Bounded Symmetric Domains in Banach Spaces 9811214107, 9789811214103

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Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

Library of Congress Cataloging-in-Publication Data Names: Chu, Cho-Ho, author. Title: Bounded symmetric domains in Banach spaces / Cho-Ho Chu, Queen Mary, University of London, UK. Description: New Jersey : World Scientific, [2020] | Includes bibliographical references and index. Identifiers: LCCN 2020025418 | ISBN 9789811214103 (hardcover) | ISBN 9789811214110 (ebook for institutions) | ISBN 9789811214127 (ebook for individuals) Subjects: LCSH: Banach spaces. | Symmetric domains. Classification: LCC QA322.2 .C48 2020 | DDC 515/.732--dc23 LC record available at https://lccn.loc.gov/2020025418

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

Copyright © 2021 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the publisher.

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Contents Preface

vii

Chapter 1. Introduction

1

1.1 Holomorphic maps in Banach spaces

1

1.2 Banach manifolds

11

1.3 Symmetric Banach manifolds

34

Chapter 2. Jordan and Lie algebraic structures

51

2.1 Jordan algebras

51

2.2 Jordan triple systems

68

2.3 Lie algebras and Tits-Kantor-Koecher construction

102

2.4 Jordan and Lie structures in Banach spaces

114

2.5 Cartan factors

133

Chapter 3. Bounded symmetric domains

145

3.1 Algebraic structures of symmetric manifolds

145

3.2 Realisation of bounded symmetric domains

153

3.3 Rank of a bounded symmetric domain

184

3.4 Boundary structures

195

3.5 Invariant metrics, Schwarz lemma and dynamics

210

3.6 Siegel domains

254

3.7 Holomorphic homogeneous regular domains

273

3.8 Classification

300

Chapter 4. Function theory

313

4.1 The class S

313

4.2 Bloch constant and Bloch maps

318

v

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Contents 4.3 Banach spaces of Bloch functions

340

4.4 Composition operators

355

Bibliography

367

Index

385

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Preface

This book discusses aspects of bounded symmetric domains and their algebraic structures. The focus is on infinite dimensional domains and some recent advances in the geometric and analytic theory of these domains, of which the dissemination has been confined to research papers so far and a self-contained monographic exposition seems timely. With a concise bibliography, it is intended as a convenient reference for a broad readership including research students. ´ Cartan, symmetric spaces in finite Since the seminal work of E. dimensions have been intensively studied and constitute an important area of research in geometry and literature abounds. Lie theory has been an important tool in the investigation of these manifolds and their classification. In finite dimensions, bounded symmetric domains are exactly the class of Hermitian symmetric spaces of non-compact type, via the Harish-Chandra realisation. The finite dimensional concept of a symmetric domain can be extended naturally to infinite dimension and in recent decades, Jordan algebras and Jordan triple systems have gradually become a significant part of the theory of bounded symmetric domains due to the successful application of them in providing a unified and fruitful treatment of both finite and infinite dimensional symmetric domains. Nevertheless, this remarkable success also hinges on the close relationship between Jordan and Lie algebras. A far-reaching accomplishment is the discovery that every bounded symmetric domain can be realised as the open unit ball of a complex Banach space equipped with a Jordan structure. This led vii

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to fertile interactions with infinite dimensional analysis and operator theory. In this book, we present an introduction to some basic theory of infinite dimensional Jordan and Lie algebras (but not excluding the finite dimensional ones) and explain in detail how they are used to show that a bounded symmetric domain is biholomorphic to the open unit ball of a Banach space with a Jordan structure. We discuss various applications of this realisation of a bounded symmetric domain. The most important ones concern the classification and geometric function theory of these domains. The book begins with some basic concepts and notations in complex analysis in Banach spaces and a brief review of Banach manifolds. Symmetric Banach manifolds, which generalise finite dimensional Hermitian symmetric spaces, are introduced, together with the associated Lie structures. As alluded to earlier, Jordan and Lie algebras play an important role in the theory of bounded symmetric domains. We discuss Jordan and Lie structures in Chapter 2, with a view for later applications, which requires a dimension-free setting: infinite dimensional algebras are not excluded. We first discuss the structures of Jordan algebras and their generalisation, Jordan triple systems. The Tits-Kantor-Koecher construction is shown in Section 2.3, which establishes the one-one correspondence between Jordan triple systems and a subclass of Lie algebras. This will be vital later for constructing a holomorphic embedding of a bounded symmetric domain in a complex Banach space equipped with a Jordan triple structure, alias JB*-triple. The concept of JB*-triples is introduced in Section 2.4, where a number of examples are given. These include complex spaces of matrices, operators, Hilbert spaces and interestingly, C*-algebras. This hints at a close connection with functional analysis. The fundamental examples of JB*-triples, known as Cartan

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factors, are discussed in Section 2.5. They are the building blocks of JB*-triples, but crucially, the classifying spaces of bounded symmetric domains. Chapter 3 is the highlight of the book, where a full discussion of bounded symmetric domains takes place. We begin by discussing the Jordan and Lie structures of symmetric Banach manifolds, which are of fundamental importance. This paves the way for the major task in Section 3.2 of proving one of the most important theorems in the book, due to Kaup [99], which asserts, in Theorem 3.2.18 and Theorem 3.2.20, that a domain in a complex Banach space is a bounded symmetric domain if and only if it is biholomorphic to the open unit ball of a JB*-triple. A finite dimensional precursor has been shown by Loos [125]. Kaup’s result can be viewed as a version of the Riemann mapping theorem for all dimensions since the complex plane is a one-dimensional JB*-triple. It is hard to exaggerate the consequences of this theorem. It offers us, in addition to other methods, a Jordan approach to geometry of symmetric manifolds and we do just that in ensuing sections, by identifying a bounded symmetric domain as the open unit ball of a JB*triple. We show in Section 3.3 that the rank of a bounded symmetric domain can be defined by the ambient JB*-triple structure. This leads to a classification of finite-rank bounded symmetric domains in terms of JB*-triples. In Section 3.4, we study the boundary structures of a bounded symmetric domain and describe its boundary components in terms of Jordan structures. The success of the Jordan approach relies on two fundamental bounded linear operators on a JB*-triple, namely, the Bergman operator and the left multiplication, called a box operator. They can be used to describe the M¨obius transformations and automorphisms of a bounded symmetric domain. Using this device, we study the Carath´eodory metric and Kobayashi metric on bounded symmetric domains in Section 3.5 and inevitably, the Schwarz lemma

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comes to the fore. We discuss variants of the Schwarz lemma, leading to the Denjoy-Wolff theorem and the investigation of iteration of holomorphic self-maps on a bounded symmetric domain. The holomorphic equivalence of the open unit disc and the upper half-plane in C, via the Cayley transform, is fundamental in complex analysis. In Section 3.6, we discuss the finite and infinite dimensional generalisations of the upper half-plane. These are called the Siegel domains, which are defined over a positive cone. The main question is, regarding bounded symmetric domains as a generalisation of the open unit disc in C, when are they holomorphically equivalent to Siegel domains? We provide an answer and generalise the Cayley transform in this section, where Jordan algebras play an important role. Siegel domains of the first kind over a cone C in a Hilbert space is biholomorphic to a symmetric domain exactly when C is a linearly homogeneous self-dual cone. In finite dimensions, Koecher [109] and Vinberg [170] have shown that these cones are in one-one correspondence with the formally real Jordan algebras. One can find in Section 2.4 an infinite dimensional extension of this correspondence. In Section 3.7, as a further application of Jordan theory, we determine completely which bounded symmetric domains are holomorphic homogeneous regular manifolds. The notion of these manifolds in finite dimensions has been introduced by Liu, Sun and Yau [120] in connection with the estimation of several canonical metrics on the moduli and Teichm¨ uller spaces of Riemann surfaces. We extend this concept to infinite dimension and show its connection with the rank of a bounded symmetric domain. The last section of this chapter is devoted to the classification of bounded symmetric domains. We discuss a Jordan ap´ Cartan’s proach to the classification of these domains and show that E. seminal classification of finite dimensional bounded symmetric domains can be viewed as a special case of this Jordan classification. In the last chapter, the realisation of bounded symmetric domains

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in JB*-triples is used again to render an effective and unified treatment of various topics in function theory on these domains, in all dimensions. The main discussions include distortion theorems, Bloch functions and the Bloch constant in the first three sections, as well as composition operators on Banach spaces of Bloch functions in the last section. These are familiar topics in several complex variables, but have only been studied recently in infinite dimension. However, the content of the chapter is by no means exhaustive and needless to say, the geometric function theory of infinite dimensional bounded symmetric domains has yet to be fully developed. It is hope that this ‘handbook’ may help to facilitate the enterprise. To conclude, I wish to thank my wife for her constant support and encouragement, without which this book would not have materialised.

Acknowledgement

This work was partly supported by a research grant (EP/R044228/1) from the Engineering and Physical Sciences Research Council in UK.

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Chapter 1

Introduction 1.1

Holomorphic maps in Banach spaces

To facilitate ensuing discussions, we begin with an introduction to some basic concepts in complex analysis and geometry as well as notations. In this first section, we discuss holomorphic maps in finite and infinite dimensional settings. Throughout the book, the open unit disc in the complex plane C is denoted by D = {z ∈ C : |z| < 1} and a domain in a Banach space is meant to be a non-empty open connected set. The open unit ball of a complex Hilbert space will be called a Hilbert ball. A domain D in a complex Banach space is called circular if z ∈ D implies eiθ z ∈ D for all θ ∈ R; it is called balanced if λD ⊂ D for all λ ∈ D. A balanced domain contains the origin 0. As usual, we denote the closure of a set E in a Banach space by E, the interior of E by E 0 or int E, and the topological boundary of E by ∂E. It would be impossible for the latter notation to be confused with partial differentiation. Let V and W be (real or complex) Banach spaces, and let U be an 1

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Introduction

open subset of V . A function f : U −→ W is said to be differentiable at a point a ∈ U if there is a continuous linear map f 0 (a) : V −→ W satisfying kf (a + h) − f (a) − f 0 (a)(h)k = 0. h→0 khk lim

The map f 0 (a) is unique and called the (Fr´echet) derivative of f at a, which is sometimes denoted by Da f or dfa .

The function f is called

differentiable in U if it is differentiable at every point in U, in which case the derivative f 0 is a mapping f 0 : U −→ L(V, W ) where L(V, W ) is the Banach space of continuous linear operators from V to W . We say that f is continuously differentiable in U, or of class C 1 , if the derivative f 0 is continuous on U. One defines k-times continuously differentiable functions, or C k -functions, for k ∈ N, by iteration. A smooth function on U is one that is infinitely differentiable, that is, it is in the class C k for all k ∈ N. A basic rule in differentiation is the chain rule which states that the composite f ◦ g of two differentiable functions f and g, whenever well-defined, is differentiable and the derivative is given by f 0 (g(a)) ◦ g 0 (a) for a in an open set U where g is defined. A very useful theorem on differentiation is the following mean value theorem (cf. [57, 8.5.4]). Theorem 1.1.1. Let f : U −→ W be a differentiable function on an open set U which contains the segment {a + sh : a, h ∈ U, 0 ≤ s ≤ 1}. Then we have kf (a + h) − f (h)k ≤ khk sup kf 0 (a + sh)k. 0≤s≤1

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1.1 Holomorphic maps in Banach spaces

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In finite dimensions, the complex d-dimensional Euclidean space Cd

is often identified with the real 2d-dimensional Euclidean space R2d

via the real linear isomorphism (z1 , . . . , zj , . . . , zd ) ∈ Cd 7→ (x1 , y1 , . . . , xj , yj , . . . , xd , yd ) ∈ R2d where we usually write zj = xj + iyj ∈ R + iR, and the real linear isomorphism z = (z1 , . . . , zj , . . . , zd ) ∈ Cd 7→ iz = (iz1 , . . . , izj , . . . , izd ) ∈ Cd gives rise to a real linear isomorphism J

: (x1 , y1 , . . . , xj , yj , . . . , xd , yd ) ∈ R2d

(1.1)

7→ (−y1 , x1 , . . . , −yj , xj , . . . , −yd , xd ) ∈ R2d such that −J 2 is the identity map on R2d . We call J the canonical complex structure on R2d . Differentiability of a complex function in Cd can be characterized by the Cauchy-Riemann equations, which can be described in terms of the differential operators   ∂ 1 ∂ ∂ = −i ∂zj 2 ∂xj ∂yj

and

∂ 1 = ∂z j 2



∂ ∂ +i ∂xj ∂yj

 .

By a well-known theorem of Hartogs, a function f : U −→ C having continuous partial derivatives at each point of an open set U ⊂ Cd is differentiable if and only if it satisfies the Cauchy-Riemann equations ∂f =0 ∂z j

(j = 1, . . . , d).

If V and W are complex Banach spaces, a differentiable function f : U −→ W is smooth and is usually called holomorphic. In addition, it has a local power series expansion which is made precise below.

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Introduction First, for n ∈ N, the vector space Ln (V, W ) of all continuous n-

linear maps F :V · · × V} −→ W | × ·{z n-times is a Banach space in the norm kF (v1 , . . . , vn )k . vk 6=0 kv1 k · · · kvn k

kF k = sup

We define L0 (V, W ) = W . If W is the underlying scalar field of the Banach space V , then L1 (V, W ) is just the dual space V ∗ of V . Definition 1.1.2. A continuous map p : V −→ W between Banach spaces is called a homogeneous polynomial of degree n if there exists P ∈ Ln (V, W ) such that (v ∈ V ).

p(v) = P (v, . . . , v)

If P is chosen to be symmetric, that is, invariant under permutation of variables, then P is uniquely determined by p and is called the polar form of p (cf. [84]). We denote by P n (V, W ) the vector space of all homogeneous polynomials of degree n from V to W , and equip it with the norm kpk = sup v6=0

kp(v)k kvk

(p ∈ P n (V, W )).

We note that P n (V, W ) is a closed subspace of the space C(V, W ) of continuous maps from V to W , in the pointwise topology. Definition 1.1.3. Let V and W be Banach spaces. A power series from V to W is a formal sum ∞ X n=0

pn

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where pn ∈ P n (V, W ). Its radius of convergence is defined to be the largest non-negative number R ≤ ∞ such that the series ∞ X

pn (v)

(v ∈ V )

n=0

converges uniformly for kvk ≤ r and r < R. As in the scalar case, the radius of convergence R can be obtained by R=

1 . lim supn kpn k1/n

Likewise, one can define the radius of convergence R for the series where Fn ∈ Ln (V, W ), for which we also have R= so that

∞ X

Fn

n=0

1 , lim supn kFn k1/n

∞ X

Fn (v1 , . . . , vn )

n=0

converges uniformly whenever max(kv1 k, . . . , kvn k) ≤ r and r < R. A function f : U −→ W from an open set U in a Banach space V to another one W is said to be analytic at a point a ∈ U if it can be expressed as a convergent power series about a which means that there P is a power series n pn with positive radius of convergence such that f (v) =

∞ X

pn (v − a)

n=0

for each v in some neighbourhood of a. An analytic function f : U −→ W is one that is analytic at every point in U. If an analytic function f : U −→ W is bijective and the inverse f −1 is analytic, then f is called bianalytic.

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Introduction Analytic functions are smooth but not vice versa. For instance,

the function

 f (x) =

exp(−x−1 ) (x > 0) 0 (x ≤ 0)

is smooth on R but not analytic at the origin 0. (We note that the same example has been given in [37, p. 64], but a minus sign is missing there.) For complex Banach spaces, however, holomorphic functions are analytic and the term “biholomorphic” is a synonym of “bianalytic”. P If an analytic function f has a power series representation n pn about a ∈ U with pn (v) = Pn (v, . . . , v), then its n-th derivative at a is given by f (n) (a) = n!Pn ∈ Ln (V, W ). Given a bounded holomorphic function f : U −→ W on a bounded open set U with a ∈ U, we have the Cauchy inequality kf 0 (a)k ≤

1 sup{kf (z)k : z ∈ U} R

(1.2)

where R is the distance between a and the topological boundary of U. Example 1.1.4. For any Banach spaces V , W and Z over F = R, C, a bilinear map f : V × W −→ Z is analytic and the derivative f 0 (a, b) : V × W −→ Z at any point (a, b) ∈ V × W is given by f 0 (a, b)(x, y) = f (a, y) + f (x, b). The following two fundamental properties of analytic functions are frequently used. We refer to [168, Theorem 1.11, Theorem 1.23] for the proofs. Theorem 1.1.5. (Principle of analytic continuation) Let f : U −→ W be an analytic function on an open connected set U such that f (x) = 0 for all x in some non-empty open set S ⊂ U. Then f is identically 0.

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Theorem 1.1.6. (Inverse function theorem) Let f : U −→ W be an analytic function on an open set U such that the derivative f 0 (a) : V −→ W has a continuous inverse for some a ∈ U. Then f is bianalytic from a neighbourhood of a onto a neighbourhood of f (a). Another useful fact is Cartan’s uniqueness theorem which states that two holomorphic maps f and g, one of which is biholomorphic, from a bounded domain U to another domain in a complex Banach space must coincide if f (p) = g(p) and f 0 (p) = g 0 (p) for some p ∈ U. In [33], H. Cartan’s proof of this result is for two-dimensional domains, but it can be extended readily to domains in Banach spaces. We will make use of the following two useful consequences. Lemma 1.1.7. Let D be a bounded domain in a complex Banach space V . Given a holomorphic map f : D −→ V such that f (p) = 0 and f 0 (p) = 0 for some p ∈ D, then f is identically zero on D. Proof. Apply Cartan’s uniqueness theorem to the identity map id : D −→ D and the holomorphic map id + f . Lemma 1.1.8. Let D be a circular bounded domain in a complex Banach space V containing the origin 0. A biholomorphic map f : D −→ D satisfying f (0) = 0 is (the restriction of ) a linear map (on V ). Proof. Pick θ ∈ R\{0}. Define a holomorphic map F : D −→ D by F (z) = f −1 (e−iθ f (eiθ z))

(z ∈ D).

Then we have F (0) = 0 and F 0 (0) = (f −1 )0 (0)f 0 (0) which is the identity map on V . Hence Cartan’s uniqueness theorem gives f (eiθ z) = eiθ f (z)

(z ∈ D).

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Introduction

Since f (0) = 0, the power series of f about 0 has the form f=

∞ X

pk

k=1

and we have ∞ X

iθ

iθ

iθ

e pk (z) = e f (z) = f (e z) =

∞ X

k=1

iθ

pk (e z) =

k=1

∞ X

eikθ pk (z).

k=1

Hence pk = 0 for k 6= 1 and f is linear. We end this section with a review of different modes of convergence of sequences of holomorphic maps. We are concerned with three kinds of convergence, namely, pointwise convergence, uniform convergence on compact sets and locally uniform convergence. Let U be a non-empty open subset in a complex normed vector space V and let C(U, W ) denote the complex vector space of all continuous maps from U to a complex normed vector space W . Given two (non-empty) sets A and B in V , we write d(A, B) = inf{kx − yk : x ∈ A, y ∈ B} and for p ∈ V , simply write d(p, B) for d({p}, B) which is the distance from p to B. A subset K of U is said to be strictly contained in U if d(K, V \U) > 0. Consider the following three families of subsets of U: K1 = {F : F is a finite subset of U}, K2 = {K : K is a compact subset of U}, K3 = {B : B is a finite union of closed balls strictly contained in U} where a closed ball is a closed set of the form B(a, r) := {v ∈ V : kv − ak ≤ r}

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for some a ∈ V and r > 0, and as usual, B(a, r) := {v ∈ V : kv − ak < r} is called an open ball (centred at a, with radius r). These three families determine the locally convex topologies for the kinds of convergence just mentioned. The topology Tj determined by the family Kj (j = 1, 2, 3) can be defined by a local base. A local base for the topology Tj , is given by {U (K, ε) : K ∈ Kj , ε > 0} where U (K, ε) = {h ∈ C(U, W ) : sup{kh(x)k : x ∈ K} < ε}. A sequence (fn ) in C(U, W ) converges to a function f ∈ C(U, W ) in the topology T1 means that the convergence is pointwise. The topology T2 is the topology of uniform convergence on compact sets in U, and T3 is the topology of locally uniform convergence. Evidently, we have T1 ⊂ T2 ⊂ T3 . The topology T2 is also called the compact-open topology. We refer to [63, Proposition IV.3.1, Lemma IV.3.2, Lemma IV.3.3] for a proof of the following fundamental result. Proposition 1.1.9. Let V , W be complex normed vector spaces and U a non-empty open subset of V . In the space C(U, W ) of complex continuous functions from U to W , the topology T3 of locally uniform convergence is equivalent to the topology T2 of uniform convergence on compact sets if and only if dim V < ∞. The subspace H(U, W ) of holomorphic maps in C(U, W ) is closed in both topologies T2 and T3 . Definition 1.1.10. A family F of functions in H(U, W ) is called a normal family if every sequence (fn ) in F admits a locally uniformly convergent subsequence.

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Introduction

This notion of normality is a natural extension of the finite dimensional one (cf. [7]). The well-known Montel’s theorem asserts that, for an open set U ⊂ Cm , a family F ⊂ H(U, C) is normal if (and only if) it is uniformly bounded on compact subsets of U (see, for example, [134, p. 8]). This condition for normality is still true for a family F ⊂ H(U, Cd ) of Cd -valued functions. Lemma 1.1.11. For U ⊂ Cm , a family F ⊂ H(U, Cd ) is normal if it is uniformly bounded on compact subsets of U. Proof. Let (fn )∞ n=1 be a sequence in F. We can write fn (z) = (fn1 (z), . . . , fnd (z)) ∈ Cd for each z ∈ U, where the C-valued holomorphic functions {fnj : n = 1, 2, . . . ; j = 1, . . . , d} are uniformly bounded on compact subsets of U. Applying Motel’s theorem repeatedly, one can find a locally uniformly convergent subsequence (fn11 ) of (fn1 ), then there is a locally uniformly convergent subsequence (fn22 ) of (fn21 ), and so on, finally one finds a locally uniformly convergent subsequence (fndd ) of (fndd−1 ). It can be seen readily that the subsequence fnd = (fn1d , . . . , fndd ) of (fn ) is locally uniformly convergent. An infinite dimensional version of the Vitali theorem states that a sequence fn : D1 −→ D2 of holomorphic maps between bounded domains in Banach spaces converges locally uniformly to a holomorphic map f : D1 −→ D2 whenever it converges uniformly to f on an open ball strictly contained in D1 (cf. [146, Theorem 2.13]). Notes. There are several books devoted to complex analysis in Banach spaces, the ones we will refer to occasionally in the sequel are [63] and [133]. One can find a proof in [63, Theorem II.3.4] of the

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1.2 Banach manifolds

11

maximum principle for holomorphic maps f in H(U, W ), which states that the function x ∈ U 7→ kf (x)k cannot have a maximum at x0 ∈ U unless kf (·)k is constant on a neighbourhood of x0 . Further, if U is a domain and W is strictly convex, then the map x ∈ U 7→ kf (x)k cannot achieve its maximum in U unless f is constant [63, Corollary III.1.5].

1.2

Banach manifolds

Infinite dimensional symmetric domains are Banach manifolds. We discuss some basic properties of Banach manifolds in this section. Definition 1.2.1. Let F = R or C. A Banach manifold M over F is a Hausdorff topological space with a family A = {(Uϕ , ϕ, Vϕ )} of local charts (Uϕ , ϕ, Vϕ ) satisfying the following conditions: (i) Uα is an open subset of M and M =

S

ϕ Uϕ ;

(ii) ϕ : Uϕ −→ Vϕ is a homeomorphism onto an open subset of a Banach space Vϕ over F; (iii) the local charts are compatible, that is, the change of charts ψϕ−1 : ϕ(Uϕ ∩ Uψ ) −→ ψ(Uϕ ∩ Uψ ) is bianalytic; (iv) the family A is maximal relative to conditions (i), (ii) and (iii), that is, if (U, ϕ, V ) is a local chart compatible with all the local charts in A, then (U, ϕ, V ) ∈ A. A family {(Uϕ , ϕ, Vϕ )} satisfying conditions (i), (ii) and (iii) above is called an atlas of M or an analytic structure on M . Since an atlas can always be extended to a maximal one satisfying condition (iv), it is often sufficient to exhibit an atlas for a topological space to be a

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Introduction

Banach manifold. A local chart (or system of local coordinates) at a point p ∈ M is a chart (Uϕ , ϕ, Vϕ ) with p ∈ Uϕ . Note that the Banach spaces Vϕ in an atlas need not be the same space, but Vϕ and Vψ are isomorphic if Uϕ ∩ Uψ contains a point p, in which case the derivative (ψϕ−1 )0 (ϕ(p)) : Vϕ −→ Vψ is an isomorphism. If all Banach spaces Vϕ in the atlas are isomorphic, we can always find an equivalent atlas in which they are all equal to some Banach space V , in which case, we say that the manifold M is modelled on the Banach space V and that V is a model space for M . We define the dimension dim M of M to be that of V if dim V < ∞. If V is infinite dimensional, we define dim M = ∞. For a local chart (U, ϕ, V ) of a Banach manifold M , the set {p ∈ M : ∃ a local chart (Uψ , ψ, Vψ ) at p and Vψ ' V } is open and closed in M . Consequently, on a connected component of M , we can choose an atlas in which all Vϕ are the same space. In particular, if M is connected, then we can find a model space for M . We call a manifold real or complex according to the underlying scalar field of the spaces Vϕ . Since a complex Banach space can be viewed as a real Banach space when the scalar multiplication is restricted to the real field, a complex Banach manifold can be viewed as a real Banach manifold with the underlying real analytic structure. In particular, a complex manifold modelled on a d-dimensional complex Euclidean space Cd can be viewed as a real analytic manifold modelled on the 2d-dimensional Euclidean space R2d . If, in Definition 1.2.1, the Banach spaces Vϕ are real and the coordinate transformations ψϕ−1 : ϕ(Uϕ ∩ Uψ ) −→ ψ(Uϕ ∩ Uψ )

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are only smooth, then we call M a (real) smooth Banach manifold and, its atlas a differentiable structure. Assuming analytic structures in our definition of Banach manifolds has the advantage of unifying both real and complex cases. Of course, we can (and will) always regard a Banach manifold (according to our definition) as a smooth Banach manifold. Sometimes we speak of analytic Banach manifolds to highlight the underlying analytic structure. Example 1.2.2. Every Banach space over F is itself a Banach manifold over F, with the analytic structure given by the identity map. Also, an open subset U of a Banach manifold M is a Banach manifold, endowed with the atlas {(U ∩ Uϕ , ϕ|U ∩Uϕ , Vϕ )} derived from the atlas {(Uϕ , ϕ, Vϕ )} of M . In particular, a domain in a Banach space V is a Banach manifold modelled on V . Example 1.2.3. Given two Banach manifolds M and N over the same field with analytic structures {(Uϕ , ϕ, Vϕ )} and {(Vψ , ψ, Wψ )} respectively, the Cartesian product M × N is a Banach manifold with the natural analytic structure {(Uϕ × Vψ , ϕ × ψ, Vϕ × Wψ )}. Example 1.2.4. Let V be a real Banach space and let S(V ) = {(λ, v) ∈ R × V : λ2 + kvk2 = 1} be the unit sphere in the product manifold R × V . Let p = (1, 0) and q = (−1, 0). On S(V ), define two local charts (S(V )\{p}, ϕ, V ) and (S(V )\{q}, ψ, V ) by v v ; ψ(λ, v) = . 1−λ 1+λ We have ϕ(−1, 0) = 0 =ψ(1, 0). For v ∈ ϕ(S(V )\{p})\{0}, we have  ϕ(λ, v) =

ϕ−1 (v) =

kvk2 −1 , 2v kvk2 +1 kvk2 +1

ψ ◦ ϕ−1 (v) =

v kvk2

. Therefore

(v ∈ ϕ(S(V )\{p} ∩ S(V )\{q})).

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wsbsdbschu

Introduction

Hence the above charts define an analytic structure on S(V ). If V = Rn , we adopt the usual notation S n for the unit sphere S(Rn ) in Rn+1 . A mapping f : M −→ N between Banach manifolds (over the same field F) is called analytic if, for each x ∈ M , there are charts (U, ϕ, V ) of M and (V, ψ, W ) of N such that x ∈ U, f (U) ⊂ V and the composed map ψ ◦ f ◦ ϕ−1 : ϕ(U) −→ ψ(V) ⊂ W is analytic. A bijection f : M −→ N is called bianalytic if both f and the inverse f −1 are analytic, in which case M is said to be bianalytic to N . The coordinate map ϕ : U −→ ϕ(U) of a chart (U, ϕ, V ) is bianalytic. Holomorphic and biholomorphic maps on complex Banach manifolds are defined likewise. Holomorphic maps on complex Banach manifolds are analytic and in the sequel, the terms “holomorphic” (respectively, biholomorphic) and “analytic” (respectively, bianalytic) are interchangeable for complex manifolds. Occasionally, a biholomorphic map is also called a biholomorphism. One can also define smooth maps between real smooth Banach manifolds M and N in the same manner, and a smooth map f : M −→ N is called a diffeomorphism if it is bijective and the inverse f −1 is also smooth, in which case we say that M is diffeomorphic to N . A bianalytic map f : M −→ M is called an automorphism of M . The automorphisms of M form a group, with composition product, called the automorphism group of M and is denoted by Aut M . We define tangent vectors on a manifold as directional derivatives along smooth curves. A smooth curve in a Banach manifold M over F is a smooth map γ : (−c, c) −→ M on some open interval in R, where c > 0. Let p ∈ M and let γ : (−c, c) −→ M be a smooth curve such that

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γ(0) = p. Let (U, ϕ, V ) be a local chart at p. Then γ(t) ∈ U for t near 0 and the derivative (ϕ ◦ γ)0 (0) : R −→ V is (identified with) a vector in V . If γ1 : (−c1 , c1 ) −→ M is another smooth curve such that γ1 (0) = p and (ϕ ◦ γ)0 (0) = (ϕ ◦ γ1 )0 (0), then for any chart (Uψ , ψ, Vψ ) at p, we have (ψ ◦ γ)0 (0) = (ψ ◦ ϕ−1 )0 (ϕ(p))(ϕ ◦ γ)0 (0) = (ψ ◦ γ1 )0 (0). Hence we can define an equivalence relation ∼ on smooth curves γ in M with γ(0) = p by γ ∼ γ1 if (ϕ ◦ γ)0 (0) = (ϕ ◦ γ1 )0 (0) for a local chart (U, ϕ, V ) at p. An equivalence class [γ]p is called a tangent vector to M at p. We define the tangent space of M at p to be the set Tp M of all tangent vectors [γ]p at p. For the local chart (U, ϕ, V ) at p, the map [γ]p ∈ Tp M 7→ (ϕ ◦ γ)0 (0) ∈ V is a bijection. Indeed, given a vector v ∈ V , there is a smooth curve γv (t) = ϕ−1 (ϕ(p) + tv) in M from some interval in R such that (ϕ ◦ γv )0 (0) = v. Hence Tp M can be identified with V via the above bijection and is equipped with a Banach space structure. In particular, if M is an open subset of a Banach space W and p ∈ M , then Tp M = W via the identification [γ]p ∈ Tp M 7→ γ 0 (0) ∈ W . In the above chart, if γ(t) ∈ U is defined, then (ϕ ◦ γ)0 (t) ∈ V is a tangent vector at γ(t). Indeed, (ϕ ◦ γ)0 (t) = (ϕ ◦ α)0 (0) with [α]γ(t) ∈ Tγ(t) M , where α(s) is the curve α(s) = ϕ−1 (ϕ(γ(t)) + s(ϕ ◦ γ)0 (t)) defined on some interval in R. Remark 1.2.5. In view of the above identification of tangent vectors, we often denote by γ 0 (t) the tangent vector to a curve γ at γ(t).

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wsbsdbschu

Introduction Let f : M −→ N be an analytic map between Banach manifolds

and let p ∈ M . The map dfp : [γ]p ∈ Tp M 7→ [f ◦ γ]f (p) ∈ Tf (p) N is a continuous linear map and is called the differential of f at p. The differential of a smooth map is defined in the same manner. Example 1.2.6. Let M and N be open subsets of Banach spaces V and W respectively. Let f : M −→ N be an analytic map and let p ∈ M . Identifying the tangent spaces Tp M and Tf (p) N with V and W as above, the differential dfp : Tp M −→ Tf (p) N is the derivative f 0 (p) : V −→ W . Indeed, given v ∈ V with v = γ 0 (0) for some [γ]p ∈ Tp M , we have dfp ([γ]p ) = [f ◦ γ]f (p) = (f ◦ γ)0 (0) = f 0 (γ(0))γ 0 (0) = f 0 (p)v. Example 1.2.7. Let (U, ϕ, V ) be a local chart of a Banach manifold M . For p ∈ U, the differential dϕp : Tp U −→ Tϕ(p) V is given by dϕp ([γ]p ) = [ϕ ◦ γ]ϕ(p) . Under the identification [ϕ ◦ γ]ϕ(p) ∈ Tϕ(p) V 7→ (ϕ ◦ γ)0 (0) ∈ V 7→ [γ]p ∈ Tp M, we have dϕp ([γ]p ) = [γ]p . The following result is an important consequence of the inverse function theorem for Banach spaces. Theorem 1.2.8. Let f : M −→ N be an analytic (a smooth) map between (smooth) Banach manifolds such that the differential dfp : Tp M −→ Tf (p) N is bijective at some point p ∈ M . Then f is locally bianalytic (diffeomorphic) at p, that is, f is bianalytic (diffeomorphic) from a neighbourhood of p onto a neighbourhood of f (p). A closed subspace E of a Banach space V is called a complemented subspace if there is a closed subspace E c of V such that V = E ⊕ E c

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is the direct sum of E and E c , or equivalently, if there is a continuous linear projection from V onto E. If dim V < ∞, then every subspace of V is complemented. Definition 1.2.9. Let f : M −→ N be an analytic (a smooth) map between (smooth) Banach manifolds M and N . Let p ∈ M and let dfp : Tp M −→ Tf (p) N be the differential of f at p. The map f is called an immersion at p if dfp is injective and its image is a complemented subspace of Tf (p) N . We call f a submersion at p if dfp is surjective and its kernel is a complemented subspace of Tp M . Compared with the case of bijective differential dfp where the inverse function theorem implies that f is locally invertible, the weaker condition of injectivity or surjectivity of dfp alone entails the existence of one-sided inverse of f locally. More precisely, we have the following characterizations of immersion and submersion. Lemma 1.2.10. An analytic (a smooth) map f : M −→ N between (smooth) Banach manifolds is a submersion at p ∈ M if, and only if, there is an open neighbourhood V of f (p) ∈ N and an analytic (a smooth) map g : V −→ M such that g(f (p)) = p and f ◦g is the identity map on V. Proof. We consider the analytic case. The proof for the smooth case is verbatim. Let f be a submersion at p ∈ M . Then the differential dfp : Tp M −→ Tf (p) N is surjective with complemented kernel K. Therefore Tp M = K ⊕ K c and dfp is bijective on K c . Let (U, ϕ, Tp M ) be a chart at p with ϕ(p) = 0 and (Vψ , ψ, Tf (p) N ) a chart at f (p), with ψ(f (p)) = 0 and f (U) ⊂ Vψ , so that ψ ◦ f ◦ ϕ−1 is analytic on ϕ(U). On an open neighbourhood B of (0, 0) ∈ K × K c ≈ Tp M , we can

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wsbsdbschu

Introduction

define an analytic map F : B −→ K × Tf (p) N by F (u, v) = (u, ψf ϕ−1 (v))

((u, v) ∈ B)

where v ∈ K c ∩ ϕ(U). The derivative F 0 (0, 0) : K × K c −→ K × Tf (p) N is bijective since F 0 (0, 0)(u, v) = (u, (ψf ϕ−1 )0 (0)(v)) where (ψf ϕ−1 )0 (0) = dfp : K c −→ Tf (p) N is an isomorphism. By the inverse function theorem, F is bianalytic from a neighbourhood of (0, 0) ∈ K × K c onto a neighbourhood S × O of (0, ψf (p)) ∈ K c × Tf (p) N , where S and O are open. The inverse F −1 induces a welldefined analytic map h : ψf ϕ−1 (v) ∈ O 7→ v ∈ K c ∩ ϕ(U). Let V = ψ −1 (O ∩ ψ(Vψ )) and define g : V −→ U ⊂ M by g(y) = ϕ−1 ◦ h ◦ ψ(y)

(y ∈ V)

which satisfies g(f (p)) = p. For each y ∈ V, we have ψ(y) = ψf ϕ−1 (v) for some v ∈ K c ϕ(U) and f (g(y)) = f (ϕ−1 (h(ψ(y)))) = f (ϕ−1 (v)) = y

(y ∈ V).

Conversely, let f have local right inverse g : V −→ M . Then the differential dfp ◦ dgf (p) is the identity map on Tf (p) N . Hence the differential dfp is surjective and dgf (p) is injective. It follows that dgf (p) ◦ dfp : Tp M −→ Tp M is a continuous projection with kernel (dfp )−1 (0) which is complemented in Tp M .

