) and w E W := L(H1,H2) Then 2 = UxVxW Let C be the topologically regular open convex cone of the unital C*-algebra U (over the center K of D ) and define a C-positive mapping 0 : VxV + U by

.

.

Since

.

Hence it follows that 1vI2 < 0 ( v , v ) 6 2(v12 topologically regular. Now define '4 : VxW + V

Then Q(Y(b,w),v) = blw*v 2 + v 1w*b2 satisfied. Further,

.

Q

is

by

Therefore 16.0.2

is

TUBE DOMAINS AND S I E G E L DOMAINS

It follows that w(w) has operator norm carries the compatible norm 16.7.1. Put B : = { w E W : IwI < 1 } Then

< IwI

269

,

where

V

.

U

vl)

D := { (

E

< 1 , Re(2u-Qw(v,v)) >

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<*(DCxB) Since D is also invariant under the action Since D is a domain, it of H< , 16.19.1 implies D = 0 follows that C is connected. Since u + (u,O,O) gives a holomorphic embedding of the domain DC into D , the Carathgodory tangent norm on DC is compatible. The real Banach L i e group for

.

.

acts analytically on DC I and this action is locally transie Then 12.17 and 12.18 imply that HC a c t s

t i v e at

.

(locally) transitively on DC and that C and convex. Hence DC is homogeneous under affine

DC

are

TUBE DOMAINS AND SIEGEL DOMAINS

279

transformations, whereas

C is homogeneous under the group of linear transformations. Since D~ is connected, it follows that

< exp('/rO) >

is also a domain. Since (u,v) + (u,v,O) gives a holomorphic embedding of D' into D , the Carathgodory tangent norm on D'

is compati le.

The real Ranach Lie group :=

H'

acts analytica ly on

D'

< exp(h-l

,

h-1,2

@I

1 ho

fB

)

>

and this action is locally

.

(e,O) E D ' By 12.18, it follows that For every v E V transitively on D' acts ( locally) transitive at

12.10 imply

.

(e+tLa(v,v)/2,v) 2

that @(v,v) + 2e/(t -1) E C we get @(v,v) E Hence

.

E

if

D'

t > 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

<

G :=

g

.

G :=

<

independent of PROOF.

Then t h e o r b i t

>

exp p ( g )

-

Put

p(g) >

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

<

:=

-

containing

N

is a n i d e a l i n

-

.

>

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) : =

<*a,(c)2 maps I, into c n/2'I-x and therefore

ZRZ(vn The real-analytic mappings T K and 2/n u , (u) E B i K -1 T are inverse to each other on an open neighborhood of , + 0 E QZ" By 3.1, T~ is real-bianalytic with inverse By 20.4, the real-analytic mapping

.

.

.

a;':

3

-1 u + p,

( T ~ U )E

is well-defined.

By 5.23 and 20.4.2, in an open neighborhood of 0 E 2

u

for all

u

E

-K

Qz

MK

20.6.1 holds for all By 3.1,

.

and 20.6.1 follows.

O.E.D.

DEFINITION. A hermitian Ranach Jordan triple is called positive if every u E 2 satisfies

20.7

K

Z

A positive hermitian Banach Jordan triple Z over called a JB*-triple if every u E 2 satisfies ( u n u * l = IuI

K

.

2

over

is

20.7.2

For a positive hermitian Ranach Jordan triple Z , we + have Bz = Z and 0 , = { u E 2 : Iuau*I < 1 } If Z is a JB*-triple, then Q 2 coincides with the open unit ball of Z Note the formal analogy between 20.7.1 and 20.7.2 and the "Gelfand-Neumark" axioms 15.1.1 and 15.15.1 for C*-algebras. For any JR*-triple 2 , we have Aut(2) c Ug(Z) and aut(Z) c u g ( 2 ) , Hence l g - = aut(2) C TO(Z)

.

.

.

20.8 u

E

LEMMA, 2

Let

2

K

be a JR*-triple over

.

Then every

satisfies ({uu*u}I = lul

3

.

20.8.1

PROOF. Let W he the closed Jordan subtriple of generated by u and put v := {uu*u} E W Then

.

2

BOUNDED SYMMETRIC DOMAINS AND JB*-TRIPLES

3 since W v o v * I W = ( u o u * ) IW Since 2 is hermitian, we get IVI2 =

337

is abelian by 19.25.

Ivo v*( = Sup ILz(vov*) = sup CLw(vo v*)

= (sup ELw(uou*))3

=

(sup CLz(UDu*))3 =

(U"U*l3 =

( u (6

.

Q.E.D.

20.9 COROLLARY. A closed Jordan subtriple JB*-triple 2 is a JR*-triple.

W

of a

PROOF. By 18.14, W is -hermitian. By 2.13, we have CLW(un v*+vo u*) C R and CLW(un u*) > 0 for all Further, 20.2 implies u,v E W sup C LW ( u o u * ) = sup L ' ~ ~ ( U O U * ) Since 20.8 implies

.

.

luou*IWI = I U D U * ~= ( u I 2

,

the assertion follows.

O.E.D.

.

By 20.10 EXAMPLE. Let 2 be a C*-algebra over K 18.15, 2 is a hermitian Banach Jordan triple with respect to the triple product 18.4.1 and the operator norm. Further,

2 ( U P V*+VO u*) = L(uv*+vu*) + R(v*u+u*v) for all u,v E 2 , where L(x)z := xz and R(y)z := z y denote the left and right multiplication operators on 2 respectively. Therefore 15.1.1 implies

,

2

.

2 1 u n ~ * (< ( L ( u u * ) ~+ IR(u*u)( < 1uu*1 + lu*ul = 2 IuI Since 15.1.1 and 15.15.1

imply

I{uu*u}(2 = (uu*u12 = 1(u*u)31 =

(sup E,(u*u))3

=

Iu*u13 = lu

.

=

sup Cz(u*u) 3

6 I

Iuou*l = 1uI2 By 15 18, the operators are hermitian on the complex C*-algebra Zc : = 2 0 C whenever x* = x and the operators L(uu*) K and R(u*u) are positive. Therefore 14.30 implies 20.3.1 and 14.31 shows that u o u* is a positive hermitian operator on ZC Therefore ZLz(u u*) > 0 and 14.30 implies it follows that L(x) and R ( x )

.

SECTION 2 0

338

Hence every C*-algebra over

K

is a JR*-triple.

Ry 20.9,

every JC*-triple is a JB*-triple. 20.11 THEOREM. Suppose Z is a positive hermitian Ranach Jordan triple over K such that

-

{ u E Z : Iuau*I < 1 } is bounded. connected symmetric normed Banach manifold ilz =

with

Z

can be identified with

-

Qz

Then the simplyMassociated via the canonical chart.

PROOF. Let (P,p,Z) be the canonical chart of N about o and let M denote the orbit of o E N under the group < exp p ( g - 1 > Then M is a domain in N and, by 20.4, we -1 have 0 := ilic p(MnP) We first show that p ( R ) = M Since 0 is a bounded domain in Z = p(P) , the boundary of NOW assume p - l ( ~ )# M p -1 (13) c N is contained in P Since M is connected and p-l(F3) is open, there exists a point m E MnP such that u := p(m) E a R , i.e., u*l = 1 Hence tu E R for 0 < t < 1 By 20.6,

.

.

.

.

.

.

IUD

.

1/2) u t := tanh-l(t(ua u*) U E Z 1/2 ( u o u*) is well-defined. Since m t := p -1 (tu) for 0 < t < 1 defines a smooth curve in M n P with m o = o , there exists -1 a smooth curve y in n (MnP) such that y ( 0 ) = o , y(t) E Pfor o < t < 1 and p-(y(t)) = tu for 0 < t < 1 Here n : M + M is the universal covering and By 20.6, we have y(t) = gt(o) for t < 1 , p- := pon where

.

.

By 18.24, there exists a unique tangent norm is invariant under G- and satisfies bo all v E T ~ ( M - ) Since ~~(~)(p-)(;(t)) that

.

b

on

M-

which

BOUNDED SYMMETRIC DOMAINS AND JB*-TRIPLES

.

for t < 1 Let W 2 generated by u

.

denote the closed Jordan suhtriple of Then 5.25 and 19.25 imply

-

aaz ( p - o g t o p - l ) ~ ( ~ )=l ~ exp((ut-fzu;z})-)'(o)~w Since 2.13 implies h since

Y(t)

I idW-t2

EkW(uo u*) C CkZ(uou*)

(;(t))

339

=

2 I(idw-t u o u * )-1u1

=

,

.

it follows that

> (1-t2 ) -1 lul ,

u o u*l < 1 - t2 1uo u*l = 1 - t2

.

idW -t2 u o u *

for all s < 1 For s + 1 , it follows that infinite arc length. This contradiction shows Hence M is simply-connected and p- : M + R bianalytic.

.

Hence

y has -1 p (R) = M is

.

O.E.D.

20.12 COROLLARY. Let 2 be a positive hermitian Ranach Jordan triple over K such that

M-

-

: = .Q2

=

f

u

E

2 :

1uo u*( < 1 }

is bounded. Then G- : = < exp(g-) > C Aut(M-) is a real Banach Lie group with Lie algebra g , acting analytically There exists a unique and (locally) transitively on Mtangent norm on M- which coincides with the given norm 1 . 1 on 2 = TO(M-) such that M- becomes a G--symmetric normed Banach manifold, with symmetry j : = -id about 0 E M-

.

.

-

PROOF. The analytic action P of g on M- is topologically faithful and locally transitive. Further, j is a symmetry of M- about 0 such that there is a commuting diagram

where Ad(j)l'g= amid for u = 21 Banach Lie group with Lie algebra g

. ,

Ry 7 . 4 , Gis a real acting analytically

340

SECTION 2 0

.

and (locally) transitively on M Since Mis simplyis connected, the isotropy Subgroup K connected and Gof G- at 0 E M- is a connected Banach Lie subgroup of Now apply 12.31 and 17.7. with Lie algebra '9-c u a ( Z )

.

G-

Q.E.D.