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Remark 1.2.11. The above lemma implies that a submersion f : M −→ N is an open map. If f : M −→ N is an immersion at p ∈ M , then we have Tf (p) N = V ⊕ V c where V = dfp (Tp M ) is the image of the differential dfp . With local charts (U, ϕ, Tp M ) at p and (Vψ , ψ, Tf (p) M ) at f (p), as in the above proof, we can define an analytic map F : Tp M ×V c −→ V ×V c ≈ Tf (p) N locally by F (u, v) = ((ψf ϕ−1 )(u), v) for (u, v) in a neighbourhood of (0, 0) ∈ Tp M × V c . Again, with similar arguments as before, the derivative F 0 (0, 0) is bijective and the local inverse F −1 induces an analytic map g : V −→ M from a neighbourhood V of f (p) such that g ◦ f is the identity map on a neighbourhood of p. This gives the following characterization of an immersion. Lemma 1.2.12. An analytic (a smooth) map f : M −→ N between (smooth) Banach manifolds is an immersion at p ∈ M if, and only if, there are open neighbourhoods U and V of p and f (p) ∈ N respectively, and an analytic (a smooth) map g : V −→ M such that f (U) ⊂ V and g ◦ f is the identity map on U. A topological subspace N of a Banach manifold M is called a submanifold of M if N is itself a Banach manifold and the inclusion map ι : N ,→ M is an immersion. In this case, the tangent space Tp N of N at a point p ∈ N identifies as a complemented subspace of the tangent space Tp M of M at p via the differential dιp : Tp N −→ Tp M . An equivalent way of saying that N is a submanifold of M is that at each point p ∈ N , there is a chart (Uϕ , ϕ, Vϕ ) of M and a complemented subspace Wϕ ⊂ Vϕ such that ϕ(N ∩ Uϕ ) = Wϕ ∩ ϕ(Uϕ ) in which case, {(N ∩ Uϕ ), ϕ|(N ∩Uϕ ) , Wϕ } forms an atlas of N .

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wsbsdbschu

Introduction Every open topological subspace of a Banach manifold M is a

submanifold of M . Let M be a Banach manifold over F and let T (M ) =

[

Tp M

p∈M

be the union of tangent spaces of M . Then T (M ) is a Banach manifold over F with an analytic structure induced from that of M . Indeed, let S {(Uϕ , ϕ, Vϕ )} be an atlas of M and write T Uϕ = p∈Uϕ Tp M . Define a combine map T (ϕ) : T Uϕ −→ Vϕ × Vϕ by T (ϕ)[γ]p = (ϕ(p), (ϕ ◦ γ)0 (0))

( [γ]p ∈ Tp M ).

The image of T Uϕ is ϕ(Uϕ ) × Vϕ . Define a topology on T (M ) in which the open sets O are those such that T (ϕ)(O ∩ T Uϕ ) is open in Vϕ × Vϕ . Then {(T Uϕ , T (ϕ), Vϕ × Vϕ )} is an analytic structure on T (M ). The Banach manifold T (M ) is a vector bundle on M , with bundle projection π : [γ]p ∈ T M 7→ p ∈ M, and is called the tangent bundle of M . An analytic vector field on M is an analytic map X : M −→ T M such that X(p) ∈ Tp M , that is, X is a section of the tangent bundle T (M ). We often write Xp for X(p). The analytic vector fields on M form a vector space XM under pointwise addition and scalar multiplication. Given an analytic map f : M −→ N between Banach manifolds M and N . We can define the tangent map df : T (M ) −→ T (N ) as the combine map df (Xp ) = dfp (Xp )

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which in turn induces the map df ◦ X : M −→ T (N ) for each X ∈ XM . Analytic vector fields can be regarded as differential operators in the following way. Given an analytic map f : U −→ W from an open subset U of M to a Banach space W (over the same field of M ), and given X ∈ XM , one can define an analytic map Xf : U −→ W by Xf (p) = dfp (Xp )

(p ∈ M ).

The above mapping is written as Xf = df ◦ X. We can define the n-th iterate X n of X inductively by X 0 f = f,

X n f = X(X n−1 f )

(n = 1, 2, . . .).

(1.3)

Example 1.2.13. Let U be an open set in the n-dimensional Euclidean space Cn , considered as a Banach manifold, with local chart given by the inclusion map ϕ : U ,→ Cn . Let X : U −→ T U be a holomorphic vector field. For each z = (z1 , . . . , zn ) ∈ U, we have X(z) = [γz ]z ∈ Tz U for some smooth curve γz = ((γz )1 , . . . , (γz )n ) : (−c, c) −→ U with γz (0) = z. Identify Tz U with Cn via [γz ]z 7→ γz0 (0) and define a holomorphic map h : U −→ Cn by h(z) = γz0 (0) = ((γz )01 (0), . . . , (γz )0n (0)) and write h(z) = (h1 (z), . . . , hn (z)). Then for each holomorphic function f : U −→ C, we have  ∂f ∂f Xf (z) = dfz (X(z)) = (z), . . . , (z) (γz0 (0)) ∂z1 ∂zn n n X X ∂f ∂f = (γz )0j (0) (z) = hj (z) (z). ∂zj ∂zj 

j=1

j=1

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wsbsdbschu

Introduction

Therefore we can write, as a differential operator, X=

n X j=1

hj

∂ ∂zj

∂ and write for short X = h ∂z .

In view of this example, we shall represent a holomorphic vector field X, in a local chart Uϕ of a manifold M modelled on a complex Banach space V , by a holomorphic function h : Uϕ −→ V and denote ∂ , and sometimes write, by abuse of notation, X(z) = this by X = h ∂z ∂ h(z) ∈ V . One can consider X as the vector field X = h ∂z on ϕ(Uϕ ),

where h is the holomorphic map h ◦ ϕ−1 : ϕ(Uϕ ) −→ V . Given two analytic vector fields X, Y on a Banach manifold over F, there is a unique analytic vector field [X, Y ] on M , called the commutator (or brackets) of X and Y , such that for each open set U ⊂ M , we have [X, Y ]f = X(Y f ) − Y (Xf ) for all analytic maps f from U to a Banach space W (cf. [168, p. 60]). Using the above formula, one can verify easily that the brackets [·, ·] : XM × XM −→ XM is antisymmetric, bilinear and satisfies the Jacobi identity [X, [Y, Z]] + [Y, [Z, X]] + [Z, [X, Y ]] = 0

(X, Y, Z ∈ XM ).

It follows that XM is a Lie algebra in the above brackets. Let (Uϕ , ϕ, Vϕ ) be a local chart at p ∈ M . A tangent vector [γ]q at a point q ∈ Uϕ identifies with the vector (ϕ ◦ γ)0 (0) ∈ Vϕ . On this chart, an analytic vector field X can be viewed as an analytic function X : Uϕ −→ Vϕ and Xϕ = dϕ ◦ X = X by Example 1.2.7. It follows that [X, Y ]ϕ = X(Y ϕ) − Y (Xϕ) = dY ϕ ◦ X − dXϕ ◦ Y = dY ◦ X − dX ◦ Y

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and [X, Y ](p) = dYp (Xp ) − dXp (Yp )

(p ∈ U).

∂ Example 1.2.14. Given holomorphic vector fields X = h ∂z and Y = ∂ k ∂z on an open set D in a complex Banach space V , we have [X, Y ] = ∂ f ∂z where

f (z) = k 0 (z)h(z) − h0 (z)k(z)

(z ∈ D).

A bianalytic map f : M −→ N between Banach manifolds induces a map f∗ : XM −→ XN

(1.4)

defined by (f∗ X)f (p) = dfp (Xp )

(X ∈ XM, p ∈ M ).

If g is another bianalytic map on N , then we have the tangent map d(g ◦ f ) = dg ◦ df and it follows that (g ◦ f )∗ = g∗ ◦ f∗ . In particular, taking g = f −1 implies that f∗ is bijective. Further, it is straightforward to verify that f∗ is in fact a Lie algebra isomorphism (cf. [168, 4.5]). Given an analytic vector field X on a Banach manifold M , using the existence theorem for differential equations in Banach spaces, one can find an open interval Ip in R containing 0 and an analytic curve γp : Ip −→ Uϕ satisfying the differential equation dγp (t) = X(γp (t)) ∈ Vϕ dt and the initial condition γp (0) = p (cf. Remark 1.2.5). Since the solution also depends analytically on the initial point, the theory of differential equations gives further the following result (cf. [140, §I.7] and [118, Chapter IV]).

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wsbsdbschu

Introduction

Theorem 1.2.15. Given an analytic vector field X on a Banach manifold M and p ∈ M , there is an open neighbourhood U of p and, an open interval I in R containing 0, such that for all q ∈ U, the curve γq satisfying γq0 (t) = X(γq (t)) and γq (0) = q is defined on I. The map α : (t, q) ∈ I × U 7→ γq (t) ∈ M is analytic and satisfies α(s + t, q) = α(s, α(t, q)) for s, t, s + t ∈ I and α(t, q) ∈ U. For t ∈ I, let αt : U −→ M be the analytic map αt (q) = α(t, q) = γq (t)

(q ∈ U).

Since α0 (p) = p ∈ U, we have αt (p) ∈ U for t near 0 and hence the differential (dα−t )αt (p) sends a tangent vector Yαt (p) to a tangent vector in Tp M . Noting that X(p) = [γp ] ∈ Tp M and for an analytic map f from U to a Banach space, we have Xf (p) = [f ◦ γp ] = (f ◦ γp )0 (0) = 1 lim (f (αt (p)) − f (p)), a direct computation gives t→0 t 1 (dα−t (Yαt (p) ) − Yp ) t→0 t

[X, Y ](p) = lim

(1.5)

which expresses in some sense the rate of change of Y in the direction of X, known as the Lie derivative of Y in the direction of X (cf. [118, p. 121]). Definition 1.2.16. The map α : (t, q) ∈ I×U 7→ γq (t) ∈ M in Theorem 1.2.15 is called a local flow or an integral curve of the vector field X. If the flow α is defined on R × M , then X is called a complete analytic vector field. In a local chart (Uϕ , p, V ) of a Banach manifold modelled on a complex Banach space V , with ϕ(p) = 0, a local flow α(t, q) of an ∂ analytic vector field X = h ∂z on M can be written more explicitly

in the following way (cf. [168, Theorem 5.11]). One can find an open

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neighbourhood B of 0 in ϕ(Uϕ ) and some δ > 0 such that the local flow ∂ α(t, z) of the vector field X = (hϕ−1 ) ∂z on ϕ(Uϕ ) is given by

α(t, z) =

∞ n X t n=0

n!

X n ι(z),

(t, z) ∈ (−δ, δ) × B

(1.6)

where ι : ϕ(Uϕ ) −→ ϕ(Uϕ ) is the identity map and X n ι is as defined in (1.3). By the uniqueness theorem in differential equations (cf. [118, p. 88]), there is only one flow α : R × M −→ M of a complete analytic vector field X, satisfying α(0, p) = p. For each t ∈ R, we define an analytic map exp tX : M −→ M by exp tX(p) = α(t, p) for each p ∈ M . The notation is suggested by the property exp (s + t)X = exp sX ◦ exp tX

(s, t ∈ R)

which also implies that each map exp tX is bianalytic since exp tX ◦ exp −tX = exp 0X is the identity map on M . We call the homomorphism t ∈ R 7→ exp tX ∈ Aut M the one-parameter group of X. In this notation, we have d X(p) = exp tX(p) (p ∈ M ). dt t=0 Denote by aut M the set of all complete analytic vector fields on a Banach manifold M . The map X ∈ aut M 7→ exp 1X ∈ Aut M is denoted by exp.

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wsbsdbschu

Introduction Let g : M −→ N be a bianalytic map between Banach manifolds.

Then the induced Lie algebra isomorphism g∗ : XM −→ XN , as defined in (1.4), maps aut M to aut N and for X ∈ aut M , we have the commutative diagram (t, p) ∈ R × M −→   yι × g (t, g(p)) ∈ R × N −→

exp tX(p) ∈ M   yg exp tg∗ X(g(p)) ∈ N

where g(exp tX(p)) = exp tg∗ X(g(p)) gives g ◦ exp tX ◦ g −1 = exp tg∗ X.

(1.7)

Remark 1.2.17. For a smooth Banach manifold M , the tangent bundle T (M ) is a smooth Banach manifold in which case one considers smooth vector fields X : M −→ T (M ), smooth flows and so on. A parallel theory for smooth Banach manifolds can be developed along with analytic manifolds, we suppress the repetition but will make use of the results without more ado. We now discuss two important classes of Banach manifolds, namely, the Riemannian manifolds and the Lie groups. First, Riemannian manifolds. In what follows, by a Riemannian manifold, we shall mean a real smooth Banach manifold M , modelled on a real Hilbert space and equipped with a Riemannian metric. A Riemannian metric g on M is a ‘smooth’ choice of an inner product g(p) on the tangent space Tp M for each p ∈ M , where smoothness refers to g as a function of p. Let us make this precise below. Recall that L2 (V, F) denotes the Banach space of continuous bilinear forms on a Banach space V over F = R, C. If V is finite dimensional, we have the identification L2 (V, F) = V ∗ ⊗ V ∗ where the latter

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is equipped with the injective tensor norm. Hence bilinear forms on V are the so-called (0, 2)-type tensors or covariant tensors of order 2. If V ˆ )∗ is infinite dimensional, we have the identification L2 (V, F) = (V ⊗V which only contains the algebraic tensor V ∗ ⊗ V ∗ as a subspace, where ˆ denotes the projective tensor product. Let L20 (V, F) be the closed ⊗ subspace of L2 (V, F), consisting of all symmetric bilinear forms on V . One can view L20 ( · , F) as a functor on the category of Banach spaces over F. Let M be a smooth manifold modelled on a real Banach space V . A symmetric bilinear form g ∈ L20 (V, R) is called positive definite if g(x, x) > 0

for all

x ∈ V \{0}

in which case, g is often called an inner product. We call g completely positive definite if g is a complete inner product on V . By the open mapping theorem for Banach spaces, g is completely positive definite on V if, and only if, there is a constant cg > 0 such that g(x, x) ≥ cg kxk2

(x ∈ V ).

Of course, if dim V < ∞, then complete positive definiteness is the same as positive definiteness. Now let V be a real Hilbert space with inner product h·, ·i. Then by the Riesz representation theorem, L20 (V, R) is linearly isometric to the closed subspace L(V )s of L(V ), consisting of all symmetric operators in L(V ). The linear isometry g ∈ L20 (V, R) 7→ Lg ∈ L(V )s is implemented by g(x, y) = hLg x, yi

(x, y ∈ V )

where g(x, x) ≥ 0 for all x ∈ V if, and only if, Lg ≥ 0. In this case, positive definiteness of g is equivalent to the symmetric operator Lg being positive and injective. Complete positive definiteness of g is equivalent

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Introduction

to the existence of a constant cg > 0 such that g(x, x) ≥ cg hx, xi

(x ∈ V )

which is also equivalent to Lg being positive and invertible. We can apply the functor L20 ( · , R) to the tangent bundle π:

[

Tp M −→ M

p∈M

and form a vector bundle L20 (π) :

[

L20 (Tp M, R) −→ M.

p∈M

A Riemannian metric on M is a smooth section g : M −→

[

L20 (Tp M, R)

p∈M

of this bundle such that each g(p) : Tp M × Tp M −→ R is completely S positive definite. The differential structure on p∈M L20 (Tp M, R) can be described as follows (cf. [118, p. 170]). Let {(Uϕ , ϕ, V )} be an atlas of M where each tangent space Tp M identifies with the model space V = {(ϕ ◦ γ)0 (0) : [γ]p ∈ Tp M }, and g(p) ∈ L20 (Tp M, R) identifies with gϕ (p) ∈ L20 (V, R), defined by gϕ (p)( (ϕ ◦ α)0 (0), (ϕ ◦ β)0 (0) ) = g(p)([α]p , [β]p ). Write L20 (M ) =

S

p∈M

L20 (Tp M, R) and L20 Uϕ =

S

p∈Uϕ

L20 (Tp M, R).

Define an injective map L(ϕ) : L20 Uϕ −→ V × L20 (V, R) L(ϕ)(g(p)) = (ϕ(p), gϕ (p))

(g(p) ∈ L20 (Tp M, R))

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The image of L(ϕ) is ϕ(Uϕ ) × L20 (V, R). Then {(L20 Uϕ , L(ϕ), V × L20 (V, R))} is an atlas on L20 (M ) whose open sets are the sets O for which L(ϕ)(O ∩ L20 Uϕ ) is open in V × L20 (V, R). Smoothness of the Riemannian metric g : M −→ L20 (M ) means that in local charts (Uϕ , ϕ, V ), the map gϕ : Uϕ −→ L20 (V, R) = L(V )s

(1.8)

is smooth. If a smooth manifold M modelled on a real Hilbert space V admits a Riemannian metric g, we call (M, g), or M (if g is understood), a Riemannian manifold. We usually denote the Riemannian metric g(p) on Tp M by h·, ·ip , and if confusion is unlikely, the symbol g is also often used to denote a diffeomorphism between manifolds. It is well known that every finite dimensional paracompact smooth manifold admits a Riemannian metric. We recall that a topological space M is paracompact if every open covering of M has a locally finite refinement. Metric spaces are paracompact. We refer to [118, p. 36, 171] for a proof of the following existence theorem. Theorem 1.2.18. Let M be a paracompact smooth manifold modelled on a separable Hilbert space. Then M admits a Riemannian metric. Definition 1.2.19. Let M and N be Riemannian manifolds. A diffeomorphism g : M −→ N is called an isometry if it satisfies the condition hXp , Yp ip = hdgp (Xp ), dgp (Yp )ig(p)

(p ∈ M ; Xp , Yp ∈ Tp M ).

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wsbsdbschu

Introduction By polarization, the preceding condition is equivalent to hXp , Xp ip = hdgp (Xp ), dgp (Xp )ig(p)

(p ∈ M ; Xp ∈ Tp M ).

The isometries of a Riemannian manifold M form a group G(M ) under composition. We call G(M ) the isometry group of M . We say that G(M ) acts transitively on M if given two points p, q ∈ M , there exists g ∈ G(M ) such that g(p) = q. We end this section with a review of Lie groups. A (real or complex) Banach manifold G is called a (resp. real or complex) Banach Lie group if G is a group and the group operations are analytic, that is, the multiplication (x, y) ∈ G × G 7→ xy ∈ G and the inverse map x ∈ G 7→ x−1 ∈ G are analytic. In practice, to verify analyticity of the group operations, it suffices to verify the following three conditions (cf. [21, III.1.1]): (i) the left translation `a : g ∈ G 7→ ag ∈ G is analytic, for all a ∈ G; (ii) the mapping (x, y) ∈ G × G 7→ xy −1 ∈ G is analytic in a neighbourhood of (e, e) where e is the identity of G; (iii) the conjugation g ∈ G 7→ aga−1 ∈ G is analytic in a neighbourhood of e, for all a ∈ G. Let G be a Banach Lie group with identity e. For each a ∈ G, the right translation ra : g ∈ G 7→ ga ∈ G is an analytic map. An analytic vector field X : G −→ T (G) is called left invariant (respectively, right invariant) if (d`a )g (Xg ) = Xag (respectively, (dra )g (Xg ) = Xga ) for all a, g ∈ G. This can be written as d`a ◦ X = X ◦ `a and dra ◦ X = X ◦ ra in terms of the tangent maps d`a , dra : T (G) −→ T (G). An invariant vector field X is determined entirely by its value at the identity e since Xa = (d`a )e (Xe ) for all a ∈ G if X is left invariant.

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Given two left invariant analytic vector fields X and Y on G, it can be verified that the commutator [X, Y ] is also left invariant. Therefore the vector space L of all left invariant analytic vector fields on G forms a Lie algebra in the product [X, Y ]. Given a tangent vector [γ]e in the tangent space Te G at the identity e, the vector field X defined by X(g) = (d`g )e ([γ]e ) is left invariant and X(e) = [γ]e . It follows that the map X ∈ L 7→ Xe ∈ Te G is a linear isomorphism since two left invariant vector fields are identical if they have the same value at e. Therefore the tangent space Te G is a Lie algebra in the brackets [Xe , Ye ] := [X, Y ]e

(X, Y ∈ L).

We call g = Te G = {Xe : X ∈ L} the Lie algebra of G. Lemma 1.2.20. A left invariant analytic vector field X on a Banach Lie group G is complete. Proof. By left invariance of X, given X(e) = [γ]e ∈ Te G, we have X(a) = [aγ]a ∈ Ta G. If γe : I −→ G satisfies dγe (t) = X(γe (t)) dt

and γe (0) = e,

then the curve γa = aγe solves dγa (t) = X(γa (t)) dt

and γa (0) = a.

Let αe : I ×U −→ G be a local flow of X satisfying α(0, p) = p for p in a neighbourhood U of e. Then the map αa : (t, ap) ∈ I ×aU 7→ aαe (t, p) ∈ G is a local flow of X satisfying αa (0, ap) = ap for all ap ∈ aU. Taking S the union a∈G I × aU and applying the uniqueness theorem, we arrive at a local flow α : I × G −→ G of X satisfying α(0, a) = a for a ∈ G. Since α(s + t, ·) = α(s, α(t, ·))

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Introduction

for s, t, s + t in I, one can extend α to a map αX : R × G −→ G by defining n-times z   }|  { t t t αX (t, ·) = α ,α , ··· α ,· n n n

for sufficiently large n, which can be seen to be well-defined. Since αX (s + t, a) = αX (s, αX (t, a)) and αX is analytic on I × G, it follows that αX is analytic on R × G and is the flow of X. A one-parameter subgroup of a Banach Lie group G is an analytic homomorphism θ : (R, +) −→ G such that θ(0) = e. Given such a homomorphism θ, we have dθ0 (1) ∈ Te G. Conversely, for each left invariant vector field X on G, the homomorphism t ∈ R 7→ exp tX(e) ∈ G is a one-parameter subgroup of G satisfying d(exp tX(e))0 (1) = Xe . The exponential map exp : Te G −→ G is defined by exp(Xe ) = exp X(e)

(X ∈ L).

The exponential map need not be surjective, but if it is, then G is called exponential. Definition 1.2.21. Let G be a Banach Lie group with Lie algebra g = Te G. The adjoint representation of G is the homomorphism Ad : G −→ Aut g defined by the differential of the inner automorphism: Ad(g) = d(rg−1 `g )e : g −→ g.

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We note that Ad is a homomorphism since r(gh)−1 `gh = rg−1 `g ◦ rh−1 `h . In the notation of (1.4), we have Ad(g) = (rg−1 `g )∗ and by (1.7), exp Ad(g)tXe = exp tAd(g)Xe = g(exp tXe )g −1

(X ∈ g).

(1.9)

The automorphism group Aut g is contained in the Banach space L(Te G) of continuous linear operators on the tangent space g = Te G. Considering Ad as the map Ad : G −→ L(Te G), we can take its differential which defines the adjoint representation ad : g −→ L(Te G) of g: ad(Xe ) = d(Ad)e (Xe ). Lemma 1.2.22. Given two left invariant analytic vector fields X and Y on a Banach Lie group G, we have [X, Y ]e = ad(Xe )(Ye ). Proof. We have Ad(g)(Ye ) = (drg−1 )g (d`g )e (Ye ) = (drg−1 )g (Yg ) by left invariance. Let αt (·) = αX (t, ·) be the flow of X in Lemma 1.2.20. Since X is left invariant, we have `a ◦ αt = αt ◦ `a for a ∈ G. Hence we have α−t (a) = α−t (`a (e)) = `a (α−t (e) = rα−t (e) (a). It follows from (1.5) that 1 ((dα−t )αt (e) Yαt (e) − Ye ) t 1 = lim ((drα−t (e) )αt (e) Yαt (e) − Ye ) t→0 t 1 = lim (Ad(αt (e))Ye − Ye ) t→0 t = ad(Xe )(Ye ).

[X, Y ](e) = lim

t→0

Let G be a Banach Lie group with identity e and Lie algebra g. A subgroup K of G is called a Banach Lie subgroup if it is a submanifold

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wsbsdbschu

Introduction

of G, in which case K is closed and a Banach Lie group in the induced topology of G [168, p. 128] and it can be shown that the subalgebra k = {X ∈ g : exp tX ∈ K, ∀t ∈ R}

(1.10)

of g identifies with the Lie algebra of K. Further, the left coset space G/K = {gK : g ∈ G}

(1.11)

carries the structure of a Banach manifold and the quotient map π : G −→ G/K is a submersion [168, Theorem 8.19]. Manifolds of the form G/K are called homogeneous spaces. Let p = π(e) = K. Then the differential dπe : g −→ Tp (G/K) has kernel ker dπe = k and gives the canonical isomorphism g/k ≈ Tp (G/K). Notes. The basic material of Banach manifolds in this section is extracted from the author’s book [37]. There is a substantial literature on infinite dimensional manifolds, Lie groups and Lie algebras including, for example, [21, 106, 118, 140, 168].

1.3

Symmetric Banach manifolds

Symmetric Banach manifolds are an infinite dimensional generalisation of Hermitian symmetric spaces in Cd . Bounded symmetric domains form a special class of these manifolds, which need not possess a Riemannian structure. Let M be a (smooth or analytic) Banach manifold, with tangent bundle T M = {(p, v) : p ∈ M, v ∈ Tp M }. A mapping ν : T M −→ [0, ∞) is called a tangent norm if ν(p, ·) is a norm on the tangent space Tp M for each p ∈ M . We call ν a compatible tangent norm if it satisfies the following two conditions.

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(i) ν is continuous. (ii) For each p ∈ M , there is a local chart (U, ϕ, V ) at p, and constants 0 < c < C such that ckdϕa (v)kV ≤ ν(a, v) ≤ Ckdϕa (v)kV

(a ∈ U, v ∈ Ta M ).

Remark 1.3.1. A compatible tangent norm satisfying certain smoothness and convexity conditions is known as a Finsler metric or Finsler function (cf. [3, 13]). Given an analytic Banach manifold M with a tangent norm ν, a bianalytic map f : M −→ M is called ν-isometric or a ν-isometry if it satisfies ν(f (p), dfp (·)) = ν(p, ·)

for all

(p, ·) ∈ T M.

(1.12)

For a real smooth Banach manifold M , ν-isometries of M are defined to be diffeomorphisms satisfying condition (1.12). Example 1.3.2. A Riemannian manifold (M, g) modelled on a real Hilbert space V , with inner product h·, ·i, admits a compatible tangent norm ν : T M −→ [0, ∞) defined by ν(p, v) := g(p)(v, v)1/2

(p ∈ M, v ∈ Tp M ≈ V )

where, as noted before, g(p)(v, v) = hLg(p) v, vi for some symmetric operator Lg(p) in L(V )s and, complete positivity of g(p) implies g(p)(v, v) ≥ cg(p) hv, vi for some cg(p) > 0. Smoothness of the metric g implies that ν is continuous and for each p ∈ M , there is a local chart (Uϕ , ϕ, V ) at p such that the smooth map gϕ : Uϕ −→ L(V )s in (1.8) satisfies kLgϕ (a) − Lgϕ (p) k ≤ cgϕ (p) /2 for all a, p ∈ Uϕ , by continuity.

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wsbsdbschu

Introduction It follows from hgϕ (p)v, vi ≥ cgϕ (p) kvk2 and |hgϕ (p)v, vi − hgϕ (a)v, vi| = |hLgϕ (p) v, vi − hLhgϕ (a) v, vi| cg (p) ≤ kLgϕ (a) − Lgϕ (p) kkvk2 < ϕ kvk2 2

that  cg (p)  cgϕ (p) kvk2 ≤ hgϕ (a)v, vi ≤ Lkgϕ (p) k + ϕ kvk2 2 2

(v ∈ V )

where ν(a, v) = hgϕ (a)v, vi1/2 . The ν-isometries of M are exactly the isometries of M with respect to the Riemannian metric g. Example 1.3.3. Let M be a finite dimensional complex manifold modelled on Cd . Then there is a canonical almost complex structure J on M , viewed as a 2d-dimensional real manifold (cf. [80, Chapter VIII, §1]). The almost complex structure J is a map p ∈ M 7→ Jp ∈ L(Tp M ) such that Jp : Tp M −→ Tp M is a complex structure on the tangent space Tp M , that is, Jp is a real linear isomorphism such that −Jp2 is the identity map, and for each smooth vector field X on M , the vector field JX is smooth, where JX(p) = Jp (Xp ) for p ∈ M . If we identify Tp M with Cd = R2d , then Jp is the canonical complex structure given in (1.1). We call J a Hermitian structure on M if M admits a Riemannian metric g satisfying gp (Jp u, Jp v) = gp (u, v)

(u, v ∈ Tp M, p ∈ M )

in which case, g is called Hermitian with respect to J. A finite dimensional complex manifold with a Hermitian structure is called a Hermitian manifold. In view of Example 1.3.2, a Hermitian manifold admits a compatible tangent norm defined by the Riemannian metric g.

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We now show that a finite dimensional bounded domain D in Cd carries the structure of a Hermitian manifold. For this, we first introduce the Bergman kernel of D. Let λ be the Lebesgue measure on Cd and let H2 (D) be the complex vector space of square λ-integrable holomorphic functions on D. Then H2 (D) is a separable Hilbert space with inner product Z hf, gi =

f (z)g(z)dλ(z) D

which is called the Bergman space of D. For each w ∈ D, the map f ∈ H2 (D) 7→ f (w) ∈ C is a continuous linear functional on H2 (D). By the Riesz representation theorem, there is a unique function k(·, w) ∈ H(D) such that Z f (w) = hf, k(·, w)i =

f (z)k(z, w)dλ(z)

(f ∈ H2 (D)).

D

The function k(z, w) on D × D is called the Bergman kernel of D. Let {ϕn }∞ n=1 be an orthonormal basis in H2 (D). Then we have k(·, w) =

X X hk(·, w), ϕn iϕn = ϕn (w)ϕn . n

n

From this we deduce that k(z, w) = k(w, z) and that k(z, ·) is holomorphic on D. By considering the holomorphic polynomials on D, it can be seen that k(w, w) > 0 for all w ∈ D. The domain D ⊂ Cd is a complex manifold modelled on Cd . Using the Bergman kernel k(z, w), one can define a Hermitian (Riemannian) metric on D as follows. For each z ∈ D, define hz (u, v) :=

d X i,j=1

∂2 log k(z, z)ui v j ∂zi ∂z j

for u = (u1 , . . . , ud ), v = (v1 , . . . , vd ) ∈ Cd . Then hz (·, ·) is a positive definite Hermitian bilinear form on the tangent space Tz (D) ≈ Cd and

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Introduction

its real part gz (u, v) := Re

d X i,j=1

∂2 log k(z, z)ui v j ∂zi ∂z j

is an inner product on Tz (D) ≈ R2d and g is a Riemannian metric on D, which is Hermitian with respect to the canonical almost complex structure J on D. With this Hermitian structure, a bounded domain in Cd is therefore endowed with a compatible tangent norm. The metric g (or h) is known as the Bergman metric on D. The biholomorphic maps between bounded domains are invariant under the Bergman metric. We refer to [80, Chapter VIII, Proposition 3.5] for a proof of the following basic result. Lemma 1.3.4. Let f : D1 −→ D2 be a biholomorphic map between two bounded domains D1 and D2 in Cd . Then f is an isometry with respect to the Bergmann metric, that is, hz (u, v) = hf (z) (f 0 (z)u, f 0 (z)v)

(z ∈ D, u, v ∈ Cd ).

Besides the tangent norm defined by the Bergman metric on a bounded domain D in Cd , one can define another tangent norm on any bounded domain in a complex Banach space. This is given in Example 1.3.6 below. Example 1.3.5. Given a complex Banach manifold M , define a mapping C : T M −→ [0, ∞) by C(p, v) = sup{|f 0 (p)(v)| : f ∈ H(M, D) and f (p) = 0} for (p, v) ∈ T M , where H(M, D) denotes the set of all holomorphic maps from M to D. Then clearly C(p, ·) is a semi-norm on the tangent

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space Tp M . Given a holomorphic map h : M −→ M and (p, v) ∈ T M , we have C(h(p), dhp (v)) = sup{|dfh(p) (dhp (v))| : f ∈ H(M, D) and f (h(p)) = 0} = sup{|d(f ◦ h)p (v)| : f ∈ H(M, D) and (f ◦ h)(p) = 0} ≤ sup{|dfp (v)| : f ∈ H(M, D) and f (p) = 0} = C(p, v). In particular, if h : M −→ M biholomorphic, we have C(h(p), dhp (v)) = C(p, v). The mapping C : T M −→ [0, ∞) is called the Carath´eodory tangent semi-norm on M . The Carath´eodory tangent semi-norm on M need not be a compatible tangent norm. However, it is indeed such if M is a bounded domain in a Banach space. Example 1.3.6. Let D be a bounded domain in a complex Banach space V . Then the Carath´eodory tangent semi-norm C : T D −→ [0, ∞) defined in the preceding example is a compatible tangent norm on D. Indeed, pick p ∈ D and a chart (U, ϕ, V ) at p with r = d(U, ∂D) > 0. Then for each holomorphic map f : D −→ D, the Cauchy inequality (1.2) implies r|f 0 (a)v| ≤ rkf 0 (a)kkvk ≤ kvk

(a ∈ U, v ∈ V )

and hence rC(a, v) ≤ kvk for (a, v) ∈ U ×V . On the other hand, let D be contained in some open ball B(0, s). For each linear functional h ∈ V ∗ with khk < 1, the holomorphic function f = satisfies f (a) = 0 and hence |h(v)| =

1 2s (h

|2sf 0 (a)v|

− h(a)) : D −→ D

≤ 2sC(a, v). It follows

that kvk ≤ 2sC(a, v) for a ∈ U and v ∈ V . In particular, C(a, ·) is a norm on V , which is equivalent to the norm of V . Moreover, it can be seen that C is continuous on T D (cf. [168, 12.23]).

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Introduction The mapping C is called the Carath´eodory tangent norm on D and

we sometimes write k · ka for the norm C(a, ·) on V . Definition 1.3.7. Let M be a complex Banach manifold equipped with a compatible tangent norm ν. We denote by Aut(M, ν) = {f ∈ Aut M : f is a ν-isometry} the subgroup of all ν-isometries in the automorphism group Aut M of M. Lemma 1.3.8. Let D be a bounded domain in a complex Banach space V , with the Carath´eodory tangent norm ν. Then we have Aut(D, ν) = Aut D. Proof. This follows from Example 1.3.5. Given a Banach manifold M with a compatible tangent norm ν, one can define a distance function on M by the integrated form of ν. Let γ : [0, 1] −→ M be a piecewise smooth curve. The length `(γ) of γ is given by Z `(γ) =

1

ν(γ(t), γ 0 (t))dt.

0

We define the integrated form dν : M × M −→ [0, ∞) of ν by dν (a, b) = inf {`(γ) : γ(0) = a, γ(1) = b}

(1.13)

γ

where γ : [0, 1] → M is a piecewise smooth curve. It can be verified that (M, dν ) is a metric space. Example 1.3.9. Consider the Carath´eodory tangent norm C(p, v) = sup{|f 0 (p)(v)| : f ∈ H(D, D) and f (p) = 0}

(p ∈ D, v ∈ C)

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on the complex open unit disc D, where we have |f 0 (p)(v)| = |f 0 (p)||v| ≤

1 − |f (p)|2 |v| |v| ≤ 1 − |p|2 1 − |p|2

by the Schwarz-Pick lemma. On the other hand, the M¨obius transformation g−p (z) =

z−p 1 − pz

(z ∈ D)

0 (p) = maps the point p to the origin 0, with derivative g−p

follows that C(p, v) =

1 1−|p|2

and it

|v| 1 − |p|2

which is the infinitesimal Poincar´e metric on D. In this case, the integrated distance dC (a, b), the Poincar´e distance, is given by a−b 1 + a−b 1 1−ab = log dC (a, b) = tanh−1 1 − ab 2 1 − a−b

(a, b ∈ D).

(1.14)

1−ab

Remark 1.3.10. On a bounded domain D in a complex Banach space, the Carath´eodory distance cD is defined by cD (a, b) = sup{dC (f (a), f (b)) : f ∈ H(D, D)}

(a, b ∈ D)

which is invariant under biholomorphic maps on D and will be studied in Section 3.5. Although cD = dC is the integrated distance of the Carath´eodory tangent norm on D, the Carath´eodory distance cD need not coincide with the integrated distance of the Carath´eodory tangent norm on other domains D. Let M be a connected manifold modelled on a Banach space V , with a compatible tangent norm ν. One can define the topology of locally uniform convergence on Aut(M, ν) using the metric dν . Choose

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Introduction

a chart (U, ϕ, V ) at p ∈ M such that ϕ(U) is bounded in V and ϕ(p) = 0. Let r > 0 so that Bdν (p, 4r) := {x ∈ M : dν (x, p) < 4r} ⊂ U. Let Bp = Bdν (p, r/2) := {x ∈ M : dν (x, p) < r/2} and define ρBp (f, h) = sup{dν (f (x), h(x)) : x ∈ Bp }

(f, h ∈ Aut(M, ν)).

Then ρBp is a well-defined metric on Aut(M, ν) by the principle of analytic continuation and moreover, the metric ρBq defined by another chart (V, ψ, V ) at q ∈ M induces the same topology as the one induced by ρBp . This topology is called the topology of locally uniform convergence on Aut (M, ν). The following fundamental result is due to Upmeier [167]. Theorem 1.3.11. Let M be a connected complex Banach manifold equipped with a compatible tangent norm ν. A subgroup G of Aut(M, ν), which is closed in the topology of locally uniform convergence, can be topologised to a real Banach Lie group of which the Lie algebra is given by g = {X ∈ aut M : exp tX ∈ G, ∀t ∈ R}. In particular, the group Aut(M, ν) of ν-isometric automorphisms of M carries the structure of a real Banach Lie group with Lie algebra aut(M, ν) = {X ∈ aut M : exp tX ∈ Aut(M, ν), ∀t ∈ R}, which is a real Banach Lie algebra where the norm is determined by ∂ the preceding chart (U, ϕ, V ) at some p ∈ M . The norm of X = h ∂z ∈

aut(M, ν) is defined by kXk = sup{kh(x)k : x ∈ Bp }.

(1.15)

We refer to [168, 13.14, 13.17] for a detailed proof of this theorem. In the special case where M is a bounded domain in a Banach

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space, equipped with the Carath´eodory tangent norm ν on T M , we have Aut(M, ν) = Aut M from Lemma 1.3.8 and hence the following useful result. Corollary 1.3.12. Let M be a bounded domain in a complex Banach space. Then its automorphism group Aut M admits a real Banach Lie group structure and its Lie algebra is the Lie algebra aut M of complete holomorphic vector fields on M . Remark 1.3.13. With the topology of locally uniform convergence, the automorphism group Aut D of a bounded domain D in an infinite dimensional Banach space need not be a Lie group, as shown in [172, § 2.4]. The topology making Aut D into a Lie group is finer than the topology of locally uniform convergence (cf. [168, 13.6]), but they coincide if D is a symmetric domain as defined in Definition 1.3.15 (cf. [168, p. 228]), in which case the preceding corollary has also been proved by Vigu´e [172]. For finite dimensional bounded domains, it is a well-known result of H. Cartan [34] that their automorphism groups are Lie groups in the compact-open topology. It seems appropriate here to amend a slip in the Notes of the author’s book [37, p. 169] where it should have been said that ‘Aut M is a Lie group in a topology finer than the topology of locally uniformly convergence’. We now introduce the fundamental concept of a symmetric Banach manifold. It is convenient to begin with the notion of a symmetry of a manifold. Let M be a Banach manifold endowed with a compatible tangent norm ν and let p ∈ M . A symmetry at p is a ν-isometry s : M −→ M satisfying the following two conditions: (i) s is involutive, that is, s2 is the identity map on M ,

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Introduction

(ii) p is an isolated fixed-point of s, in other words, p is the only point in some neighbourhood of p satisfying s(p) = p. Lemma 1.3.14. Let M be a Banach manifold and let s : M −→ M be a symmetry at p ∈ M . Then the differential (ds)p : Tp M −→ Tp M is the map −id, where id is the identity map of Tp M . Proof. By [168, 17.1], on can find a local chart (U, ϕ, Tp M ) at p with ϕ(p) = 0 and a continuous linear map ` : Tp M −→ Tp M such that the following diagram commutes. s

U ∩ s−1 (U) −→ U ∩ s−1 (U)     ϕ y yϕ `

Tp M −→ Tp M Since s2 is the identity map, so is `2 and hence there is a direct decomposition Tp M = {Xp : `(Xp ) = Xp } ⊕ {Xp : `(Xp ) = −Xp }. Moreover, {Xp : `(Xp ) = Xp } = {0} because p is an isolated fixed-point of s. It follows that −` is the identity map on Tp M and so is (−ds)p . It follows from Lemma 1.3.14 and Cartan’s uniqueness theorem that there can only be one symmetry at a point p in a bounded domain in a complex Banach space, viewed as a Banach manifold equipped with the Carath´eodory tangent norm. For a connected Riemannian manifold, a symmetry is also unique if it exists (cf. [37, p. 101]). Definition 1.3.15. By a symmetric Banach manifold, we mean a connected Banach manifold M , equipped with a compatible tangent norm ν, such that there is a symmetry sp : M −→ M at each p ∈ M .

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Given a symmetric Banach manifold M modelled on a complex Banach space V , it has been shown in [168, Theorem 17.16] that the map  : h ∈ Aut(M, ν) 7→ h(a) ∈ M is a submersion at the identity e of Aut(M, ν), for each a ∈ M . In particular, the differential d of  at e, which is the evaluation map X ∈ aut(M, ν) 7→ X(a) ∈ V,

(1.16)

is surjective, and we have the following equivalent definition of a complex symmetric Banach manifold. Lemma 1.3.16. Let M be a connected Banach manifold with a compatible tangent norm ν, modelled on a complex Banach space V . The following conditions are equivalent. (i) M is a symmetric Banach manifold. (ii) There is a symmetry sp at some point p ∈ M and the map h ∈ Aut(M, ν) 7→ h(a) ∈ M is a submersion at e for each a ∈ M . Proof. The preceding remark establishes (i) ⇒ (ii). Conversely, condition (ii) implies that Aut(M, ν) acts transitively on M , that is, given p, q ∈ M , there exists h ∈ Aut(M, ν) with h(p) = q. Indeed, the submersion h ∈ Aut(M, ν) 7→ h(a) ∈ M is an open map by Remark 1.2.11 and hence the orbit O(a) = {h(a) : h ∈ Aut(M, ν)} is open and closed in M . Since M is connected and a disjoint union of orbits, we must have M = O(a), which implies transitivity of the action. It follows that there is a symmetry at every point a ∈ M , given by h−1 ◦ sp ◦ h, where h ∈ Aut(M, ν) satisfies h(a) = p.

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wsbsdbschu

Introduction The preceding proof shows that a complex symmetric Banach man-

ifold (M, ν) is a homogeneous space of Aut(M, ν). Let K = {h ∈ Aut(M, ν) : h(a) = a} be the isotropy subgroup at a. Then K is closed in the topology of locally uniform convergence and hence, by Theorem 1.3.11, a real Banach Lie group with Lie algebra k = {X ∈ aut(M, ν) : exp tX(a) = a, ∀t ∈ R}. Further, the inclusion map ι : K ,→ Aut(M, ν) is an immersion as its differential at the identity is the inclusion map k ,→ aut(M, ν) and k is a direct summand of aut(M, ν), by Lemma 3.1.1. Hence K is a Banach Lie subgroup of Aut(M, ν) and as in (1.11), the left coset space Aut(M, ν)/K is a Banach manifold and the quotient map π : Aut(M, ν) −→ Aut(M, ν)/K is a submersion. By transitivity, the map ρ : hK ∈ Aut(M, ν)/K 7→ h(a) ∈ M is a bijection and since submersions are open maps, the following commutative diagram implies that ρ is a homeomorphism. ρ

Aut(M, ν)/K −→ M π% Aut(M, ν)

(1.17)

Further, ρ is bianalytic since the submersions π and  have local left inverses by Lemma 1.2.10. For instance, the submersion π has a local left inverse fπ on some neighbourhood V of each point in Aut(M, ν)/K which implies ρ|V =  ◦ fπ is analytic. Likewise, ρ−1 is analytic. Therefore, we can identity M with the homogeneous space Aut(M, ν)/K via ρ. However, we note that a homogeneous space of Aut (M, ν) need not be a symmetric manifold in general [143], but if it admits one symmetry sp , then it is a symmetric manifold since sp can be moved to a symmetry

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at any point by the transitive action of Aut (M, ν) (cf. Example 1.3.22 below). Example 1.3.17. A (finite dimensional) Hermitian manifold M with a Hermitian metric g is called a Hermitian symmetric space if each point p ∈ M is an isolated fixed-point of an involutive biholomorphic map sp : M −→ M which is isometric with respect to g (cf. [80, p. 372]). In view of Examples 1.3.2 and 1.3.3, Hermitian symmetric spaces are complex symmetric Banach manifolds and we can regard the latter as an infinite dimensional generalisation of the former. In finite dimensions, Riemannian symmetric spaces are real symmetric Banach manifolds modelled on Rd . Definition 1.3.18. A connected Riemannian manifold (M, g) modelled on a real Hilbert space is called a Riemannian symmetric space if it is a symmetric Banach manifold with respect to the tangent norm defined by g. By definition, a Hermitian symmetric space is a finite dimensional Riemannian symmetric space. Example 1.3.19. A real Hilbert space V is a Riemannian symmetric space. The symmetry sp at p ∈ V is given by sp (x) = 2p − x. An important class of complex symmetric Banach manifolds are the bounded symmetric domains. Definition 1.3.20. A bounded symmetric domain is defined to be a bounded domain D in a complex Banach space such that each point p ∈ D is an isolated fixed-point of an involutive biholomorphic map sp : D −→ D. We call sp a symmetry of D. In Section 3.6, we will discuss an interesting class of real symmetric Banach manifolds associated to bounded symmetric domains.

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Introduction

By considering the Carath´eodory tangent norm and Lemma 1.3.8, we see that bounded symmetric domains are complex symmetric Banach manifolds. In view of Lemma 1.3.16, they are (strictly) homogeneous domains according to the following definition. Definition 1.3.21. A domain D in a complex Banach space is called homogeneous if the automorphism group Aut D acts transitively on D. If D satisfies the stronger condition that the map h ∈ Aut D 7→ h(a) ∈ D is a submersion (at the identity) for some a ∈ D, then it is called strictly homogeneous. When a finite dimensional bounded symmetric domain is viewed as a Hermitian manifold with the Bergman metric, it is a Hermitian symmetric space by Lemma 1.3.4. In fact, finite dimensional bounded symmetric domains are Hermitian symmetric spaces of the so-called non-compact type, this says they have non-positive sectional curvature ´ Cartan [32] and Harish-Chandra [78] [32]. The seminal works of E. show that a Hermitian symmetric space of the non-compact type is biholomorphic to a bounded symmetric domain in Cd (cf. [80, Chapter VIII, Theorem 7.1]). In this way, one can identify the class of Hermitian symmetric spaces of the non-compact type with the class of finite dimensional bounded symmetric domains, and regard bounded symmetric domains in Banach spaces as an infinite dimensional generalisation of Hermitian symmetric spaces of the non-compact type. Example 1.3.22. The open unit disc D ⊂ C is the simplest example of a bounded symmetric domain. Evidently, the reflection `(z) = −z is a symmetry at 0. Given a ∈ D, the M¨obius transform g−a (z) =

z−a 1 − az

(z ∈ D)

(induced by −a) is biholomorphic on D, satisfying g−a (a) = 0, and the −1 biholomorphic map g−a ◦ ` ◦ g−a is a symmetry at a.

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One can extend the previous example to higher dimensions. Indeed, all finite and infinite dimensional (complex) Euclidean unit balls, that is, open unit balls of complex Hilbert spaces, are bounded symmetric domains. Let D = {v ∈ H : kvk < 1} be the open unit ball of a Hilbert space H with inner product h·, ·i. Then, as before, `(v) = −v is a symmetry at 0 and for a ∈ H, the M¨obius transform induced by −a is defined by p   p 1 − kak2 a 2 v + ( 1 − kak − 1)hv, ai g−a (v) = −a+ 1 − hv, ai kak2

(v ∈ H)

−1 and g−a ◦ ` ◦ g−a is a symmetry at a. We will explain in Chapter 3 how

one arrives at the above M¨ obius transform g−a in the general setting of bounded symmetric domains in Banach spaces. Finite dimensional bounded symmetric domains were classified by ´ E. Cartan [32] using Lie algebras and Lie groups. This classification can be expressed in terms of Jordan algebraic structures and thereby extended to a larger class of infinite dimensional bounded symmetric domains. To achieve this, we first need to develop the relevant theory of Jordan algebras and Jordan triples, which is the task of the next chapter. Details of the classification will be discussed in Section 3.8. Notes. The books by Helgason [80] and Satake [155] are classics on finite dimensional symmetric spaces. Satake’s book also discusses the related Jordan triple structures, so does Loo’s book [125], which focuses on finite dimensional bounded symmetric domains. These references are relevant to our later discussions.

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Jordan and Lie algebraic structures 2.1

Jordan algebras

´ Cartan’s seminal work, Lie theory has played a pivotal role Since E. in the study of finite dimensional bounded symmetric domains. It has been found relatively recently that the closely related Jordan theory offers a transparent algebraic description of these domains as well as their geometric structures, which is also accessible in infinite dimension. Indeed, bounded symmetric domains are biholomorphic to the open unit balls of JB*-triples, which are complex Banach spaces equipped with a Jordan triple structure. This fundamental result and many of its applications will be discussed later, including Cartan’s classification of finite dimensional bounded symmetric domains in terms of JB*-triples. To prepare for this endeavour, we commence a chapter on the relevant basics of Jordan and Lie algebras. We begin with Jordan algebras, which are closely related to Lie algebras. One important feature of these algebras is that the multiplication need not be associative. 51

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Jordan and Lie algebraic structures By an algebra we mean a vector space A over a field, equipped

with a bilinear product (a, b) ∈ A2 7→ ab ∈ A. We do not assume associativity of the product. If the product is associative, we call A associative. Homomorphisms and isomorphisms between two algebras are defined as in the case of associative algebras. An antiautomorphism of an algebra A is a linear bijection ϕ : A −→ A such that ϕ(ab) = ϕ(b)ϕ(a) for all a, b ∈ A. If A is over the field F = R or C, an antiautomorphism ϕ is called an involution if ϕ(ϕ(a)) = a and ϕ(λa) = λϕ(a), where λ denotes the usual conjugate of λ ∈ F. We call an algebra A unital if it contains an identity which will always be denoted by 1, unless stated otherwise. As usual, one can adjoint an identity 1 to a non-unital algebra A to form a unital algebra A1 , called the unit extension of A. A Jordan algebra is a commutative algebra over a field F and satisfies the Jordan identity (ab)a2 = a(ba2 )

(a, b ∈ A).

We always assume that F is not of characteristic two, but in later sections, F is usually either R or C. To avoid confusion, homomorphisms and isomorphisms between Jordan algebras are sometimes called Jordan homomorphisms and Jordan isomorphisms to distinguish them from the ones for other algebraic structures. The concept of a Jordan algebra was introduced by P. Jordan, J. von Neumann and E. Wigner [94], with the aim to formulate an algebraic model for quantum mechanics. They introduced the notion of an r-number system which is, in modern terminology, a finite-dimensional, formally real Jordan algebra. In fact, the term “Jordan algebra” first appeared in a paper by Albert [9]. It denotes an algebra of linear trans-

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formations closed in the product 1 A · B = (AB + BA). 2 Although Jordan algebras were motivated by quantum formalism, unexpected and important applications in algebra, geometry and analysis have been discovered. In particular, the applications of Jordan theory to symmetric manifolds are the subject of our discussions. On any associative algebra A, one can define a product ◦ by 1 a ◦ b = (ab + ba) 2

(a, b ∈ A)

where the product on the right-hand side is the original product of A. The algebra A becomes a Jordan algebra with the product ◦. We call this product special. A Jordan algebra is called special if it is isomorphic to, and hence identified with, a Jordan subalgebra of an associative algebra A with respect to the special Jordan product ◦. Otherwise, it is called exceptional. It is often convenient to express the Jordan identity as an operator identity. Given an algebra A and a ∈ A, we define a linear map La : A −→ A, called the left multiplication by a, as follows La (x) = ax

(x ∈ A).

The Jordan identity can be expressed as [La , La2 ] = 0

(a ∈ A)

where [·, ·] is the usual commutator product of linear maps. Given a, b ∈ A, we define the quadratic operator Qa : A −→ A and box operator a b : A −→ A by Qa = 2L2a − La2 ,

a b = Lab + [La , Lb ].

(2.1)

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These operators are fundamental in Jordan theory, as well as the linearization of the quadratic operator: Qa,b = Qa+b − Qa − Qb . Let A be an algebra and let a ∈ A. We define a0 = 1 if A is unital, a1 = a,

an+1 = aan

(n = 1, 2, . . .).

The following power associative property depends on the assumption that the scalar field F for A is not of characteristic two. Theorem 2.1.1. A Jordan algebra A is power associative, that is, am an = am+n

(m, n = 1, 2, . . .)

for each a ∈ A. In fact, we have [Lam , Lan ] = 0. Proof. For any α, β in the underlying field F, we have [La+αb+βc , L(a+αb+βc)2 ] = 0 for all a, b, c ∈ A. Expand the product, we find that the coefficient of the term αβ is 2[La , Lbc ] + 2[Lb , Lca ] + 2[Lc , Lab ] which must be 0. Since F is not of characteristic 2, we have [La , Lbc ] + [Lb , Lca ] + [Lc , Lab ] = 0. Applying the above operator identity to an element x ∈ A and using

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commutativity of the Jordan product yields (La Lbc + Lb Lca + Lc Lab )(x) = (Lbc La + Lca Lb + Lab Lc )(x) = Lbc Lx (a) + Lbx Lc (a) + Lxc Lb (a) = (Lbc Lx + Lcx Lb + Lxb Lc )(a) = (Lx Lbc + Lb Lcx + Lc Lxb )(a) = (L((bc)a) + Lb La Lc + Lc La Lb )(x). Put b = an and c = a in the above identity, we obtain a recursive formula Lan+2 = 2La Lan+1 + Lan La2 − Lan L2a − L2a Lan which implies that each Lan is a polynomial in La and La2 which commute. It follows that Lan commutes with Lam for all m, n ∈ N. In particular, we have Lan La (am ) = La Lan (am ) and power associativity follows from induction. Corollary 2.1.2. Let A be a Jordan algebra and let a ∈ A. The subalgebra A(a) generated by a in A is associative. In fact, we have the following deeper result. We omit the proof which can be found, for instance, in the books [92, 128]. Shirshov-Cohn Theorem. Let A be a Jordan algebra and let a, b ∈ A. Then the Jordan subalgebra B generated by a, b (and 1, if A is unital) is special.

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Jordan and Lie algebraic structures One can use the Shirshov-Cohn theorem to establish various iden-

tities in Jordan algebras. For instance, in any Jordan algebra A, we have the following identity: 2L3a − 3La2 La + La3 = 0

(2.2)

for each a ∈ A. In other words, we have 2a(a(ab)) − 3a2 (ab) + a3 b = 0 for a, b ∈ A. To see this, let B be the Jordan subalgebra of A generated by a and b. Then it is special and hence embeds in some associative algebra (A0 , ×) with 1 ab = (a × b + b × a). 2 In B, we have 2a(a(ab)) = 3a2 (ab) =

1 3 (a × b + 3a2 × b × a + 3a × b × a2 + b × a3 ) 4 1 (3a3 × b + 3a2 × b × a + 3a × b × a2 + 3b × a3 ) 4

which, together with 2a3 b = a3 × b + b × a3 , verifies the identity. Example 2.1.3. The Cayley algebra O, known as the Octonions, is a complex non-associative algebra with a basis {e0 , e1 , . . . , e7 } and satisfies a2 b = a(ab) ,

ab2 = (ab)b

(a, b ∈ O)

(2.3)

where e0 is the identity of O and e2j = −e0 for j 6= 0. We will denote by O the real Cayley algebra which is the real subalgebra of O with basis {e0 , . . . , e7 }. Historically, octonions were discovered by a process of duplicating the real numbers R. Indeed, the complex numbers arise from R as the product R × R with the multiplication (a, b)(c, d) = (ac − db, bc + da)

(a, b, c, d ∈ R).

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The real associative quaternion algebra H can be constructed by an analogous duplication process. One can define H as C × C, with the multiplication ¯ b¯ (a, b)(c, d) = (ac − db, c + da)

(a, b, c, d ∈ C),

which is isomorphic to the following real non-commutative algebra of 2 × 2 matrices: 

  a b : (a, b) ∈ C × C . −¯b a ¯

In the identification with this algebra, H has a      0 i 0 1 0 , j= , i= 1= −1 0 −i 0 1

(2.4)

basis    0 i 1 , k= i 0 0

satisfying i2 = j2 = k2 = ijk = −1,

ij = −ji = k.

Likewise, O can be defined as the product H×H with the multiplication ¯ b¯ (a, b)(c, d) = (ac − db, c + da)

(a, b, c, d ∈ H)

where the conjugate c¯ of a quaternion c = α1 + xi + yj + zk is defined by c¯ = α1 − xi − yj − zk so that the real part of c is Re c = 21 (c + c¯) = α1. A positive quaternion is one of the form α1 for some α > 0. The basis elements of H × H are e0 = (1, 0), e1 = (i, 0), e2 = (j, 0), e3 = (k, 0), e4 = (0, 1), e5 = (0, i), e6 = (0, j), e7 = (0, k). The algebras C, H and O are quadratic, that is, each element x satisfies the equation x2 = αx + β1 for some α, β ∈ R, where 1 denotes the identity of the algebra. If x = (a1 , a2 ) ∈ H × H with an = αn 1 + xn i + yn j + zn k

(n = 1, 2),

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then we have x2 = 2α1 x − (x21 + y12 + z12 + x22 + y22 + z22 )e0 . Example 2.1.4. Let F = R, C or H, with the usual conjugation c ∈ F 7→ c ∈ F (which is just the identity map if F = R). An n × n matrix (aij ) with aij ∈ F is called hermitian if (aij ) = (aji ). Real hermitian matrices are usually called symmetric. For n ≥ 2, the real vector space Hn (F) of n × n hermitian matrices over F is a special Jordan algebra in the Jordan product 1 (aij ) ◦ (bij ) = ((aij )(bij ) + (bij )(aij )) 2 where the product on the right-hand side is the usual matrix product. Example 2.1.5. Besides the Jordan matrix algebras introduced in the previous example, another important class of special real Jordan algebras are the real spin factors. They are defined as a Hilbert space direct sum H ⊕ R, where H is a real Hilbert space with inner product h·, ·i, equipped with the Jordan product (a ⊕ α)(b ⊕ β) := (βa + αb) ⊕ (ha, bi + αβ)

(a, b ∈ H, α, β ∈ R).

We note that the Jordan matrix algebra H2 (R) has a basis       1 0 0 1 1 0 , , 0 −1 1 0 0 1 and the 3-dimensional spin factor R2 ⊕R is Jordan isomorphic to H2 (R) via the isomorphism       1 0 0 1 1 0 (a, b) ⊕ α 7→ a +b +α . 0 −1 1 0 0 1

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Example 2.1.6. A well-known example in [8] of an exceptional Jordan algebra is the 27-dimensional real algebra H3 (O) = {(aij )1≤i,j≤3 : (aij ) = (e aji ), aij ∈ O} of 3 × 3 matrices over O, hermitian with respect to the involution e in O defined by (α0 e0 + · · · + α7 e7 )e= α0 e0 − · · · − α7 e7 . The Jordan product is given by 1 A ◦ B = (AB + BA) (A, B ∈ H3 (O)) 2 where the multiplication on the right is the usual matrix multiplication. We refer to [92, 128] for a more detailed analysis of H3 (O). The exceptionality of H3 (O) involves the so-called s-identities which are valid in all special Jordan algebras, but not all Jordan algebras. One such identity was first found by Glennie [66]: 2Qx (z)Qy,x Qz (y 2 ) − Qx Qz Qx,y Qy (z) = 2Qy (z)Qx,y Qz (x2 ) − Qy Qz Qy,x Qx (z) which does not hold in H3 (O). Definition 2.1.7. Two elements a and b in a Jordan algebra A are said to operator commute if the left multiplications La and Lb commute. The centre of A is the set Z(A) = {z ∈ A : Lz La = La Lz , ∀a ∈ A}. We observe that La Lb = Lb La if, and only if, (ax)b = a(xb) for all x ∈ A. Evidently, the centre Z(A) = {z ∈ A : (za)b = z(ab), ∀a, b ∈ A} is an associative subalgebra of A. Definition 2.1.8. An element e in an algebra A is called an idempotent if e2 = e. Two idempotents e and u are said to be orthogonal (to each other) if eu = ue = 0. An element a ∈ A is called nilpotent if an = 0 for some positive integer n.

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Lemma 2.1.9. Let A be a unital Jordan algebra with an idempotent e. Let a ∈ A. The following conditions are equivalent. (i) a and e operator commute. (ii) Qe (a) = Le a. (iii) a and e generate an associative subalgebra of A. Proof. (i) ⇒ (ii). We have Qe (a) = 2(L2e − Le )(a) = 2e(ea) − ea = 2e2 a − ea = ea. (ii) ⇒ (iii). Let B be the subalgebra generated by a and e. By the Shirshov-Cohn theorem, B is isomorphic to a Jordan subalgebra B 0 of an associative algebra (A0 , ×) with respect to the special Jordan product. Identify a and e as elements in B 0 . Then 1 Le a = (e × a + a × e) = Qe (a) = e × a × e 2 since e = e2 = e×e. Multiply the above identity on the left by e, we get e × a = e × a × e. Multiplying on the right by e gives a × e = e × a × e. Hence e × a = a × e and ea = e × a. Hence (B 0 , ×) is a commutative subalgebra of (A0 , ×) and the special Jordan product in B 0 is just the product × and is, in particular, associative. (iii) ⇒ (i). In the proof of Theorem 2.1.1, we have the operator identity [Le , Lbc ] + [Lb , Lce ] + [Lc , Leb ] = 0 for all b, c ∈ A. Put c = e, we have [Le , Lbe ] + [Lb , Le ] + [Le , Leb ] = 0 which gives 2[Le , Lbe ] = [Le , Lb ].

(2.5)

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Since e2 = e, in the special Jordan algebra A(a, e, 1) generated by a, e and 1, it can be verified easily that a = Qe (a) + Q1−e (a) and Q1−e (a)e = 0 as well as Qe (a)e = Qe (a). Substitute Q1−e (a) for b in (2.5), we get [Le , LQ1−e (a) ] = 0. Putting b = Qe (a) in (2.5) gives [Le , LQe (a) ] = 0. It follows that [Le , La ] = [Le , LQe (a) ] + [Le , LQ1−e (a) ] = 0.

Lemma 2.1.10. Let A be a finite dimensional associative algebra containing an element a which is not nilpotent and not an identity. Then A contains a non-zero idempotent which is a polynomial in a, without constant term. Proof. We may assume A has an identity 1. Finite dimensionality implies that there is a non-zero polynomial p of the least degree and without constant term such that p(a) = 0. Write p(x) = xk q(x) where k ≥ 1 and q is a polynomial such that q(0) 6= 0. The degree deg q of q is strictly positive since a is not nilpotent. There are then polynomials q1 and q2 with deg q1 < deg q and xk q1 (x) + q2 (x)q(x) = 1 where the non-zero polynomial g(x) = xk q1 (x) has no constant term and deg g < deg p. Hence e = g(a) 6= 0. We have e2 = e since a2k q1 (a) + ak q2 (a)q(a) = ak and g(a)2 − g(a) = a2k q1 (a)2 − ak q1 (a) = ak q2 (a)q(a)q1 (a) = 0.

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Lemma 2.1.11. Let A be a Jordan algebra. Then an element a ∈ A is nilpotent if, and only if, the left multiplication La : A −→ A is nilpotent. Proof. If La is nilpotent, then an+1 = Lna (a) implies that a is nilpotent. Conversely, for any a ∈ A with an = 0, we show that La is nilpotent by induction on the exponent n. The assertion is trivially true if n = 1. Given that the assertion is true for n, we consider an+1 = 0. We have (a2 )n = 0 = (a3 )n and therefore La2 and La3 are nilpotent by the inductive hypothesis. It follows from the identity 2L3a = 3La2 La − La3 that La is nilpotent. Given an idempotent e in a Jordan algebra A, the left multiplication Le : A −→ A satisfies the equation 2L3e − 3L2e + Le = 0

(2.6)

by the identity (2.2). Hence an eigenvalue α of Le is a root of 2α3 − 3α2 + α = 0 and is 0, 12 or 1. If A is associative, then L2e = Le and

1 2

is not an

eigenvalue of Le . Nevertheless, we denote the eigenspaces of 2Le by Ak (e) = {x ∈ A : 2ex = kx}

(k = 0, 1, 2)

and call Ak (e) the Peirce k-space of e. The above remark implies A1 (e) = {0} if A is associative. We define two linear operators Qe : A −→ A and Q⊥ e : A −→ A by Qe = 2L2e − Le ,

2 Q⊥ e = 4(Le − Le ).

(2.7)

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Evidently Le commutes with both Qe and Q⊥ e . Using the equation (2.6), one can easily establish Le Qe = Qe = Q2e ,

1 ⊥ 1 ⊥ 2 Le Q⊥ e = Qe = (Qe ) , 2 2

Le (I − Qe − Q⊥ e)=0

where I is the identity operator on A and, Qe and Q⊥ e are mutually orthogonal. It follows that A2 (e) = Qe (A),

A1 (e) = Q⊥ e (A),

A0 (e) = (I − Qe − Q⊥ e )(A) (2.8)

which gives rise to the following Peirce decomposition of A: A = A0 (e) ⊕ A1 (e) ⊕ A2 (e). We will return to the Peirce decomposition with more details in the more general setting of Jordan triple systems. We note for the time being that the Peirce spaces A0 (e) and A2 (e) are Jordan subalgebras of A as shown below. We also note that A2 (e) never vanishes. Lemma 2.1.12. The Peirce spaces of an idempotent e in a Jordan algebra A satisfy A0 (e)A0 (e) ⊂ A0 (e), A1 (e)A1 (e) ⊂ A0 (e)⊕A2 (e), A2 (e)A2 (e) ⊂ A2 (e). Proof. We first prove the second inclusion. Let x, y ∈ A1 (e) and let xy = a0 + a1 + a2 be the Peirce decomposition of xy. We have 0 = [Ly , Lex ] + [Le , Lxy ] + [Lx , Lye ] 1 1 = [Ly , Lx ] + [Le , Lxy ] + [Lx , Ly ] 2 2 = [Le , Lxy ]. In particular, [Le , Lxy ](e) = 0 gives 0 = e(xy) − e(e(xy))   1 1 = a2 + a1 − e a2 + a1 2 2 1 1 1 = a2 + a1 − a2 − a1 = a1 . 2 4 4

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Hence xy ∈ A0 (e) ⊕ A2 (e). Let x, y ∈ Aj (e) where j = 0, 2. Then we have [Le , Lx2 ] = 0 by the first equation in the proof above. Using this and expanding ((x + e)y)(x + e)2 = (x + e)(y(x + e)2 ), we obtain 2(xy)(xe) + (xy)e + 2(ey)(xe) = 2x(y(xe)) + x(ye) + 2(ey)(xe) which gives (xy)e = 2j xy. This proves the first and the last inclusions.

Definition 2.1.13. An idempotent e in a Jordan algebra A is called maximal if the Peirce 0-space A0 (e) is {0}. A non-zero idempotent e is called primitive if there are no non-zero orthogonal idempotents u and v satisfying e = u + v. Lemma 2.1.14. Let A be a finite dimensional Jordan algebra which contains no non-zero nilpotent element. Then A contains a maximal idempotent. Proof. Ignore the trivial case A = {0}. Applying Lemma 2.1.10 to an associative subalgebra of A generated by a non-zero element, one finds a non-zero idempotent e. If A0 (e) 6= {0}, then again one can pick a non-zero idempotent u ∈ A0 (e). Then e0 = e + u is an idempotent and A0 (e) ⊂ A0 (e0 ). Since u ∈ A0 (e0 )\A0 (e), we have dim A0 (e) < dim A0 (e0 ). By finite dimensionality of A, this process of increasing dimension must stop, yielding a maximal idempotent. Proposition 2.1.15. Let A be a finite dimensional Jordan algebra which contains no non-zero nilpotent element. Then A has an identity. Proof. By Lemma 2.5.3, A contains a maximal idempotent e so that A = A1 (e) ⊕ A2 (e).

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We show A1 (e) = {0}. Let x ∈ A1 (e). Then x2 ∈ A0 (e)⊕A2 (e) = A2 (e) and we have 1 3 x = (xe)x2 = x(ex2 ) = x3 2 which implies x = 0. Hence A = A2 (e) in which e is the identity. A real Jordan algebra A is called formally real if a21 + · · · + a2k = 0 implies a1 = · · · = ak = 0 for a1 , . . . , ak ∈ A. By Proposition 2.1.15, a finite dimensional formally real Jordan algebra has an identity. The real spin factor H ⊕ R and the Jordan matrix algebras Hn (F) are formally real for F = R, C and H. Example 2.1.16. The exceptional Jordan algebra H3 (O) is formally real since for each a = (aij ) ∈ H3 (O), we have 2

Trace(a ) =

3 X

aij aji =

i=1

3 X

aij e aij

i=1

where aij e aij = (α02 + · · · + α72 )e0 for aij = α0 e0 + · · · + α7 e7 ∈ O. Theorem 2.1.17. Let A be a finite dimensional real Jordan algebra. The following conditions are equivalent. (i) A is formally real. (ii) a2 + b2 = 0 ⇒ a = b = 0 for any a, b ∈ A. (iii) Each x ∈ A admits a decomposition x = α1 e1 + · · · + αn en where α1 , . . . , αn ∈ R and e1 , . . . , en are mutually orthogonal idempotents in A. (iv) The bilinear form (x, y) ∈ A2 7→ Trace (x y) ∈ R is positive definite.

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Proof. (ii) ⇒ (iii). Let x ∈ A. Then the subalgebra A(x) generated by x is associative and has an identity e by Proposition 2.1.15. Finite dimensionality implies that e = e1 + · · · + en for some mutually orthogonal primitive idempotents in A(x). Hence we have, by associativity of A(x), x = e1 xe1 + · · · + en xen . We show that the associative algebra ej A(x)ej reduces to Rej . Indeed, by primitivity, there is no non-zero idempotent in ej A(x)ej other than ej , and it follows from Lemma 2.1.10 that each a ∈ ej A(x)ej \{0, ej } gives rise to a polynomial g(a) without constant term satisfying g(a) = ej which implies that a is invertible in ej A(x)ej . Hence ej A(x)ej is a field over R and must be either R or C. Since C is not formally real, we conclude ej A(x)ej = Rej and therefore x=

X

αj e j

j

for some α1 , . . . , αn ∈ R. (iii) ⇒ (iv). Given x = α1 e1 + · · · + αn en for some mutually orthogonal idempotents e1 , . . . , en , we have Trace (x x) =

X

αj2 Trace (ej

ej ) ≥ 0.

j

If Trace (x x) = 0, then Trace (ej

ej ) = 0 and ej = 0 since ej

ej =

Lej has eigenvalues 0, 1/2 or 1, for all j. (iv) ⇒ (i). Let a21 + · · · + a2k = 0. Then Trace (a1 a1 + · · · + ak aj = 0 for all j.

P

j

ak ) = 0. Hence Trace (aj

Trace (aj

aj ) =

aj ) = 0 implies

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An element a in a Jordan algebra A with identity e is called invertible if there exists an element a−1 ∈ A (which is necessarily unique) such that aa−1 = e and (a2 )a−1 = a. This is equivalent to the invertibility of the quadratic operator Qa , in which case a−1 = Q−1 a (a). If the left multiplication La : A −→ A is invertible, then a is invertible with inverse a−1 = L−1 a (e). A subspace J of a Jordan algebra A is called an ideal if a ∈ A and x ∈ J imply ax ∈ J, in which case J is also an ideal in the unit extension A1 of A and, the quotient space A/J is a Jordan algebra with the natural product (a + J)(b + J) = ab + J

(a, b ∈ A).

The kernel ϕ−1 (0) of a Jordan homomorphism ϕ : A −→ B is an ideal of A. Given an ideal J of a Jordan algebra A, the quotient map q : A −→ A/J is a homomorphism with kernel q −1 (0) = J. A Jordan algebra A with non-trivial multiplication, that is, ab 6= 0 for some a, b ∈ A, is called simple if the only ideals of A are {0} and A itself. Finite dimensional formally real Jordan algebras have been classified in the seminal work of [94]. They are a finite direct sum of simple real Jordan algebras of the following types: Hn (R), Hn (C), Hn (H) (3 ≤ n < ∞), H3 (O), H ⊕ R

(2.9)

where dim H < ∞. Jordan algebras of any dimension have been classified by Zelmanov [183, 184]. An important connection of finite dimensional formally real Jordan algebras to geometry has been discovered by Koecher [109] and Vinberg [170]. They established the one-one correspondence between these algebras and a class of Riemannian symmetric spaces, namely, the linearly homogeneous self-dual cones (cf. Definition 2.4.11), which play

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a useful role in the study of automorphic functions on bounded homogeneous domains in complex spaces [144, 170]. Given a finite dimensional formally real Jordan algebra A, the set C = {a2 : a ∈ A} forms a proper cone, its interior is the corresponding linearly homogeneous self-dual cone. Here, by a cone C in a real vector space V , we mean a non-empty subset of V satisfying (i) C + C ⊂ C and (ii) αC ⊂ C for all α > 0. A cone C is called proper if C ∩ −C = {0}. The partial ordering on V induced by a proper cone C will be denoted by ≤C , or by ≤ if C is understood, so that x ≤ y whenever y − x ∈ C. Conversely, if V is equipped with a partial ordering ≤, we let V+ = {v ∈ V : 0 ≤ v} denote the corresponding proper cone. We will discuss the result of Koecher and Vinberg in more detail, as well as its infinite dimensional generalisation in Section 2.4. Notes. The basic results of Jordan algebras presented in this section, as well as Jordan triple systems in the following section, are classical and overlap largely with those given in the author’s book [37]. Further details can be found in the books by Braun and Koecher [23], Jacobson [92], Schafer [156] and McCrimmon [128].

2.2

Jordan triple systems

In this section, we introduce a generalisation of Jordan algebras, namely, the Jordan triple systems. These are real or complex vector spaces equipped with a Jordan triple structure. We discuss both real and complex cases although we will mainly be concerned with complex Jordan triple systems later.