20.13 COROLLARY. The open unit ball M- of a JR*-triple Z over K is a bounded symmetric domain, homogeneous under the group G- of bianalytic transformations. In particular, the open unit ball of a C*-algebra over K and, more generally, of a JC*-triple over K is a bounded symmetric domain. It will now be shown that bounded symmetric domains in complex Banach spaces can actually be characterized as the open unit balls of complex JB*-triples, up to biholomorphic equivalence. 20.14 PROPOSITION. Suppose W is an abelian hermitian complex Banach Jordan triple. Then the closed linear subspace C(W) :=

c <

,

waw*

idw > C L(w)

generated by idW and the operators u v* for u,v E W an abelian C*-algebra. Hence there exist a compact space and an isometric *-isomorphism +w : C(W) + C ( S , C )

is S

PROOF. By 19.20, the commutative set W o W* C L(W) is closed under the operator product. Hence C(W) is an abe ian unital closed subalgebra of L(W) Since

.

4 unv* =

X(utXv) 0 (U+XV)* 20.14.1 A 4=1 for all u,v E W , the Vidav-Palmer Theorem 15.27 implies that C(W) is a C*-algebra under the involution (x+iy)* : = x-iy f o r all x,y E C(W) A H&(W) By 20.14.1, we have

.

(unv*)* = v n u *

.

The last assertion follows from 15.13.

Q.E.D.

*

BOUNDED SYMMETRIC DOMAINS AND JB -TRIPLES

341

For abelian Ranach Jordan triples generated by a single element, the compact space S can he identified more explicitly.

is a hermitian complex Ranach Z generated by u E Z Then there exists an isometric *-isomorphism +U : C(w) + c(S,C) such that S := Cllw(uu u*) is a compact subset of R and 20.15

PROPOSITION.

Jordan triple and

Suppose

W

.

2

is the closed Jordan subtriple of

+,(u

u*lW) = idS

.

PROOF. By 19.25, C(W) is the closed unital subalgehra of L(W) generated by x := u o u*lW The space S in 20.14

.

consists of all continuous unital homomorphisms s : C(W) and carries the topology of pointwise convergence. Hence

+ C

f(s) := s(x) defines a continuous mapping f : S + C which is injective since x generates C(W) Since f(S) = Cfiw(uo u*) by [17; 17.41, f is a homeomorphism. Let

.

denote the adjoint isometric *-isomorphism and put -1 +u := (f*) 00, 20.16

COROLLARY.

Suppose

Jordan triple such that Then 2 is positive.

2

Q.E.D.

is a hermitian complex Ranach

Zllz(uou*) < 0

implies

u = 0

PROOF. Let W be the closed Jordan subtriple of generated by u Assume S := Cllw(u o u*) $fR, exists a non-zero function f E C(S,R) such that

.

.

.

.

2

Then there

f ISnR+ = 0 Then y := +il(f) E C(W) belongs to the closed real subalgebra of L(W) generated by x := u u* I W Since W is abelian, we have (yu)p (YU)* Hence

.

=

y2 u o u* = y 2x

E

L(W)

-

SECTION 20

342

@,((yu)o

(yu)*) = +,(y)

2

$u(x) = f2-idg

0

.

.

Therefore EL,((yu)O (yu)*) < 0 By 20.2, Eg,((yu)o (yu)*) < 0 Hence yu = 0 by assumption. Since f2*ids # 0 I this is a contradiction. Therefore S > O and

.

20.2 implies that

2

Q.E.T).

is positive.

20.17 PROPOSITION. Suppose Z is a positive hermitian complex Banach Jordan triple. Then )uI, : = luo u * )1/2 defines a continuous semi-norm on

Z

20.17.1 satisfying

< lul,

sup lEaz(uov*)l

lvl,

.

Now Then idz - u u u * and idz - v o v* belong to the convex cone X+ = H k + ( Z ) of all Since positive hermitian operators on 2 (u-v) 0 (u-v)* E X + by assumption on Z , it follows that PROOF. It is clear that u + IuI, suppose (ul, 1 and (vl, < 1

.

is continuous.

.

4*idZ - (u+v)n (u+v)* = 2(idZ-un u*)

+

(u-v)o (u-v)*

+ 2(id Z- v o v*) E

x+

.

.

Hence sup ZIZ((u+v)n (u+v)*) c 4 and (u+vI, < 2 It For any state f follows that I * I m is a semi-norm on Z Hence of L ( Z ) (cf. 14.33), 14.34 implies f(Hk+(Z)) C R+ f(uo u*) > 0 , f(un v*+vou*) E R and f(uo v*-vo u*) E iR I since Therefore u n v * - v o u* = i(uo (iv)*+(iv)O u*) f(uo v*)* = f(vo u*) , and the Cauchy-Schwarz inequality [2R; 6.2.11 implies

.

.

.

If(U0 v*)I < f(uou*l ll2 f(vo Now assume all states f

.

lulo. = 1 = lvl, Therefore 14.35.1

.

V*)1/2

Then If(uo v*)I < 1 implies

for

*

BOUNDED SYMMETRIC DOMAINS AND JB -TRIPLES

343

The continuous semi-norm I * I m defined by 20.17.1 is called the spectral semi-norm of the positive hermitian

.

complex Banach Jordan triple 2 If U n u * = 0 implies is a norm on 2 , called u = 0 for every u E Z , then I I m We will now use Ranach Jordan the spectral norm of 2

.

triples to study bounded symmetric domains in complex Ranach spaces. Let us recall the definition. 20.18 DEFINITION. A bounded domain R in a complex Ranach space 2 is called symmetric if for every z E R there 2 = idR and ex i sts a I' symme try I' j z E Aut(B) satisfying j z having z as isolated fixed point. Since the Carathgodory metric bB and the Carathgodory tangent norm 6, are invariant under Aut(R) , it follows that every bounded symmetric domain B is a symmetric metric Banach manifold with respect to tiR and a symmetric normed In particular, the real Banach manifold with respect to 6, Banach Lie group Aut(B) acts (locally) transitively on R

.

.

20.19 LEMMA. Suppose p is an analytic action of a Banach Lie algebra g on a connected Banach manifold M and let p"

.

be the universal covering manifold of M Let be the lifted action of g on M" Then every deck

n : M" + M

.

transformation g E Aut(M-) (satisfying g,(p"x) = p " ~ for all x E g

.

PROOF.

For

t

E

R

,

nog = n

)

we have

Therefore 3.1 implies gOexp(t0p"x) = exp(t*p"X)og this is true for t = 0

.

20.20

THEOREM.

satisfies

Every bounded symmetric domain

A

,

since O.E.D.

in a

SECTION 2 0

344

complexRanach space

Z

canonical chart of €3 hiholomorphic mapping symmetric domain

D

is simply-connected, and the

about q : R in Z

.

o +

E

D

has an extension to a onto a balanced bounded R

.

PROOF. We may assume o = 0 Since M1 : = R is a symmetric normed Banach manifold with respect to the Carathgodory tangent norm 8, , 18.17 implies that Z = TO(13) becomes a hermitian Aanach Jordan triple such that p1 : = q , defined on a neighborhood P1 of 0 E R , satisfies

(P1

g :=

Consider the where k C T o ( Z ) and p is given by 18.12.3. homogeneous Banach manifold N = H/H+ and the domain M 2 := M C N associated with Z Let (P,p,Z) he the canonical chart of N about o := H+ and define

.

.

For k E {1,2} , let and P2 : = PIP2 he the universal covering manifold and choose k with n k ( o ) = o Consider the canonical chart

P 2 := P n M 2

n k : M;

o

E

Mi

+ M

.

(Pi,p;,Z) of M i about o , where P i : = P k o n k By 17.16, 18.21 and 18.22, there exist locally transitive, analytic actions p k Of g on Mk Such that (pk)*(pkX) = X for all X E g Let denote the "lifted" locally transitive, analytic action of g on M i satisfying (pi)*(piX) = X for a l l X E g , Since the

.

PL

n

act ions p k are topologically faithful, it follows from 7.4 that there exists a simply-connected real Ranach Lie group G with Lie algebra g By [21; 1 1 1 . 6 . 3 , T h . 3 1 , G i s uniquely determined up to hianalytic group isomorphism. Hence 6.14 implies that there exist (locally) transitive, analytic actions ri of G on M L with differential p; Since M; is simply-connected and the canonical mapping G / K = M i is real-hianalytic, the isotropy subgroup

.

.

K =

{ g

E

G :

rL(g,o) = o } = < exp k >

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

<

1213 = I { z z * z } I = ( z o ( z * o z ) ~< 1ZI.IZ*OZ It follows t h a t

A

carries a C * - i n v o l u t i o n .

z1212* By 1 5 . 8 ,

a C*-algebra. 20.34 part

LEMMA.

X

.

Then

=

(ZI

A

3

is

O.E.D.

Let Mx

Z E

be a JB*-algebra

Hi(Z)

whenever

with self-adjoint

x

E

X

.

.

BOUNDED SYMMETRIC DOMAINS AND JB*-TRIPLES

Let

PROOF.

f

E

f(exp(itx)) = 1

L(Z,C)

.

I f 1 = 1 = f(e)

satisfy

351

Then

+ it f(x) + X(t) , where

.

lim X(t)/t = 0 t+O exp(itx)l = 1 for all t E R Hence By 20.33 and 1 5 . 8 , 1t-l + i f(x) + X(t /t I < l/t , showing that For t + 0 , we Re(i f(x) + X(t)/t) < 0 for all t > 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 <~ ~lUl.lU*OUI

< Iu121u*I

= IUI 3

. .

Hence u + u* is a C*-involution on U, By 21.11 and M = y o e* = (yoe*+eoy*)/2 has real spectrum 20.3.1, Y whenever y* = y It follows that U x is an abelian unital In case K = R , every element in U, C*-algebra over K

.

is self-adjoint.

.

Now apply 15.13.

O.E.D.

It will now be shown that tripotents give rise to Siege1 domain realizations (of the third kind) of bounded symmetric domains via the so-called Cayley transformations. We will

361

SYMMETRIC SIEGEL DOMAINS

restrict our attention to the complex case and assume in the following that Z is a Complex JB*-triple with open unit ball R = MLet M+ denote the simply-connected symmetric normed Banach manifold which is "dual" to the bounded symmetric domain Y(cf. Sect. 20). For K = t , let g K

.

denote the real Banach Lie algebra of all infinitesimal holomorphic isometries of M K Via the canonical coordinate,

.

g K can be realized as a subalgehra of T ( 2 ) and there exist l K a - l K g , where multiplicative gradations g K = g lgK = aut(z) = u a ( z ) c ~ ~ ( 2 ) and -1g

= { X z : U E Z } .

K

Here XI1 is defined by 20.4.1. Now let e be a tripotent of 2 with corresponding Peirce splitting Z = IJ x V x W , where U := Zl(e) is a Jordan *-algebra with unit element and W := Z0(e) are Jordan subtriples e and V := 2 1/2(e) of 2 By 20.30, the involution of U is isometric. Let X = { x E U : x* = x } denote the self-adjoint part of IJ

.