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Given a complex vector space V , we denote by V the conjugate of V , which is the complex vector space obtained from V itself, but with the scalar multiplication replaced by (λ, x) ∈ C × V 7→ λ · x := λx ∈ V . For a real vector space V , its conjugate V is defined to be itself. A real or complex vector space V is called a (real or respectively, complex) Jordan triple system if it is equipped with a triple product {·, ·, ·} : V 3 −→ V called a Jordan triple product, which is linear and symmetric in the outer variables, but conjugate linear in the middle variable, and satisfies the following identity: {a, b, {x, y, z}} = {{a, b, x}, y, z} − {x, {b, a, y}, z} + {x, y, {a, b, z}} (2.10) where a conjugate linear map on a real vector space is just a linear map. By a Jordan triple system, we mean a real or complex Jordan triple system. We note that the Jordan triple product is trilinear in a real Jordan triple system. In the sequel, we will be concerned with mainly complex Jordan triple systems and henceforth, by a Jordan triple, we mean a complex Jordan triple system. Remark 2.2.1. We should point out that a different terminology is used in [37], where the term Hermitian Jordan triple is used for a complex Jordan triple system defined here. A vector subspace W of a Jordan triple system V is called a subtriple if x, y, z ∈ W implies {x, y, z} ∈ W . Given subsets A, B and C of V , we define {A, B, C} = {{a, b, c} : a ∈ A, b ∈ B, c ∈ C}.

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Jordan and Lie algebraic structures A linear map f : V −→ W between two Jordan triple systems

V and W is called a (Jordan) triple homomorphism if it preserves the triple product: f {a, b, c} = {f (a), f (b), f (c)}

(a, b, c ∈ V ).

A triple homomorphism is called a (Jordan) triple monomorphism if it is injective. A bijective triple homomorphism is called a (Jordan) triple isomorphism. We call the identity (2.10) the main triple identity of a Jordan triple system. It is often written as {{a, b, x}, y, z} − {x, {b, a, y}, z} = {a, b, {x, y, z}} − {x, y, {a, b, z}}. (2.11) Example 2.2.2. The complex space Cn can be given various Jordan triple structures, for instance, one can define a Jordan triple product by {x, y, z} = xyz

(x, y, z ∈ Cn )

where y is the complex conjugate of y and the multiplication on the right is the coordinatewise multiplication. We can also equip Cn with the Jordan triple product {x, y, z} = hx, yiz + hz, yix where h·, ·i denotes the usual inner product. The concept of a Jordan triple system was originally derived from a generalisation of Jordan algebras relating to Lie algebras and differential geometry. We will show in Section 2.3 that they are in one-one correspondence with a class of Lie algebras via the Tits-Kantor-Koecher construction.

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In a real Jordan algebra A, we define a canonical Jordan triple product by {a, b, c} = (ab)c + a(bc) − b(ac)

(2.12)

for a, b, c ∈ A. If A is a complex Jordan algebra equipped with an algebra involution ∗, the canonical Jordan triple product is defined by {a, b, c} = (ab∗ )c + a(b∗ c) − b∗ (ac).

(2.13)

Equipped with one of the above triple products, the Jordan algebra A is a Jordan triple system. We first prove three basic identities in a Jordan triple system. Lemma 2.2.3. Given x, y, z in a Jordan triple system, we have {{x, y, x}, y, z}} = {x, {y, x, y}, z}.

(2.14)

Proof. This follows by putting a = x and b = y in the above triple identity. Lemma 2.2.4. Given x, y, z is a Jordan triple system, we have {x, y, {x, z, x}} = {x, {y, x, z}, x}.

(2.15)

Proof. Applying the triple identity repeatedly yields {x, y, {x, z, x}}={{x, y, x}, z, x}} − {x, {y, x, z}, x} + {x, z, {x, y, x}} = 2{x, z, {x, y, x}} − {x, {y, x, z}, x} = 2{{x, z, x}, y, x} − 2{x, {z, x, y}, x} + 2{x, y, {x, z, x}} − {x, {y, x, z}, x} = 4{x, y, {x, z, x}} − 3{x, {y, x, z}, x} = {x, {y, x, z}, x}.

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Jordan and Lie algebraic structures The above proof also yields the identity {x, y, {x, z, x}} = {{x, y, x}, z, x}.

(2.16)

Lemma 2.2.5. Given x, y, z is a Jordan triple system, we have {{x, y, x}, z, {x, y, x}} = {x, {y, {x, z, x}, y}, x}.

(2.17)

Proof. Adding the two triple identities {y, x, {y, x, z}} = {{y, x, y}, x, z} − {y, {x, y, x}, z} + {y, x, {y, x, z}} {z, x, {y, x, y}} = {{z, x, y}, x, y} − {y, {x, z, x}, y} + {y, x, {z, x, y}} we obtain {y, {x, y, x}, z} = 2{y, x, {y, x, z}} − {y, {x, z, x}, y}. It follows that {{x, y, x}, z, {x, y, x}} = 2{{{x, y, x}, z, x}, y, x} − {x, {z, {x, y, x}, y}, x} = 2{{{x, y, x}, z, x}, y, x} − 2{x, {y, x, {y, x, z}}, x} + {x, {y, {x, z, x}, y}, x} = {x, {y, {x, z, x}, y}, x} where the last identity follows from repeated applications of (2.14): {{{x, y, x}, z, x}, y, x} = {{x, {y, x, z}, x}, y, x} = {x, {{y, x, z}, x, y}, x} = {x, {y, {x, z, x}, y}, x}.

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In place of the triple identity (2.10), one can also use the basic identities (2.14), (2.15) and (2.17) as the defining identities for a Jordan triple system, as in [124], in which case the triple identity can be derived. The arguments in the proof of Lemma 2.2.5 can be repeated to yield the identities below. Lemma 2.2.6. Given a, x, y, z in a Jordan triple system, we have 2{x, a, {y, a, z}} = {x, {a, y, a}, z} + {x, {a, z, a}, y}

(2.18)

2{a, x, {a, y, z}} = {{a, x, a}, y, z} + {a, {x, z, y}, a}

(2.19)

2{a, {x, a, y}, z} = {{a, x, a}, y, z} + {{a, y, a}, x, z}. (2.20) Proof. The first identity follows from adding the two triple identities {y, a, {x, a, z}} = {{y, a, x}, a, z} − {x, {a, y, a}, z} + {x, a, {y, a, z}} {z, a, {x, a, y}} = {{z, a, x}, a, y} − {x, {a, z, a}, y} + {x, a, {z, a, y}}. We obtain the third identity by adding the two triple identities {a, x, {a, y, z}} = {{a, x, a}, y, z} − {a, {x, a, y}, z} + {a, y, {a, x, z}} {a, y, {a, x, z}} = {{a, y, a}, x, z} − {a, {y, a, x}, z} + {a, x, {a, y, z}}. The second identity follows from the triple identity {{a, x, a}, y, z} = {a, x, {a, y, z}} − {a, {x, z, y}, a} + {{a, y, z}, x, a}.

In a Jordan triple system V , we define the odd powers of an element x by induction as follows: x1 = x,

x3 = {x, x, x},

x2n+1 = {x, x2n−1 , x}

(n = 2, 3, . . .).

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By (2.15) and induction, we have x2n+1 = {x2n−1 , x, x}.

(2.21)

One often makes use of the following polarization in a Jordan triple system V : 2{x, y, z} = {x + z, y, x + z} − {x, y, x} − {z, y, z}.

(2.22)

It follows that the triple product in a Jordan triple system is completely determined by the symmetrized product {x, y, x}. If V is a complex Jordan triple system, we also have the polarization identity 4{z, y, z} = (y + z)3 + (y − z)3 − (y + iz)3 − (y − iz)3

(2.23)

and the triple product in V is determined by the cubes x3 . Given a real Jordan triple system (V, {·, ·, ·}), it can be complexified to a complex Jordan triple system. Let Vc = V ⊕ iV be the complexification of the vector space V . Then the following triple product turns Vc into a complex Jordan triple system: {x ⊕ iu, y ⊕ iv, x ⊕ iu}c = ({x, y, x} − {u, y, u} + 2{x, v, u}) ⊕ i (−{x, v, x} + {u, v, u} + 2{x, y, u}). (2.24) We shall call (Vc , {·, ·, ·}c ) the complexification of the Jordan triple system V . Jordan triple systems are equivalent to Jordan pairs with involution. The concept of a Jordan pair was introduced by Loos [124] in connection with a pair of mutually dual Riemannian symmetric spaces. A pair (V−1 , V1 ) of real or complex vector spaces is called a Jordan pair if there are two trilinear maps {·, ·, ·}α : Vα × V−α × Vα −→ Vα

(α = ±1)

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which are symmetric in the outer variables and satisfy {a, b, {x, y, z}α }α = {{a, b, x}α , y, z}α − {x, {b, a, y}−α , z}α + {x, y, {a, b, z}α }α (2.25) for a, x, z ∈ Vα and b, y ∈ V−α . An involution θ of a Jordan pair (V−1 , V1 ) is a conjugate linear isomorphism θ : V−1 → V1 satisfying θ{x, θy, x}−1 = {θx, y, θx}1

(x, y ∈ V−1 ).

Of course, θ is linear for a real Jordan pair. Evidently, if (V−1 , V1 ) is a Jordan pair with involution θ, then V−1 is a Jordan triple system with triple product {x, θy, z}−1 for x, y, z ∈ V−1 . Conversely, given a real Jordan triple system (V, {·, ·, ·}), let V−1 = V1 = V . Then (V−1 , V1 ) is a Jordan pair with trilinear maps {x, y, z}α = {x, y, z} for α = ±1 and the identity map θ : V−1 → V1 is an involution. For a complex Jordan triple system (V, {·, ·, ·}), we let V−1 = V and V1 = V , the latter being the conjugate of V defined at the beginning of this section. Then the identity map θ : V−1 −→ V1 is a conjugate linear involution of the Jordan pair (V−1 , V1 ) with the trilinear map {x, y, z}α = {x, θ±α y, z}. Example 2.2.7. Let Mm,n (C) be the complex vector space of m × n complex matrices. Then (Mm,n (C), Mn,m (C)) is a Jordan pair with trilinear maps 1 {A, B, C}α = (ABC + CBA) 2

(A, C ∈ V−α , B ∈ Vα )

where V−1 = Mm,n (C) and V1 = Mn,m (C). The pair has an involution θ : Mm,n (C) −→ Mn,m (C)

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given by θ(B) = B ∗ , where B ∗ = ( bji ) is the adjoint of the matrix B = (bij ). The complex vector space Mm,n (C) is a Jordan triple with triple product 1 {A, B, C} = {A, θ(B), C}−1 = (AB ∗ C + CB ∗ A) 2 and Mmm (C) is a Jordan algebra with involution * and Jordan product A ◦ B = (AB + BA)/2. We shall write Mm (C) for Mmm (C) in the sequel. One can consider an infinite dimensional extension of the above example. Let H and K be complex Hilbert spaces and let L(H, K) be the Banach space of bounded linear operators between H and K. Then it is a complex Jordan triple in the triple product 1 {R, S, T } = (RS ∗ T + T S ∗ R) 2

(R, S, T ∈ L(H, K))

where S ∗ : K −→ H is the adjoint of S. Henceforth we will write L(H) for L(H, H). In fact, for any real associative algebra A with special Jordan product a ◦ b = (ab + ba)/2, the canonical Jordan triple product of the Jordan algebra (A, ◦) is given by 1 {a, b, c} = (abc + cba). 2 If A is over C and equipped with an involution ∗ , then the canonical Jordan triple product is given by 1 {a, b, c} = (ab∗ c + cb∗ a). 2 Example 2.2.8. Let M1,2 (O) = {(z1 , z2 ) : z1 , z2 ∈ O} be the complex vector space of 1 × 2 matrices over O. Given y = (y1 , y2 ) with y1 =

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α1,k ek and y2 =

0

7 X

77

α2,k ek , we define yj ∗ = αj,0 e0 −

7 X

αj,k ek for

1

0

j = 1, 2, and 

∗

y =

y1 ∗ y2 ∗

 .

The space M1,2 (O) is a 16-dimensional Jordan triple in the following triple product:

1 {x, y, z} = (x(y ∗ z) + z(y ∗ x)). 2

Example 2.2.9. Let H3 (O) be the complex vector space of 3×3 matrices over the Cayley algebra O, hermitian with respect to the standard involution in O, that is, (aij ) belongs to H3 (O) if, and only if, (aij ) = (e aji ) where the usual linear involution e on O is defined by 7 X

!e αk ek

= α0 e0 −

P7

k=0 αk ek

αk ek

k=1

k=0

for

7 X

∈ O. We always equip H3 (O) with the Jordan product

1 A ◦ B = (AB + BA) 2

(A, B ∈ H3 (O))

where the multiplication on the right is the usual matrix multiplication. The product ◦ makes H3 (O) into a complex exceptional Jordan algebra. There is a natural conjugate linear involution \ on O defined by 7 X k=0

!\ αk ek

=

7 X

αk ek

k=0

which induces a conjugate linear involution ∗ on H3 (O): (aij )∗ := (a\ij ).

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We always equip H3 (O) with the triple product {A, B, C} = (A ◦ B ∗ ) ◦ C + A ◦ (B ∗ ◦ C) − B ∗ ◦ (A ◦ C). With this triple product, H3 (O) becomes a Jordan triple. The exceptional real Jordan algebra H3 (O) in Example 2.1.6 is the real form of H3 (O) with respect to the involution ∗ : H3 (O) −→ H3 (O), that is, H3 (O) = {(aij ) ∈ H3 (O) : (aij )∗ = (aij )} and H3 (O) = H3 (O) + iH3 (O) is the complexification of H3 (O). We can embed M1,2 (O) as a subtriple of H3 (O) via the triple monomorphism 

 0 z1 z2 (z1 , z2 ) ∈ M1,2 (O) 7→ ze1 0 0  ∈ H3 (O). ze2 0 0 For each element a in a Jordan triple system V , we define a binary product ◦a in V by x ◦a y = {x, a, y}

(x, y ∈ V ).

The above product is clearly commutative. It also satisfies the Jordan identity which follows from the identities (2.10), (2.18) and (2.15): x2 ◦a (x ◦a y) = {{x, a, x}, a, {x, a, y}} = {{{x, a, x}, a, x}, a, y} − {x, {a, {x, a, x}, a}, y} + {x, a, {{x, a, x}, a, y}} = x ◦a (x2 ◦a y)

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where, by identities (2.18) and (2.15), we have {x, {a, {x, a, x}, a}, y} = 2{{x, a, {x, a, x}}, a, y} − {{x, a, x}, {a, x, a}, y} = 2{{x, a, {x, a, x}}, a, y} − 2{x, {{a, x, a}, x, a}, y} + {{x, {a, x, a}, x}, a, y} = 3{{x, a, {x, a, x}}, a, y} − 2{x, {a, {x, a, x}, a}, y} = {{x, a, {x, a, x}}, a, y}. Definition 2.2.10. Let V be a Jordan triple system and a ∈ V . The a-homotope V (a) of V is defined to be the Jordan algebra (V, ◦a ). Using identities (2.14) and (2.18), we see that the canonical triple product {·, ·, ·}a in the a-homotope V (a) = (V, ◦a ) is given by {x, y, x}a = 2(x ◦a y) ◦a x − (x ◦a x) ◦a y = 2{{x, a, y}, a, x} − {{x, a, x}, a, y} = {x, {a, y, a}, x}. In particular, if {a, y, a} = y for all y ∈ V , then we have {x, y, x}a = {x, y, x} for all x, y ∈ V and the Jordan triple system (V (a) , {·, ·, ·}a ) is just V itself. Therefore Jordan triple systems generalise Jordan algebras in that unital real Jordan algebras and complex Jordan algebras with involution are Jordan triples containing a unit element, that is, an element e satisfying {e, y, e} = y for all elements y. One can deduce power associativity in V via homotopes. Lemma 2.2.11. Let V be a Jordan triple system and let a ∈ V . For odd natural numbers m, n and p, we have {am , an , ap } = am+n+p .

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Proof. Let V (a) be the a-homotope and x(n) the n-th power of x in the Jordan algebra (V (a) , ◦a ). By (2.21) and induction, we have a2n−1 = a(n) . By (2.14) and power associativity in Jordan algebras, we have {am , {a, a, a}, ap } = {{am , a, a}, a, ap } = a(m+3/2) ◦a a(p+1/2) = a(m+3+p+1/2) = am+3+p . One concludes the proof by induction, using the identity {am , {a, a2k−1 , a}, ap } = {{a, a, am }, a2k−1 , ap } + {am , a2k−1 , {a, a, ap }} − {a, a, {am , a2k−1 , ap }}.

Definition 2.2.12. Let V be a Jordan triple system and x, y ∈ V . Extending the definition of a box operator on a Jordan algebra, we define the box operator x y : V −→ V by (x y)(v) = {x, y, v}

(v ∈ V ).

Given two subsets A and B of V , we shall write A B = {a b : a ∈ A, b ∈ B}. The triple identity (2.11) can be written in terms of Lie brackets of box operators: [a b, x y] = {a, b, x} y − x {y, a, b}.

(2.26)

Definition 2.2.13. A Jordan triple system V is called abelian or commutative if [a b, x y] = 0

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for all a, b, x, y ∈ V , in other words, if the operators a b and x y commute for all a, b, x, y ∈ V . A subspace F of V is called flat if x y = y x for all x, y ∈ F . Lemma 2.2.14. Let V be a Jordan triple system. Then V is abelian if and only if {a, b, {x, y, z}} = {a, {b, x, y}, z} = {{a, b, x}, y, z} for all a, b, x, y, z ∈ V . Proof. By the triple identity (2.26), the above identities are equivalent to [x y, z b](a) = 0 = [x b, a y](z) for all a, b, x, y, z ∈ V . Example 2.2.15. Let V be a Jordan triple system and let a ∈ V . It is plain from power associativity in Lemma 2.2.11 that the linear span V (a) of odd powers of a is the subtriple of V generated by a, that is, the smallest subtriple of V containing a. Moreover, power associativity implies that V (a) is abelian. If V is real, then V (a) is flat, a consequence of power associativity again. Example 2.2.16. Flat Jordan triple systems must be abelian. However, an abelian Jordan triple system need not be flat. For instance, C with the triple product {x, y z} = xyz is abelian but not flat. Apart from the box operator, there are two important operators on Jordan triple systems, namely, the quadratic operator and the Bergman operator. Let V be a Jordan triple system and let a, b ∈ V . The quadratic operator Qa : V −→ V , induced by a, is defined by Qa (x) = {a, x, a}

(x ∈ V )

(2.27)

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which is linear if V is real, but conjugate linear if V is a complex Jordan triple system. The Bergman operator B(a, b) : V −→ V , induced by (a, b), is defined by B(a, b)(x) = x − 2(a b)(x) + Qa Qb (x)

(x ∈ V ).

(2.28)

In terms of quadratic operators, the identity (2.15) can be formulated as Qa (b a) = (a b)Qa

(a, b ∈ V ).

(2.29)

One also deduces from (2.15) that B(a, b)Qa = Qa−Qa (b)

(a, b ∈ V ).

The identity (2.17) can be formulated as QQa (b) = Qa Qb Qa

(a, b ∈ V ).

(2.30)

We need to derive a few more identities for later applications. Given a, b ∈ V , let us define the map Q(a, b) : V −→ V by Q(a, b)(x) = {a, x, b}

(x ∈ V ).

We have of course Qa = Q(a, a) and it is easy to verify that Q(a + tx, a + tx) = Qa + 2tQ(a, x) + t2 Qx for all scalars t. Hence we have, from the identity (2.30), Q(Qa+tx (b), Qa+tx (b)) = Qa+tx Qb Qa+tx = (Qa + 2tQ(a, x) + t2 Qx )(Qb Qa + 2tQb Q(a, x) + t2 Qb Qx ). Comparing the coefficients of t on both sides above, we obtain 2Q(Qa (b), {a, b, x}) = Q(a, x)Qb Qa + Qa Qb Q(a, x).

(2.31)

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Also, comparing the coefficients of t2 on both sides of the previous identity, we obtain 2Q(Qa (b), Qx (b)) + 4Q({a, b, x}, {a, b, x}) = Qa Qb Qx + Qx Qb Qa + 4Q(a, x)Qb Q(a, x).

(2.32)

Replacing b by b + c in (2.31) and expand, we get Q(Qa (b), {a, c, x}) + Q(Qa (c), {a, b, x}) = Qa Q(b, c)Q(a, x) + Q(a, x)Q(b, c)Qa .

(2.33)

Lemma 2.2.17. For a, c and x in a Jordan triple system V , we have 2Q(Qa Qc (x), {a, c, x}) = Qa Qc Qx (c a) + (a c)Qx Qc Qa . Proof. Substitute Qc (x) for b in (2.33), we have 2Q(Qa Qc (x), {a, c, x}) = 2Qa Q(Qc (x), c)Q(a, x) + 2Q(a, x)Q(Qc (x), c)Qa − 2Q(Qa (c), {a, Qc (x), x}) = 2Qa Qc (x c)Q(a, x) + 2Q(a, x)(c x)Qc Qa − 2Q(Qa (c), {a, c, Qx (c)}) where the second identity follows from (2.14). Applying (2.19) and (2.20) to the first two terms, the last formula becomes Qa Qc (Q(Qc (x), a) + Qx (c a)) + (Q(Qc (x), a) + (a c)Qx )Qc Qa − 2Q(Qa (c), {a, c, Qx (c)}) = Qa Qc Qx (c a) + (a c)Qx Qc Qa + Qa Qc Q(Qc (x), a) + Q(Qc (x), a)Qc Qa − 2Q(Qa (c), {a, c, Qx (c)}) = Qa Qc Qx (c a) + (a c)Qx Qc Qa where the last three terms in the second formula sum to 0 by (2.31).

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Lemma 2.2.18. For a, c and x in a Jordan triple system V , we have 2Q(x, Qa Qb (x)) + 4Q({a, b, x}, {a, b, x}) = Qa Qb Qx + Qx Qb Qa + 4(a b)Qx (b a). Proof. Consider the last term above and apply (2.20), we have 2(a b)Qx (b a) = 4Q(x, a)(b x)(b a) − 2Q(Qx (b), a)(b a) which, by (2.19), equals 2Q(x, a)(Qb (x) a) + 2Q(x, a)Qb Q(x, a) − 2Q(Qx (b), a)(b a) and by (2.20) again, the above is equal to Q(x, Qa Qb (x)) + (x Qb (x))Qa + 2Q(x, a)Qb Q(x, a) − Q(Qa (b), Qx (b)) − (Qx (b) b)Qa = Q(x, Qa Qb (x)) + 2Q(x, a)Qb Q(x, a) − Q(Qa (b), Qx (b)) where the last identity follows from (2.14). It follows from (2.32) that 4(a b)Qx (b a) = 2Q(x, Qa Qb (x)) + 4Q(x, a)Qb Q(x, a) − 2Q(Qa (b), Qx (b)) = 2Q(x, Qa Qb (x)) + 4Q({a, b, x}, {a, b, x}) − Qa Qb Qx − Qx Qb Qa which completes the proof. We are now ready to prove an important identity for the Bergman operator. Theorem 2.2.19. Let V be a Jordan triple system and let x, y, z ∈ V . The Bergman operator B(x, y) satisfies Q(B(x, y)z, B(x, y)z) = B(x, y)Qz B(y, x).

(2.34)

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Proof. The identity can be proved by comparing the expansions of both sides. Indeed, the left-hand side is equal to Qz − 4Q(z, {x, y, z}) + 2Q(z, Qx Qy (z)) + 4Q({x, y, z}, {x, y, z}) − 4Q(Qx Qy (z), {x, y, z}) + Q(Qx Qy (z), Qx Qy (z)) whereas the right-hand side equals Qz− 2(x y)Qz − 2Qz (y x) + Qx Qy Qz + Qz Qy Qx + 4(x y)Qz (y x) − 2Qx Qy Qz (y x) − 2(x y)Qz Qy Qx + Qx Qy Qz Qy Qx which is identical to the left-hand side by the triple identity (2.10), Lemma 2.2.5, Lemma 2.2.17 and Lemma 2.2.18. Definition 2.2.20. A Jordan triple system V is called non-degenerate if Qa = 0 =⇒ a = 0 for each a ∈ V . Lemma 2.2.21. Let V be a non-degenerate Jordan triple system and let a ∈ V . If x a = 0 for all x ∈ V , then a = 0. Proof. We have 0 = {x, a, {y, a, y}} = {{x, a, y}, a, y} − {y, {a, x, a}, y} + {y, a, {x, a, y}} = −{y, {a, x, a}, y} for all y ∈ V which implies Q{a,y,a} = 0 for all y ∈ V . Hence {a, y, a} = 0 for all y ∈ V and a = 0. Lemma 2.2.22. Let V be a non-degenerate Jordan triple system and let a ∈ V . If {x, a, x} = 0 for all x ∈ V , then a = 0.

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Proof. We have {x + v, a, x + v} = 0 for all x, v ∈ V which implies x a = 0 for all x ∈ V . Hence a = 0. Let V be a finite dimensional Jordan triple system over F = R, C. We define a bilinear form h ·, · i : V × V −→ F by hx, yi = Trace (x y). If V is a finite dimensional complex Jordan triple system, then hx, yi = Trace (x y) is a complex sesquilinear form. We call h · , · i the trace form of V . If the trace form of a Jordan triple system V is non-degenerate, that is, hx, yi = 0 for all y ∈ V implies x = 0, then every linear map T : V −→ V has an adjoint T ∗ : V −→ V with respect to the trace form: hT x, yi = hx, T ∗ yi

(x, y ∈ V ).

In this case, the quadratic form q(x) = Trace (x x) is also called the trace form as q determines h·, ·i completely. We call q, or h·, ·i, positive definite if q(x) > 0 for all x 6= 0. Lemma 2.2.23. Let V be Jordan triple system which admits a nondegenerate trace form. Then we have, for every a, b ∈ V , (a b)∗ = b a and hence the trace form h ·, · i is symmetric if V is real. If V is complex, the trace form h ·, · i is Hermitian. Proof. By triple identity (2.26), we have h(a b)x, yi − hx, (b a)yi = Trace ((a b)x y) − Trace (x = Trace [a b, x y] = 0.

(b a)y)

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Hence ha, bi = Trace (a b) = Trace (a b)∗ = Trace (b a) = hb, ai if V is real, whereas Trace (a b) = Trace (a b)∗ = Trace (b a) if V is complex. Lemma 2.2.24. Let V be a real Jordan triple system. The following conditions are equivalent. (i) The trace form is non-degenerate. (ii) The bilinear form (x, y) ∈ V 2 7→ Trace (x y + y x) is nondegenerate. Proof. (i) ⇒ (ii). This follows from Lemma 2.2.23. (ii) ⇒ (i). The bilinear form  x, y  = Trace (x y + y x) on V is non-degenerate and symmetric. Using Trace [x y, u v] = 0 and the triple identity (2.26) as before, we have  (x y)u, v  = Trace ((x y)u v + v = Trace (u

(x y)u)

(y x)v + (y x)v u)

=  u, (y x)v  . Hence the box operator y x is the adjoint of x y with respect to the bilinear form  ·, · . Therefore Trace (y x) = Trace (x y) and (i) follows. Definition 2.2.25. A Jordan triple system V is called semisimple if for each a ∈ V , we have a x is nilpotent for all x ∈ V =⇒ a = 0. Definition 2.2.26. A Jordan triple system V is called anisotropic if {x, x, x} = 0 implies x = 0 for each x ∈ V . Lemma 2.2.27. Let V be a Jordan triple system which admits a positive definite trace form. Then V is anisotropic.

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Proof. Let x ∈ V and {x, x, x} = 0. Using the identities (2.18) and (2.16), we deduce that the box operator x x is nilpotent, in fact, (x x)3 = 0. Therefore Trace (x x) = 0 which implies x = 0 since the trace form is positive definite.

For finite dimensional Jordan triple systems, semisimplicity is equivalent to non-degeneracy of the trace form. Lemma 2.2.28. Let V be a finite-dimensional Jordan triple system. The following conditions are equivalent. (i) V is semisimple. (ii) The trace form of V is non-degenerate. The above conditions imply that V is non-degenerate. Proof. We show that {a ∈ V : a x is nilpotent ∀x ∈ V } = {a ∈ V : Trace (a x) = 0 ∀ x ∈ V } from which the equivalence of (i) and (ii) follows immediately. Indeed, if a x is nilpotent, then Trace (a x) = 0. If the operator a x is not nilpotent for some x ∈ V , then the left multiplication La : V (x) −→ V (x) on the x-homotope V (x) is not nilpotent as the two operators are identical. Therefore a is not nilpotent in the Jordan algebra V (x) by Lemma 2.1.11. From Lemma 2.1.10, there is an idempotent e in the subalgebra A(a) of V (x) generated by a, and e is of the form P e = k αk a(nk ) . It follows that Trace (e x) = Trace Le > 0 and the

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triple identity gives Trace (e x) =

* X

+ αk a

(nk )

,x

k

=

X

αk h(a(nk −1) x)(a), xi

k

* =

a,

+ X

(nk −1)

αk (x a

)(x)

k

! = Trace

a

X

αk (x a(nk −1) )(x)

k

and hence a does not belong to the above set on the right. This proves the equality of the two sets. Assume condition (i) and let Qa = 0 for some a ∈ V . For every x ∈ V , we have 2(a x)2 = Qa (x) x + Qa Qx = 0 by the identity (2.19). Hence a x is nilpotent for all x ∈ V and a = 0 by semisimplicity. This proves non-degeneracy of V . Definition 2.2.29. A finite dimensional Jordan triple system V is called positive if the box operator x x : V −→ V has a non-negative spectrum for each x ∈ V . Example 2.2.30. A positive Jordan triple system need not be nondegenerate. Let V =

   0 α : α, β ∈ R 0 β

be equipped with the triple product {a, b, c} = abc + cba

(a, b, c ∈ V )

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where the product on the right is the matrix product. Then V is a   0 α positive real Jordan triple system. Indeed, for x = , the eigen0 β 2 2 values  of the  box operator x x are β and 2β . However, Qa = 0 for 0 1 a= . 0 0 The role of idempotents in a Jordan algebra is played by tripotents in a Jordan triple system. Definition 2.2.31. An element e in a Jordan triple system V is called a tripotent if e = {e, e, e}. A tripotent e is said to be orthogonal to a tripotent f if e f = 0. For tripotents e and f in a Jordan triple system V , it will be shown that the condition e f = 0 is equivalent to f

e = 0. Two elements

a, b ∈ V are defined to be orthogonal to each other if a b = b a = 0. Lemma 2.2.32. Let V be an anisotropic Jordan triple system and let a, b ∈ V . Then a and b are orthogonal to each other if, and only if, a b = 0. Proof. Let a b = 0. By the triple identity (2.26), we have x {b, a, y} = {a, b, x} y − [a b, x y] = 0

(x, y ∈ V )

which gives {b a(y), b a(y), b a(y)} = 0

(y ∈ V ).

Hence b a = 0 by anisotropicity. Lemma 2.2.33. Let e be a tripotent in a Jordan triple system V . Then the box operator e e : V −→ V has eigenvalues in {0, 1/2, 1}.

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Proof. Let V (e) be the e-homotope of V . Then e is an idempotent in the Jordan algebra V (e) with the Jordan product a ◦e b = {a, e, b}. The box operator e e : V −→ V is the left multiplication operator Le : V (e) −→ V (e) which has eigenvalues 0, 1/2 or 1 by (2.6). We now show the existence of tripotents. Theorem 2.2.34. Let V be a finite dimensional Jordan triple system. The following conditions are equivalent. (i) V is semisimple and positive. (ii) The trace form q(x) = Trace (x x) is positive definite. (iii) Each non-zero x ∈ V admits a unique decomposition x = α1 e1 + · · · + αn en where 0 < α1 < · · · < αn and e1 , . . . , en are mutually orthogonal tripotents in V . Proof. (i) ⇒ (ii). We have q(x) ≥ 0 by positivity of V . If Trace (x x) = 0, then all eigenvalues of x x are zero and we must have x x = 0 since x x is self-adjoint with respect to the trace form on V , by Lemma 2.2.23. It follows that x is nilpotent in the a-homotope V (a) for each a ∈ V since in the Jordan algebra V (a) , we have x(4) = {{x, a, {x, a, x}}, a, x} = {({x, a{x, x, a}} + {x, {a, x, x}, a} − {x, x, {x, a, a}}) , a, x} = 0. Hence the left multiplication Lx : V (a) −→ V (a) is nilpotent by Lemma 2.1.11. It follows that Trace (x a) = Trace Lx = 0 and x = 0 by semisimplicity of V .

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be the subtriple of V generated by x. Then V (x) is a finite dimensional inner product space with the trace form q. Let L be the real linear span of the box operators {a b|V (x) : a, b ∈ V (x)}. Then L is a commutative algebra with respect to composition as V (x) is abelian, as noted in Example 2.2.15. By Lemma 2.2.23, L consists of self-adjoint operators on V (x) and can therefore be simultaneously diagonalised. In other words, there is a basis {v1 , . . . , vn } in V (x) such that Lvk ∈ Rvk for all L ∈ L and k = 1, . . . , n. In particular, we have (vk

vk )(vk ) = λk vk

(k = 1, . . . , n)

where λk 6= 0 since V is anisotropic by Lemma 2.2.27. Moreover vk /λk is an idempotent in the vk -homotope V (vk ) on which the left multiplication Lvk /λk has eigenvalues 0, 1/2 or 1. The latter says the same of the box operator (vk

vk )/λk . It follows that Trace (vk

vk ) is a positive

multiple of λk and hence λk > 0. √ Let ek = vk / λk for k = 1, . . . , n. Then e1 , . . . , en are mutually orthogonal tripotents in V since (ei ej )(ek ) = (ek

ej )(ei ) ∈ Rek ∩ Rei = {0}

for i 6= k. After permutation and sign change, we can write x = α1 e1 + · · · + αn en

(0 ≤ α1 ≤ · · · ≤ αn ).

By orthogonality, we have xp = α1p e1 + · · · + αnp en for odd powers xp of x. Since V (x) is spanned by these odd powers, we must have 0 < α1 < · · · < αn .

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To see uniqueness of the decomposition, let x = µ1 u1 + · · · + µm um where 0 < µ1 < · · · < µm and u1 , . . . , um are mutually orthogonal tripotents. Then we have 2k+1 µ2k+1 u1 + · · · + µm um = x(2k+1) ∈ V (x) 1

for k = 1, 2, . . .. Hence each uj is in V (x) and u1 , . . . , um form a basis P of V (x) with m = n. We have uj = k βk ek and X uj = {uj , uj , uj } = βk3 ek k

which imply βk = 0 or ±1. It follows that uj equals ±ek for some k; but the inequalities on the coefficients imply uj = ej and µj = αj for each j . (iii) ⇒ (i). Given x = α1 e1 + · · · + αn en with orthogonal tripotents e1 , . . . , en , we have x x=

n X

αk2 (ek

ek )

k=1

where each ek

ek has spectrum in {0, 1/2, 1}. Hence x x has non-

negative spectrum. If Trace (x x) = 0, then αk2 = 0 for all k and x = 0. This proves semisimplicity of V . Definition 2.2.35. Let V be a semisimple positive finite dimensional Jordan triple system and let x ∈ V . The decomposition x = α1 e1 + · · · + αn en in Theorem 2.2.34 (iii) is called the spectral decomposition of x. We define the triple spectrum of x to be the set s(x) = {α1 , . . . , αn }.

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V induced by a tripotent u ∈ V . Let V (u) be the u-homotope of V . As noted before, u is an idempotent in the Jordan algebra V (u) with Jordan product a ◦u b = {a, u, b} and the box operator u u : V −→ V is the left multiplication Lu : V (u) −→ V (u) , with eigenvalues in {0, 1/2, 1}. Definition 2.2.36. Let u be a tripotent in a Jordan triple system V . The eigenspaces  Vk (u) =

k z ∈ V : (u u)(z) = z 2

 (k = 0, 1, 2)

are called the Peirce k-spaces of u, and the eigenspace decomposition V = V0 (u) ⊕ V1 (u) ⊕ V2 (u) is called the Peirce decomposition of V . By (2.7) and (2.8), the Peirce k-space is the range of the Peirce k-projection Pk (u) : V −→ V given by, using (2.18), P2 (u)(z) = 2L2u (z) − Lu (z) = 2{u, u, {u, u, z}} − {u, u, z} = Q2u (z) P1 (u) = 4(Lu − L2u ) = 2(Lu − (2L2u − Lu )) = 2(u u − Q2u ) P0 (u) = I − 2(u u − Q2u ) − Q2u = I − 2u u + Q2u = B(u, u) where I : V −→ V is the identity operator. Trivially, if u = 0, then P0 (u) = I and V = V0 (u). We usually consider non-zero tripotents in Peirce decompositions. By Lemma 2.1.12, the Peirce 2-space V2 (u) of a tripotent u ∈ V is a Jordan subalgebra of the u-homotope V (u) , containing the identity u, with Jordan product a ◦u b = {a, u, b}

(a, b ∈ V ).

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Lemma 2.2.37. Given a tripotent u in an abelian Jordan triple system V , we have V = V0 (u) ⊕ V2 (u) where (V2 (u), ◦u ) is an (abelian) associative algebra. Proof. Since V is abelian, the box operator u u : V −→ V is a projection, that is (u u)2 = u u and it follows that 2{u, u, v} = v implies v = 0. Hence V1 (u) = {0}. The associativity of (V2 (u), ◦u ) follows directly from the abelian condition on the triple product. Definition 2.2.38. A non-zero tripotent u in a Jordan triple system V is called maximal or complete if V0 (u) = {0}. It is called minimal if V2 (u) = Cu, and called unitary if V2 (u) = V , given a complex Jordan triple V . Both maximal and unitary tripotents cannot be 0 unless V = {0}. We see from Lemma 2.2.37 that an abelian Jordan triple system becomes an abelian associative algebra if it admits a maximal tripotent. Example 2.2.39. Consider the Jordan triple M2 (C) of 2 × 2 complex matrices. For the tripotent  u=

1 0 0 0



the Peirce k-projections are given by     a b 0 0 P0 (u) = c d 0 d  P1 (u)  P2 (u)

a b c d



a b c d





0 b c 0





a 0 0 0



=

=

.