21.13

LEMMA.

The vector field

-

xe

:=

a

(e-{ze*z})G

E

g-

induces an additive gradation

where the eigenspaces given as follows:

-

gs =

{ Y

E

g

-

:

[Y,X,]

=

2sY }

are

.

SECTION 21

362

and -1 PROOF.

-

g-0 = g 0 n g

For

u

E

2

-1

,

-

-

={xc

: c e X @ W } .

we have

.

= 2 (uo e*-e o u * )a E

[x;,x:]

Therefore 21.11 and 21.9 imply

and

whenever a E iX , b commutes with ’9;

E

.

V and c E X @ W It follows that

[Y,X,l

.

.

-

.

-

- [x;,x;l

a1[X;‘X;l

[x~,x;I / 2

Xe

= 2SY

for all Y E g s and s E {0,+1/2,+1j NOW let 6 Then 6e = 2 {ee*(Ge)} + {e(Ge)*e} implies 6e = a It follows that where a E ix and b E V 6

-

By 18.10,

E

1g o

-

lg + b E

.

a

-

~ + xe I / 2 -( x ~ -+[xita, + (Xa-a-[Xa-a‘Xel/2) E: g 1 a,a E iX and $911 for 2(X;f[X-,X-l) ‘‘L3-b’ x-1) e - b 2 = (X;+b+[X;+b,Xel 1 + for all 8,b E V , the assertion follows. E g 1/2 @ g

Since

-

2 ( X-+ a-

-

=

-

0.E.D.

The vector field

xet

:=

(e+{ze*z/)--aa z

E

g

t

on M+ is called the Cayley vector field associated with the tripotent e E 2 and

+ ge := exp(5 Xe)

E

Aut(M+)

SYMMETRIC SIEGEL DOMAINS

363

is called the corresponding Cayley transformation. Note that -1 4, - 4-e is the Cayley transformation associated with the tripotent -e Let (ge)* be the Lie algebra automorphism of T(M+) induced by 3, Then 5.16 implies

.

.

+

(ge)*Y = exp(l4 ad(Xe))Y

-

.

for all Y E We will now describe the action of (gel* - constructed in 21.13. on each of the eigenspaces g s of 21.14 PROOF. shows

LEMMA.

= 2E = 2 ~a -

(g,)*X,

+

Since

+

az

a

V-

+

[Xer[XerXe]] = -4 Xe

av

.

an induction argument

ad(XL)2nXe = (-4)" Xe and

+ 2n+l ad(Xe) for all

21.15

n > 0

.

COROLLARY.

h := (ge)*(g

-)

xe-

=

(-4)" [:

+ eIxe

It follows that

The real Banach Lie algebra has an additive gradation

S E C T I O N 21

364

21.16.1 21.16.2

and = -2 -li2

(ge)*lx;'x,1 PROOF.

Since

:X

x; +

commutes with

-

Xa

-1/2 + + IXe'Xb1 2

,

-

Xc

and

6

. (by 21.11,

21.9.3 and 1 8 . 1 0 ) , the first three assertions follow from

+

+

.

€ o r all Y E gSince 2 A2n = (-1)"X; A Xi = -Xb , an induction argument shows b' - -- (-1)"AX; for a l l n > 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 ]<

21.24

1

.

LEMMA.

O.E.D. For

w

E

W

,

define a conjugate-linear operator

Y(w) on V by putting Y(w)b := 2 {eb*w} idV + Y(W) is invertible whenever ( w ( < 1 PROOF.

By 21.9.2 and 21.9.3, Qe+,b

.

Then

.

we have

= {(e+w)b*(e+w)}

= 2 {wb*e} = Y(w)h

.

.

If lwl < 1 , 21.23 implies (e+wl < 1 By 19.25, the closed Jordan subtriple Z z generated by z := e+w is abelian. Since

( z o z*lZz( < 1

by 20.7.2,

20.2 implies

370

SECTION 21

It follows that €or every a > 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'

<

.

Conversely, let

In fact (u,v,w)

b : = -(idV+Y(w))

-1

v

E

V

.

D'C D

.

D

E

implies

Then

H<-D'C D

IwI < 1

.

and, by 21.24,

exists and satisfies

g O , b ( U l v l w )E Dn(UxW) = D'

.

By 21.21,

ge(BnUxRnW)

=

D'

.

Since 21.23 implies since ge(u,O,w) = (ge(u),O,w) BnU x BnW = Hn(UxW) , it follows that D' C ge(F3) Since h, c aut(ge(R)) by 21.18, this implies D c ge(F3) In particular, D is biholomorphically equivalent to a bounded domain. Since g-e(ulO,w) = (g-e(u)lO,w) by 21.9.3, we have

.

D' = ge(B) n (UXW) For every

Y

-1

h o w 1ho

,

.

.

21.25.1

is invariant under -1 g := exp(Y) since g(u,O,w) = (u,O,gw) for Y E h0 1 whereas the assertion for Y E h o follows from 21.17. E

D'

Since

[holh,l c h , , 5.16 implies that D is also invariant under g It follows that the real Ranach Lie algebra aut(D) conta ns the closed subalgebra

.

h,

h, ~ ~

: = h -1 fB k 1 / 2 fB h ,

*

aut (D) be the corresponding analytic action. Since ( h= iX - ~I ~ ~ ( h - ~ ,= ~V) I pe( l h ) = W and pe('ho) = X , h, acts (locally) transitively at e E D Since the connected manifold ge(B) carries a compatible Let

p

:

+

.

7

tangent norm invariant under ge(B) NOTES.

=

D C 2

.

p

,

12.18 implies

By 21.25.1, we can now apply 16.19.

Q.E.D.

The principal result 21.25 appears in C911 (for Siege1

domains of the second kind corresponding to tripotents with vanishing Peirce 0-space). The finite dimensional case has been studied in C103; 5 101 and, from a Lie theoretic point of view, by A. Korhnyi and J. Wolf. For the basic properties

of the Peirce decomposition, including 21.8, see C 1021.

SECTION 22

372

22.

AUTOMORPHISM GROUPS OF JORDAN STRUCTURES

The Jordan structures (i.e., Jordan algebras and Jordan triples) associated with symmetric Banach manifolds give rise to various groups of automorphisms which can be interpreted in geometric or analytic terms and are also related to certain symmetric real Banach manifolds. In the following, let 2 be a JB*-triple over K E {R,C}

.

22.1 Z

E

Z

LEMMA. Suppose u . Then u = O .

PROOF.

By 20.8.1,

Let

Aut(Z,Z)

E

we have

satisfies

Z

1uI3 =

gQz = Qgzg

of (2,Z)

(QuUl

g#

E

.

GR(2)

#

Q.E.D.

satisfying the

22.2.1

is uniquely determined by

g#

= 0

for all

denote the set of all transformations

g E G k ( Z ) such that there exists operator identity

By 22.1,

QZu = 0

g

.

The elements

Aut(Z,Z) are called automorphisms of the "Jordan pair" associated with Z [102; 1.31.

22.2 PROPOSITION. G := Aut(Z,Z) is a subgroup of G&(Z) , and g g# is a symmetry of G (i.e., an automorphism of -f

.

period 2 ) Hence g* := (g#)-l defines an involution of G (i.e., an anti-automorphism of period 2 ) .

It follows that

Qgug # Qv = gQ,Q,

=

Q # g gu 4 v

Q

and, by 22.1,

373

JORDAN AUTOMORPHISM GROUPS

.

g

By definition, this implies

g'

E

and

G

Q.E.D. 22.3 EXAMPLE. In analogy to 9.17, a pair (u,v) E 2 x 2 is called quasi-invertible i f the Bergmann operators B(u,v*) and B(v,u*) are invertible. In this case, 21.8 implies B(u,v*) E Aut(Z,Z) and B(u,v*)* = B(v,u*) The group

.

Int(2,Z) := < B(u,v*) : (u,v) quasi-invertible > generated by these inner automorphisms of the Jordan pair (2,Z) is a normal subgroup of Aut(2,Z) since g B(u,v*) g-' = B(gu,(g # v)*) for all g E Aut(2,Z)

.

Let aut(2,Z) denote the set of all operators g a ( z ) such that there exists 6' E g k ( 2 ) satisfying the operator identity 6

E

. 6 .

[6,uo v*l = (6u)o v* + u a ( 6 # v)* By 22.1, 6' of aut(Z,Z) (2,Z)

.

22.4.1

is uniquely determined by The elements are called derivations of the "Jordan pair"

22.4 PROPOSITION. The set g := aut(Z,Z) is a subalgebra of g k ( Z ) , and 6 + 6' is an anti-linear automorphism of g having period 2 Hence 6 * := -6' defines an involution of g (i.e., an anti-linear anti-automorphism of period 2 ).

.

PROOF.

We have

2 {(Su){xy*z}*u} + QU 6#{xy*z}

6{{yx*u}z*u} - 6{yx*{uz*u}} = 2{(6{YX*U})Z*U} + 2{{YX*U}(6#Z)"U} + 2{{yx*u}z*(6u)} { (6Y)X*{UZ"U}} - {Y(G#X)*{UZ"U}} - {yx*6{uz*u}} = 2{ { (6y)x*u}z*u} + 21 {Y(6#X)*U}Z*U} + 21 {yx*(6u)}z*u} = 6{U{XY*Z}*U}

= 2

-

+

-

2{{YX*U}(6'Z)*U}

-

+ 2{ {yx"u}z*(Gu)}

#

{ (6Y)X*{UZ*U}}

tY(6 x)*{..*.}} - 2{YX*{ (6U)Z*U}} - {yx*{u(6#z)*u}} # = {U{X(SY)*Z}*U} + {u{ ( 6 x)y*z}*u} + {u{xy*(6#z,}*u} + 2({{YX*(6U)}Z*U} + {{YX*U}Z*(6U)} - {yx*{ (6U)Z*U}j) = Qu({(6

#

#

X)Y*Z} + {x(~Y)*z} + { x Y * ( ~z ) } ) + 2{(6u){xy*z}*u}

.

374

SECTION 2 2

By 22.1, we have 6' 2.14.1 implies that all 6 , c~ g

c

g

and

6''

[6,r1] E 9

.

= 6

and

.

The Jacobi identity [6,r1]# = [6#,n#] for 0.E.D.

Similar as in 2.15, it is shown that the exponential mapping exp : gE(2) + GR(2) satisfies exp(aut(2,Z))c Aut(2,Z) and exp(6*) = (exp 6 ) * for all 6 c aut(2,Z)

.

22.5 EXAMPLE. Suppose E and F are Hilbert spaces over D c {R,C,E} and consider the JB*-triple 2 := L(E,F) over -1 the center K of D Then g a l d z : = azd defines a group homomorphism

.

.

satisfying g:,d = ga*,d* Similarly, 6a,dz := az-zd defines a continuous Lie algebra homomorphism gR(F) x gR(E) 3 (a,d) satisfying

6:,d

.

=

+

6a,d

E

aut(Z,Z)

BY 6.28, these mappings yield a

commuting diagram

22.6 EXAMPLE. Let u,v E Z , By 18.8.3, we have u V* E aut(2,Z) and ( u n v*)* = v o u * The linear span

.

int(2,Z) := K < U O V * : u,v

E

2

>

generated by these inner derivations of the Jordan pair ( 2 , Z ) is an ideal in aut(2,Z) (by 22.4.1). It is clear that int(2,Z) is invariant under the involution 6 + 6*

.

Our next aim is to compute the exponential of inner derivations. 22.7

LEMMA.

Suppose

2

is a Jordan algebra and

a

and

b

375

JORDAN AUTOMORPHISM GROUPS

belong to the subalgebra A in 2 Then Papb - Paob

.

PROOF.

.

Specializing 19.6.1

generated by a single element

to

u = x

or

z = x

,

respectively, we get the operator identity M ( X ~ ) M+~ 2 ~ ( x o y =) M~ ~ M(X 2

+ 2 ~ ~ ~ ( x o y )

Y

+ M(xLoy)

= 2 M M M X Y X

on

Z

.

By 19.8.1,

it follows that

2 P(x ,Y) =

Putting

x = y = a

-

M(x2) M ( X ~ ) M ~+

,

z = b

P(aob,a) = Pa Mb

and using 19.22, we get

+ [P(b,a),Ma]

Polarizing this identity, we get for all P(aob,c)

M(x 2oy)

MY

+ P(cob,a)

=

= PaMb

.

a,b,c

2 P(a,c) Mb

A

E

.

Therefore Papb = 2 PaMi = P((aob)ob,a)

-

PaM(b 2 ) = 2 P(aob,a)Mb

+ P(aob,aob)

-

-

P(aob2,a)

P(aob2,a) = Paob

.

Q.E.D.

22.8

Then

COROLLARY. Let 2 be a unital Banach Jordan algebra. exp(2 Mx) = P(exp x) for all x E Z

.

PROOF. The closed unital subalgebra x contains m

ut : = exp(tx) =

(txIk 1 7

k=O

A

of

Z

generated by

376

SECTION 2 2

for all t all s , t

E

.

.

R By 22.7, P(uS) P(Ut) = P(Us+t) Since uo = e and dut dt

-(O)

=

x

E

L (2)

for

,

it follows that

Q.E.D.

22.9 PROPOSITION. For every JB*-triple 2 , we have exp(int(2,Z)) C Int(Z,Z) (uniform closure). PROOF. By 6.7, it suffices to show that exp(uo v*) E Int(2,Z) With respect to the Jordan algebra product xoy : = {xv*y} on 2 , we have MU = u v* and Pu - QUOv , Let Z ' denote the unitization of 2 , with unit element e , and define

.

m

exp(u) : =

(ktv)

U 1 7 E 2' ,

k=O

where u(krv) = ( u a v*) k - l ~ denotes the k-th power of in Z ' For x := exp(u) - e E 2 , we have

.

B(x,-v*) = Qe+xQv = Qexp(u)Qv = 'exp(u) by 22.8. Since u o v* E aut(2,Z) Hence B(v,-x*) = exp(2 v m u*) invertible.

.

,

exp(2

=

UR

u

v*)

21.8 implies (x,-v) is quasiQ.E.D.

The automorphism group of a JR*-triple

Z

can be written

as Aut(2) = { g

E

Aut(2,Z) : g*g = id 2

Similarly, the derivation algebra of aut(2) = { 6 By 18.11, Aut(2) algebra aut(2)

.

E

Z

aut(Z,Z) : 6 * + 6

1 .

22.10.1

can be written as =

o

}

.

is a real Aanach Lie group with Lie In case K = c , we have

22.10.2

JORDAN AUTOMORPHISM GROUPS

aut(2,~) = aut(2) C since 6 - 6* 6 E aut(2,Z)

and

i(6+6*)

.

belong to

aut(2)

int(2) c int(2,Z)

It is clear that =

1 u 1. n

v

j By 22.10.2,

*

1

E

whenever

Then

.

D

22.10.3

t

22.10 LEMMA. Let 2 be a JB*-triple. int(2) = int(2,Z) A aut(2) PROOF.

377

.

Now suppose

int(z,z)n aut(2)

.

we have

Hence

4D =

1 j

((uj-vj)o (u.+v.)* - (u.+v.)o (u -v.)*) 3 7 7 3 j i

In case

K

, we

= C

int(2,Z) = int(2)'

have

22.11 EXAMPLE. Suppose E and F D E {R,C,FI} and put 2 := L(E,F) analytic group homomorphism UI1(F)

X

U R ( E ) 3 (a,d)

+

.

E

int(2)

.

O.E.D.

.

are Hilbert spaces over Then 22.5 defines an

ga,d

E

Aut(2)

whose differential is the continuous Lie algebra homomorphism U ( F ) x U ( E ) 3 (a,d)

+

E

aut(2)

. .

An element u E 2 is called unitary if u u u* = idz In the following, we assume that 2 is a JB*-triple over K containing unitary elements. For every unitary e E 2 , 2 becomes a unital involutive Banach Jordan algebra with product xoy := {xe*y} and involution z* = Qe z = {ez*e} Then Pz - QzQe for all z E 2 By 19.18, u E 2 is invertible if and only if invertible and

Pu

Gk(Z)

E

=

u*

.

. .

.

By 19.27, every unitary

u

is

378

SECTION 2 2

22.12 LEMMA. Suppose Int(Z,Z) contains QuQv PROOF. imply

w

Put

=

u,v

are invertible.

Z

E

and

x

-1

Q,

PuQe

=

Quw = u

,

Q,Q(w,x) Since

Qwu

=

w

,

QuQw

=

-

Then 18.8.4 and 19.7

Q,

) =

QuQw = idZ

is bijective, we get

QUOv - QUQw-, Since

.

w-v

:=

QUQwQu = Q(Quw) = Q(PUu Since

.

= B(v*-~-u,v*)= (QVQU)*

B(u,(u*-'-v)*)

:= u * - l

Then

2 Q,Q(w,X)

.

Now

+ QUOx

18.2.7 implies = 2

( U O ~ " ) ( U O x*)

18.8.1 and 18.2.1

-

uo

X*

.

imply

u o w* = ( u o w*)QuQw = QU(wo u*)Qw

Q, Q(Qwu,w)

=

QuQw = idz

=

.

Hence Q, Q(w,x) = U D x* and therefore QuQv Similarly, w := v* and y := w-u satisfy w o v* = id and Q(w,y)Qv = y o v* , i.e.,

-'

.

B(u,x*) QwQv = idz

=

,

z

Since QUOv = Qw-y Q, = QwQv - 2 Q(w,y)Q, + QyQv = B(Y,V*) B(u,x*) = QuQv and B(x,u*) = QvQu are invertible, it follows that (u,u*-'-v) is quasi-invertible. Q.E.D. 22.13

COROLLARY.

Suppose

u,v

are unitary.

Z

E

Then

QuQv = B(u,(u-v)*) = B(v-u,v*) belongs to

Aut(2) n Int(Z,Z)

.

Int(Z) := < QuQv : u,v of all inner automorphisms of PROOF.

Since

0,' =

Q:

=

idz

(QUQv)* = QvQu = (QUQv)-'

.

2

The group E

Z

unitary >

is a normal subgroup of

, we

get Hence 22.10.1

implies

2

.

J O R D A N AUTOMORPHISM GROUPS

E

Aut(Z)

g 0,g-l

= Qgu

Q uQ v

Aut(2)

.

22.14

LEMMA.

. .

For every

Hence

For P

For

t

g

g

Qe

put

GR(Z) : g e

E

for a l l

g

u

E

Z

t

,

it follows t h a t

.

ge

Then

i n v e r t i b l e and

.

Aut(2,z)

E

and

:= g - l P

gt

,

22.2

implies

.

= 0 Q = gQzg*Qe = gPZ(0,g*Qe) gz e

gz

Qeg*Qe = g - l p P = gPzg gz

,

GR(Z)

E

Aut(2,Z)

E

Pe = i d Z

Since

is u n i t a r y and is a n o r m a l s u b g r o u p o f gu

Q.E.D.

g* = Qe g

PROOF.

,

Aut(Z)

E

Int(Z)

Aut(Z,Z) = { g

and

g

379

is i n v e r t i b l e a n d

ge

Conversely, suppose ge ' for a l l z Then

i s i n v e r t i b l e and

ge

.

t t Qgz = Pg z Qe = gPzg Qe = g Q z ( Q e g Q e ) Hence

g

Aut(2,Z)

22.15

COROLLARY.

E

t g* = Oeg Qe

and

.

Q.E.D.

is a c l o s e d s u b g r o u p o f

Aut(Z,Z)

GR(2)

a n d a Banach L i e g r o u p i n t h e o p e r a t o r norm t o p o l o g y . PROOF.

P

gz

Then

Let

g

GR(2)

E

= g P z g -1 Pge

id2 = P

g = P gP(g-'e) ge f o r some v E 2

I

P(g

injective.

.

z ) = g-lPzgp(g-le)

22.15.2

= g- l p

. .

g P ( g -'e) and t h e r e f o r e ge Hence P is s u r j e c t i v e and ge Now 1 9 . 9 . 1 i m p l i e s

i d Z = Pe = P ( P g e v ) = P Aut(2,Z)

satisfy the operator identities

Therefore

e = P v ge

is a l s o P P Hence Pge ge v , g e g e is i n v e r t i b l e . By 22.14,

is c h a r a c t e r i z e d by 22.15.2

and is t h e r e f o r e

.

< 3 in (9,g-l) By 7 . 1 4 , Aut(2,Z) is a Banach L i e g r o u p i n t h e o p e r a t o r norm t o p o l o g y . Q.E.D.

a l g e b r a i c of d e g r e e

22.16

COROLLARY.

invertible.