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Example 2.2.40. In the Jordan triple H3 (O), introduced in Example 2.2.9, the Peirce 1-space of the tripotent   1 0 0 u = 0 0 0 0 0 0 is none other than M1,2 (O) = P1 (u)(H3 (O)). More generally, consider the Jordan triple L(H, K) in Example 2.2.7, an operator u ∈ L(H, K) is a tripotent if, and only if, u = uu∗ u, that is, u is a partial isometry. The operators ` = uu∗ and r = u∗ u are projections on the Hilbert spaces K and H respectively. They can be represented, with suitable orthonormal bases, by square block matrices     1K O 1H O , r =   `= O O O O where 1H and 1K are identities. In this representation, each operator in the Peirce 2-space L(H, K)2 (u) = P2 (u)L(H, K) = {` T r : T ∈ L(H, K)} has a rectangular matrix representation   [` T r] O . P2 (u)T = ` T r =  O O The other two Peirce projections of u are given by   O O , P0 (u)T = (1K − `)T (1H − r) =  O [(1K − `)T (1H − r)]   O [` T (1H − r)] . P1 (u)T = `T (1H − r) + (1K − `)T r =  [(1K − `)T r] O

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The matrix form of the Peirce decomposition of T ∈ L(H, K) is given by  T =

[` T r]

[` T (1H − r)]



. [(1K − `)T r] [(1K − `)T (1H − r)]

Let u be a tripotent in a Jordan triple system V . For any real scalar t 6= 0, the Bergman operator B(u, (1 − t)u) : V −→ V is invertible. Indeed, a simple calculation gives B(u, (1 − t)u) = P0 (u) + tP1 (u) + t2 P2 (u)

(2.35)

and therefore B(u, (1 − t)u) has inverse 1 1 B(u, (1 − t−1 )u) = P0 (u) + P1 (u) + 2 P2 (u) t t by mutual orthogonality of the Peirce projections. Since B(u, (1 − t)u)v =

2 X

tk Pk (u)v

(2.36)

k=0

for each v ∈ V , it follows that, for t 6= 1, we have B(u, (1 − t)u)v = tk v if, and only if, v ∈ Vk (u) for k = 0, 1, 2. However, for k ∈ / {0, 1, 2}, we have B(u, (1 − t)u)v = tk v if, and only if, v = 0. We now derive the basic Peirce multiplication rules. Theorem 2.2.41. Let u be a tripotent of a Jordan triple system V . Then the Peirce k-spaces Vk (u) satisfy {V0 (u), V2 (u), V } = {V2 (u), V0 (u), V } = {0} {Vi (u), Vj (u), Vk (u)} ⊂ Vi−j+k (u) where Vα (u) = {0} for α ∈ / {0, 1, 2}.

(2.37)

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Proof. Let x ∈ V2 (u) and y ∈ V0 (u). We first observe that {u, y, u} = Q3u y = Qu P2 (u)y = Qu P2 (u)P0 (u)y = 0. Hence by (2.18), we have {z, u, y} = {z, {u, u, u}, y} = 2{z, u, {u, u, y}} − {z, {u, y, u}, u} = 0 for any z ∈ V . It follows from (2.20) that {x, y, z} = {{u, {u, x, u}, u}, y, z} = 2{u, {y, u, {u, x, u}}, z} − {{u, y, u}, {u, x, u}, z} = 0 for all z ∈ V . Likewise {V0 (u), V2 (u), V } = {0}. To show (2.37), we make use of invertibility of B(u, (1−t)u) for real scalars t 6= 0, with inverse B(u, (1 − t−1 )u), and deduce from Theorem 2.61 that B(u, (1−t)u){z, x, z} = {B(u, (1−t)u)z, B(u, (1−t−1 )u)x, B(u, (1−t)u)z} for x, z ∈ V . By polarization in (2.22), we have B(u, (1 − t)u){x, y, z} = {B(u, (1 − t)u)x, B(u, (1 − t−1 )u)y, B(u, (1 − t)u)z} (2.38) for x, y, z ∈ V . In particular, for vα ∈ Vα (u), the above remarks imply B(u, (1 − t)u){vi , vj , vk } = {B(u, (1 − t)u)vi , B(u, (1 − t−1 )u)vj , B(u, (1 − t)u)vk } = {ti vi , t−j vj , tk vk } = ti−j+k {vi , vj , vk } and {vi , vj , vk } ∈ Vi−j+k (u). The Peirce multiplication rules reveal immediately that the Peirce k-spaces Vk (u) of a tripotent u in a Jordan triple system V are subtriples of V . These rules also entail the following useful results.

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Corollary 2.2.42. Let u, e be tripotents in a Jordan triple system V such that u ∈ V2 (e). Then we have V2 (u) ⊂ V2 (e) and V0 (e) ⊂ V0 (u). Proof. We have V2 (u) = P2 (u)(V ). Each x ∈ V has a Peirce decomposition x = x0 + x1 + x2 ∈ V0 (e) ⊕ V1 (e) ⊕ V2 (e) with respect to e and the Peirce rules imply P2 (u)(x) = P2 (u)(x2 ) ∈ V2 (e). The Peirce rules also imply (u u)(V0 (e)) = {0}. Corollary 2.2.43. Let u, v be tripotents in a Jordan triple system V . The following conditions are equivalent. (i) u and v are orthogonal to each other. (ii) v u = 0. (iii) {u, u, v} = 0. (iv) {v, v, u} = 0. Proof. This follows easily from Theorem 2.2.41. Indeed, orthogonality implies (iv) which in turn implies that u is in the Peirce 0-space V0 (v) of v and therefore v u = 0 by Theorem 2.2.41. By the same token, (ii) is equivalent to (iii). Given two mutually orthogonal tripotents e1 and e2 in a Jordan triple system V , it is evident that e1 + e2 is also a tripotent. One can form a joint Peirce decomposition of V with respect to e1 and e2 . By orthogonality, we have B(ej , tej )ek = ek

(j, k ∈ {1, 2}, j 6= k and t ∈ R)

and a direct computation using (3.57) yields B(e1 , (1−t)e1 )B(e2 , (1−s)e2 ) = B(e2 , (1−s)e2 )B(e1 , (1−t)e1 ). (2.39)

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Since B(e1 , (1 − t)e1 ) = P0 (e1 ) + tP1 (e1 ) + t2 P2 (e1 ) and B(e2 , (1 − s)e2 ) = P0 (e2 ) + sP1 (e2 ) + s2 P2 (e2 ), comparing coefficients in the equation (2.39), one finds that the Peirce projections Pj (e1 ) and Pk (e2 ) commute. Therefore we have the decomposition V =

M

Vj,k = V0,0 ⊕ V0,1 ⊕ V0,2 ⊕ V1,1 ⊕ V1,2 ⊕ V2,2

0≤j≤k≤2

where Vk,k = V2 (ek ) for k 6= 0 and V0,0 = V0 (e1 ) ∩ V0 (e2 ) is the range of the projection P0 (e1 )P0 (e2 ), and V0,1 = V0 (e2 ) ∩ V1 (e1 ),

V0,2 = V0 (e1 ) ∩ V1 (e2 ),

V1,2 = V1 (e1 ) ∩ V1 (e2 )

are ranges of mutually orthogonal projections Pj (ej 0 )Pk (ek0 ) for suitably chosen indices j, j 0 , k and k 0 . More generally, given a family {e1 , . . . , en } of mutually orthogonal tripotents in a Jordan triple system V , one can form the joint Peirce decomposition of V as follows. For i, j ∈ {0, 1, . . . , n}, the joint Peirce space Vij is defined by Vij := Vij (e1 , . . . , en ) = {z ∈ V : 2{ek , ek , z} = (δik + δjk )z for k = 1, . . . , n}, where δij is the Kronecker delta and Vij = Vji . The decomposition M V = Vij 0≤i≤j≤n

is called a joint Peirce decomposition. More verbosely, V00 =V0 (e1 ) ∩ · · · ∩ V0 (en ), Vii =V2 (ei )

(i = 1, . . . , n),

Vij =Vji = V1 (ei ) ∩ V1 (ej ) \ Vi0 =V0i = V1 (ei ) ∩ V0 (ej ) j6=i

(1 ≤ i < j ≤ n), (i = 1, . . . , n).

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101

The Peirce multiplication rules {Vij , Vjk , Vk` } ⊂ Vi`

and Vij

Vpq = {0}

for i, j ∈ / {p, q}

hold. The contractive projection Pij (e1 , . . . , en ) from V onto Vij = Vij (e1 , . . . , en ) is called a joint Peirce projection which satisfies  0 (i 6= j) Pij (e1 , . . . , en )(ek ) = (2.40) δik ek (i = j). We shall simplify the notation Pij (e, . . . , en ) to Pij if the tripotents e1 , . . . , en are understood.

For a single tripotent e ∈ V , we have

P11 (e) = P2 (e), P10 (e) = P1 (e) and P00 (e) = P0 (e). Let M = {0, 1, . . . , n} and N ⊂ {1, . . . , n}. The Peirce k-spaces of P the tripotent eN = i∈N ei are given by M V2 (eN ) = Vij , (2.41) i,j∈N

V1 (eN ) =

M

Vij ,

(2.42)

i∈N j∈M \N

V0 (eN ) =

M

Vij .

(2.43)

i,j∈M \N

More details of the construction of the preceding joint Peirce decomposition can be found in [124, 5.14]. A subspace J of a Jordan triple system V is called a triple ideal, or simply, an ideal, if it satisfies the condition {J, V, V } + {V, J, V } ⊂ J. If a subspace J ⊂ V satisfies only {J, V, J} ⊂ J, then it is called an inner ideal. The concept of an inner ideal is important in modern Jordan structure theory. Inner ideals are substitutes

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for one-sided ideals, the latter are absent in Jordan triple systems. Actually, every left or right ideal, or their intersection, in an associative algebra A is an inner ideal in the special Jordan algebra (A, ◦) with the Jordan product a ◦ b = (ab + ba)/2. So is any subspace of the form aAb. In a Jordan triple system V , the subspace {v, V, v} is an inner ideal, called the principal inner ideal determined by v. Given an ideal J of a Jordan triple system V , the quotient space V /J is naturally a Jordan triple system with the triple product {x + J, y + J, z + J} = {x, y, z} + J. The kernel ϕ−1 (0) of a triple homomorphism ϕ : V −→ W is an ideal of V . On the other hand, an ideal J of a Jordan triple system V is the kernel of the quotient map q : V −→ V /J which is a triple homomorphism. Let u be a tripotent in a Jordan triple system V . Applying the Peirce multiplication rules in Theorem 2.2.41 to the Peirce decomposition V = V0 (u) ⊕ V1 (u) ⊕ V2 (u), one deduces the following fact readily. Proposition 2.2.44. Given a tripotent u of a Jordan triple system V , the Peirce spaces V0 (u) and V2 (u) are inner ideals of V .

2.3

Lie algebras and Tits-Kantor-Koecher construction

We now show an important connection between Jordan triple systems and Lie algebras via the Tits-Kantor-Koecher construction. Lie algebras play an important role in geometry and this connection provides us with a useful link to apply Jordan theory to geometry. We will only be concerned with real or complex Lie algebras which, however, can be infinite dimensional.

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103

In what follows, a Lie algebra is a real or complex vector space g of any dimension, with a bilinear multiplication, called the Lie brackets, (x, y) ∈ g × g 7→ [x, y] ∈ g satisfying [x, x] = 0 and the Jacobi identity [[x, y], z] + [[y, z], x] + [[z, x], y] = 0 for all x, y, z ∈ g. We note that the multiplication is not associative but is anticommutative: [x, y] = −[y, x]. On any associative algebra a, one can define the Lie brackets by commutation: [x, y] = xy − yx where the product on the right-hand side is the original product in a. Then (a, [·, ·]) is a Lie algebra. Unlike Jordan algebras, it follows from a theorem of Poincar´e, Birkhoff and Witt [21, I.2.7] that any Lie algebra can be obtained in this way from an associative algebra. Given subspaces h and k of a Lie algebra g, we define [h, k] = {[x1 , y1 ] + · · · + [xn , yn ] : x1 , . . . , xn ∈ h; y1 , . . . , yn ∈ k} which is a subspace of g. We note that [h, k] = [k, h]. A Lie algebra g is called abelian if [g, g] = 0. An ideal of g is a subspace h of g satisfying [g, h] ⊂ h. For instance, [g, g] is an ideal of g. Given an ideal h of a Lie algebra g, the quotient space g/h is a Lie algebra in the product [x + h, y + h] = [x, y] + h

(x + h, y + h ∈ g/h).

A (Lie) homomorphism between two Lie algebras g and h is a linear map θ : g −→ h satisfying θ[x, y] = [θx, θy] for all x, y ∈ g.

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Given an ideal h of g, the quotient map θ : x ∈ g 7→ x + h ∈ g/h is a homomorphism. A bijective homomorphism between Lie algebras is called an isomorphism or a Lie isomorphism. An isomorphism from g onto itself is called an automorphism, or a Lie automorphism, of g. Let g be the conjugate of the vector space g. Then g is a Lie algebra with the multiplication of g. An automorphism θ : g −→ g is called involutive if θ2 is the identity map. In other words, for a complex Lie algebra g, an involutive automorphism is a conjugate linear bijection θ : g −→ g which preserves the Lie brackets and θ2 is the identity map. An involutive automorphism of a Lie algebra g is also called an involution. We will adopt this terminology although an algebra involution defined at the beginning of Section 2.1 is meant to be an antiautomorphism. However, confusion should be unlikely from the context. Definition 2.3.1. Let g be a complex Lie algebra and let gR be its real restriction, that is, g itself considered as a Lie algebra over the real field R. A real form of g is a subalgebra gr of gR such that g = gr + igr . If gr is the real form of g, the map σ : X +iY ∈ gr +igr 7→ X −iY ∈ gr + igr is called the conjugation of g with respect to gr . It is readily seen that σ is conjugate linear on g and a Lie automorphism of the real restriction gR . Definition 2.3.2. Let g be a Lie algebra. The set Aut g of all automorphisms θ : g −→ g forms a group with composition as group product, called the automorphism group of g. A derivation of a Lie algebra g is a linear map δ : g −→ g satisfying δ[x, y] = [δx, y] + [x, δy]

(x, y ∈ g).

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105

The vector space aut g of all derivations of g is a Lie algebra in the Lie brackets [δ, γ] = δγ − γδ. For each element x ∈ g, the map ad(x) : g −→ g defined by (y ∈ g)

ad(x)(y) = [x, y]

is a derivation of g and the Jacobi identity implies that the map ad : g −→ aut g is a homomorphism, called the adjoint representation of g. The kernel of ad is the centre of g: z(g) = {x ∈ g : [x, y] = 0 ∀y ∈ g}. The range of ad, denoted by ad g, is an ideal of aut g since [δ, ad(x)] = ad(δx)

(δ ∈ aut g, x ∈ g).

The elements of ad g are called the inner derivations of g. A Lie algebra g is called solvable if its derived series g ⊃ g(1) = [g, g] ⊃ g(2) = [g(1) , g(1) ] ⊃ · · · ⊃ g(n+1) = [g(n) , g(n) ] ⊃ · · · eventually terminates, that is, g(n) = {0} for some n. Given a finite-dimensional Lie algebra g, the symmetric bilinear form β : g × g −→ F

(F = R or C)

defined by β(x, y) = Trace (ad(x)ad(y)) is called the Killing form of g. The Killing form β is invariant, that is, β([x, y], z) = β(x, [y, z])

(x, y, z ∈ g)

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which is equivalent to β([x, y], z) + β(y, [x, z]) = 0

(x, y, z ∈ g).

The latter condition says that ad(x) is skew-symmetric with respect to β. A Lie algebra g is called semisimple if g contains no non-zero abelian ideal which is equivalent to the condition that g contains no non-zero solvable ideal. We prove below the Cartan-Killing criterion for semisimplicity. Theorem 2.3.3. A finite dimensional Lie algebra g is semisimple if and only if its Killing form β is non-degenerate. Proof. Let the Killing form β(x, y) = Trace (ad(x)ad(y)) be non-degenerate. Let h be an abelian ideal of g. We show that h = {0}. Let x ∈ h and let y ∈ g. Since h is an ideal, we have ad(x)ad(y)(g) ⊂ h and hence (ad(x)ad(y))2 (g) ⊂ ad(x)ad(y)(h) ⊂ [x, h] = {0} since h is abelian. This shows that the linear map ad(x)ad(y) : g −→ g is nilpotent and therefore Trace (ad(x)ad(y)) = 0. Non-degeneracy of β gives x = 0. Conversely, let g be semisimple. Let k = {x ∈ g : β(x, g) = {0}}. We need to show k = {0}. Since β is invariant, k is an ideal of g. We have Trace (AB) = 0 for A, B ∈ ad(k). It follows from Cartan’s solvability criterion that ad(k) is a solvable ideal in aut g. Semisimplicity of g implies that the homomorphism ad has zero kernel and therefore k is a solvable ideal in g. Hence k = {0} by semisimplicity again.

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The Tits-Kantor-Koecher construction originally relates Jordan algebras to finitely graded Lie algebras. This construction has been extended to Jordan triple systems by Meyberg [130]. We will describe the construction for Jordan triple systems. Let Z be the ring of integers. By a Z-grading of a Lie algebra g, we mean a decomposition of g into a direct sum of vector subspaces: g=

M

gn

n∈Z

such that [gn , gm ] ⊂ gn+m . The grading is said to be finite if the set {n : gn 6= 0} is finite. It is said to be non-trivial if ⊕n6=0 gn 6= 0. Lie algebras with a non-trivial finite Z-grading have been classified by Zelmanov [185] in which the Tits-Kantor-Koecher construction plays an important part. If A ⊂ Z, a Lie algebra g=

M

gα

α∈A

is said to be graded if [gα , gβ ] ⊂ gα+β where gα+β = {0} if α + β ∈ / A. L L A Lie homomorphism ψ : α∈A gα −→ α∈A hα is called graded if it respects the grading, that is ψ(gα ) ⊂ hα . There is a one-to-one correspondence between 3-graded Lie algebras g−1 ⊕ g0 ⊕ g1 and Jordan pairs [128]. The 3-graded Lie algebras with an involution correspond to Jordan triple systems. By an involutive Lie algebra (g, θ), we mean a Lie algebra g equipped with an involutive automorphism θ. We will always denote by k the 1eigenspace of θ, and by p the (−1)-eigenspace of θ so that g has the decomposition g = k ⊕ p where [k, k] ⊂ k,

[p, p] ⊂ k and

[k, p] ⊂ p.

If (g, θ) is finite dimensional, the involution θ is called a Cartan invo-

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lution if the symmetric bilinear form βθ defined by βθ (x, y) = −β(x, θy)

(x, y ∈ g)

is positive definite. Example 2.3.4. Let V be a normed vector space and gl(V ) the normed algebra of continuous linear self-maps on V . Then gl(V ) is a Lie algebra in the usual Lie brackets [X, Y ] = XY − Y X

(X, Y ∈ gl(V )).

If n = dim V < ∞, we often denote gl(V ) as gl(n, F) where F = R or C. If V is a Hilbert space, then the subspace glhs (V ) of gl(V ), consisting of all Hilbert-Schmidt operators, is an ideal and is equipped with a natural complete inner product hX, Y i2 = Trace (XY ∗ )

(X, Y ∈ glhs (V )).

Of course, glhs (V ) = gl(V ) if dim V < ∞. We can define an involution θ : gl(V ) −→ gl(V ) by θ(X) = −X ∗

(X ∈ gl(V ))

where X ∗ : V −→ V denotes the adjoint operator of X. If dim V < ∞, then θ is a Cartan involution since −β(X, θX) = −Trace (ad(X)ad(θX)) = Trace (ad(X)ad(X ∗ )) = Trace (ad(X)ad(X)∗ ) ≥ 0 where ad (X)∗ : gl(V ) −→ gl(V ) is the adjoint operator of ad (X) with respect to the inner product h·, ·i2 . In fact, the Killing form of gl(V ) can be computed explicitly, it is given by β(X, Y ) = 2nTrace (XY ) − 2Trace (X)Trace (Y )

(X, Y ∈ gl(V )).

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Definition 2.3.5. A graded Lie algebra g = g−1 ⊕ g0 ⊕ g1 is called a Tits-Kantor-Koecher Lie algebra or TKK Lie algebra if g admits an involution θ, which is negatively graded, that is θ(gα ) = g−α . We call θ the TKK-involution of g and note that the map x ∈ g−1 7→ θ(x) ∈ g1 is a linear isomorphism. We call g canonical if [g−1 , g1 ] = g0 . We define the canonical part of g to be the Lie subalgebra gc = g−1 ⊕ [g−1 , g1 ] ⊕ g1 which is also a TKK Lie algebra with the restriction of θ as the TKKinvolution. There is a one-one correspondence between Jordan triple systems and TKK Lie algebras. To show this, we prove a lemma first. Lemma 2.3.6. Let V be a non-degenerate Jordan triple system and let P P P P j aj bj = k uk vk . Then we have j bj aj = k vk uk . Proof. We have    X X  aj bj , x y  =  (aj j

bj )x

uk

vk , x y =

k

 aj  y

j

# X

y−x

 X  bj

j

" =



! X (uk

vk )x

! y−x

k

X

vk

uk

y

k

 P  aj y = x v u y for all x, y ∈ V . k k k P P By Lemma 2.2.21, we conclude j bj aj = k vk uk .

which gives x

P

j bj

We now show the Tits-Kantor-Koecher construction of a Lie algebra from a non-degenerate Jordan triple system.

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Theorem 2.3.7. Let V be a non-degenerate Jordan triple system. Then there is a canonical Tits-Kantor-Koecher Lie algebra L(V ) with grading L(V ) = L(V )−1 ⊕ L(V )0 ⊕ L(V )1 and an involution θ such that L(V )−1 = V = L(V )1 and {x, y, z} = [[x, θy], z] for x, y, z ∈ L(V )−1 . Proof. Form the algebraic direct sum L(V ) = V−1 ⊕ V0 ⊕ V1 where V−1 = V and V1 = V , which is the conjugate of V and is just V if it is real, and V0 is the linear span of V

V in the space L(V ) of

linear self-maps on V . The Jordan triple identity (2.26) implies that V0 is a Lie algebra in the bracket product [h, k] = hk − kh. By Lemma 2.3.6, the mapping x y∈V

V 7→ y x ∈ V

V

is well-defined and extends to a conjugate linear map

\

: V0 −→ V0

satisfying [x y, u v]\ = −[y x, v u]. This enables us to define an involutive conjugate linear map θ : L(V ) −→ L(V ) by θ(x ⊕ h ⊕ y) = y ⊕ −h\ ⊕ x

(x ⊕ h ⊕ y ∈ V−1 ⊕ V0 ⊕ V1 )

where we also write (x, h, y) for x ⊕ h ⊕ y.

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Evidently, θ2 is the identity map and also, θ is conjugate linear since θ(α(x, a b, y)) = θ(αx, α(a b), α · y) = (αy, −α(b a), α · x) = α(y, −(b a), x). By identifying Vα naturally as subspaces of L(V ), we see immediately that θ(Vα ) = V−α for α = 0, ±1. Equip L(V ) with the multiplication [x ⊕ h ⊕ y, u ⊕ k ⊕ v] = (h(u) − k(x), [h, k] + x v − u y, k \ (y) − h\ (v)). (2.44) With this multiplication, one can show that L(V ) becomes a Lie algebra. The proof of this has been given in [37, Theorem 1.3.8]. We suppress the details. It is routine to verify that θ is an automorphism. Given x ∈ V−1 and y ∈ V1 , we have [x, y] = [(x, 0, 0), (0, 0, y)] = (0, x y, 0) and hence [V−1 , V1 ] = V0 , that is, L(V ) is canonical. We also have [V−1 , V−1 ] = [V1 , V1 ] = 0. To facilitate the computation of the Lie product (2.44) in the TitsKantor-Koecher construction, one can make use of matrix notation (cf. [185]). The TKK Lie algebra L(V ) = V−1 ⊕ V0 ⊕ V1 can be written in the following matrix form: P  j aj  L(V ) =  y

bj

with the Lie multiplication



x

 : aj , bj , x, y ∈ V −

P

j bj

aj

  

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

  0 x v−u y  ,   (2.45)  :=  y 0 0 −v x + y u v 0 0 x

 

0 u



     0 0 −a b(x) a b  :=   . (2.46)  ,  0 −b a b a(y) 0 y 0 

0 x

The construction in Theorem 2.3.7 translates the non-degeneracy of a Jordan triple V into the following property of its TKK Lie algebra L(V ): [[a, θy], a] = 0

for all a, y ∈ L(V )−1 =⇒ a = 0

which is equivalent to the condition (ad a)2 = 0 =⇒ a = 0

(a ∈ L(V )−1 )

(2.47)

since (ad a)2 (x ⊕ h ⊕ y) = −Qa (y) for a ∈ L(V )−1 and x ⊕ h ⊕ y ∈ L(V ). Definition 2.3.8. A TKK Lie algebra g = g−1 ⊕ g0 ⊕ g1 is called non-degenerate if (ad a)2 = 0 =⇒ a = 0 for a ∈ g−1 . Two TKK Lie algebras (g, θ) and (g0 , θ0 ) are said to be isomorphic if there is a graded isomorphism ψ : g −→ g0 which commutes with involutions: ψθ = θ0 ψ. Given a TKK Lie algebra g = g−1 ⊕ g0 ⊕ g1 with TKK involution θ, there is a natural way to construct a Jordan triple system V such that L(V ) = g−1 ⊕ [g−1 , g1 ] ⊕ g1 , which is the canonical part gc of g. Indeed, it suffices to take V = g−1 and define a triple product on V by {a, b, c} := [[a, θ(b)], c]

(a, b, c ∈ V ).

Then the Jacobi identity in g and the condition [g−1 , g1 ] = 0 implies that V , together with the above triple product, is a Jordan triple system. Further, g is non-degenerate if and only if L(V ) is so.

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Hence the preceding construction V 7→ L(V ) establishes the correspondence between non-degenerate Jordan triple systems and TKK-Lie algebras. This correspondence is one-to-one in the sense that two Jordan triple systems V and V 0 are triple isomorphic if and only if the corresponding TKK-Lie algebras (L(V ), θ) and (L(V 0 ), θ0 ) are isomorphic, that is, there is a graded Lie isomorphism ϕ e : L(V ) −→ L(V 0 ) satisfying ϕθ e = θ0 ϕ. e Indeed, given a triple isomorphism ϕ : V −→ V 0 , one can define the corresponding graded isomorphism ϕ e : L(V ) −→ L(V 0 ) by ϕ(a e ⊕ h ⊕ b) = ϕa ⊕ ϕhϕ−1 ⊕ ϕb

(a ⊕ h ⊕ b ∈ L(V )).

Conversely, given a graded isomorphism ψ : L(V ) −→ L(V 0 ) satisfying ψθ = θ0 ψ, the restriction ψ|V : V −→ V 0 defines a triple isomorphism. Notes. The construction of Lie algebras from Jordan algebras was discovered independently by Tits [165], Kantor [115, 97] and Koecher [110, 111]. Meyberg introduced the concept of a Jordan triple system in [130] and extended the construction of Koecher to the wider class of Jordan triple systems. The TKK construction in this section is essentially the same as the one given in [37, Theorem 1.3.8]. However, the construction in Theorem 2.3.7 includes the case of complex Jordan triples, which are not considered in [37, Theorem 1.3.8], where the triple product in a ‘Jordan triple’ as defined there is linear in the middle variable.

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Jordan and Lie structures in Banach spaces

To apply Jordan and Lie theory to bounded symmetric domains of all dimensions, we need to consider infinite dimensional Jordan triples and Lie algebras which carry the structure of a Banach space. Unless stated otherwise, all Banach spaces in the sequel are over the complex field C. Definition 2.4.1. A complex Jordan triple system V is called a JB*triple if it is a complex Banach space in which the Jordan triple product {·, ·, ·} is continuous and for each a ∈ V , the continuous box operator a a : V −→ V satisfies the following conditions: (i) a a is hermitian, that is, k exp it(a a)k = 1 for all t ∈ R; (ii) a a has a non-negative spectrum; (iii) ka ak = kak2 where the linear exponential operator exp it(a a) : V −→ V is defined by exp it(a a)(x) = x +

∞ X 1 (it)n (a a)n (x) n!

(x ∈ V )

n=1

and (a a)n is the n-fold product of a a in the Banach algebra L(V ) of bounded linear operators on V . By [20, p. 46], a a is a hermitian element in L(V ) if and only if it has real numerical range. The spectrum of a a is the set σ(a a) := {λ ∈ C : λ1 − a a is not invertible in L(V )} and condition (ii) states that σ(a a) ⊂ [0, ∞). Condition (iii) in Definition 2.4.1 can be replaced by k{a, a, a}k = kak3

(2.48)

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(cf. [37, Lemma 3.1.3]). It follows that JB*-triples are non-degenerate and anisotropic. The fundamental role of JB*-triples in bounded symmetric domains will unfold in ensuing sections. Here are some examples. Example 2.4.2. The complex Euclidean space Cn is a JB*-triple in the Euclidean norm and coordinatewise triple product {x, y, z} := (x1 y 1 z1 , . . . , xn y n zn ) for x = (x1 , . . . , xn ), y = (y1 , . . . , yn ), z = (z1 , . . . , zn ) ∈ Cn , where ‘− ’ denotes the complex conjugate as usual. Example 2.4.3. Let {Vα }α∈Λ be a family of normed vector spaces. We `∞ M Vα to be the direct sum define their `∞ -sum α∈Λ `∞ M

Vα = {(vα ) ∈ ×α Vα : sup kvα k < ∞}, α

α∈Λ

equipped with the `∞ -norm k(vα )k∞ := sup kvα k. α

If Λ = {1, 2, . . . } and Vα = F for all α ∈ Λα , where F = C or R, then `∞ M Vα is just the Banach space `∞ of bounded sequences in F . α∈Λ

Let Vα be a JB*-triple for each α ∈ Λ. One can verify that the `∞ `∞ M sum Vα , with the `∞ -norm, is a JB*-triple with the coordinatewise α

triple product. Infinite dimensional JB*-triples include C*-algebras. Example 2.4.4. A C*-algebra is a norm closed subalgebra A of the Banach ∗-algebra L(H) of bounded linear operators on a Hilbert space

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H such that a ∈ A implies a∗ ∈ A, where a∗ is the adjoint of the operator a. We refer to [154] for a comprehensive exposition of C*algebras. In a C*-algebra A, an operator a ∈ A is hermitian, i.e. k exp itak = 1 for all t ∈ R, if and only if it is self-adjoint, i.e. a = a∗ (cf. [20, p. 47]). Since A is an associative algebra with involution, it is a complex Jordan algebra with involution in the special Jordan product 1 a ◦ b = (ab + ba) 2

(a ∈ A)

and also a complex Jordan triple system in the canonical Jordan triple product 1 {a, b, c} = (ab∗ c + cb∗ a). 2

(2.49)

Given a ∈ A, we can write the box operator a a : A −→ A as the sum 2a a = Laa∗ + Ra∗ a of the left multiplication Laa∗ : x ∈ A 7→ aa∗ x ∈ A and the right multiplication Ra∗ a : x ∈ A 7→ xa∗ a ∈ A. We have k exp itLaa∗ k = sup{k exp it(aa∗ x)k : kxk ≤ 1} ≤ k exp it(aa∗ )k = 1 for all t ∈ R. Hence Laa∗ is a hermitian operator on A and likewise, Ra∗ a is hermitian. It follows that 2a a is hermitian since its numerical range is contained in the sum of those of Laa∗ and Ra∗ a (cf. [20, p. 15]). Further, the spectrum σ(La∗ a ) of La∗ a : A −→ A coincides with the spectrum of a∗ a in the C*-algebra A, which is contained in [0, ∞). Likewise σ(Raa∗ ) ⊂ [0, ∞). By [20, p. 53], both Laa∗ and Ra∗ a have positive numerical range, which implies that a a has positive numerical range. This implies that σ(a a) ⊂ [0, ∞) since the spectrum of an operator is contained in its numerical range (cf. [20, p. 19]). It follows

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that, in the canonical triple product (2.49), a C*-algebra A is JB*-triple since the identity (2.48) is just the C*-identity k{a, a, a}k2 = kaa∗ ak2 = k(aa∗ a)(a∗ aa∗ )k = kaa∗ k3 = kak6 . Tripotents in a C*-algebra A are precisely the partial isometries, which are elements e ∈ A such that e = ee∗ e, equivalently, ee∗ (and hence e∗ e) is a projection. The Peirce k-spaces of the tripotent e are given by A0 (e) = (1 − ee∗ )A(1 − e∗ e), A1 (e) = (1 − ee∗ )Ae∗ e + ee∗ A(1 − e∗ e), A2 (e) = ee∗ Ae∗ e. Example 2.4.5. We will denote by C0 (Ω) the Banach algebra of complex continuous functions vanishing at infinity on a locally compact Hausdorff space Ω, equipped with the involution defined by pointwise complex conjugation: f ∗ (ω) := f (ω)

(f ∈ C0 (Ω), ω ∈ Ω).

It is well-known that C0 (Ω) is isometrically ∗-isomorphic to a commutative C*-algebra in some L(H). Hence C0 (Ω) is a complex Jordan triple system in the canonical Jordan triple product {f, g, h}(ω) = f (ω)g(ω)h(ω)

(f, g, h ∈ C0 (Ω), ω ∈ Ω).

The celebrated Gelfand-Naimark theorem asserts that, via an isometric ∗-isomorphism, commutative C*-algebras are of the form C0 (Ω) and also, a Banach algebra A with an involution ∗ satisfying kx∗ xk = kxk2 for all x ∈ A is (isometrically ∗-isomorphic to) a C*-algebra. Given a normed vector space E, we will denote its dual space by E∗

as usual.

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Definition 2.4.6. A JB*-triple V is called a JBW*-triple if it is a dual Banach space, that is, V = E ∗ for some complex Banach space E, in which case E is called a predual of V . In fact, the predual E is unique in the sense that, if V = E ∗ = F ∗ , then E = F when they are canonically embedded into V ∗ (cf. [87, (3.21)]). Henceforth, we will denote by V∗ unambiguously the predual of a JBW*-triple V and refer to the weak topology w(V, V∗ ) as the weak* topology of V . The triple product of a JBW*-triple V is separately weak* continuous (cf. [37, Theorem 3.3.9]) and a Jordan triple isomorphism between two JBW*-triples is necessarily weak* continuous (cf. [87, (3.22)]). The second dual V ∗∗ of a JB*-triple V is a JBW*-triple and V identifies as a subtriple of V ∗∗ via the canonical embedding V ,→ V ∗∗ [37, Corollary 3.3.5]. Example 2.4.7. A C*-algebra A is called a von Neumann algebra if it has a predual A∗ , in which case A∗ is unique [154, Corollary 1.13.3]. In the canonical triple product (2.49), von Neumann algebras are JBW*triples. In particular, L(H) is a JBW*-triple. Its predual is the Banach space T (H) of trace-class operators on H. A von Neumann algebra A always contains an identity and an abundance of projections, which are elements p ∈ A satisfying p = p∗ = p2 . In fact, A is the norm closed linear span of its projections. Lemma 2.4.8. A closed subtriple of a JB*-triple is itself a JB*-triple in the inherited triple product. Proof. Let W be a closed subtriple of a JB*-triple V and let a ∈ W . The restriction a a|W to W of the box operator a a : V −→ V is the box operator x ∈ W 7→ {a, a, x} ∈ W on W in the inherited triple product and is hermitian since k exp it(a a|W )k ≤ k exp it(a a)k = 1

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for all t ∈ R. Let A := {T ∈ L(V ) : T (W ) ⊂ W }. Then A is a closed subalgebra of the Banach algebra L(V ) and a a ∈ A. As an element in L(V ), the spectrum σ(a a) is contained in [0, ∞). In particular, the set C\σ(a a) is connected and by [153, p. 239], the spectrum σA (a a) of a a in the algebra A coincides with σ(a a). Since the map T ∈ A 7→ T |W ∈ L(W ) is an algebra homomorphism, we have σ(a a|W ) ⊂ σA (a a) ⊂ [0, ∞). Finally, the identity k{a, a, a}k = kak3 clearly holds in the subtriple W . This concludes the proof. Example 2.4.9. The Jordan triple L(H, K) of bounded linear operator between Hilbert spaces H and K is a JBW*-triple. Indeed, L(H, K) is the dual space of the Banach space of trace-class operators between H and K. Moreover, L(H, K) can be identified as a closed subtriple of the JBW*-triple L(H ⊕ K) of bounded linear operators on the direct sum H ⊕ K of H and K, via the embedding   0 0 a ∈ L(H, K) 7→ ∈ L(H ⊕ K) a 0 which is an isometric triple homomorphism. Given a Hilbert space (H, h·, ·i), the linear isometry x ∈ L(C, H) 7→ x(1) ∈ H identifies the two spaces and induces a JB*-triple structure on H. The adjoint x∗ of x ∈ L(C, H) is given by x∗ (h) = hh, x(1)i for h ∈ H and hence the triple product in H can be expressed as 1 {x, y, z} = (hx, yiz + hz, yix). 2

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For any a, b ∈ H, the adjoint of the box operator a b : H −→ H is b a, by the associativity of the inner product h·, ·i, that is, h{a, b, x}, yi = hx, {b, a, y}i

(x, y ∈ H).