Then

Suppose gu

g

E

Aut(2,Z)

is i n v e r t i b l e ,

Pu

and E

u

E

2

Aut(Z,Z)

is and

3 80

SECTION 2 2

.

t P* = P(u*) The mapping g + gt Pu = Pu I U automorphism of Aut(2,Z) of period 2 and (gu1-l = (gt)-l(u-l)

is an anti-

.

.

PROOF. Since pU E G ~ ( z ), pgU = gPugt E GQ(Z) 19.9.1 implies P(Puz) = PuPzPu and PUe = u2 is

invertible. By 22.14, it follows that and Pi = Pu Hence 18.2.4 implies

.

: P

=

Pu

E

Further,

G := Aut(Z,Z)

QePuQe = QeQuQe 2 = Q(Qeu)Qe = P(U*)

For g E G , we have If h E G , we get

P ge

E

G

and hence

gt = g-lPge

G

E

.

2 t t (gh)t = Qe(gh)* Qe = Qeh*g*Qe = Qeh*Qeg*Qe = h g and hence

Further,

(gt)t = p (9-l)t = P (gt)-l = P P-1g = g ge ge ge ge t -1 t -1 -1 -1 implies P-' = (gPug = (9 pu g gu -1

22.17

< Pu

O.E.D.

COROLLARY, : u

E

2

The group

invertible > = < QuQv : u,v

is a normal subgroup of PROOF.

.

For invertible

Aut(2,Z) u,v

E

QuQv = PuQePvQe = PUP:

2

contained in

,

2

E

invertible

>

Int(2,Z).

we have

=

PuP(v*)

.

It follows that the transformations Pu and QuQv generate the same subgroup of Int(2,Z) Now suppose g E Aut(2,Z) Then gu is invertible by 22.16 and

.

gPug

.

-1 = gPug t Pge -1 = P

By 22.13, we get

P-l

g u ge

Int(2) = < PU : u

. Q.E.D. E

2

unitary >

.

JORDAN AUTOMORPHISM GROUPS

381

22.18 PROPOSITION. The set S := { u E 2 : U D u* = idz } of all unitary elements in 2 is a closed real submanifold of

PROOF. For any e E S , 2 becomes a unital Jordan algebra with product zow = {ze*w} and involution z * = {ez*e] For K = +_ , put XK : = { z E 2 : z * = K Z } Then $ ( z ) := zoz* + satisfying defines a real-analytic mapping Q : 2 .+ X

.

.

$'(Z)V = zov* + voz* = (VOZ*)*

+

.

voz*

22.18.1

Since Me = Pe = idz , there exists an open neighborhood fi of e E 2 such that G X ( 2 ) contains M(z*) and P(z) for + is surjective all z E fi It follows that ~ ' ( z :) 2 + X and its null-space M( z*)-l(X-) has the topological -1 + + is a complement M(z*) (X ) Hence Q : P X submersion. Since S n P = { u E P : $ ( u ) = e } by 19.27, 8.9 implies the assertion. Q.E.D.

.

.

.+

22.19 PROPOSITION. Let e E Z be unitary. Then the real Banach Lie algebra k + := aut(2) has the multiplicative gradat ion

k+ = lk+

-lk+

@

22.19.1

where

lk+

=

{ 6

E

k+

: 6e = 0

}

and -1 k+ = ( M y : y * = - y } = { u o e * - e n u * : u

PROOF. By 21.11, we have 2M = y o e * - e o y * y+ whenever y* = -y If 6 E k , then

.

showing that (be)* = -6e 6 - M elk+. Y 22.20

COROLLARY.

.

The group

Put

y := 6e

.

E

z }

E

aut(z)

Then O.E.D.

.

SECTION 22

382

K+ : =

< exp( k')

> 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<

exp(

.

Aut(C) acts The group

h ) > 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

.

<

1

1

It follows that Iztzl < ~ z t ~ =* (zI2 ~ z =~ z*z whenever z*z = 2 Since p(P) = Z , it

6 p(QjnP)

.

SECTION 2 3

410 follows that

Since

and

is s t a r l i k e a n d t h e r e f o r e c o n n e c t e d , w e g e t

B

p(M-1

n,nP

M-C

.

= B

hK

b i n a r y Banach L i e a l g e b r a

K

a

=

{ (az-zd)-az

hl = { (zcz-ct

c fi)a 2 az '

ho

X + X#

The homomorphism

g# = l!h

:=

: a = -at

gK

maps

K

,

h t fB h:

@

L(z) ,

E

onto the

A

E

where

K }

and

hK

The r e a l B a n a c h L i e a l g e b r a s 1

}

2t

E

-

hK = { ( a z - z d ) - aa z : a = a = -a

t

= { { z b * z }a=

hK

:=

g#" =

,

d * = -d

: b @

E

Z

-'hK

}

.

satisfy

} c aut(2)

and

-1 hK = { ( b + w { z b * z } ) & Since

E

on

,

M-

.

M-

18.20,

Z

E

.

G-

acts l o c a l l y t r a n s i t i v e l y a t

12.18 i m p l i e s t h a t

G-

acts ( l o c a l l y ) t r a n s i t i v e l y

By 20.36,

is a J B * - t r i p l e

2

<

e x p ( h-)

>

a n d , b y 20.13 a n d a c t s (locally)

C Aut(2)

t r a n s i t i v e l y on t h e o p e n u n i t b a l l o f coincides with

Then

M-

Since

G-

t h e group

suppose

.

}

is b o u n d e d , t h e r e e x i s t s a c o m p a t i b l e m e t r i c o n

B

i n v a r i a n t under

o

: b

g

E

GK

0

.

,

g i v e n b y 23.3.6,

g # ( O ) = b/d

-1 9

= 0

.

is convex.

B

satisfies

!'.)

d # 0

Hence

a b tt

(;t

T = 9

In particular,

which t h e r e f o r e

2

and

Now

r(g,o) = o b = 0

.

.

Since

=

bIt satisfies g

E

GK

,

r ( g-1 ,o) = o t h i s implies

,

we g e t

c = 0

,

a

# 0

b' = 0

and

c' = 0

and hence

.

Since

CLASSICAL BANACH MANIFOLDS

411

It follows that

B = y = O .

g =

0

d-l (0 0

a

UL(Z)

E

0

x

.

rJ(K)

d

By 12.31, the JB*-norm on Z = To(p) To(MK) induces a GK -invariant compatible tangent norm on MK Further, r(j) is a symmetry of MK about o The commuting diagram 23.1.5 and 17.7, applied to the analytic action p # ( X # ) := pX of hK on MK , show that M K is a symmetric normed GK -manifold. O.E.D.

.

.

The manifolds MK are called classical Banach manifolds of type IV. Similarly, Z is called a classical JB*-triple of type IV. In order to describe the Cayley transformation for classical manifolds of type IV, let e E Z be a vector satisfying Since

for all

z

(ele) = 2

E

xe

2

:=

,

-

and

e = ae

it follows that t

(;-: i)

e

E

for some

a

is unitary.

E

U(K)

.

The matrix

g+

-e *

+

induces the Cayley vector field p(Xe) E aut(M ) satisfying 3 Since Xe = -4 Xe , an induction argument shows 23.1.6. 2 for all n > 0 X2n+2 = ( - 4 ) n Xe It xZn+l = (-4)"Xe and e fol lows that

.

2 exp(tXe) = id + (1-cos(2t))Xe/4

+

sin(2t)Xe/2

+

.

Therefore the Cayley transformation r(ge) E Aut(M ) associated with the tripotent e is induced by the matrix

ge = exp($xe)

=

+\

5 -1

-e*

1

SECTION 2 3

412

satisfying

-

.

e = e

Now assume that

Since

I(-!

je := gej g-e =

0

;

-1

ee:-l

E

G+

it follows that (ge)# maps the open unit ball bianalytically onto the tube domain D := { z

The symmetry on

E

2 :

t Re(z z )

about

D

e t

.

+ ((elz)l2 -

, B

(212)

of

>

2

0

}

.

is given by

(je)#(z) = 2(e(elz)-z)/z z The classical Banach manifolds are of particular We will now show that the importance in case K = C holomorphic automorphism groups of these manifolds are (essentially) given by collineations.

.

23.4 LEMMA. Suppose M+ is a classical complex Banach manifold of "compact type". Let ( P , p , Z ) be a canonical + Then for every X E T ( M + ) I chart about o E M p*X E ~ ( 2 ) is a polynomial vector field of degree < 2

.

.

PROOF. We first choose a suitable coordinate transformation 2 Suppose first that g with r(g ) = id and P A r(g)P # 9 2 = L(E,F) is of type I. If dim(F) > dim(E) , define

.

0

g := (i;E where

F = (:)

.

0

i;H

;

idE

1

E

uL(L)

Then 23.4.1

for a l l u E Q := G L ( E ) C U := L(E) If d i m ( F ) c dim(E) , define

and

v

E

V := L ( E I H )

.

413

CLASSICAL BANACH MANIFOLDS

0 g :=

where

E =

.

(,)H

Then g,(v,u)

for a l l

u

E

G L ( F ) C U :=

:=

SZ

= (-u-lv,u-l)

L(F)

is of t y p e IIeVen ( u = l )

If

and

23.4.2 v

E

V :=

L(H,F)

.

(0=-1) , p u t

or o f t y p e 111

Then

-1 g#(z) = u * z for a l l

23.4.3

i n t h e non-empty o p e n s e t

z

B := Z

A

GR(E)

.

If

Then

23.4.4 for a l l v

E

u

E

u

GL(H) n

8 =

.

V := L ( H , C )

If

Z 0

0

c

u

:=

{ u e L (HI : ut

= -u

1

and

i s of t y p e I V , p u t

-1

Then

g # ( z ) = 2z/(z t z ) forall

z e ~ : = { z ~ ~ t: z z # O } .

Now let

p,x

23.4.5

X

E

T(M)

= Ad(g#)(p,Y) ,

two c a s e s .

and d e f i n e

Y := r ( g ) * X

E.

T(M)

.

Then

To c o m p l e t e t h e p r o o f , w e d i s t i n g u i s h

SECTION 23

414

Case 1: series

about 23.4.3

Types IIeven

0 E 2 , with implies

,

hi

E

Q.