(2.50)

Besides C*-algebras, there is another class of Jordan algebras which carry the structure of a JB*-triple. We first introduce the real forms of these algebras. A real Jordan algebra B is called a JB-algebra if it is also a Banach space and the norm satisfies kabk ≤ kakkbk,

ka2 k = kak2 ,

ka2 k ≤ ka2 + b2 k

for all a, b ∈ B. Idempotents in JB-algebras are also called projections. A JB-algebra A is called a JBW-algebra if it is a dual Banach space in which case the predual of A is unique [77, Theorem 4.4.16], the weak* topology on A is unambiguous and A must have an identity [77, Lemma 4.1.7]. The second dual A∗∗ of a JB-algebra A is a JBWalgebra in which A identifies as a closed Jordan subalgebra [77, Theorem 4.4.3]. JB-algebras are formally real Jordan algebras [77, Corollary 3.3.8]. In particular, finite dimensional JB-algebras are Hilbert spaces in the trace form ha, bi = Trace(a b) by Theorem 2.1.17. A real Jordan algebra H is called a JH-algebra if it is also a real Hilbert space in which the inner product h·, ·i is associative, that is, hab, ci = hb, aci

(a, b, c ∈ H).

Real spin factors H ⊕R are JH-algebras. Finite dimensional JH-algebras are exactly the Euclidean Jordan algebras introduced in [60].

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Lemma 2.4.10. Let A be a finite dimensional real Jordan algebra. The following conditions are equivalent. (i) A is formally real. (ii) A is a JH-algebra with an identity 1. (iii) A is a JB-algebra. Proof. (i) ⇒ (ii). By Lemma 2.2.23, a finite dimensional formally real Jordan algebra, equipped with the trace form as an inner product, is a JH-algebra. (ii) ⇒ (i). Given a2 + b2 = 0 in a JH-algebra with identity 1, we have ha, ai + hb, bi = ha2 , 1i + hb2 , 1i = ha2 + b2 , 1i = 0 which gives a = b = 0. (i) ⇒ (iii). In a finite dimensional formally real Jordan algebra A, the set {a2 : a ∈ A} forms a proper cone, which induces a partial ordering ≤ in A and the following norm kak := inf{λ > 0 : −λ1 ≤ a ≤ λ1} makes A into a JB-algebra [77, Corollary 3.1.7]. The celebrated result of Koecher and Vinberg, stated at the end of Section 2.1, establishes the one-one correspondence between finite dimensional formally real Jordan algebras and linearly homogeneous self-dual cones. In view of Lemma 2.4.10, an appropriate infinite dimensional generalisation of formally real Jordan algebras is the notion of a unital JH-algebra. Indeed, it has been shown recently in [40] that unital JH-algebras are in one-one correspondence with linearly homogeneous self-dual cones of any dimension (see Theorem 2.4.14 below), generalising the Koecher-Vinberg result to infinite dimension.

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Definition 2.4.11. Let V be a real Hilbert space with inner product h·, ·i. An open cone Ω ⊂ V is called self-dual homogeneous if it satisfies the following conditions: (i) (self-duality) Ω = Ω∗ , where Ω∗ := {v ∈ V : hv, xi > 0 ∀x ∈ Ω\{0}} is called the dual cone of Ω; (ii) (homogeneity) Ω is linearly homogeneous, that is, given x, y ∈ Ω, there is a continuous linear isomorphism h : V −→ V such that h(Ω) = Ω and h(x) = y. Evidently, the preceding concept of homogeneity can be defined for cones in real Banach spaces, which will be discussed in Section 3.6. Remark 2.4.12. A self-dual homogeneous cone is also called a symmetric cone in literature (e.g. [60]). We use the former terminology in this book lest the latter be confused with the notion of a symmetric domain in complex Banach spaces. Remark 2.4.13. To avoid confusion with the notion of homogeneity introduced in Definition 1.3.21 for bounded domains, we shall henceforth adopt the terminology ‘linearly homogeneous self-dual cone’ instead of ‘self-dual homogeneous cone’. We refer to [40] for a proof of the following extension of the aforementioned result of Koecher and Vinberg. Theorem 2.4.14. Let Ω be an open cone in a real Hilbert space. Then Ω is a linearly homogeneous self-dual cone if, and only if, it is of the form Ω = int {a2 : a ∈ H} for a unique unital JH-algebra H.

(2.51)

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Remark. The finite dimensional result of Koecher and Vinberg has also been shown in [60, 155]. The derivation of the Jordan identity in the proof of Theorem 3.1 in [40, p. 369] can be replaced by verbatim arguments in [60, p. 50], using the associativity of the inner product. Example 2.4.15. Let Ω be the interior of the closed cone {x2 : x ∈ H ⊕ R} = {a ⊕ α : α ≥ kak} in a real spin factor H ⊕ R. Then it is self-dual and linearly homogeneous. For H = Rn (n ≥ 2), the cone Ω is known as the Lorentz cone, it can be written as Ω = {(x1 , . . . , xn+1 ) ∈ Rn+1 : xn+1 > 0 and x2n+1 > x21 + · · · + x2n }. It is also known as the second-order cone in optimization theory. Interestingly, for 1 < p < ∞ and p 6= 2, the so-called p-order cone Cp := {(x1 , . . . , xn+1 ) ∈ Rn+1 : xn+1 > 0 and xpn+1 > xp1 + · · · + xpn } is not linearly homogeneous in Rn ⊕ R, nor self-dual [91]. Let Ω be a linearly homogeneous self-dual cone in a real Hilbert space, identified with int {a2 : a ∈ H} in a unital JH-algebra H with inner product h·, ·i. Then it follows from Corollary 2.4.20 and Lemma 3.6.12, shown later, that each element in Ω is invertible and, as shown in [37, Theorem 2.3.19], one can define a Riemannian metric on Ω by gω (u, v) = h{ω −1 , u, ω −1 }, vi

(ω ∈ Ω, u, v ∈ H)

which turns Ω into a Riemannian symmetric space. For a finite dimensional irreducible linearly homogeneous self-dual cone Ω, where irreducibility means that Ω is not a direct product of two non-trivial linearly homogeneous self-dual cones, this metric is proportional to the

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one defined by the so-called characteristic function of the cone Ω, as follows (see, for example, [60]). Let λ be the Euclidean measure on a finite dimensional Hilbert space V with inner product h·, ·i, and Ω an open cone in V . The characteristic function ϕ : Ω −→ [0, ∞) is defined by Z exp −hx, yidλ(y) (x ∈ Ω). ϕ(x) = Ω∗

One can define a Riemannian metric on Ω by gx (u, v) = Du Dv log ϕ(x)

(x ∈ Ω, u, v ∈ V )

(2.52)

where Du and Dv are directional derivatives. The metric in (2.52) is called the canonical Riemannian metric on Ω. Positive cones will feature in the discussion of Siegel domains in the next chapter, where a classification of unital JH-algebras is given. We now introduce the complexification of JB-algebras, which are complex Jordan algebras with involution and form a class of JB*-triples in the canonical Jordan triple product. Definition 2.4.16. A complex Jordan algebra B with an involution ∗ is called a JB ∗ -algebra or Jordan C ∗ -algebra if it is also a Banach space in which the norm satisfies kabk ≤ kakkbk,

ka∗ k = kak,

k{a, a, a}k = kak3

for all a, b ∈ B, where {·, ·, ·} denotes the canonical Jordan triple product defined in (2.13). A JB*-algebra is called a JBW*-algebra if it has a predual, which is necessarily unique. Evidently, a C*-algebra is a JB*-algebra in the special Jordan product. Given a JB*-algebra B, its self-adjoint part Bsa = {a ∈ B : a∗ = a}

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forms a JB-algebra in the inherent Jordan product [77, 3.8.2] and we have B = Bsa + iBsa . Conversely, it has been shown in [179] that a unital JB-algebra A can be complexified to a JB*-algebra A = A + iA so that A identifies with the self-adjoint part Asa of A. The norm on A is given by the Minkowski functional of the convex hull of the set {exp ia : a ∈ A} ⊂ A and it coincides with the original norm on A. Example 2.4.17. The exceptional real Jordan algebra H3 (O) in Example 2.1.16 is formally real and by Lemma 2.4.10, it is a unital JB-algebra in the order-unit norm kak = inf{λ > 0 : −λ1 ≤ a ≤ λ1}. Its complexification is the exceptional Jordan algebra H3 (O) = H3 (O)+ iH3 (O) which is a JB*-algebra. Example 2.4.18. For any JB*-triple V which admits a non-zero tripotent u, the u-homotope of the Peirce 2-space P2 (u)V is a JB*-algebra in the inherited norm, with involution defined by a∗ := {u, a, u} and u becomes the identity of the algebra. Unital JB*-algebras are in one-one correspondence with the class of bounded symmetric domains realisable as tube domains. We will discuss the detail in Section 3.6.

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Lemma 2.4.19. A JB*-algebra B, with the canonical Jordan triple product {a, b, c} = (ab∗ )c + a(b∗ c) − b∗ (ac), is a JB*-triple. In particular, a JBW*-algebra is a JBW*-triple. Proof. For each a ∈ B, it has been shown in [181, Theorem 6] that a = a∗ if, and only if, the left multiplication La : B −→ B is hermitian. Given x ∈ B, we can write x = a + ib with a = a∗ and b = b∗ . Since x x = a a + b b + 2i(Lb La − La Lb ) where the left multiplications La and Lb are hermitian, x x is hermitian (cf. [20, p. 47]). The closed subalgebra A generated by a in B is associative and is therefore an abelian C*-algebra since each z ∈ A satisfies kzk3 = k(zz ∗ )zk ≤ kzz ∗ kkzk ≤ kzk2 kz ∗ k = kzk3 . Identifying A as the algebra C0 (Ω) of complex continuous functions vanishing at infinity on a locally compact Hausdorff space Ω (cf. Example 2.4.5), it is readily seen that the operator a a|A : A −→ A has non-negative spectrum since it is just the left multiplication by |a|2 . The closed subtriple V (a) generated by a in A is a JB*-triple by Lemma 2.4.8 and hence σ(a a|V (a) ) ⊂ [0, ∞). The JB*-algebra B is a complex Jordan triple system with continuous triple product. Hence by Lemma 3.2.10 (cf. Section 3.2), we have

1 σ(a a) ⊂ (S + S) ⊂ [0, ∞) 2 where S = σ(a a|V (a) ) ∪ {0}.

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Finally, observe that x∗ x∗ = ∗ ◦ (x x) ◦ ∗. Hence σ(x∗ x∗ ) = σ(x x). Now x x + x∗ x∗ = 2a a + 2b b implies σ(x x) ⊂ [0, ∞). By the preceding lemma, the exceptional Jordan algebra H3 (O) is a JBW*-triple. The finite dimensional Jordan triple M1,2 (O) identifies as a closed subtriple of H3 (O) and is therefore, by Lemma 2.4.8, a JBW*-triple. Given a JB-algebra A, the set {x2 : x ∈ A} forms a proper closed cone [77, Lemma 3.3.7] which induces a partial ordering ≤ in A. If A has an identity e, then the norm of A satisfies kak = inf{λ > 0 : −λe ≤ a ≤ λe}

(a ∈ A)

(cf. [77, Proposition 3.3.10]). The identity e lies in the interior int {x2 : x ∈ A} of the cone since ka − ek < 1/2 implies − 21 e ≤ a − e ≤ particular, 0 ≤

1 2e

1 2e

and in

≤ a.

Corollary 2.4.20. Let A be a unital JB-algebra and C = int {x2 : x ∈ A}. Then each element z ∈ A + iC is invertible in the JB*-algebra A = A + iA.

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Proof. Let z = v + ix2 ∈ A + iC. By a remark following Theorem 2.1.17 in Section 2.1, it suffices to show that the left multiplication Lz : A −→ A is invertible in the unital Banach algebra L(A) of bounded linear operators on A, that is, the spectrum σ(Lz ) does not contain 0. Let 1 be the identity of L(A) and let B ⊂ L(A) be a maximal abelian subalgebra containing Lz . Then the spectrum σB (Lz ) of Lz in B coincides with σ(Lz ). Hence for each λ ∈ σ(Lz ), there is a unital algebra homomorphism ϕ : B −→ C such that ϕ(Lz ) = λ. We have kϕk = 1 and by the Hahn-Banach theorem, ϕ extends to a continuous linear functional ϕ e ∈ L(A)∗ with kϕk e = 1. Observe that for each element c ∈ C, the left multiplication Lc : A −→ A is invertible. Indeed, c − C is an open neighbourhood of 0 ∈ A and hence the identity e ∈ A is in α(c − C) for some α > 0, which gives c = α−1 + a2 for some a ∈ A and Lc = α−1 1 + La2 . We have σ(La2 ) = σ(a a) ⊂ [0, ∞) by Lemma 2.4.19 and therefore σ(Lc ) ⊂ α−1 + [0, ∞). We have Lz = Lv +iLx2 . As noted in the proof of Lemma 2.4.19, Lv is a hermitian operator on A as well as Lx2 . Hence the numerical range of Lx2 is the convex hull of its spectrum σ(Lx2 ) [20, p. 58]. It follows that for each ψ ∈ L(A)∗ with ψ(1) = 1 = kψk, we have ψ(Lv ) ∈ R [20, p. 46] and ψ(Lz ) = ψ(Lv ) + iψ(Lx2 ) ∈ R + i(0, ∞) 6= {0}, which gives 0∈ / σ(Lz ). In the proof of Lemma 2.4.19, one can actually say more about the closed subtriple V (a) in A. We refer to [37, Theorem 3.1.12] for a proof of the following fundamental result. Theorem 2.4.21. Let V be a JB*-triple and a ∈ V \{0}. Then there is a locally compact Hausdorff space Sa ⊂ (0, ∞), the triple spectrum of a, such that the closed subtriple V (a) generated by a is isometrically triple isomorphic to the JB*-triple C0 (Sa ) of continuous functions on

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Sa vanishing at infinity. We now discuss Lie algebras which carry Banach space structures. A real or complex Lie algebra g is called a normed Lie algebra if g is a normed vector space and the Lie product is continuous: k[X, Y ]k ≤ CkXkkY k

(X, Y ∈ g)

for some C > 0. A Banach Lie algebra is a normed Lie algebra g, which is also a Banach space in the given norm. There are many natural examples of Banach Lie algebras. For instance, a C*-algebra A is a Banach Lie algebra in the usual Lie brackets [a, b] = ab − ba

(a, b ∈ A).

In what follows, we will only be concerned with those Banach Lie algebras induced by bounded symmetric domains. They are related to the TKK Lie algebras of JB*-triples. Let V be a JB*-triple and let L(V ) = V−1 ⊕ V0 ⊕ V1 be its TKK Lie algebra, as constructed in Section 2.3. We will equip L(V ) with the norm k(x, h, y)k := kxk + khk + kyk,

(x, h, y) ∈ V−1 ⊕ V0 ⊕ V1

which makes L(V ) into a normed Lie algebra. We recall that V−1 = V , V1 = V and V0 =

 k X 

j=1

aj

bj : a1 , . . . , ak , b1 , . . . , bk ∈ V

 

.



If V is finite dimensional, then L(V ) is a finite dimensional Banach Lie algebra.

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Banach Lie group is a Banach Lie algebra. We show that, in local coordinates, the Lie product can be expressed by continuous bilinear maps and is therefore continuous.

Theorem 2.4.22. The Lie algebra g of a Banach Lie group G is a Banach Lie algebra.

Proof. We already know that g = Te G is a Lie algebra and also, it identifies with the Banach space V in a local chart (U, ϕ, V ) at e. We show that the Lie product (Xe , Ye ) ∈ Te G × Te G 7→ [Xe , Ye ] ∈ Te G = V is continuous. Let Xe = [α] ∈ Te G and Ye = [β] ∈ Te G. By Lemma 1.2.22, We have

[Xe , Ye ] = ad([α])(Ye ) 1 = lim (Ad(α(t))Ye − Ad(α(0))Ye ) t→0 t d (Ad(α(t))Ye ) = dt t=0
d = d(rα(t)−1 `α(t) )e Ye dt t=0   d d −1 = (ϕ(α(t)β(s)α(t) )) dt t=0 ds s=0

where a tangent vector [γ] ∈ Te G identifies with (ϕ ◦ γ)0 (0) ∈ V . Considering ϕ(α(t)β(s)α(t)−1 ) = F (t, s) as a function from some

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open neighbourhood of (0, 0) ∈ R2 to V , we have the Taylor expansion F (t, s) = F (0, 0) + F 0 (0, 0)(t, s) +

1 00 F (0, 0)((t, s), (t, s)) + · · · 2!

1 (D1 D1 F (0, 0)(t, t) 2 + 2D1 D2 F (0, 0)(t, s) + D2 D2 F (0, 0)(s, s)) + · · · 1 = (ϕ ◦ β)0 (0)s + (D1 D1 F (0, 0)(t, t) + 2D1 D2 F (0, 0)(t, s) 2 + D2 D2 F (0, 0)(s, s)) + · · · .

= D1 F (0, 0)t + D2 F (0, 0)s +

It follows that   d d F (t, s) = D1 D2 F (0, 0)(1, 1) = f ((ϕ ◦ α)0 (0), (ϕ ◦ β)0 (0)) dt t=0 ds s=0 where f : V × V −→ V is a continuous bilinear map. It follows that [Xe , Ye ] = f (Xe , Ye ) and the Lie product is continuous. Remark 2.4.23. In contrast to the finite dimensional case, a Banach Lie algebra need not arise as the Lie algebra of a Banach Lie group [173]. However, if the centre z(g) = {X ∈ g : [X, g] = 0} of a Banach Lie algebra (g, [·, ·]) is trivial, then g is the Lie algebra of a connected Banach Lie group. We refer to [149] for a proof. Example 2.4.24. A Banach algebra A with identity 1 is a Banach Lie algebra in the commutator product [a, b] = ab − ba

(a, b ∈ A).

The set G of invertible elements is open in A and forms a group in the associative multiplication of A, and G is a Banach Lie group with Lie algebra T1 G = A.

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Example 2.4.25. The infinite dimensional analogue of the general linear groups is the group GL(H) of invertible elements in the Banach algebra L(H) of bounded linear operators on a Hilbert space H. The group GL(H) is norm open in L(H) and is a Banach Lie group modelled on L(H) which is a Banach Lie algebra in the commutator product. The exponential map exp : L(H) −→ GL(H) is the usual one: exp A = 1 + A +

A2 + ··· 2!

(A ∈ L(H))

where 1 is the identity operator in L(H). The two-sided ideal K(H) of compact operators in L(H) is a Lie ideal and is the Lie algebra of the Banach Lie group GLc (H) = {A ∈ GL(H) : 1 − A ∈ K(H)}. In the operator norm topology, the group GL2 (H) = {A ∈ GL(H) : 1 − A is Hilbert-Schmidt} is a Banach Lie group and its Lie algebra is the Hilbert space L2 (H) of Hilbert-Schmidt operators, equipped with the commutator product and the Hilbert-Schmidt norm !1/2 kAk2 =

X

kA(eα )k2

(A ∈ L2 (H))

α

where {eα } is an orthonormal basis of H. If H is complex, the unitary group U (H) = {u ∈ GL(H) : uu∗ = u∗ u = 1} is a real Banach Lie group in the norm topology and its Lie algebra is the real Banach space u(H) = {A ∈ L(H) : A + A∗ = 0}

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of skew-hermitian operators, equipped with the commutator product. For a real Hilbert space, the orthogonal group O(H) = {t ∈ GL(H) : tt∗ = t∗ t = 1} is a Banach Lie group with Lie algebra o(H) = {A ∈ L(H) : A + A∗ = 0}, consisting of skew-symmetric operators.

Notes. We refer the readers to the recent book [30] for a comprehensive list of references for JB*-algebras and JB*-triples. One can find more details of JB*-triples in the books [37] and [64], the latter contains a chapter devoted to JBW*-triples and their classification.

2.5

Cartan factors

We introduce in this section a fundamental class of JB*-triples, called the Cartan factors, which are the building blocks of JB*-triples. We will see in Sections 3.3 and 3.8 that finite dimensional Cartan factors are the classifying spaces of finite dimensional bounded symmetric domains. There are six types of Cartan factors, listed below. Type I Type II

L(H, K),

(dim H ≤ dim K)

{z ∈ L(H) : z t = −z},

Type III {z ∈ L(H) : z t = z}, Type IV Type V Type VI

spin factor, M1,2 (O) = {1 × 2 matrices over the Cayley algebra O}, H3 (O) = {3 × 3 hermitian matrices over O},

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where L(H, K) is the JBW*-triple of bounded linear operators between Hilbert spaces H and K, and z t denotes the transpose of z in the JBW*triple L(H) := L(H, H) of bounded linear operators on H. The Jordan triple product in the first three types is given by 1 {x, y, z} = (xy ∗ z + zy ∗ x) 2 where y ∗ denotes the adjoint of y. The transpose on L(H) is defined by a conjugation j : H −→ H, which is a conjugate linear isometry such that j 2 is the identity map. Given z ∈ L(H), the transpose z t is defined by z t = jz ∗ j. It is evident that the Type II and Type III Cartan factors are weak* closed subtriple of L(H) and hence they are JBW*-triple by Lemma 2.4.8. In fact, L(H) and Type III Cartan factors are JBW*-algebra in the special Jordan product x ◦ y = (xy + yx)/2. A spin factor is a JB*-triple V equipped with a complete inner product h·, ·i and a conjugation ∗ : V → V satisfying hx∗ , y ∗ i = hy, xi and {x, y, z} =


1 hx, yiz + hz, yix − hx, z ∗ iy ∗ . (2.53) 2

We have already seen that H3 (O) is a JB*-algebra with Jordan product 1 x · y = (xy + yx) 2 where the product on the right-hand side is the usual matrix product. The Cartan factor M1,2 (O) can be identified as a subtriple of H3 (O). The open unit balls of the finite dimensional Cartan factors are ´ exactly the six types of irreducible bounded symmetric domains in E. Cartan’s classification. This explains the etymology of Cartan factor. The open unit ball of a spin factor is known as a Lie ball. For J = I, II, III, IV, V and VI, a bounded symmetric domain is called a Type J domain if it is biholomorphic to the open unit ball of a Type J Cartan factor.

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We will discuss the classification in more detail in Section 3.8. For now, we study the structures of spin factors, which play an important role in physics (cf. [64]). Let V be a spin factor and let k · kh be the inner product norm of V . The spin factor norm k · k satisfies kak3 = k{a, a, a}k = kha, aia − ha, a∗ ia∗ /2k

(2.54)

where |ha, a∗ i| ≤ kakh ka∗ kh = kak2h . Actually, the two norms k · k and k · kh are equivalent. Indeed, we have 1 3 1 kak2h kak ≤ kak2h kak − |ha, a∗ i|ka∗ k ≤ kak3 ≤ kak2h kak 2 2 2 and kakh ≤

√

2kak ≤

√

3kakh

(a ∈ V ).

(2.55)

In particular, (V, k · k) is a reflexive Banach space and V is a JBW*triple. We denote by Vh the Hilbert space (V, h·, ·i) equipped with the triple product 1 {a, b, c}h = (ha, bic + hc, bia) 2

(a, b, c ∈ V ).

The two Banach spaces V and Vh have very different geometry. Let D and Dh be the open unit balls of V and Vh respectively. By (2.55) and √ (2.58) below, we have Dh ⊂ D ⊂ 2Dh . Lemma 2.5.1. Let V be a spin factor and Vh the underlying Hilbert space. Then (i) the tripotents of V are either minimal or maximal; (ii) the minimal tripotents of V , with dim V > 1, are exactly the extreme points v of the closed unit ball Dh of Vh , satisfying hv, v ∗ i = 0.

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Also, we have v v ∗ = 0 if v is a minimal tripotent. Proof. (i) Let a ∈ V be a tripotent. Then we have 1 a = {a, a, a} = ha, aia − ha, a∗ ia∗ . 2 If ha, a∗ i = 0, then kak2h = ha, ai = 1 and {a, V, a} = CV . Hence a is a minimal tripotent of V and an extreme point of Dh . If ha, a∗ i = 6 0, then a∗ = λa where 2(ha, ai − 1) = 1. |λ| = ha, a∗ i √ √ ¯ and Let e = λa. Then we have a = λe p p √ ¯ ∗ = λλa ¯ = λa = e. e∗ = λa If v ∈ Z0 (a), then {a, a, v} = 0 and 0 = 2{e, e, v} = he, eiv + hv, eie − he, v ∗ ie∗ = he, eiv implies v = 0. Hence a is a maximal tripotent of V . (ii) Given an extreme point v ∈ Dh with hv, v ∗ i = 0, we have hv, vi = 1 and therefore 1 {v, v, v} = hv, viv − hv, v ∗ iv ∗ = v. 2 Hence v is a tripotent of V and is minimal since {v, V, v} = C v. Conversely, given a minimal tripotent v ∈ V , we first show hv, v ∗ i = 0. Since dim V > 1, we can pick v⊥ ∈ V \{0} such that hv, v⊥ i = 0. By minimality, there exists α ∈ C such that 1 1 ∗ ∗ αv = {v, v⊥ , v} = hv, v⊥ iv − hv, v ∗ iv⊥ = − hv, v ∗ iv⊥ 2 2 and hence

1 ∗ αhv, v ∗ i = − hv, v ∗ ihv⊥ , v ∗ i = 0. 2

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It follows that hv, v ∗ i = 0. Further, the proof in (i) reveals that kvk2h = hv, vi = 1, that is, v is an extreme point of Dh . Finally, given a minimal tripotent v and any x ∈ V , we have 1 {v, v ∗ , x} = (hv, v ∗ ix + hx, v ∗ iv − hv, x∗ iv) = 0. 2

Triple orthogonality in V should not be confused with orthogonality with respect to the inner product h·, ·i. However, two triple orthogonal tripotents e and u must be orthogonal in the Hilbert space Vh since he, ui = h{e, e, e}, ui = he, {e, e, u}i = 0. Given two triple orthogonal tripotents e and u in V , we note that u is a scalar multiple of e∗ . Indeed, {e, e, u} = 0 implies he, eiu − he, u∗ ie∗ = 0 and u = he, u∗ ie∗ . It follows that in a spin factor V , there are at most two mutually triple orthogonal tripotents. Let a ∈ V \{0}. By Theorem 2.4.21, the closed subtriple V (a) generated by a identifies with the JB*-triple C0 (Sa ) of continuous functions vanishing on the triple spectrum Sa . Since V (a) is reflexive, Sa must be a finite set in which case, each indicator function χt of {t} ⊂ Sa , where χt (t) = 1 and χt (x) = 0 for x 6= t, is a tripotent in C0 (Sa ) and these indicator functions are mutually triple orthogonal. It follows from the above remark that Sa reduces to a set of at most two points. We therefore infer that there are two mutually orthogonal minimal tripotents e, u ∈ V such that a = αe + βu

(α ≥ β ≥ 0)

(2.56)

and kak = α. This decomposition is called a spectral representation of a. As u is a scalar multiple of e∗ , it can be written as a = α1 e + α2 e∗

(α2 ∈ C, kak = α1 ≥ |α2 |).

(2.57)

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This representation is unique if α1 > |α2 |. We note that α2 can be 0 and kak2h = α12 + |α2 |2 ≥ α12 = kak2 .

(2.58)

Also, {a, a, a} = α1 |α1 |2 e + α2 |α2 |2 e∗ implies that a is a tripotent if, and only if, α1 = α13 and α2 = α2 |α2 |2 . We can express the spin factor norm in terms of the inner product. Given v = α1 e+α2 e∗ in (2.57), we have hv, vi = α12 +|α2 |2 and hv, v ∗ i = 2α1 α2 . This gives hv, vi +

p

hv, vi2 − |hv, v ∗ i|2 = 2α12 = 2kvk2 .

The Lie ball D can therefore be written as p 1 D = {v ∈ V : (hv, vi + hv, vi2 − |hv, v ∗ i|2 ) < 1}. 2

(2.59)

Since |hv, v ∗ i| ≤ hv, vi, one can also represent D in the form D = {v ∈ V : 1 − hv, vi +

1 1 |hv, v ∗ i|2 > 0, |hv, v ∗ i| < 1}. 2 2

Remark 2.5.2. Under the condition (2.53), C is not a spin factor in the usual triple product {a, b, c} = abc, with involution z 7→ z and the standard inner product. However, in some literature, the factor 1/2 is not included in the defining condition (2.53) for a spin factor (e.g. [64]), in which case C becomes a ‘spin factor’ in the usual triple product. If one adopts this definition, some dimension restriction and straightforward scaling would have to be made to our results above. For instance, the fraction 1/2 should be dropped from the description of the Lie ball in (2.59). Let n > 2 and equip Cn with the standard inner product and involution z = (z1 , . . . , zn ) ∈ Cn 7→ z = (z 1 , . . . , z n ) ∈ Cn . If one defines the spin triple product without 1/2: {z, z, z} = 2hz, ziz − hz, ziz

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then Cn is a spin factor in this alternative definition and the open unit ball D has the form D = {z ∈ Cn : 1 − 2hz, zi + |hz, zi|2 > 0, |hz, zi| < 1} which is another common representation of a finite dimensional Type IV domain (cf. [90, 108]). According to (2.57), a maximal tripotent v has the form v = e + αe∗ where e is a minimal tripotent and |α| = 1. It follows that v ∗ = αv √ and kvkh = 2. We note that the above representation of a maximal tripotent is not unique. Lemma 2.5.3. Let a ∈ V \{0} satisfy a + λa∗ = 0 for some |λ| = 1. Then a/kak is a maximal tripotent. Proof. Let a have the spectral decomposition a = α1 e + α2 e ∗ as in (2.57). Then we have 0 = a + λa∗ = (α1 + α2 λ)e + (α2 + α1 λ)e∗ which implies α2 = −α1 λ and a = α1 (e − λe∗ ) where e − λe∗ is a maximal tripotent and α1 = kak. Lemma 2.5.4. Let v ∈ V be a maximal tripotent. Then v v : V −→ V is the identity map and v is a unitary tripotent. Proof. As noted earlier, we have v ∗ = λv for some |λ| = 1. For each a ∈ V , we have v v(a) = {v, v, a} 1 1 1 = hv, via + ha, viv − hv, a∗ iv ∗ 2 2 2 1 λ ∗ = a + ha, viv − ha, v iv = a. 2 2

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Jordan and Lie algebraic structures Hence P1 (v) = 4(v v − (v v)2 ) = 0 and v is unitary. Using the spectral decomposition of v = α1 e + α2 e∗ , we have a

useful expression of the Bergman operator B(v, v). First, the triple orthogonal minimal tripotents e and e∗ give rise to a joint Peirce decomposition of V : V =

M

Vij

0≤i≤j≤2

where each Peirce space Vij is the range of a joint Peirce projection Pij : V −→ V . In fact, we have P00 = P0 (e)P0 (e∗ ) ; P11 = P2 (e) ;

P01 = P0 (e∗ )P1 (e) ;

P12 = P1 (e)P1 (e∗ ) ;

P02 = P0 (e)P1 (e∗ )

P22 = P2 (e∗ )

(cf. [37, p. 38]). For a minimal tripotent e ∈ V , the Peirce projections are given by P0 (e) = h· , e∗ ie∗ ;

P2 (e) = h· , eie

P1 (e) = I − h· , eie − h· , e∗ ie∗ = P1 (e∗ ) and they are self-adjoint operators on the Hilbert space Vh . Hence for any z ∈ V , we have |hz, ei| = kP2 (e)zk ≤ kzk in the spin norm. We note that P2 (e∗ ) = P0 (e) and (P1 (e)v)∗ = P1 (e)(v ∗ )

(v ∈ V ).

(2.60)

Since he, e∗ i = 0, a simple computation gives P00 = P01 = P02 = 0 and P12 = P1 (e). Hence the Bergman operator B(v, v) for v = α1 e + α2 e∗ has the simple form X B(v, v) = (1 − |αi |2 )(1 − |αj |2 )Pij 1≤i≤j≤2 2 2

= (1 − |α1 | ) P2 (e) + (1 − |α1 |2 )(1 − |α2 |2 )P1 (e) + (1 − |α2 |2 )2 P0 (e).

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In particular, we have B(v, v)1/2 (v) = (1 − |α1 |2 )α1 e + (1 − |α2 |2 )α2 e∗ α2 e∗ α1 e + . B(v, v)−1/2 (v) = 1 − |α1 |2 1 − |α2 |2

(2.61) (2.62)

Example 2.5.5. Let A be a JB-algebra with identity 1 such that its complexification A = A + iA is a spin factor. Then A is a real spin factor H ⊕ R1 for some real Hilbert space H. To see this, we note that A is a JB*-algebra with involution a + ib 7→ (a + ib)− = a − ib, and a spin factor with inner product h·, ·i and conjugation ∗. Since 1 is a maximal tripotent, we have 1∗ = λ1 for some λ ∈ C with |λ| = 1, and (2.54) implies h1, 1i = 2. We have the orthogonal decomposition A = H ⊕ R1 where H = {a ∈ A : ha, 1i = 0}. For each a ∈ H, we have 1 a = {1, a, 1} = (h1, ai1 + h1, ai1 − h1, 1∗ ia∗ = −λa∗ 2 and 1 a2 = {a, a, 1} = ha, ai1. 2 For a, b ∈ H, we have ha, bi = hb∗ , a∗ i = h−λb, −λai = hb, ai. It follows that H is a real Hilbert space with the inner product hh·, ·ii := 12 h·, ·i and the Jordan product of A = H ⊕ R1 satisfies (a ⊕ α1)(b ⊕ β1) = αb + βa + αβ1 + ab = (αb + βa) ⊕ (αβ + hha, bii)1. Hence A is a real spin factor.

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Jordan and Lie algebraic structures We end this section with representations of the TKK-Lie algebras

of Cartan factors. It has been shown by Tits [165] that the TKK-Lie algebra of the Cartan factor H3 (O) is the exceptional Lie algebra E7 and therefore the TKK-Lie algebra of M1,2 (O) identifies with a Lie subalgebra of E7 . For Cartan factors of Type I, II, III and IV, the TKK-Lie algebras can be represented as Lie algebras of matrices. Given a C*-algebra A ⊂ L(H), the tensor product M2 (C) ⊗ A identifies with the matrix C*-algebra    a b M2 (A) = : a, b, c, d ∈ A c d  ∗  ∗ ∗ a b a c with the usual matrix product and involution = b∗ d∗ c d (cf. [161, p. 192]). It is a Banach Lie algebra in the Lie brackets [A, B] = AB − BA

(A, B ∈ M2 (A)).

In particular, M2 (L(H)) is a Banach Lie algebra. Given a subset E ⊂ M2 (L(H)), we define E ∗ = {A∗ : A ∈ E}. Theorem 2.5.6. Let V ⊂ L(H) be a Cartan factor of Type I, II, III or IV. Then its TKK Lie algebra L(V ) = V ⊕ V0 ⊕ V is Lie isomorphic to the following Lie subalgebra of M2 (L(H)):  Pn  ∗ x   j=1 aj bj  : x, y, a1 , . . . , an , b1 , . . . , bn ∈ V B(V ) =  P   y∗ − nj=1 b∗j aj which is a TKK Lie algebra with gradings Pn ∗ 0  j=1 aj bj  B(V )0 = P  0 − nj=1 b∗j aj

B(V )−1

   0 x = :x∈V , 0 0

  : aj , bj ∈ V

B(V )1 = B(V )∗−1 .

  

(2.63)

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The TKK involution is given by P   P y ∗ x − j bj a∗j j aj bj 2  7→  θe :  P P ∗ y∗ − j b∗j aj 2x∗ j aj bj

 .

Proof. Let L(V ) = V ⊕ V0 ⊕ V be the TKK Lie algebra of V , with TKK-involution θ, where V is contained in the JBW*-triple L(H) of bounded linear operators on a Hilbert space H. Given a1 , . . . , an and b1 , . . . , bn in V , it has been proved in [48, Theorem 4.4] that the following two conditions are equivalent: (i)

n X

aj

bj = 0

(ii)

j=1

n X j=1

a∗j bj

=

n X

b∗j aj = 0.

j=1

This enables us to define a map Φ : L(V ) → B(V ) by 1 P x ∗ j aj bj 2 2 X Φ(x, aj bj , y) =  P j y∗ − 12 j b∗j aj

 .

Evidently, Φ is a linear bijection. Further, one can verify readily that Φ preserves the Lie multiplication in (2.45) and (2.46). Hence Φ induces the gradings B(V )−1 = Φ(V ) and B(V )0 = Φ(V0 ) in (2.63), as well as B(V )1 = Φ(V ) = B(V )∗−1 . Also, the TKK involution θe of B(V ) is induced from the involution θ of L(V ) via Φ: e θ(Φ(x,

X

aj

bj , y)) := Φ(θ(x,

j

X j

aj

bj , y)) = Φ(y, −

X

bj

aj , x).

j

This completes the proof. Using Theorem 2.5.6, one can derive a matrix representation of TKK-Lie algebras of JB*-triples, shown in [48, Corollary 4.13], where the TKK-Lie algebras of JB*-triples have also been characterised among complex Lie algebras.