1

Since

5

-1 +

For types IIeVen and 111,

)z

is holomorphic on

hX(5z)

F o r type IV, 23.4.5

E Q ,

.

n=O y

# 0

,

C

and

and

.

implies t

zzt h (-) 22

z2z -

z z

Since

hX(5z)

5 +

m

1

and

-0

hX(Z) = h y ( T22) for all z satisfies

C

m

gn hi(zl =

2 0 hX(Z) = -ozhyZ

for

Consider the power

1 5 2-n z hy(aZ n -1 ) Z n=O n=O 5 # 0 , 3.1 implies that h i = 0 for n > 2 m

for

.

.

p"(2,Z)

E

hX(z) = -az hy(az for all z satisfies

.

I11 and IV

m

1

yn hi(z) =

5

n=O 3.1 implies that

Case 2: Types I and IIodd expansion p*X = hX(z)G a

+ k

.

2-n

zt2 is holomorphic on

n 22 z tz (h (-)T 2%

hi = 0

for

n 22 hy(T)) z z n > 2 and -ZZ

t

Consider the power series

a

(2)-

av

=

m

1

(hi(z)=a

+

kn X ( Z )aE )

n=O

about ( 0 , O ) E UxV = 2 , with !h E p(Z,U) and !k E p"(2,V) If 2 is of type I and dim(F) > dim(E) (x=l) or if 2 is of type II,dd ( x = - l ) , 23.4.1 and 23.4.4 imply

.

23.4.6

CLASSICAL BANACH MANIFOLDS

-’

X*ky(u,Xv) = kX (u for all u E B 5 + hy(5u,Xv) satisfy

and and

v

- v hx(u-’,vu-’)u

,vu -l)u

23.4.7

.

Since the mappings ky(5u,Xv) are holomorphic on V

E

5 +

OD

hy(5u,Xv) =

415

5 2-n

-1

and

C

n -1 ,vU -1 )U u hX(U

n=O and

for

5 # 0

,

3.1 implies

hi = 0

and

kn X = 0

n > 2

for

.

Further, kX(u,vu) 2 = v hX(u,vu) 2

Now suppose that 23.4.2

2

.

23.4.8

.

dim(F) < dim(E)

is of type I and

Then

implies hy(v,u) = -u h (-u-1 v,U -1 )u X

and ky(v,u) = -u k

-

(-U-’V,U-’)

X

u hX(-u -1 v,u -1 )v

.

for all u E Q and v E V Since the mappings 5 + hy(vt5u) and 5 + ky(v,
C

and

m

hy(v,5u) =

-1

5: 2-n u hX(-u n -1 V,U -1 )U

n=O and

for 5 # 0 , 3.1 implies kX(uv,u) 2 = hX(uv,u)v 2

.

h; = 0

and

ki = 0

for

n > 2

and

Q.E.D.

E be a Hilbert space over D E: {R,C,H) , Then every continuous derivation 6 of the Jordan algebra U := L ( E ) (over the center K of D ) has the form 23.5

LEMMA.

Let

SECTION 23

416

6z = [w,z] := wz - zw

.

If 6 is a *-derivation (i.e., w E U 6(z*) = (6z)* for all z E IJ ) , we may assume w* = -w

for some

Assume

PROOF.

where z

E

where

H U

A

Applying

dimD(E)

> 1

and write

is a Hilbert space over D can be written as matrices

L(H)

E

,

v

E

H

,

f

E

to the identity

6

.

L(H,D)

.

Then the elements

and

(l)L = (1)

a

, we

E

D

.

Define

get

.

Hence we may assume where v E H and f E UH,D) 6(l) = 0 Applying 6 to the identities and (a)2 = (a ) , we get 2(a) = (a)(l) + (l)(a

.

for all a E D , where rl is a K-linear Jordan derivation of D Since D is a division ring, it follows from [ 6 9 , p. 31 that rl is a derivation of D (as an associative ring). By the Theorem of Skolem-Noether [155; p. 1661, there exists b E D such that qa = [b,a] for all a Hence &(a) = [(b),(a)] and we may assume &(a) = 0 Applying 6 to the identities (v) = 2(v)o(l) , ( v ) =~ 0 and (va) = 2(v)o(a) , we get

.

.

for all v all a E D identities

E

,

.

.

H , where 4 E L(H) Since [($),(a)] = 0 for we may assume 6(v) = 0 for all v Then the

(A)o(l) = 0

and

(Av) = 2(A)o(v)

.

imply

CLASSICAL BANACH MANIFOLDS

417

.

for all A E L(H) Finally, the identities (f) = 2(f)o(l) , ( f ) * = 0 and

6(A) = 0

imply 6 ( f ) = 0 for all f proves the first assertion. (6Z)*

for all

z

E

=

.

U

E

.

L(H,D) Hence Now assume

6 = 0

.

This

(wz-ZW)" = 6(Z*) = wz* - z*w Then

dz = [(w-w*)/~,z]

.

O.E.D.

.

23.6 LEMMA. Let E be a Hilbert space over D E {R,C,E} Then every continuous derivation 6 of the real Jordan algebra X := H & ( E ) has the form 6x = [w,x] , for some skewadjoint element w E L ( E )

.

.

PROOF. The assertion is trivial if dimD(E) < 1 Now assume dimD(E) = 2 Then X = kk2(D) is isomorphic to a real spin factor R @ Y (cf. 19.41, where Y : = { x E X : trace(x) = 0 } Let {el, e n } be an orthonormal basis of Y under the scalar product (xly) := trace(xy) Since 6 ( Y ) c Y and 61Y is skewsymmetric, we get n n 2 6 = 1 [M(6eu),M(eu)l = ad( 1 [6eu,eu1/4 ) u=1 u=l

.

.

...,

.

.

In fact, for

l~ =

,

l,...,n

we have

,.

= 2 6 e

Since eu and 6eu are self-adjoint, it follows that n w := E l [6eu,eul/8 E $(D) is skew-adjoint. This proves the assertion f o r dimD(E) = 2 Now assume dimD(E) > 3 Then there exists a finite number 2: > 3 such that E = Fr (Hilbert sum), where F is a Hilbert space over D It follows that L(E) = L(F)rXr

.

.

.

.

SECTION 23

418

By [ 6 9 ; p. 143, Cor. 11, 6 has an extension to a real-linear In case D = C , this extension *-derivation 6 of L(E) is complex-linear. Now apply 23.5. O.E.D.

.

LEMMA. Let p : L(E,H) + L(E,K) be a continuous linear for all v E L ( E , H ) and ng satisfying ~ ( v u ) = (pv u maPP such all u E L(E) of rank 1. Then there exists d E L ( H , K ) that uv = dv 23.7

.

PROOF. Let h E E be non-zero. Then vh = 0 implies p(v)h = 0 , since any projection u : E + Ch satisfies p(v)u = ~ ( v u ) = p(0) = 0 Hence d(vh) : = (pv)h defines a continuous linear mapping d : H + K For any u E L(E) of rank 1, we have

.

It follows that

p(v) = dv

.

.

O.E.D.

23.8 LEMMA. Let 2 be a classical complex -*-triple. aut(2) = 'hK and aut(2,Z) = hoC

Then

.

PROOF. Since aut(2,Z) = aut(2)' and h,c = 1 hK suffices to prove the first assertion. Clearly, For all u,v E 2 , we have 'hK c aut(2)

0

R

C

,

it

.

2K(U

0

V*-V

0

Now assume 6 E aut(2) distinguish two cases:

U*) = [xv,xc] K

.

E

'hK

.

23.8.1

For the rest of the proof, we

.

Case 1: Types IIeven, 111 and IV In this case, Z is a JB*-algebra with unit element e , Let X be the selfadjoint part of 2 Then aut(2) = Mix d aut(X) by 22.24.2. Since Mix c 'hK by 23.8.1 and 21.11, we may assume 6 E aut(X) For types IIeven or 111, we have X = H ( E ) , where E is a Hilbert space over D = H or D = R , respectively. By 23.6, it follows that 6 E 'hK For type IV, X = R @ Y is a real spin factor and t E L(Y) 6 = -6 Hence 6 c 1 hK

.

.

.

.

.

419

CLASSICAL BANACH MANIFOLDS

Case 2:

Types I and IIodd.

In this case, let

be the Peirce splitting of Z with respect to the tripotent e Then U is a JB*-algebra with unit element e Let X be the self-adjoint part of U Then 6e = ix + b I where By 23.8.1, we may assume 6e = 0 This x E X and b E V implies

.

.

where

h

aut(X)

E

.

.

and

p

E

satisfy

L(V)

p{ue*v} = {(Xu)e*v}

.

+ {ue*(pv)}

23.8.2

and

.

for all u E U and v,v1,v2 E V For type I, we have X = H Q ( E ) and V = L(E,H) , where E and H are complex such Hilbert spaces. By 23.6, there exists a = -a* E L ( E ) that 6x = [a,xl Since

.

we may assume

h =

~ ( v u )= (pv)u and u(v) = dv for all that

0

.

,

and 23.8.3

.

imply

p(vl)*v2 + v*p(v2) = 0 By 23.7, 1 v , where d = -d* E L(H) It follows

For type IIoddl we have V = L(H,C)

Then 23.8.2

U =

{ u

E

.

L(H) : u t+u = 0 }

and

where

is an even-dimensional complex Hilbert space. tripotent

Choose the

SECTION 2 3

420

-idK

0

e := (

0

1

idK and write the elements of

where

u

E

and

U

v

E

V

E "

as matrices

2

.

Since

is a unitary matrix, the left multiplication $ 2 := g z defines a Jordan triple isomorphism from 2 onto the closed complex Jordan subtriple

Here E denotes the quaternion Hilbert space with underlying complex Hilbert space H Further, gX = H ~ ( E ) and $e = idE Hence := $A$-' is a continuous algebra derivation of HL(E) By 23.6, there exists

.

.

.

a = (

a

-

B

-) = -a* E L(E)C L ( H ) -B a such that rlw = [a,w] for all w E HP.(E)c and fit = B Therefore -Ax = (eae)x + xa (eaelt = a Since

.

.

we may assume

.

Hence and

a* = -a

.

By 23.8.2 and 23.8.3, J, := + p $ - l E L(V) satisfies J,(vw) = ($JV)W and +(vl)*v2 + vf $(v2) = o for all w E H I ~ ( E ) ' and vIv1,v2 E V Since the algebra generated by HI1(EIC acts irreducibly on H , it follows that there exists d = -d* E C such that J,v = dv for all v E V Therefore pv = dv and X = 0

.

.