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Chapter 3

Bounded symmetric domains 3.1

Algebraic structures of symmetric manifolds

We begin this Chapter by discussing the Jordan and Lie structures of symmetric Banach manifolds which are of fundamental importance. Our goal is to use these structures to show that bounded symmetric domains in Banach spaces can be realised as the open unit balls of JB*-triples. Let M be symmetric Banach manifold modelled on a complex Banach space V and equipped with a compatible tangent norm ν. Then the group Aut(M, ν) of ν-isometries is a real Banach Lie group, by Theorem 1.3.11. In the case of a bounded symmetric domain D in a Banach space, we have Aut(D, ν) = Aut D, by Lemma 1.3.8, where ν is the Carath´eodory tangent norm and the topology of the real Banach Lie group Aut D is that of locally uniform convergence. Write G = Aut(M, ν) and let sp : M −→ M be a symmetry at some point p ∈ M . Then sp is an element of the ν-isometry group G = Aut(M, ν), which is a real Banach Lie group in a topology finer 145

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than the topology of locally uniform convergence, by Corollary 1.3.12. The real Banach Lie algebra g of G is given by g = aut(M, ν) = {X ∈ aut M : exp tX ∈ Aut(M, ν), ∀t ∈ R} where aut M is the set of complete holomorphic vector fields on M . If M is a bounded symmetric domain, then g = aut M . The adjoint representation θ := Ad(sp ) : g −→ g

(3.1)

is the differential dσe at the identity e ∈ G, where σ(g) = sp gsp

(g ∈ G).

Since s2p = e, we have (dσ 2 )e = (dσ)σ(e) ◦ dσe = dσe ◦ dσe = θ2 where (dσ 2 )e is the identity map I : g −→ g and hence θ is an involution. It follows that g decomposes into a direct sum, called the canonical decomposition of g, g=k⊕p

(3.2)

of ±1-eigenspaces of θ: k = {X ∈ g : θ(X) = X},

p = {X ∈ g : θ(X) = −X}

(3.3)

satisfying [k, k] ⊂ k,

[k, p] ⊂ p,

[p, p] ⊂ k.

Since θ is an automorphism of the Lie algebra g, we see that k is a Lie subalgebra of g. Also, [p, p] is an ideal of k. Lemma 3.1.1. In the canonical decomposition (3.2) induced by the symmetry sp at p ∈ M , we have k = {X ∈ g : X(p) = 0} = {X ∈ g : exp tX(p) = p, ∀t ∈ R}.

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Proof. In the notation of (1.7), we have θ = (sp )∗ and exp tθX = sp (exp tX)sp for all t ∈ R. If θX = X, then exp tX = sp exp tXsp and exp tX(p) = sp (exp tX(p)) implies exp tX(p) = p for sufficiently small t since p is an isolated fixed point of sp . It follows that d Xp = exp tX(p) = 0. dt t=0 On the other hand, given Xp = 0, we have d d exp tX(p) = exp(s + t)X(p) dt t=s dt t=0 d = exp sX ◦ exp tX(p) dt t=0 = d(exp sX)p (Xp ) = 0 for all s ∈ R. Therefore exp tX is constant in t and we have exp tX(p) = p for all t ∈ R. In turn, the last condition implies θX = X. Indeed, by Lemma 1.3.14, the differential (dsp )p is minus the identity map on Tp M which gives d(sp (exp tX)sp )p = (dsp )p ◦ d(exp tX)p ◦ (dsp )p = d(exp tX)p and it follows from Cartan’s uniqueness theorem that exp tX = sp (exp tX)sp for all t ∈ R. Hence d d θX(·) = exp tθX(·) = sp (exp tX)sp (·) dt t=0 dt t=0 d = exp tX(·) = X(·). dt t=s

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Bounded symmetric domains It follows from the preceding lemma that the evaluation map X ∈ p 7→ X(p) ∈ V

(3.4)

is a real linear injection. As noted in (1.16), the map X ∈ g 7→ X(p) is surjective and hence the map in (3.4) is a real linear isomorphism. Now we are going to manufacture a Jordan structure in V using the Tits-Kantor-Koecher construction and the evaluation map in (3.4). This reveals the close connection between the Jordan and Lie structures of the symmetric manifold M . First, we define a complex structure J : p −→ p. Given X ∈ p, we have iX(p) ∈ V and by (3.4), iX(p) = JX(p) for a unique JX ∈ p. This defines J which is real linear and −J 2 is the identity map on p. We note that X ∈ g does not imply iX ∈ g and in fact, k is purely real in the sense of the following lemma. Lemma 3.1.2. In the canonical decomposition (3.2), we have k ∩ ik = {0}. Proof. Let Y ∈ k and Y = iZ ∈ ik. We show Y = 0. For s + it ∈ C, the map (exp sY )(exp tZ) = exp(s − it)Y : M −→ M is a biholomorphic isometry and therefore the map F : s + it ∈ C −→ (d exp(s − it)Y )p ∈ L(Tp M ) is a bounded holomorphic map, where L(Tp M ) is the complex Banach space of continuous linear self-maps on Tp M . By Liouville theorem, ϕ ◦ F : C −→ C is constant for each continuous linear functional ϕ on L(Tp M ). Hence F is constant and (d exp Y )p is the identity map. Since exp Y (p) = p, Cartan’s uniqueness theorem implies that exp Y itself is an identity map on M , that is, Y = 0. Nevertheless, we have Jθ = θJ,

[JX, Y ](p) = dYp (JX(p)) − d(JX)p (Y (p)) = i[X, Y ](p)

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for X ∈ p, Y ∈ k since Y (p) = 0. Hence (X ∈ p, Y ∈ k).

J[X, Y ] = [JX, Y ]

Further, J satisfies the condition in the following lemma. A seemingly short derivation of this condition in [37, Lemma 2.5.4] requires some details, which are given below. Lemma 3.1.3. For X, Y ∈ p, we have [JX, JY ] = [X, Y ]. Proof. Let s = sp be the symmetry at p ∈ M . By Lemma 1.3.14, one can find a local chart (V, ϕ, V ) at p with ϕ(p) = 0 ∈ V such that the following diagram commutes: s

V −→   yϕ V

V   yϕ

−id

−→ V

where s(V) = V and we may assume D = ϕ(V) is a bounded domain in V. ∂ In the local chart, write X = hX ∂z , Y = hY

JY = hJY

∂ ∂z ,

∂ ∂z ,

∂ JX = hJX ∂z and

where hX , hY , hJX and hJY are holomorphic functions

from V to V . ∂ Consider a holomorphic vector field Z = hZ ∂z on M as the vector ∂ field Z = hZ ∂z on D = ϕ(V), where the holomorphic map hZ : D −→ V

is given by hZ = hZ ◦ ϕ−1 . If Z ∈ p, then we have θZ = −Z and s ◦ exp tZ ◦ s = exp ts∗ Z = exp tθZ = exp t(−Z)

(t ∈ R)

which gives ϕ ◦ s ◦ exp tZ ◦ s ◦ ϕ−1 = ϕ ◦ exp t(−Z) ◦ ϕ−1

(t ∈ R)

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from the above commutative diagram, where ϕs = −ϕ and sϕ−1 (·) = ϕ−1 (− ·). It follows that − exp tZ ◦ s ◦ ϕ−1 = exp t(−Z) ◦ ϕ−1

(t ∈ R)

and hence Z((ϕ−1 (−v)) = Z(ϕ−1 (v))

(v ∈ D).

Therefore hZ (−v) = hZ ◦ ϕ−1 (−v) = hZ (ϕ−1 (v)) = hZ (v)

(v ∈ D)

and so h0Z (0) = 0. ∂ In particular, for [X, Y ] = h[X,Y ] ∂z with h[X,Y ] = h[X,Y ] ◦ ϕ−1 , we

have, as in Example 1.2.14, h[X,Y ] (v) = h0Y (v)hX (v) − h0X (v)hY (v)

(v ∈ D)

which entails h[X,Y ] (0) = 0 as well as h0[X,Y ] (0) = h00Y (0)hX (0) + h0Y (0)h0X (0) − h00X (0)hY (0) − h0X (0)h0Y (0) = h00Y (0)hX (0) − h00X (0)hY (0). Let Z = [JX, Y ] + [X, JY ] − i[X, Y ] + i[JX, JY ]. Then we have hZ (0) = h[JX,Y ] (0) + h[X,JY ] (0) − ih[X,Y ] (0) + ih[JX,JY ] (0) = 0 and repeating the preceding computation renders h0Z (0) = h0[JX,Y ] (0) + h0[X,JY ] (0) − ih0[X,Y ] (0) + ih0[JX,JY ] (0) = 0. By Lemma 1.1.7, we have hZ = 0 and hence hZ = 0 (by the principle of analytic continuation for M (cf. [168, Theorem 3.1])). Hence Z = 0, which implies [JX, Y ] + [X, JY ] = i[X, Y ] − i[JX, JY ] ∈ k ∩ ik. By Lemma 3.1.2, we conclude [X, Y ] − [JX, JY ] = 0.

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We note from the previous proof that [JX, Y ] = −[X, JY ]

(X, Y ∈ p).

Let pc be the complexification of p and extend J naturally to a complex linear map on pc . Let p+ = {X ∈ pc : JX = iX},

p− = {X ∈ pc : JX = −iX}

(3.5)

so that pc = p+ ⊕ p− and the complexification gc of g is given by gc = p+ ⊕ kc ⊕ p−

(3.6)

where kc is the complexification of k. By Lemma 3.1.3, we have [p+ , p+ ] = [p− , p− ] = 0. It follows that [pc , pc ] = [p+ , p− ] and the complexification of [p, p] ⊕ p is given by p+ ⊕ [p+ , p− ] ⊕ p− which is the canonical part of gc . With the complex structure, p is complex linear isomorphic to p+ via the map ψ : X ∈ p 7→ X − iJX ∈ p+ and hence p+ is complex linear isomorphic to V via the map ϕ : X − iJX ∈ p+ 7→ X(p) ∈ V.

(3.7)

The automorphism θ induces an involutive conjugate linear Lie automorphism σ on gc , given by σ(X + iY ) = θX − iθY

(X + iY ∈ g + ig).

We have σ(p+ ) = p− and [kc , p± ] ⊂ p± .

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Hence gc is a TKK Lie algebra with gradation in (3.6) and TKK involution σ. It follows that p+ has the structure of a Jordan triple, which induces such on V via the linear isomorphism ϕ in (3.7). The Jordan triple product {·, ·, ·}p+ of p+ is given by {X, Y, Z}p+ = [[X, σ(Y )], Z]

(X, Y, Z ∈ p+ ).

Given a, b, c ∈ V with a = X(p), b = Y (p) and c = Z(p) for X, Y, Z ∈ p, we define the triple product on V by {a, b, c} = {X(p), Y (p), Z(p)} := ϕ{ψ(X), ψ(Y ), ψ(Z)}p+ .

(3.8)

The tangent space Tp M is naturally identified with the model space V of M and we have proved the main assertion of the following far-reaching connection between symmetric manifolds and Jordan triples. Theorem 3.1.4. Let M be a symmetric Banach manifold modelled on a complex Banach space V . Then the tangent space Tp M at each p ∈ M carries the structure of a Jordan triple, induced by the symmetry sp . The Jordan triple structures of the tangent spaces obtained from (3.8) by different symmetries are all mutually isomorphic. Proof. We need only show that the Jordan triple structures of any two tangent spaces Tp M and Ta M , induced by the symmetries sp and sa in (3.8) respectively, are isomorphic. By transitivity (cf. Lemma 1.3.16), there is a biholomorphic map ψ : M −→ M such that ψ(a) = p. By uniqueness of the symmetry, we have sa = ψ −1 sp ψ. As before, let θ = Ad(sp ) and θa = Ad(sa ) be the adjoint representations, with corresponding canonical decompositions aut M = k ⊕ p = ka ⊕ pa . Then we have θψ∗ = (sp )∗ ψ∗ = (sp ψ)∗ = (ψsa )∗ = ψ∗ θa

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153

and also, ψ∗ commutes with the complex structures J : p −→ p and Ja : pa −→ pa since J(ψ∗ X)(p) = −iψ∗ X(p) = −idψa (Xa ) = dψa ((Ja X)a ) = ψ∗ (Ja X)(p) for X ∈ pa . It follows that ψ∗ is a graded Lie isomorphism between the TKK Lie algebras gc = p+ ⊕ kc ⊕ p−

and g0c = (pa )+ ⊕ (ka )c ⊕ (pa )−

and hence the Jordan triple (pa )+ is triple isomorphic to p+ , via ψ∗ |(pa )+ .

Notes. Theorem 3.1.4 is due to Kaup [98]. One can show further in this theorem that for each a ∈ V , the box operator a a : V −→ V is hermitian, that is, it has real numerical range. A complex Jordan triple with this property is called a Hermitian Jordan triple system in [98]. Conversely, given such a Hermitian Jordan triple system V , there corresponds a symmetric Banach manifold M , as constructed in [98]. We will not pursue this and its ramifications here although it is a valuable avenue to explore new grounds, but will focus on the special case of bounded symmetric domains in this correspondence. To avoid confusion, the reader should be reminded again that a complex Jordan triple in this book is called a Hermitian Jordan triple in [37].

3.2

Realisation of bounded symmetric domains

In the preceding section, we have shown that the tangent space of a symmetric Banach manifold at each point admits a Jordan triple structure, induced by the symmetry at the point and the Lie structure of the complete holomorphic vector fields on the manifold.

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Bounded symmetric domains Now we proceed to consider a bounded symmetric domain D in a

complex Banach space V , as a symmetric Banach manifold modelled on V . By Theorem 3.1.4, we can equip V with a Jordan triple structure. We aim to show that, under a suitable norm k · ksp , V becomes a JB*triple and D is biholomorphic to the open unit ball of V in this norm. To simplify our task, we make use of a result of Vigu´e in [172, Th´eor`eme 3.4.1] which asserts that every bounded symmetric domain is biholomorphic to a circular balanced bounded symmetric domain. We may and will therefore assume in what follows that D is circular and balanced, and pick the symmetry s0 at the origin 0 ∈ D. Fix D and the symmetry s0 throughout this section, and recall, in this case, the real Banach Lie algebra g := aut D of complete holomorphic vector fields on D decomposes into (±1)-eigenspaces of θ = Ad(s0 ) : aut D −→ aut D, g = aut D = k ⊕ p , and as before, the evaluation map X ∈ p 7→ X(0) ∈ V is a real linear isomorphism. Remark 3.2.1. A biholomorphic map on D of the form sp ◦ s0 , where sp is the symmetry at p ∈ D, is called a transvection. Let P = {sp ◦ s0 : p ∈ D} be the set of all transvections. It can be shown that P = exp p = {exp X : X ∈ p} (cf. [99, Proposition 4.6]) although we shall not make use of this fact in the sequel. There is a complex structure J on p satisfying JX(0) = iX(0) for X ∈ p and the complexification gc = p+ ⊕ kc ⊕ p−

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given in (3.6), is a TKK Lie algebra with involution σ(X + iY ) := θX − iθY for X, Y ∈ g. Hence p+ is a Jordan triple, which induces a Jordan triple structure on V via the complex linear isomorphism X(0) ∈ V 7→ X − iJX ∈ p+

(X ∈ p).

To compute the triple product on V , we adjust (3.8) by adding a factor

1 8

so that, for X, Y, Z ∈ p, 1 {X(0), Y (0), Z(0)} = ϕ{X − iJX, Y − iJ, Z − iJ}p+ 8

where triple product in p+ is given by

= = = =

=

1 {X − iJX, Z − iJY, X − iJX}p+ 8 1 [ [X − iJX, σ(Y − iJY )], X − iJX ] 8 1 [ [X − iJX, θ(Y + iJY )], X − iJX ] 8 1 [ [X − iJX, −Y − iJY ], X − iJX ] 8 1 1 − [ [X, Y ], X ] + [ [JX, Y ], JX ] 4 4 1 1 + i( [ [X, Y ], JX ] + [ [JX, Y ], X ]) 4 4 1 1 − [ [X, Y ], X ] + [ [JX, Y ], JX ] 4 4 1 1 − iJ(− [ [X, Y ], X ] + [ [JX, Y ], JX ]). 4 4

Hence we have 1 1 {X(0), Y (0), X(0)} = − [ [X, Y ], X ](0) + [ [JX, Y ], JX ](0) (3.9) 4 4 for X, Y ∈ p. This defines a Jordan triple product on V by polarization.

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By Lemma 3.1.1, the biholomorphic map exp tX : D −→ D satisfies exp tX(0) = 0 and hence Lemma 1.1.8 implies that exp tX is (the restriction of) a continuous linear map on D, for all t ∈ R. It follows from

d X(·) = exp tX(·) dt t=0

that h is (the restriction of) a continuous linear map on D, and by a slight abuse of notation, we may consider h as a continuous linear map on V . ∂ on D a linear vector field (respecWe call a vector field Z = f ∂z

tively, polynomial vector field) if f : D −→ V is (the restriction of) a linear map (respectively, a polynomial). Let coD be the convex hull of D and B = coD its closed convex hull. Since D contains 0 ∈ V and is bounded, open, circular and balanced, the Minkowski functional | · | : V −→ [0, ∞) defined by |x| = inf{λ > 0 : x ∈ λ(coD)}

(x ∈ V )

(3.10)

is a norm, which is equivalent to the original norm k · k of V , and we have B = {x ∈ V : |x| ≤ 1} as well as D ⊂ {x ∈ V : |v| < 1}. We are going to show that there is another equivalent norm k · ksp on V such that (V, k · ksp ) is a JB*-triple and D = {v ∈ V : kvksp < 1}, thereby achieving our goal in this section. A proof of this has been given in [37, Theorem 2.5.27] for the case when D itself is the open unit ball of V in the norm k · k. In our case, the strategy is to apply the same approach of this proof to the Jordan triple (V, | · |) instead. To avoid undue repetition in what follows, we shall suppress but refer to [37] for some verbatim arguments.

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∂ Returning to the preceding vector field X = h ∂z ∈ k where h is

linear, we have, from (1.6), exp tX(v) = exp th(v) = v + th(v) +

t3 t2 2 h (v) + h3 (v) + · · · 2! 3!

(t ∈ R)

for v ∈ D. By a slight abuse of notation, we write occasionally exp tX(v) for exp th(v), given v ∈ V . Since exp th(D) ⊂ D, we have exp th(B) ⊂ B by linearity and continuity. In other words, | exp th| ≤ 1 for all t ∈ R which, by [20, p. 46], is equivalent to saying that ih : V −→ V is a hermitian linear operator on the Banach space (V, | · |). We recall that a linear operator T in the Banach algebra L(V ) of continuous linear self-maps on V is called hermitian if it has real numerical range. We have shown that each vector field in k is linear. Let us now consider the vector fields in p. For each α ∈ D, there is a unique Xα ∈ p such that Xα (0) = α. Let 1 Yα = (Xα − iJXα ) ∈ p+ 2 where J : p −→ p is the complex structure defined earlier. Then we have Yα (0) = α. Define a holomorphic map F : D −→ D by F (z) = exp Yz (0)

(z ∈ D).

Evidently F (0) = 0. For v ∈ V and sufficiently small t ∈ R, we have tv ∈ D and d d F (0)(v) = F (tv) = exp tYv (0) = Yv (0) = v. dt t=0 dt t=0 0

Again Cartan’s uniqueness theorem implies that F is the identity map, that is, exp Yz (0) = z

(z ∈ D).

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Since [p+ , p+ ] = 0, we have 1 [Yα , Yβ ] = [ Xα − iJXα , Xβ − iJXβ ] = 0 4 for all α, β ∈ D. Therefore the Campbell-Baker-Hausdorff series (cf. [16]) gives (exp tYβ )(z) = (exp Ytβ )(exp Yz )(0) 1 1 = exp(Ytβ + Yz + [Ytβ , Yz ] + ([Ytβ , [Ytβ , Yz ]] − [Yz , [Ytβ , Yz ]]) +· · · )(0) 2 12 = tβ + z. It follows that

d Yβ (z) = exp tYβ (z) = β dt t=0 is a constant vector field. ∂ Given X = h ∂z ∈ p, the vector field [X, Yβ ] ∈ k is linear for all Yβ

which, together with the fact that Yβ has zero derivative, implies that the derivative of h is linear. Hence we must have h(·) = h(0) + p(·) where p is a homogeneous polynomial on V of degree 2. On D, we can write X(·) = X(0) + p(·).

(3.11)

∂ Remark 3.2.2. We have shown that each vector field Z = f ∂z ∈ aut D =

k ⊕ p is a polynomial vector field in which we consider f as a polynomial on V of degree at most 2. To simplify notation, we shall write Z(v) for f (v) and Z 0 (v) for f 0 (v), given v ∈ V , in the next two lemmas. Next, we compute the polynomial p in (3.11). Lemma 3.2.3. Let (V, | · |) be equipped with the Jordan triple product {·, ·, ·} in (3.9). Then in the eigenspace decomposition g = aut D = k⊕p, each X ∈ p is of the form X(z) = X(0) − {z, X(0), z}

(z ∈ D).

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Proof. The Jordan triple product of a = X(0), b = Y (0) and c = Z(0) in V is given by {a, b, c} = {X, Y, Z}p (0). In the proof of Lemma 3.1.3, we have shown that Z 0 (0) = 0 for all Z ∈ p. Let X ∈ p. By (3.11), we have X(·) = a − pa (·) where a = X(0) and pa (v) = Pa (v, v) is a homogeneous polynomial on V of degree 2, Pa being the polar form of pa . Pick any z ∈ D with z = Y (0) and Y (·) = z − pz (·), where pz (v) = Pz (v, v) is a homogeneous polynomial of degree 2. We have [Y, X](v) = X 0 (v)(Y (v)) − Y 0 (v)(X(v)) = 2Pa (v, Y (v)) − 2Pz (v, X(v)). (3.12) Hence [ [Y, X], Y ](0) = −[Y, X]0 (0)(Y (0)) = −2Pa0 (0, Y (0))(Y (0)) + 2Pz0 (0, X(0))(Y (0)) = −2Pa (0, Y 0 (0)(Y (0))) − 2Pa (Y (0), Y (0)) + 2Pz (0, X 0 (0)(Y (0))) + 2Pz (Y (0), X(0)) = −2Pa (z, z) + 2Pz (z, a). Likewise, we have [ [JY, X], JY ](0) = 2Pa (z, z) + 2Pz (z, a) which gives 1 1 {Y, X, Y }p (0) = − [ [Y, X], Y ](0) + [ [JY, X], JY ](0) = Pa (z, z). 4 4

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Therefore X(z) = X(0) − pa (z) = a − Pa (z, z) = a − {z, a, z}.

Remark 3.2.4. Since both Pa and {·, a, ·} above are symmetric bilinear maps, we have Pa (z, w) = {z, a, w}. Also, Pia = −iPa since the triple product is conjugate linear in the middle variable. Lemma 3.2.5. On the Banach space (V, | · |), the operator a b + b a : V −→ V is a hermitian operator for a, b ∈ V . In particular, a a is hermitian. Proof. Let X, Y ∈ p be such that X(0) = a and Y (0) = ib. By (3.12), we have [Y, X](v) = 2Pa (v, Y (v)) − 2Pib (v, X(v)) = 2{v, a, (ib − {v, ib, v})} − 2{v, ia, (a − {v, a, v})} = 2{v, a, ib} − 2{v, ib, a} = 2((ib) a − a (ib))(v).

(3.13)

Since [Y, X] ∈ k, we have shown before that i[Y, X] is a hermitian linear map on V . In other words, i i b a + a b = − (2((ia) a − a (ia))) = − [Y, X] 2 2 is hermitian. Remark 3.2.6. Given a complex symmetric Banach manifold M , the associated Jordan triple constructed in [98] is the tangent space Ta M at a point a ∈ M , with a unique Jordan triple product {·, ·, ·}a satisfying pa = {(α − {z, α, z}a )

∂ : α ∈ Ta M } ∂z

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in the eigenspace decomposition aut M = ka ⊕pa induced by θ = Ad(sa ) [98, (2.9)]. The triple product {·, ·, ·}a is constructed by identifying a with 0 ∈ Ta M via a local chart. In view of Lemma 3.2.3 for the bounded symmetric domain D, the above triple product {·, ·, ·}a associated to a point a ∈ D coincides with the one constructed in (3.9), using the symmetry s0 at 0 ∈ V = T0 D, by uniqueness. The preceding lemma reveals that a a is a bounded linear operi ator on (V, | · |). Let us estimate the norm of a a = − [Y, X]. We 4 recall from (1.15) that the norm k[Y, X]kaut D of the vector field [Y, X] in the Banach Lie algebra aut D is given by sup{|[Y, X](z)| : z ∈ B0 } at some neighbourhood B0 of 0 ∈ D and hence rD ⊂ B0 for some r > 0. Since, by definition of the norm in (1.15), |Z(0)| ≤ kZk for Z ∈ aut D, the evaluation map Z ∈ p 7→ Z(0) ∈ V is a real continuous linear isomorphism. By the open mapping theorem, there is a constant k > 0 such that kZkaut D ≤ k|Z(0)|

(Z ∈ p).

(3.14)

It follows that

i

ka ak = sup ka a(z)k = sup

− 4 [Y, X](z) z∈D z∈D 1 K ≤ k[Y, X]kaut D ≤ kY kaut D kXkaut D 4r 4r k2 K k2 K 2 ≤ |Y (0)||X(0)| = |a| . 4r 4r

(3.15)

Given a = X(0), b = Y (0) and c = Z(0), using similar arguments as above and the formula for the triple product in (3.9), we see that the triple product {·, ·, ·} is continuous: |{a, b, c}| ≤ C|a||b||c|

(3.16)

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where the constant C > 0 is independent of a, b and c. Lemma 3.2.7. Let D be the preceding symmetric domain in the Jordan triple V endowed with the norm | · |. Given a ∈ D\D, the operator I − a a : V −→ V is not invertible, where I is the identity operator on V. Proof. Assume that I − a a is invertible. We deduce a contradiction. For each v ∈ V , there is a unique Xv ∈ p such that Xv (0) = v. Let 1 X v = − [Xv , Xa ] + Xv ∈ k ⊕ p = aut D. 2 By Remark 3.2.2 and (3.13), we have 1 X v (a) = (− [Xv , Xa ] + Xv )(a) = (I − a a)(v). 2 Let F : V −→ V be the holomorphic map F (v) = exp X v (a). Then the differential dF0 is the evaluation map dF0 (v) = X v (a) = (I − a a)(v) which, by assumption, is a linear isomorphism. Hence, by the inverse function theorem, F maps an open neighbourhood of 0 homeomorphically onto an open subset of the image F (V ) = {exp X v (a) : v ∈ V }. This would contradict the fact that the image F (V ) is contained in the boundary ∂D = D\D of D. To see the latter, we make use of the Carath´eodory distance cD on D, introduced in Remark 1.3.10, which is invariant under biholomorphisms. In particular, the biholomorphic map exp X v : D −→ D is an isometry with respect cD . By continuity, we have exp X v (D) ⊂ D. If

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exp X v (a) ∈ D, then we have exp X v (a) = exp X v (z) for some z ∈ D. Let (an ) be a sequence in D such that limn |an − a| = 0. Then we have exp X v (an ) → exp X v (a) = exp X v (z) in D, as n → ∞. By Lemma 3.5.3, shown later, there exists α > 0 such that α|an − z| ≤ cD (an , z)

(n = 1, 2, . . .).

It follows from continuity of cD , also shown in Proposition 3.5.5 later, that 0 ≤

lim α|an − z| ≤ lim cD (an , z)

n→∞

n→∞

= lim cD (exp X v (an ), exp X v (z)) = 0 n→∞

which gives a = z ∈ D and is impossible. Hence F (V ) ⊂ ∂D. This completes the contradiction and the proof. We have already observed that the box operator a a is hermitian and therefore its spectrum σ(a a) must lie in R. We will write σ(a a) < t ∈ R to mean that λ < t for all λ ∈ σ(a a). Lemma 3.2.8. Let D ⊂ V be as in the preceding lemma. We have {a ∈ V : σ(a a) < 1} ⊂ D. Consequently σ(a a) ≤ 0 implies a = 0. Proof. Let σ(a a) < 1. Then the operator I − a a is invertible. We may assume a 6= 0 since 0 ∈ D. Let µ = inf{λ > 0 : a ∈ λD}.

(3.17)

Since D is open, we have µ > 0. Since D is balanced, we would have a ∈ D if µ < 1. Suppose, for contradiction, that µ ≥ 1. Then I − µa

a µ

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is invertible. Since

a µ

∈ D by definition of µ, we must have

Lemma 3.2.7. As D is open, we can find 0
0 and it suffices to show that f (z) = ξ for all z ∈ B(z0 , r). Pick any v ∈ B(z0 , r) and define ϕv : D −→ C by ϕv (λ) = hf (z0 + λ(v − z0 )), ξi

(λ ∈ D).

Then ϕv : D −→ D and ϕv (0) = 1 imply ϕv (D) = {1} by the maximum modulus principle. It follows that f (z0 + λ(v − z0 )) = ξ for all λ ∈ D which gives f (v) = ξ by continuity. Example 3.4.3. Let D be the open unit ball of a (complex) Hilbert space V with inner product h·, ·i. Then its topological boundary is given by ∂D = {v ∈ V : kvk = 1} = {v ∈ V : v is an extreme point of D}. Let e ∈ ∂D. Define a bounded holomorphic function f : D −→ C by f (z) =

1 2 − hz, ei

(z ∈ D).

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It can be seen that f (e) = 1 > |f (z)| for all z ∈ D\{e} and hence sup{|f (z)| : z ∈ ∂D\{e}} < sup{|f (z)| : z ∈ ∂D}. This shows that ∂D is the smallest closed set satisfying sup{|f (z)| : z ∈ ∂D} = sup{|f (z)| : z ∈ D}. Actually, the topological boundary ∂D of a bounded domain D need not be the smallest closed set where the maximum of all bounded holomorphic functions on a neighbourhood of D is achieved. Example 3.4.4. Let D = D × D be the bidisc. Then we have ∂D = (∂D × D) ∪ (D × ∂D). Applying the maximum principle, we see that ∂D × ∂D is the smallest closed subset of ∂D satisfying sup{|f (z)| : z ∈ ∂D × ∂D} = sup{|f (z)| : z ∈ D} for every continuous function f on D, which is holomorphic in D. The notion of Shilov boundary originated from the theory of commutative Banach algebras in relation to an analogue of the maximum principle and integral representations. Given a unital commutative Banach algebra A, the Shilov boundary of A exists and is the smallest closed set S in the maximal ideal space of A such that the Gelfand transform x b of each x ∈ A attains its norm on S (cf. [84, Theorem 4.15.4]). The maximal ideal space, alias spectrum or structure space, identifies with the set Ω of homomorphisms of A onto C, where Ω

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is equipped with the weak* topology w(A∗ , A) and is a weak* closed (hence compact) subset of the closed unit ball of the dual A∗ . Let D be a bounded domain in a complex Banach space V and let Cb (D) be the C*-algebra of bounded complex continuous functions on the norm closure D, with the supremum norm. Let A(D) be the closed subalgebra of Cb (D), consisting of functions which are holomorphic on D. We note that each linear functional ϕ ∈ V ∗ restricts to a function ϕ|D in A(D). The evaluation map x ∈ D 7→ εx ∈ A(D)∗ ,

ε(f ) := f (x)

(f ∈ A(D))

is continuous and bijective onto its image ε(D) = {εx : x ∈ D} ⊂ Ω, which is equipped with the weak* topology w(A(D)∗ , A(D)). We identify D with ε(D) via the evaluation map, but note that ε(D) need not be weak*-closed in Ω for otherwise, D would be compact in the weak topology w(V, V ∗ ). Indeed, if ε(D) is weak* compact, then for any sequence (xn ) in D, the sequence (εxn ) has a subsequence (εxk ) weak* convergent to some εy ∈ ε(D) and hence (ϕ(xk )) converges to ϕ(y) for each ϕ ∈ V ∗ . Let S ⊂ Ω be the Shilov boundary of A(D). For each f ∈ A(D), the norm kfbk of the Gelfand transform fb is given by sup{|ω(f )| : ω ∈ Ω} and we have kf k = sup{|εx (f )| : x ∈ D} = sup{|ω(f )| : ω ∈ Ω} = sup{|ω(f )| : ω ∈ S}

(3.34)

where |ω(f )| ≤ kωkkf k ≤ kf k for all ω ∈ Ω. Identifying D as a subset of Ω, not necessarily weak*-closed, (3.34) implies that the Shilov boundary S is contained in the weak*-closure of D in Ω. However, if D is compact, in which case V is finite dimensional, then the evaluation map x ∈ D 7→ εx ∈ ε(D) is a homeomorphism and

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we have S ⊂ D, which is the smallest (norm) closed set in D where each f ∈ A(D) attains its norm. In fact, S ⊂ ∂D since kf k = sup{|εx (f )| : x ∈ ∂D} for each f ∈ A(D) and ∂D is weak* closed in Ω. Definition 3.4.5. Let D be a bounded symmetric domain in a complex Banach space V , with norm closure D. The Bergman-Shilov boundary of D, if it exists, is the smallest (norm) closed set Σ ⊂ ∂D such that sup{|f (z)| : z ∈ Σ} = sup{|f (z)| : z ∈ D} for all bounded continuous functions f on D, which are holomorphic in D. The discussion leading to this definition reveals that the BergmanShilov boundary of a finite dimensional bounded symmetric domain always exists. The geometry of the Bergman-Shilov boundary for these domains and some other manifolds in several complex variables has been analysed in [116] and [151]. An interesting result in [116] states that a finite dimensional Hermitian symmetric space is of tube type if and only if its dimension is twice of that of its Bergman-Shilov boundary. We have shown in the previous two examples that the BergmanShilov boundary of a Hilbert ball coincides with its topological boundary, whereas the Bergman-Shilov boundary of the bidisc D×D is ∂D×∂D which is properly contained in the topological boundary. In fact, the Bergman-Shilov boundary exists in all finite-rank bounded symmetric domains and admits an appealing Jordan description. Theorem 3.4.6. Let D be a finite-rank bounded symmetric domain, realised as the open unit ball of a JB*-triple V . Then the BergmanShilov boundary of D exists and is exactly the set of maximal tripotents in V .

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Proof. We recall that a tripotent e ∈ V is maximal if its Peirce 0projection P0 (e) vanishes. By continuity of the Jordan triple product, we see that the set of maximal tripotents in V must be closed. We first show that every f ∈ A(D) achieves its norm on the set of maximal tripotents, by extending the proof in [51, Theorem 10] for finite dimensional domains. Let p be the rank of V and let z ∈ D, with spectral decomposition z = α1 e1 + · · · + αp ep where e1 , . . . , ep are mutually orthogonal minimal tripotents in V and 1 ≥ kzk = α1 ≥ · · · ≥ αp ≥ 0. By orthogonality, the closed linear span W of {e1 , . . . , ep } in V is the `∞ -sum W = Ce1 ⊕∞ · · · ⊕∞ Cep and the open unit ball of W is the intersection W ∩ D, which identifies with the polydisc W ∩ D = De1 × · · · × Dep . {z } | p-times

(3.35)

Analogous to Example 3.4.4, ∂De1 × · · · × ∂Dep is the Shilov boundary of De1 × · · · × Dep . We have z ∈ W ∩ D = De1 × · · · × Dep . Hence for any f ∈ A(D), |f (z)| ≤ sup |f | = W ∩D

sup

|f |

∂De1 ×···×∂Dep

where ∂De1 × · · · × ∂Dep is the set {β1 e1 + · · · βp ep : β1 , . . . , βp ∈ ∂D} in the identification (3.35). Each element in this set is a maximal tripotent in V . It follows that sup{|f (z)| : z ∈ D}

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is just the supremum of |f | over the set of maximal tripotents in V . To show that the maximal tripotents form the Bergman-Shilov boundary, let S be a closed subset of ∂D where every f ∈ A(D) attains its norm. We prove that S contains all maximal tripotents in V . Pick a maximal tripotent e ∈ V . Then there are mutually orthogonal minimal tripotents e1 , e2 , . . . , ep in V such that e = e1 + e2 + · · · + ep . Let P2 (ej ) be the Peirce 2-projection of ej for j = 1, . . . , p. By minimality, the Peirce 2-space V2 (ej ) = P2 (ej )(V ) = Cej is one-dimensional. Pp Define a linear map Q : V −→ V by Q = j=1 P2 (ej ). Then we Pp have Q(V ) = j=1 Cej , which can be identified with the Euclidean L p space Cp ≈ j=1 Cej , with {e1 , . . . , ep } as the standard basis. This induces an inner product h·, ·i on Q(V ) and in particular, for Q(v) = λ1 e1 + · · · + λp ep , we have hQ(v), ei =

p X

λj

and he, ei = p.

(3.36)

j=1

For v ∈ D, we have |λj | ≤ 1 for all j and hQ(v), Q(v)i ≤ p. Define a continuous function f : D −→ C by 1 f (z) = 2

  hQ(z), ei 1+ he, ei

which is holomorphic on D and f (e) = 1. Observe that |f (z)| = 1 if and only if | hQ(z),ei he,ei | = 1 since |f (z)| ≤ 1. This in turn is equivalent to p2 = |hQ(z), ei|2 ≤ hQ(z), Q(z)ihe, ei, and equivalent to Q(z) = λe for some |λ| = 1. For such λ, we have | 21 (1 + λ) | = 1 if and only if λ = 1. Hence we conclude that |f (z)| = 1 if and only if z = e and e must reside in S. This completes the proof.