6z a = ((: az

+

' ( 0O

d0))" az

E

1hK ,

O.E.D. 23.9 THEOREM. Let M+ be a classical complex Banach manifold of "compact type". Then every holomorphic vector

421

CLASSICAL BANACH MANIFOLDS field on T(M+ )

M+

is an infinitesimal collineation, i.e., C = aut(M+) = p(5 )

.

PROOF. There exists an analytic action p # of the binary C C C C complex Banach Lie algebra on hc := g+ = h i @ ho @ hml M+ such that p,(p#X) = X for all X E hc Since I E hC , 23.4 implies that u : = T(M+) has an additive for gradation u = a-l I3 u I3 ul , where p*(us) c T s ( Z ) # C every s Then P ( h s ) c us and P ( h - l ) = , since C p*(u-,) = T-l(Z) We will prove below that p*al = hl C This implies p*ao = ho In fact, let X E u o Then 6 : = p*x E T o ( Z ) For every v E Z , we have

.

# e

.

.

.

.

.

.

a C [ G , { Z V " Z } ~ l E P*[UO,Ul1 = P*Ul = hl It follows that there exists a unique

w

E

Z

. such that

6{ZV*Z} - 2{(6Z)V*Z} = }.*..I and

6# v : = w

.

defines an element 6' E g a ( Z ) Therefore C 6 E aut(Z,Z) and 23.8 implies 6 E h o In order to show n p*ul = h y , we distinguish two cases.

.

Case 1: Types IIeven, I11 and IV. By 23.4.3 and 23.4.5, the differential of the collineation r(g) E Aut(M+) satisfies C C r(g)*p(g 1 = p(g ) and r(g), a s = a _ , for all s C # C Therefore u l = r(g)*a-l = r(g),(p # h-l) = P h,

.

Case 2 : Types I and IIodd. For X E a l , put Since the mappings Y := r(g)*X = Y-l + Yo + Y1 E a + hy(cu,Xv) and 5 + ky(cu,Xv) are holomorphic (here x = 1 for type 1 and = -1 for type IIodd), it follows from 23.4.6 and 23.4.7 that hy(u,v) = h(v) is independent of u and ky(u,v) = 0 In order to show X E p # ( h C ) , it suffices to show Y E p # ( h C ) , since r(g) is a collineation. We show more generally that p # ( h C ) contains all vector fields Y E T ( M + ) satisfying

.

.

a p*Y = f(v)-au

422

SECTION 23

for some polynomial f : V + U , We may assume h E pn(V,U) where n E {0,1,2} In case n = 0 , we have C p*Y E T - l ( Z ) = h-, , In case n = 2 , 23.4.8. implies hX E p 2 (V,U) and vhX(vu) = kX(u,vu) = 0 Hence Now assume n = 1 X = r(g),Y = 0 , showing that Y = 0 For any b E V , define

.

.

.

Yb : = {zb*z}-- a = 2{ub*v}-a az au

+

,

.

{vb*v}-a av

E

h,C

.

Then

The above considerations imply €or all

v,b

f{vb*v} = 2{(fv)b*v} For type I, we have

V

=

L(E,H)

€(v) = dv

for some

d

E

V

.

23.9.1

and 23.9.1

f(vb*v) = €(v)b*v By 2 3 . 7 ,

E

implies

.

L(H,E)

and therefore

For type IIodd, we have V = L(H,C) and 23.9.1 implies $(h)(Elh) = ($h)o(kht)+(hkt)o($h) = ($h)k*ht-h*(($h)k)t €or all h,k E H , where $ = -$t E L(H) is defined by Hence $(h) = hBt - 8ht for some 8 E H $(vt := f(v) and therefore

.

23.10 COROLLARY. Let M+ be a classical complex Ranach manifold of "compact type". Then Aut(M+) is a complex Banach Lie group with Lie algebra p ( g c ) and identity component r(cC)

.

PROOF.

Since the complex Banach Lie algebra

aut(M+) acts analytically and topologically faithful on M+ it suffices to show that every g E Aut(M+) induces a linear p ( gC )

=

CLASSICAL BANACH BANACH MANIFOLDS MANIFOLDS CLASSICAL

423

. .

C C homeomorphism of ppLeft ) Let (P,p,Z) (P,p,Z) denote denote the the canonical homeomorphism of (( g4 Blank Let This Page Intentionally of M+ M+ about about o o Then the the mapping mapping chart of chart Then

induces aa complex-linear complex-linear homeomorphism homeomorphism onto onto aa closed closed subspace subspace induces -1 -1 of Z x L(Z) x t ( 2 , Z ) Here hX : = (Xp)op Now of 2 x ~ ( 2 )x f ( 2 , 2 ) Here hX : = (Xp)op Now apply 44.6. Q.E.D. apply .6. Q.E.D.

.

.

M- which which are are "dual" "dual" to The bounded bounded symmetric symmetric domains domains MThe the classical classical complex complex Banach Banach manifolds manifolds M+ M+ are are also also called the "classical". In analogy to 23.9, we have "classical". In analogy to 23.9, we have 23.11 THEOREM, THEOREM. Let Let MM- be be aa classical classical complex complex bounded bounded 23.11 symmetric domain. domain. Then Then aut(M-) aut(M-) == pp (( g-1 g-) symmetric

.

PROOF. Since Since 2Z is is the the JB*-triple JB*-triple associated associated with with MM- , , PROOF. 20.26 implies aut(M-) = k @ p , where k = aut(Z) and 20.26 implies aut(M-) = k @ p , where k = aut(2) and

-

p = { Xb : b

hi

Since kfr == h i Since

E

}

=

(hC1 @ h y ) f i h -

by 23.8, 23.8, the the assertion assertion follows. follows. by

. Q.E.D. Q.E.D.

23.12 COROLLARY. COROLLARY. Let Let MM- be be aa classical classical complex complex bounded bounded 23.12 is a real Banach Lie group group symmetric domain. Then Aut(M-) symmetric domain. Then Aut(M-) is a real Banach Lie with Lie Lie algebra algebra pp((gg--11 and and identity identity component component r(G-1 r(G-1 with

.

PROOF. Apply Apply 13.16 13.16 and and 23.11. 23.11. PROOF.

0Q.E.D. .E.D.

NOTES. The The major major results results of of this this section, section, describing describing the the NOTES. vector fields fields and and automorphisms automorphisms of of classical classical Banach Banach manifolds, manifolds, vector are due due to to W. W. Kaup Kaup (cf. (cf. C821for C 82 1 for manifolds manifolds of of type type II and and are of types 11, I11 and IV). It is possible [83] for manifolds [83] for manifolds of types 11, I11 and IV). It is possible to avoid avoid the the use use of of the the Jacobson-Rickart Jacobson-Rickart Theorem Theorem [711 [ 711 in the to in the proof of of 23.6 23.6 by by applying applying more more elementary elementary arguments arguments (cf. (cf. proof

.

c83 11 1). c83 The structure structure of of the the "classical" "classical" Banach Banach Lie Lie groups groups and and The Banach Lie algebras of Hilbert space operators has been Banach Lie algebras of Hilbert space operators has been la Harpe Harpe has has investigated in in 1531. ~ 5 3 1 .More More recently, recently, P. P. de de la investigated of "classical" "classical" Lie Lie groups groups proposed aa more more general general concept concept of proposed and Lie algebras related to von Neumann algebras and their and Lie algebras related to von Neumann algebras and their (anti-) automorphisms automorphisms C54 C541.1 . (anti-)

This Page Intentionally Left Blank

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84 8h

22

SUBJECT AND SYMBOLS INDEX

abelian 324 Abel's Theorem 237 action of Lie algebra, 86 faithful 86 analytic 86 adjoint, ad 34, 106, 107 action of (Lie) group, 98 faithful 99 continuous 99 analytic 99 adjoint, Ad 35, 106, 108 transitive 138 admissible p-ball 209 Ado's Theorem 114 algebraic group 117 analytic continuation, principle of analytic flow, 74 global 80 local 67 maximal 74 analytic mapping 6, 37 a analytic vector field h59, 60

aP

a

h(z)G

10, 37

62 anti-commutator 315 associativity condition 41 atlas 36 Aut(C) linear automorphisms of cone 387 aut(C) infinitesimal automorphisms of cone 387 Aut(M) manifold automorphisms 37 aut(M) complete analytic vector fields 80 Aut(M,d) , Aut(M,b) bianalytic isometries 187, 200 aut(M,d) , aut(M,b) infinitesimal isometries 218, 219 Aut(Z) algebra automorphisms 33, 383 aut(Z) algebra derivations 33, 383 Aut(Z) Jordan triple automorphisms 302 aut(Z) Jordan triple derivations 302 Aut(2,Z) "Jordan pair" automorphisms 372 aut(Z,2 ) "Jordan pair" derivations 373

Baire space 171 balanced (hull) D-

195 435

436

SUBJECT A N D SYMBOLS INDEX

Banach algebra 22 Banach Jordan algebra 314 Banach Jordan triple 300 Banach Lie algebra 33 Banach Lie group 92 Banach Lie subgroup 128 Banach manifold 36 Banach space 2, 25 Banach submanifold 126 Banach's open mapping theorem 18, 182, 277 Bergmann operators B(z,h) , B(u,v ) 152, 305 bianalytic 37 biholomorphic 38 binary (Banach) Lie algebra, 142 full $ ' 146 B(o;r) , Bd(o;r) open balls 187 B(S,D) bounded mappings bounded symmetric domain

25 343

*

C -algebra 250 Campbell-Hausdorff series 112 Carathgodory (pseudo-) metric 6 M

188

Carathgodory tangent (semi-) norm P M 203 Cartan's Theorem 208, 219 Cartan's uniqueness theorem 215 Casorati-Weierstrass Theorem 49 Cauchy-Hadamard formula 4 Cauchy inequalities 12 Cauchy integral formulas 11, 12 Cauchy-Schwarz inequality 240, 247, 342 Cayley transformation ge 52, 362, 399, 404, 406, 412 Cayley vector field ,'X 362 chart (about o ) , 36 canonical 44, 48, 95, 148, 292, 395, 402, 408 * linearizing 281 C -involution 247 circular 195, 294 collineation (action) 47, 103 compact-open topology 169 compatible, 2, 36, 188, 200 locally 187 commutator 33, 60 complete analytic vector field 80 complexification X c 17, 22, 314 C complex numbers 2 cone 230 conjugation 399 contraction 187, 200 convex hull co(R) 193 C(Z,W) continuous mappings 4 C,(S,D) bounded continuous mappings 25 C,(S,D) continuous mappings vanishing at infinity

25

SUBJECT AND SYMBOLS INDEX

derivation 2 3 , 5 4 , 3 0 2 , 3 8 3 derivatives f'(o) , f"(0) , af -(m) 53 aP differential of mapping T,(g)

( 0 )

, T(g) 55,

differential of Lie group action differentiable 6 directional derivative AC 55 distance dist(A,B) 11 dual Jordan triple 3 3 3 A open unit disc 4 8 D = R,C,H 2 4 dp 2 0 9

A+

,A*

7

r*

58

100, 166

373

evaluation mapping 2 0 , 5 3 , 6 7 evaluation mapping of Lie algebra action 8 6 at point m , p, 86 evaluation mapping of Lie group action 9 9 at point m , rm 9 9 exceptional Jordan algebra 3 1 5 exponential mapping exp 2 8 , 80, 9 3 E vector field 2 7 4 functional calculus fA 2 9 , 3 0 fundamental formula 2 9 8 , 3 0 1 , 3 1 9 F vector field 2 7 4

ow

261

Gauss pJane 4 8 Gelfand-Neumark-axioms 3 3 6 Gelfand-Neumark embedding theorem 2 5 4 Gelfand-Neumark-Segal construction 2 3 9 Gelfand representation 2 4 8 , 2 5 2 , 3 4 0 , 3 4 1 , 3 6 0 Godement's Theorem 1 3 0 gradation (additive, multiplicative) 6 3 Grassmann manifold Mb(L) 4 4 G(A) group of invertible elements 2 7 g(A) Lie algebra of G(A) 3 3 Ge(L) general linear operator group 2 8 g[(L) Lie algebra of Ge(L) 3 3 Gen(D) general linear matrix group 2 9 g[,(D) Lie algebra of Gtn(D) 3 3 Moebius transformation 4 1 g# ge Cayley transformation 3 6 2 , 3 9 9 , 4 0 4 , 4 0 6 , 4 1 2 g, induced Lie algebra isomorphism 61

437

438

SUBJECT AND SYMBOLS I N D E X

Hadamard's three circles theorem 2 0 9 Hahn-Banach Theorem 1 2 , 1 9 0 , 2 0 1 , 2 3 2 , hermitian 2 3 8 , 2 4 1 , 2 4 2 , 3 3 1 'hermitian 235, 3 3 1

245

-

hermitian 2 3 8 , 3 0 5 , 3 2 1 H(Z) hermitian elements 2 4 2 H+(Z) positive hermitian elements 2 4 3 H L ( E ) hermitian operators 2 4 1 H L + ( E ) positive hermitian operators 2 4 3 HL,(D) self-adjoint matrices 3 1 5 Hilbert space 4 2 Hilbert sphere 1 2 8 holomorphic 11, 3 8 homogeneous 1 9 2 homomorphism 22, 1 0 4 , 3 0 2 H quaternions 2 4

h-1 h-l

i

ho i hl h-1/2 I

142,

308

$ ,

2Ihl

ho , h, 275 K h-, h; I h: , h K H r h+ i H+ 1 4 7 Ho H1 H-1 1 4 9

,

I

1

, $

262,

274,

363

-1

I

IhK

,

-lhK

396,

403,

410

SUBJECT AND SYMBOLS INDEX identity mapping idD 18 immersion 121 infinitesimal generator 68 inner automorphism Int(g) 103 int(A) inner algebra derivations 34 int(g) inner Lie algebra derivations 34 int(X) inner Jordan algebra derivations 385 Int(Z) inner Jordan triple automorphisms 378 int(Z) inner Jordan triple derivations 303 Int(Z,Z) inner "Jordan pair" automorphisms 373 int(Z,Z) inner "Jordan pair" derivations 374 I interval with radius T 72 T inverse mapping theorem 18, 57 invertible 26, 323 involution 24, 234, 260, 319 isometry, 115, 187, 200 infinitesimal 115, 218, 219 isotropy subgroup Go 138 I vector field 65 Jacobi identity 33 Jacobson-Rickart Theorem 418 * JB,-algebra 349 JB -triple 336 * JC,-algebra 349 JC -triple 301 Jordan algebra 314 Jordan identity 314 Jordan triple (system) 299 Jordan triple identity 298, 301, 318 je 399, 404, 406, 412 jm symmetry about m 282

K = R,C 2 K 2 k(g) 174 k 96, 306 k , E 304 k' , 'k' , 'k'1k- , -112k- I

381 390

Laurent expansion 49, 193 Lx , L(x) left multiplication operator left-uniform structure 172 Leibniz' rule 23 Lb arc length 200 Lie algebra of Lie group 93 lifted action r+" , p'" 100, 101

92, 107

439

SUBJECT AND SYMBOLS INDEX

440

L(Z,W) continuous linear mappings 2 L"(z,w) continuous n-linear mappings 3 L(L) bounded operators 2 6 LJL) compact operators 4 3 Ln(D) (nxn)-matrices 2 6 Liouville's Theorem 15 locally transitive action 1 3 9 , 2 8 3 locally uniform action 8 7 , 1 5 7 local representation of mapping 37 g# local representation of Lie algebra action local representation of isotropy subalgebra local representation of Lie group action

'

'

87 p#

220,

r+ 1 5 z 2 0 1 local representation of isotropy subgroup local representation associated with p-ball 2 1 5 logarithm logA 3 1 L,(S,D) bounded measurable functions 2 5

metric Banach manifold 188 Ml(L) Grassmann manifold 4 4 Moebius transformation g 41 Montel's Theorem 170, 184 M K I M+ , M333, 400 M/R

MZ MX

130

, M(z)

Jordan multiplication operator

314

384

Neumann series 2 7 normed Banach manifold 200 no small subgroups 158 N = H/H+ 1 4 8

0 octonions

O(M,N) O,(M,W)

315 holomorphic mappings 13, 3 8 bounded holomorphic mappings

14, 3 8

0: analytic germs 5 3 open mapping theorem 15, 3 8 operator norm 3, 2 5 orbit G * o , G ( o ) 138 order of function Ordo(f) 9, 3 7 order of vector field Ordo(X) 2 2 5 order unit Banach space 2 3 2 order unit (semi-) norm 1.1, 230

Ol"(L)

, Ol(L) orthogonal groups

400,

406

290

280,

290

SUBJECT AND SYMBOLS INDEX oLU(L) RA 29

nS R 9 R, RZ

,

oL(L)

corresponding Lie algebras

441

400, 406

31 41 67 I

+

R;

I

RZ

n;

I

332, 333

1-parameter group 80, 162 Peirce Composition rules 358 Peirce splitting 354, 357 Poincarg metric w 205 polarization formula 3 polarization of identities 298, 316

P" ( Z ,W ) n-homogeneous polynomials , P ( Z ) polynomial vector fields 65 positive 232, 235, 243, 264, 336 powers 324, 327 power series (convergent) 4 projections IT^ , IT^ , ITR 1 3 0 147

f

3

IT-1

IT

+

151

(pseudo-) metric 187 P(L) , Pn-l(D) projective spaces ('F~PE,

L(E,F)) F'

(P^,p",Z)

46, 48

canonical chart of Grassmann manifold

canonical chart of

Mh

311

(PK,pK,Z) canonical chart of M K 333 Pz , P(z) quadratic representation 314 P(x,y) 318 p 296, 304

quasi-inverse zh 152 quasi-invertible 373 Q , , Q(x) quadratic representation Q(X,Y)

r

299

QXfy 355

radius of convergence R 4 4 radius of restricted convergence R R real numbers, R+ non-negative real numbers 2 Re real part 25, 260 Riemann mapping theorem 48 Riemann sphere 48 Riemann surfaces 48 right-invariant 93 Rx , R(x) right multiplication operators 92, 107

44

SUBJECT AND SYMBOLS INDEX

442

Runge's Theorem 32 Russo-Dye Theorem 256, 257 rX analytic flow 77 analytic action of g K on PK

p(f,r) p,(flr)

prr(f,r) 5 pz(f,r) 9

pQ

, pw,

restriction mappings

Rp

132

M~

333

191

Sard's Theorem 201 Schwarz' Lemma 52, 190 self-adjoint part 234, 260, 315 Siege1 domains 261, 263, 370, 399, 404, DC DCIO DCla I B 406, 412 Skolem-Noether Theorem 416 spectrum CA(x) 29, 235 spectrum of operator C&,(X) 29 spectrum space C w 237 spectral mapping theorem 31, 237, 238 spectral (semi-) norm 343 spin factor 316, 352 splitting, split subspace 42 starlike 29 state (space) of order unit space S x 232 state (space) of algebra S z 244 Stone-Weierstrass Theorem 252 subgroup, Banach Lie 128 submanifold, Banach 126 submersion 121 subtriple, Jordan 300 supremum norm IflM 141 25, 38 symmetric metric (normed) G-manifold 282, 292 symmetry 281 SR 172, 173 Sr

213

tangent bundle T(M) tangent (semi-) norm tangent space Tm(M)

58 200 54

tangent vector

h a 54 ap h-a 56 az T(M) analytic vector fields 59 Tn(Z) (n+l)-homogeneous vector fields T-l(Z) constant vector fields 64

64

SUBJECT AND SYMBOLS INDEX

To(Z)

linear vector fields

64

T (2)

quadratic vector fields 64 1 topologically faithful action 110, 172 topologically regular 231, 265 topology of locally uniform convergence triple product {xu'y} 297 {xu*y} 299, 395, 409 {xuy] 318

tripotent 355 tube domain TB , DC 196, 260 type (I-IV) 397, 403, 411 T, 110

unital, unitization 22, 314 unitary 255, 320 U(Z) 1 1 6 , 255 u(Z) Lie algebra of U(Z) 116, 256 Ut(L) linear isometries 115, 255 ut(L) infinitesimal isometries 115, 255 U.$,(D) unitary matrix group 255 utn(D) UL(F,E)

Lie algebra of

Utn(D)

255

394

(D) 394 utqIp(D) Lie algebra of U e qt P U(R) , u ( c ) 51, 281 universal covering manifold ML 38 universal covering group Grr 97 Us(B) s-neighborhood 72 Vidav-Palmer Theorem X+

positive cone

258, 340

230, 235, 253

X,* Cayley vector field

362

Xn

powers of vector field Xf 59

X..

304

xm ,

X# *66, 76, 87 xuu 299

68

209, 211

443

444

Y, ya y,

'y

S U B J E C T AND SYMBOLS I N D E X

t

I

Yb

261,

3 64

yb 364, 365 1 4 2 , 308 308