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Remark 3.4.7. We have shown in Theorem 3.3.5 that finite-rank JB*triples are reflexive Banach spaces. Using (3.36), one can endow a finite-rank JB*-triple with a concrete inner product. Maximal tripotents in a JB*-triple V are contained in the topological boundary ∂D of the open unit ball D. They can be characterised by the convex structure of D. A convex subset F of D is called a face if for all x, y ∈ D, the condition tx + (1 − t)y ∈ F for some t ∈ (0, 1) entails x, y ∈ F . A point x ∈ D is a (real) extreme point if and only if {x} is a face of D. A point u ∈ D is called a complex extreme point if whenever v ∈ V satisfies u + λv ∈ D for all λ ∈ D, we have v = 0. Evidently, a real extreme point of D is also a complex extreme point. A point u ∈ D is called a holomorphic extreme point if for every open neighbourhood Ω ⊂ C of 0 and holomorphic map f : Ω −→ D ⊂ V satisfying f (0) = u, we have f 0 (0) = 0. Plainly, a holomorphic extreme point is also a complex extreme point. Theorem 3.4.8. Let D be the open unit ball of a JB*-triple V and let u ∈ V . The following conditions are equivalent. (i) u is a (real) extreme point of D. (ii) u is a complex extreme point of D. (iii) u is a maximal tripotent. (iv) u is a holomorphic extreme point of D. Proof. (ii) ⇒ (iii). The closed subtriple V (u) generated by u is triple isomorphic to an abelian C*-algebra A in which the complex extreme points of the closed unit ball are partial isometries. It follows that u is a tripotent. One needs to show that P0 (u)V = {0}. We first observe that {z, u, z} = 0 for z ∈ P0 (u)V , by Peirce multiplication rule.

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Let X(·) = u − {·, u, ·} be the unique complete holomorphic vector field on D such that X(0) = u. One can verify that, for z ∈ P0 (u)V , the flow gt (z) = (tanh t)u + z solves the differential equation dgt (z) = X(gt (z)). dt Hence D ⊃ exp tX(P0 (u)V ∩ D) = (tanh t)u + (P0 (u)V ∩ D). Letting t → ∞, we see that u + (P0 (u)V ∩ D) ⊂ D which implies P0 (u)V = {0}. (iii) ⇒ (iv). For v ∈ V , we write v = v1 + v2 ∈ V1 (u) ⊕ V2 (u) for its Peirce decomposition. Let Ω be an open neighbourhood of 0 ∈ C and f : Ω −→ D a holomorphic map satisfying f (0) = u. Let X be the unique vector field on D such that X(0) = u. For each t ∈ R, write gt = exp(−tX). We have gt (u) = u and gt0 (u)(v1 + v2 ) = et v1 + e2t v2 . Observe that {gt ◦ f : t ∈ R} is a family of holomorphic maps from Ω to V , uniformly bounded and gt ◦ f (0) = u. By the Cauchy inequality (1.2), {gt0 (u)(f 0 (0)) : t ∈ R} is bounded in V and hence f 0 (0) = 0. (iv) ⇒ (i). By (iv), u is a complex extreme point of D and hence a maximal tripotent by (ii) ⇒ (iii). Let u + λv ∈ D for all λ ∈ [−1, 1]. We show v = 0 to complete the proof. Let v = v1 + v2 ∈ P1 (u)V ⊕ P2 (u)V be the Peirce decomposition of v. By Proposition 3.2.31, we have u + λv2 ∈ D

and u + λeit v1 + λv2 ∈ D

(t ∈ R).

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is a complex extreme point of its closed unit ball. By the result [24, Lemma 4.1] for JB*-algebras, u is an extreme point of the ball and hence v2 = 0. It follows that v1 = 0 also. It follows from the preceding result and the Krein-Milman theorem that every JBW*-triple, which is a dual Banach space, contains a maximal tripotent. Example 3.4.9. For the open unit ball D of a Hilbert (V, h·, ·i), the topological boundary ∂D is exactly the set of extreme points of the closed ball D. Given a ∈ V \{0}, the element u = a/kak is a maximal tripotent and therefore V = V1 (u) ⊕ V2 (u), where the Peirce spaces are mutually orthogonal by associativity of the inner product shown in (2.50): hx, yi = h2{u, u, x}, yi = 2hx, {u, u, y}i = 2hx, yi (x ∈ V1 (u), y ∈ V2 (u)). It follows from Example 3.2.29 that  (1 − kak2 )v (v ∈ V1 (u)) B(a, a)(v) = (1 − kak2 )2 v (v ∈ V2 (u)). To end this section, we discuss the boundary components of a bounded symmetric domain. In finite dimensions, a theory of boundary components has been developed in [176], where the boundary components in ∂D of a finite dimensional bounded symmetric domain D, contained in its compact dual X via the Borel embedding, are shown to be the orbits in X of the identity component (Aut D)0 of the automorphism group Aut D and they are classified using Lie groups. We are going to describe the boundary components of arbitrary bounded symmetric domains in terms of the Jordan structure of the underlying JB*-triples.

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Let U be a convex domain in a complex Banach space V . A holomorphic arc in the closure U is a holomorphic map γ : D −→ U . Definition 3.4.10. A subset Γ ⊂ U is called a holomorphic boundary component, or simply, a boundary component of U if the following conditions are satisfied: (i) Γ 6= ∅; (ii) for each holomorphic arc γ in U , either γ(D) ⊂ Γ or γ(D) ⊂ U \Γ; (iii) Γ is minimal with respect to (i) and (ii). Two boundary components are either equal or disjoint. The interior U is the unique open boundary component of U , all others are contained in the boundary ∂U [102]. Of interest are the boundary components in ∂U . For each a ∈ U , we denote by Γa the boundary component containing a. Lemma 3.4.11. Let D be a domain in a complex Banach space and h : D −→ U a holomorphic map. Then the image h(D) is contained entirely in one boundary component of U . Proof. Let z0 ∈ D. Then a := h(z0 ) is contained in the boundary Γa ⊂ U . We show that h(D) ⊂ Γa . Let r > 0 so that the closed ball B(z0 , r) centred at z0 is contained in D. For each v ∈ B(z0 , r)\{z0 }, the holomorphic arc γ : D −→ U defined by   λr(v − z0 ) γ(λ) = h z0 + kv − z0 k

(λ ∈ D)

0k satisfies γ(0) = a and γ( kv−z ) = h(v). Hence h(v) ∈ Γa . This proves r

h(B(z0 , r)) ⊂ Γa .

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B(b, s) ⊂ D for some s > 0, which intersects B(z0 , r). Hence h(B(b, s) ∩ B(z0 , r)) ⊂ Γh(b) ∩ Γa and h(b) ∈ Γh(b) = Γa . Let z ∈ D\B(z0 , r). We show h(z) ∈ Γa to conclude the proof. By connectedness, there is a continuous path ϕ : [0, 1] −→ D such that ϕ(1) = z and ϕ(0) = b is a boundary point of B(z0 , r). By compactness, there are a finite number of distinct points t0 , t1 , . . . , tk in [0, 1] such that ϕ[0, 1] is covered by the open balls B(ϕ(t0 ), r0 ), . . . , B(ϕ(tk ), r0 ) in D, with b ∈ B(ϕ(t0 ), r0 ) say. As before, h(b) ∈ Γa implies h(B(ϕ(t0 ), r0 )) ⊂ Γh(ϕ(t0 )) and Γh(ϕ(t0 )) ∩ Γa 6= ∅, which gives Γh(ϕ(t0 )) = Γa . Since ϕ[0, 1] is connected, there exists p ∈ [0, 1] such that kϕ(p) − ϕ(t0 )k = r0 , that is, ϕ(p) ∈ ∂B(ϕ(t0 ), r0 ). We have ϕ(p) ∈ B(ϕ(t1 ), r1 ), say. Again, this implies h(B(ϕ(t1 ), r1 )) ⊂ Γa since h(ϕ(p)) ∈ Γa . Repeat the above arguments, we conclude that h(B(ϕ(tj ), rj )) ⊂ Γa for j = 0, 1, . . . , k and in particular h(z) ∈ Γa . In Definition 3.4.10, we call the set Γ an affine boundary component of U if condition (ii) is satisfied by all affine maps α : D −→ U instead of holomorphic arcs. Lemma 3.4.12. For the open unit ball U of a Banach space, the holomorphic and affine boundary components coincide. Proof. We follow the proof in [102, 4.2]. One only needs to show that each affine boundary component Γ of U is a holomorphic boundary component. Let γ : D −→ U be a holomorphic arc such that γ(D) ∩ Γ 6= ∅. We show γ −1 (Γ) = D. Fix a ∈ γ −1 (Γ) and define f : D −→ U by   λ−a f (λ) = γ aλ − 1

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where f (0) = γ(a). By [63, Lemma III.1.2], a complex number ζ satisfies f (0) + ζ(f (λ) − f (0)) ∈ U if and only if 2|ζλ| ≤ 1 − |λ|. In particular, we have f (λ) ∈ Γ if |λ| ≤ 1/3. Equivalently, we have a−ζ a−ζ γ(ζ) = f ( 1−aζ ) ∈ Γ if | 1−aζ | ≤ 1/3. This implies γ −1 (Γ) is open

and closed in D. Theorem 3.4.13. Let D be a bounded symmetric domain realised as the open unit ball of a JB*-triple V . Given a tripotent e ∈ ∂D, the boundary component containing e is the convex set Γe = e + (V0 (e) ∩ D) ⊂ ∂D, where V0 (e) is the Peirce 0-space of e. If D is of finite rank, then all boundary components in ∂D are of this form. Proof. By Corollary 3.2.24, the convex set e + (V0 (e) ∩ D) lies in ∂D. For each v ∈ V0 (e) with 0 < kvk < 1, we see that e + v ∈ Γe by considering the holomorphic arc γ : λ ∈ D 7→ e + λv/kvk. This shows e + (V0 (e) ∩ D) ⊂ Γe . To show the reverse inclusion, let x ∈ Γe and v = x−e. We need to show v ∈ V0 (e)∩D. Let v = v2 +v1 +v0 ∈ P2 (e)V ⊕P1 (e)V ⊕P0 (e)V be the Peirce decomposition induced by the tripotent e. By the preceding lemma, Γe is also an affine boundary component. Hence there is an open neighbourhood U of 0 ∈ C such that the image of the affine map f : λ ∈ U 7→ e + λv ∈ D is contained in Γe . For each s ∈ (−1, 1), the extended M¨obius transformation gse : D −→ D is biholomorphic (cf. Remark 3.2.26) and satisfies gse (e) = e

and gse (Γe ) = Γe .

Observe that sup{kgse ◦ f (λ)k : λ ∈ U } ≤ 1 for all s ∈ (−1, 1) and by Cauchy inequality (1.2), the family {(gse ◦ f )0 (0) : s ∈ (−1, 1)} is uniformly bounded. From (3.26), it can be seen that     1 − s 1/2 1−s 0 0 (gse ◦f ) (0) = gse (e)v = v2 + v1 +v0 1+s 1+s

(−1 < s < 1).

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Since lim (1 + s) = 0, we must have v2 = v1 = 0 by uniform bounds↓−1

edness. Hence v = v0 ∈ V0 (e) and kvk ≤ ke + vk = kxk = 1. Since kvk = 1 is impossible, we have v ∈ V0 (e) ∩ D as required. Finally, if D is of finite rank, then each a ∈ ∂D has a spectral decomposition a = e1 + α1 e2 + · · · + αk ek where α1 e2 + · · · + αk ek ∈ V0 (e1 ) ∩ D and hence Γa = e1 + (V0 (e1 ) ∩ D).

Remark 3.4.14. The norm closure Γe = e + (V0 (e) ∩ D) is a face of D. To see this, let x, y ∈ D with tx + (1 − t)y ∈ e + (V0 (e) ∩ D) for some t ∈ (0, 1). Consider the Peirce decompositions x = x2 + x1 + x0 ∈ V2 (e)⊕V1 (e)⊕V0 (e) and y = y2 +y1 +y0 . There is a point v ∈ V0 (e)∩D such that e + v = tx + (1 − t)y = (tx2 + (1 − t)y2 ) + (tx1 + (1 − t)y1 ) + (tx0 + (1 − t)y0 ) ∈ V2 (e) ⊕ V1 (e) ⊕ V0 (e) which gives e = tx2 + (1 − t)y2 . Since kx2 k, ky2 k ≤ 1 and e, being the identity of the JB*-algebra V2 (e) (cf. Example 2.4.18), is an extreme point of the closed unit ball of V2 (e), we have x2 = e = y2 . As 1 = ktx + (1 − t)y)k ≤ tkxk + (1 − t)kyk ≤ 1, we must have kxk = kyk = 1. It follows from Proposition 3.2.31 that, for all θ ∈ R, ke + eiθ/2 x1 + eiθ x0 k = kP2 (e)(x) + eiθ/2 P1 (e)(x) + eiθ P0 (e)x)k = k exp −iθ(e e)(x)k = kxk = 1 and by the maximum principle, the map γ : λ ∈ D 7→ e + λ1/2 x1 + λx0 ∈ V is a holomorphic arc in D. Since γ(0) = e ∈ e + V0 (e) ∩ D, we have   1 1 1 e + x1 + x0 = γ ∈ e + V0 (e) ∩ D 2 4 4

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which implies x1 = 0. Likewise y1 = 0 and hence x, y ∈ e + V0 (e) ∩ D. Example 3.4.15. Let D be the open unit ball of a JB*-triple V . Given a tripotent e ∈ ∂D, the open unit ball De of the Peirce 0-space V0 (e) equals V0 (e) ∩ D. Since the Peirce projection P0 (e) is contractive, we see that De = P0 (e)(D) and Γe = e + De = e + P0 (e)(D). Also, the boundary ∂Γe of Γe equals e + ∂De . Each tripotent c in V0 (e) is orthogonal to e and its Peirce 0-space in V0 (e) is the eigenspace (V0 (e))0 (c) = {v ∈ V0 (e) : (c c)(v) = 0} and we have (V0 (e))0 (c) = V0 (e + c). Indeed, considering V0 (e + c) in terms of the joint Peirce spaces induced by {e, c}, we have V0 (e + c) = {v ∈ V : (e e + c c)(v) = 0} = V00 (e, c) = V0 (e) ∩ V0 (c) = (V0 (e))0 (c). Hence the boundary component in ∂De containing the tripotent c ∈ V0 (e) is of the form c + (V0 (e))0 (c) = c + V0 (e + c), and the boundary component of ∂Γe containing e+c is of the form e+c+V0 (e+c) = Γe+c , which is also a boundary component of ∂D. Example 3.4.16. For a Lie ball D in a spin factor V , introduced in Section 2.5, we have a very simple description of the boundary components of D. We first note that Ke = {e} if e is a maximal tripotent of V . For a minimal tripotent e, we have shown P0 (e) = h·, ·ie∗ which gives V0 (e) = Ce∗ and hence Ke = e + De∗ . Together with D, these are all the boundary components of D.

Notes. Lemma 3.4.2 has been shown in [47]. For finite dimensional bounded symmetric domains, Theorem 3.4.6 and Theorem 3.4.8 have

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been proved in [125, Theorem 6.5]. A list of Shilov boundaries of finite dimensional tube type bounded symmetric domains in terms of Lie groups has been appended to [51]. Theorem 3.4.13 is due to Kaup and Sauter [102]. It has also been proved by Loos [125, Theorem 6.3] for finite dimensional bounded symmetric domains.

3.5

Invariant metrics, Schwarz lemma and dynamics

It is impossible to overestimate the fundamental importance of the Poincar´e metric on the open unit disc D, and its higher dimensional generalisation, the Bergman metric, in unifying function theory and geometry for complex domains (e.g. [71]). The fact that these metrics are invariant under biholomorphic maps (and contracted by holomorphic maps) on bounded domains means that geometric invariants are preserved by biholomorphic maps which in turn, provides a powerful tool in studying automorphisms and holomorphic maps on these domains. The initial ideas go back to the Schwarz lemma and the Schwarz-Pick lemma, which can be interpreted geometrically via the Poincar´e metric. In finite dimensions, the classical invariant metrics on complex domains are well documented in literature. Our object in this section is to study two classical invariant metrics on infinite dimensional domains and some variants of the Schwarz lemma on bounded symmetric domains as well as related results in holomorphic dynamics. The Bergman metric is not available in infinite dimension. Instead, we consider the most common Carath´eodory and Kobayashi metrics on domains in complex Banach spaces. Let D be a domain in a complex Banach space V . For p ∈ D and v ∈ V , the (infinitesimal) Carath´eodory pseudo-metric of v at p is actually the Carath´eodory tangent semi-norm

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defined in Section 1.3, namely, CD (p, v) = sup{|f 0 (p)(v)| : f ∈ H(D, D) and f (p) = 0} where H(M, D) denotes the set of all holomorphic maps from M to D. If D is a bounded domain, then CD (p, v) is a compatible tangent norm, as shown in Example 1.3.6, in which case CD (p, v) is called the (infinitesimal) Carath´eodory metric, or Carath´eodory (differential) metric, on D. On a bounded domain D in a complex Banach space V , the Carath´eodory differential metric is an invariant metric, that is, it is preserved by biholomorphic maps g : D −→ D as follows: CD (g(p), dgp (v)) = CD (p, v)

(p ∈ D, v ∈ V ).

This has been shown in Example 1.3.5. Likewise, a distance function d on a domain D, i.e. a metric d : D × D −→ [0, ∞), is called an invariant distance if d(g(x), g(y)) = d(x, y) for all biholomorphic maps g : D −→ D. To avoid confusion with differential metrics on tangent bundles, a pseudo-metric d : D × D −→ [0, ∞) will be called a pseudodistance. We have also shown in Example 1.3.9 that the Carath´eodory metric on the complex unit disc D coincides with the Poincar´e metric and the integrated distance dCD of CD (p, v), given in (1.14), is the Poincar´e distance on D, which we shall denote by ρ in this section. The Carath´eodory pseudo-distance cD on a domain D in a complex Banach space V is defined by cD (z, w) := sup{ρ(f (z), f (w)) : f ∈ H(D, D)}

(z, w ∈ D).

It is called the Carath´eodory distance if cD (z, w) > 0 whenever z 6= w, which is the case if D is bounded. We note that cD is the Poincar´e distance ρ. The Carath´eodory distance is an invariant distance as holomorphic maps between domains contract (i.e. decrease) the Carath´eodory

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pseudo-distance. By definition, the Carath´eodory pseudo-distance is the smallest pseudo-distance on a domain D for which every holomorphic map f : D −→ D is a contraction (i.e. distance-decreasing). Lemma 3.5.1. Let B be the open unit ball of a complex Banach space V . Then we have cB (0, v) = ρ(0, kvk) = tanh−1 kvk for all v ∈ B. Proof. Let v ∈ B\{0} and define a holomorphic map ζ : z ∈ D 7→

zv kvk

∈

B. Then for each f ∈ H(B, D), we have ρ(f (0), f (v)) = ρ(f ◦ ζ(0), f ◦ ζ(kvk) ≤ ρ(0, kvk). On the other hand, let ϕ ∈ V ∗ be a continuous linear functional such that kϕk ≤ 1 and ϕ(v) = kvk. Then ϕ ∈ H(B, D) and ρ(ϕ(0), ϕ(v)) = ρ(0, kvk).

Given that the integrated distance dCD of the Carath´eodory metric CD (p, v) on D coincides with the Poincar´e distance, and also identical with the Carath´eodory distance cD defined above, one may ask if this is also true for all bounded domains D. This need not be the case even in finite dimensions [14] although it can be seen readily that dCD ≥ cD . We now introduce another invariant distance on complex domains. Let D be a domain in a complex Banach space V . Given two points z, w ∈ D, choose  z0 , z1 , . . . , zk ∈ D such that z0 = z and zk = w;  a1 , . . . , ak and b1 , . . . , bk in D and  holomorphic maps fj : D −→ D satisfying fj (aj ) = zj−1 and fj (bj ) = zj for j = 1, . . . , k.

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Define kD (z, w) = inf{ρ(a1 , b1 ) + · · · + ρ(ak , bk ) : a1 , . . . , ak , b1 , . . . , bk ∈ D} where the infimum is taken over all possible choices above. It can be verified readily that kD is a pseudo-distance on D. We call kD the Kobayashi pseudo-distance on D. It is called the Kobayashi distance on D if kD (z, w) = 0 implies z = w. Evidently, holomorphic maps between domains are contractions with respect to the Kobayashi pseudodistances on these domains and hence Kobayashi distance is an invariant distance. Further, the Kobayashi pseudo-distance is the largest pseudodistance on D for which every holomorphic map f : D −→ D is a contraction. In particular, we have cD (z, w) ≤ kD (z, w)

(z, w ∈ D).

Given two domains D and D0 with D ⊂ D0 , we have cD0 ≤ cD and kD0 ≤ kD . Lemma 3.5.2. On the complex open unit disc D, the Kobayashi distance kD coincides with the Poincar´e distance ρ. Proof. This follows from the fact that a holomorphic map f : D −→ D is distance-decreasing with respect to ρ. Analogous to Lemma 3.5.1, we also have kB (0, v) = ρ(0, kvk)

(3.37)

for all v in the open unit ball B of a complex Banach space. Let B(p, r) be the open ball centred at p ∈ V of radius r > 0. It follows from invariance that cB(p,r) (p, x) = kB(p,r) (p, x) = ρ(0, kx − pk/r) for all x ∈ B(p, r).

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Lemma 3.5.3. Let D be a bounded domain in a complex Banach space V . Then for each p ∈ D, there exists α > 0 such that αkx − pk ≤ cD (x, p)

(x ∈ D).

Proof. Let B(p, R) is the open ball centred at p of radius R > 0 such that D ⊂ B(p, R). Then we have cD (x, p) ≥ cB(p,R) (x, p) = ρ(0, kx − pk/R) ≥ kx − pk/R for all x ∈ D. Lemma 3.5.4. Let D be a domain in a complex Banach space V . Let p ∈ D with B(p, r) ⊂ D, where B(p, r) is the open ball centred at p of radius r > 0. Then for 0 < s < r, there exists β > 0 such that kD (x, p) ≤ βkx − pk

(x ∈ B(p, s)).

Proof. We have kD (x, p) ≤ kB(p,r) (x, p) = ρ(0, kx − pk/r) = tanh−1 (kx − pk/r) for x ∈ D. Let 0 < s < r so that s/r < 1. Since tanh−1 (0) = 0 and the derivative of tanh−1 is bounded on [0, s/r], one can find a constant c > 0 such that tanh−1 t ≤ ct for t ∈ [0, s/r]. It follows that c tanh−1 (kx − pk/r) ≤ kx − pk r for all x ∈ B(p, s). Both pseudo-distances cD and kD are continuous functions. Proposition 3.5.5. Let D be a domain in a complex Banach space V . Then both functions cD : D × D −→ [0, ∞) and kD : D × D −→ [0, ∞) are continuous.

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Proof. By Lemma 3.5.4, the functions x ∈ D 7→ cD (x, q) ∈ [0, ∞) and x ∈ D 7→ kD (x, q) ∈ [0, ∞) are continuous, for each q ∈ D. The joint continuity of cD and kD on D × D follows from the inequalities |cD (x, y) − cD (a, b)| ≤ cD (x, a) + cD (y, b) and |kD (x, y) − kD (a, b)| ≤ kD (x, a) + kD (y, b). Theorem 3.5.6. Let D be a bounded domain in a complex Banach space V . Then the Carath´eodory pseudo-distance cD and the Kobayashi pseudo-distance kD are equivalent to the norm-distance on any closed ball strictly contained in D. In particular, cD and kD are distances on D. Proof. This follows from Lemmas 3.5.3 and 3.5.4. A finite dimensional domain is called hyperbolic in [108] if its Kobayashi pseudo-distance is a distance, in which case it can be shown that the relative topology of the domain is equivalent to the one induced by the Kobayashi distance (cf. [63, Proposition IV.2.3]). According to the following definition, all bounded domains in Banach spaces are hyperbolic. Definition 3.5.7. A domain D in a complex Banach space V is called hyperbolic if the Kobayashi pseudo-distance kD is a distance and the topology defined by kD is equivalent to the relative topology of D in V . Example 3.5.8. Let Dk = D × · · · × D be the k-dimensional polydisc. Then the Carath´eodory and Kobayashi distances on Dk are given by cD ((z1 , . . . , zk ), (w1 , . . . , wk )) = kD ((z1 , . . . , zk ), (w1 , . . . , wk )) = max{ρ(z1 , w1 ), . . . , ρ(zk , wk )} which differs from the integrated distance of the Bergman metric for k > 1. We refer to [108] for a derivation of this formula, from which

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one can deduce that cM = kM for a finite dimensional bounded symmetric domain M (see [108, Chap. IV, Example 2]). In fact, we have the following general result. Theorem 3.5.9. Let D be a bounded symmetric domain. Then we have cD (x, y) = kD (x, y) = tanh−1 kg−x (y)k

(x, y ∈ D)

where g−x is the M¨ obius transformation induced by −x. Given a Cartesian product D = D1 × · · · × Dk of bounded symmetric domains, we have kD ((x1 , . . . , xk ), (y1 , . . . , yk )) = max{kD1 (x1 , y1 ), . . . , kDk (xk , yk )}. Proof. The domain D can be realized as the open unit ball of a JB*triple. Given x, y ∈ D, the M¨obius transformation g−x : D −→ D satisfies g−x (x) = 0. Hence by Lemma 3.5.1 and (3.37), we have cD (x, y) = cD (g−x (x), g−x (y)) = cD (0, g−x (y)) = kD (0, g−x (y)) = kD (x, y) where kD (0, g−x (y)) = ρ(0, kg−x (y))k) = tanh−1 kg−x (y))k. For j = 1, . . . , k, the bounded symmetric domain Dj identifies with the open unit ball of a JB*-triple Vj and the Cartesian product D = D1 × · · · × Dk identifies with the open unit ball of the `∞ -sum V1 ⊕`∞ · · · ⊕`∞ Vk . By the preceding result just proved and Example 3.2.27, we have kD ((x1 , . . . , xk ), (y1 , . . . , yk )) = max{kD1 (x1 , y1 ), . . . , kDk (xk , yk )}.

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Corollary 3.5.10. On a bounded symmetric D, realised as the open unit ball of a JB*-triple, we have cD (αx + (1 − α)y, αz + (1 − α)w) ≤ max{cD (x, z), cD (y, w)} for 0 < α < 1 and x, y, z, w ∈ D. The same inequality holds for kD . Proof. Both the Carath´eodory and Kobayashi distances are contracted by the holomorphic map h : (x, y) ∈ D2 7→ αx + (1 − α)y ∈ D.

Although the Carath´eodory distance cD need not be the integrated distance of the Carath´eodory differential metric, the Kobayashi distance kD , on the other hand, is indeed the integrated form of the Kobayashi differential metric. Let D be a domain in a complex Banach space V . The (infinitesimal) Kobayashi pseudo-metric is defined by KD (p, v) = inf{|α| : ∃ holomorphic f : D → D, f (0) = p, f 0 (0)(α) = v} for p ∈ D and v ∈ V . It is clear that KD (p, v) is a tangent semi-norm on the tangent bundle T D, as defined in Section 1.3. If KD (p, v) is a compatible tangent norm, it will be called the (infinitesimal) Kobayashi metric, or Kobayashi (differential) metric, on D. One can verify readily that the Kobayashi pseudo-metric is an invariant metric and CD (p, v) ≤ KD (p, v). Lemma 3.5.11. Let D be the open unit ball of a Banach space V . Then we have CD (0, v) = KD (0, v) = kvk

(v ∈ V ).

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Moreover, if V is a JB*-triple, then 0 CD (p, v) = KD (p, v) = kg−p (p)(v)k = kB(p, p)−1/2 (v)k

for (p, v) ∈ D × V , where g−p is the M¨ obius transformation induced by p and B(p, p) is the Bergman operator. Proof. The first assertion has been proved in [63, Lemma V.1.5]. Let V be a JB*-triple. Then we have 0 0 0 KD (p, v) = KD (g−p (p), g−p (p)v) = KD (0, g−p (p)v) = kg−p (p)vk

which is equal to kB(p, p)−1/2 (v)k by Lemma 3.2.25. Remark 3.5.12. We see from the preceding result that KD (p, v) is a compatible tangent norm on a bounded symmetric domain D. Example 3.5.13. Let D be a finite dimensional bounded domain and let hp (u, v) be the Bergman metric on D. It has been shown in [74] that CD (p, v)2 ≤ hp (v, v) for p ∈ D. If further, D is a symmetric domain realised as the open unit ball of a JB*-triple V , then we have h0 (u, v) = 2Trace(u v)

(u, v ∈ V )

by [125, Theorem 2.10]. Let g−p be the M¨obius transformation induced by −p ∈ D. By Lemma 1.3.4, we have 0 0 0 0 hp (v, v) = hg−p(p) (g−p (p)v, g−p (p)v) = h0 (g−p (p)v, g−p (p)v)

and hence 0 0 0 0 hp (v, v) ≤ 2Trace(g−p (p)v g−p (p)v) ≤ 2nkg−p (p)v g−p (p)vk.

Therefore we have 0 KD (p, v)2 = CD (p, v)2 ≤ hp (v, v) ≤ 2nkg−p (p)vk2 = 2nKD (p, v)2 .

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Let D be a domain in a complex Banach space V and let dKD (a, b) be the integrated pseudo-distance of KD , as defined in (1.13): dKD (a, b) = inf {`(γ) : γ(0) = a, γ(1) = b} γ

where γ : [0, 1] −→ D is a piecewise smooth curve. It has been shown by Royden [152] that dKD coincides with the Kobayashi pseudo-distance kD in the case of dim V < ∞. This is also true in infinite dimension, which has been proved in [63, Theorem V.4.1]. Theorem 3.5.14. Let D be a domain in a complex Banach space V and let dKD be the integrated form of the Kobayashi pseudo-metric KD on D. Then we have dKD = kD . We now turn to the question of completeness of the distances cD and kD . Lemma 3.5.15. Let D be a bounded domain in a complex Banach space V . If the Carath´eodory distance cD is complete on D, then so is the Kobayashi distance kD . The converse is false. Proof. Let (xn ) be a Cauchy sequence with respect to kD . Then it is also a cD -Cauchy sequence since cD ≤ kD . By assumption, (xn ) converges to some point a ∈ D with respect to the distance cD , and hence converges to a in the norm topology of D, by Theorem 3.5.6. Now continuity of kD gives limn→∞ kD (xn , a) = 0. For the converse, we note that the Kobayashi distance dD\{0} on the punctured disc D\{0} is complete while the Carath´eodory distance cD\{0} is not (cf. [108, p. 56]). In finite dimensions, it has been shown in [89] that a bounded domain D is a domain of holomorphy if (D, cD ) is complete. This useful result can be extended to infinite dimension. We first recall

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that an open set D in a complex Banach space V is called a domain of holomorphy [133, 10.4] if there are no open sets U and W in V satisfying the following conditions: (i) U is connected and U 6⊂ D. (ii) ∅ = 6 W ⊂ D ∩ U. (iii) For each holomorphic function f : D → C, there is a holomorphic function f˜: U → C such that f˜(z) = f (z) for each z ∈ W . Proposition 3.5.16. Let D be a bounded domain in a complex Banach space V . If D is complete with respect to the Carath´eodory distance cD , then it is a domain of holomorphy. Proof. Suppose that D is not a domain of holomorphy. Then by definition, there are open subsets U, W of V satisfying the following conditions: (i) U is connected and U 6⊂ D. (ii) ∅ = 6 W ⊂ D ∩ U. (iii) For each holomorphic function f : D → C, there is a holomorphic function f˜: U → C such that f˜(z) = f (z) for each z ∈ W . We deduce a contradiction. Without loss of generality, we may assume that U is bounded. Let W0 be a connected component of U ∩ D with W0 ∩ W 6= ∅. Then we have ∂W0 ∩ ∂D ∩ U 6= ∅. Indeed, if this is not the case, then for each p ∈ U \W0 6= ∅, either p∈ / ∂W0 or p ∈ / ∂D. If p ∈ / ∂W0 , then there is a norm-open ball Bp ⊂ U containing p such that either Bp ∩ W0 = ∅ or Bp ∩ W0c = ∅. Since

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p∈ / W0 , we must have Bp ∩ W0 = ∅. On the other hand, if p ∈ / ∂D, then there is an open ball Bp containing p such that Bp ∩ D = ∅ or Bp ∩ Dc = ∅. In either case, we have Bp ∩ W0 = ∅ since, if Bp ⊂ D, then the connected ball Bp resides in a connected component W1 of U ∩ D and we must have W1 6= W0 as p ∈ / W0 . Now the disconnection   [ Bp  U = W0 ∪  p∈U \W0

contradicts the connectedness of U . Pick a point p ∈ ∂W0 ∩ ∂D ∩ U and let (zn ) be a sequence in W0 norm-converging to p. By omitting the first few terms of the sequence if necessary, we may assume that (zn ) and p are contained in a closed ball strictly contained in U . It follows that (zn ) also converges to p with respect to the Carath´eodory distance cU . By condition (iii) above, each holomorphic function f : D → C with |f (z)| < 1 extends to a holomorphic function f˜ : U → C, which coincides with f on the connected component W0 by the identity principle. Moreover, if |f˜(u)| > 1 for some u ∈ U , then we deduce a contradiction by considering the extension to U of the function f −f1˜(u) on D. Hence we must have |f˜(u)| ≤ 1 for all u ∈ U and, by the maximum principle, |f˜(u)| < 1 for all u ∈ U . It follows that cD (zn , zm ) ≤ cU (zn , zm ) for n, m = 1, 2, . . ., where cU (zn , zm ) converges to cU (p, p) = 0 as n, m → ∞. Hence (zn ) is a Cauchy sequence in D with respect to cD . However, (zn ) does not converge in D, with respect to cD . Indeed, if (zn ) cD -converges to some point z ∈ D say, then by Lemma 3.5.3, there is a constant α > 0 such that αkzn − zk ≤ cD (zn , z) → 0

as

n→∞

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which is impossible since (zn ) does not converge in D with respect to the norm-distance. This shows that (D, cD ) fails to be complete, which is a contradiction. We therefore conclude that D is a domain of holomorphy. Proposition 3.5.17. A bounded homogeneous domain in a complex Banach space is complete in the distances cD and kD . In particular, it is a domain of holomorphy. Proof. Let D be a bounded homogeneous domain in a Banach space V . In view of Lemma 3.5.15 and Proposition 3.5.16, it suffices to show that (D, cD ) is complete. Fix p ∈ D with B(p, r) ⊂ D, where B(p, r) is the norm closed ball centred at p of radius r > 0. By Lemma 3.5.3, there exists s > 0 such that Bc (p, s) ⊂ B(p, r), where Bc (p, s) := {x ∈ V : cD (x, p) < s}. Now let (xn ) be a cD -Cauchy sequence in D. Pick xk such that cD (xn , xk ) < s for all n ≥ k. Since D is homogeneous, we can find g ∈ Aut D such that g(xk ) = p. Then the sequence (g(xn )) is a cD -Cauchy sequence and cD (g(xn ), p) = cD (g(xn ), g(xk )) = cD (xn , xk ) < s for all n ≥ k. Hence g(xn ) ∈ B(p, r) for all n ≥ k. By Theorem 3.5.6, (g(xn )) norm converges, as well as cD -converges, to some point q ∈ B(p, r). It follows that (xn ) cD -converges to g −1 (q) ∈ D, which proves completeness of cD . Now the Schwarz lemma. A familiar variant (of its main assertion) is the Schwarz-Pick lemma for holomorphic functions h : D −→ D which, as noted at the outset of the section, can be interpreted geometrically in terms of the Poincar´e metric. Indeed, the following assertion of the Schwarz-Pick lemma h(z) − h(w) z − w ≤ 1 − h(z)h(w) 1 − zw

(z, w ∈ D)

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is just saying that the holomorphic map h contracts the Poincar´e distance ρ: ρ(h(z), h(w)) ≤ ρ(z, w) = tanh

z−w 1 − zw .

−1

The Schwarz-Pick lemma for D can be extended to bounded symmetric domains via the Hahn-Banach theorem. Lemma 3.5.18. Let D be the open unit ball of a complex Banach space V and let h : D −→ D be a holomorphic map satisfying h(0) = 0. Then we have kh(z)k ≤ kzk for all z ∈ D. If V is a JB*-triple, then for any holomorphic self-map f on D, we have kg−f (w) (f (z))k ≤ kg−w (z)k

(z, w ∈ D)

where g• denotes the M¨ obius transformation. Proof. Fix z ∈ D\{0} and define a holomorphic map ζ : D −→ D by ζ(α) =

αz kzk

(α ∈ D).

For the given holomorphic map h, and for a continuous linear map ϕ ∈ V ∗ with kϕk ≤ 1, apply the Schwarz lemma to the composite map ϕ ◦ h ◦ ζ : D −→ D, which gives |ϕ ◦ h ◦ ζ(α)| ≤ |α| for all

α ∈ D.

In particular, for α = kzk, we have |ϕ(h(z))| ≤ kzk and hence kh(z)k ≤ kzk since ϕ was arbitrary. Now, if V is a JB*-triple and f : D −→ D is holomorphic, then we have g−f (w) ◦ f ◦ gw (0) = 0

(z, w ∈ D)

by composing f with M¨ obius transformations. Hence the preceding result implies kg−f (w) ◦ f ◦ gw (g−w (z))k ≤ kg−w (z)k

(z, w ∈ D)

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which gives kg−f (w) (f (z))k ≤ kg−w (z)k.

There is another assertion of the Schwarz lemma, which says that a holomorphic map h : D −→ D with h(0) = 0 must satisfy |h0 (0)| ≤ 1 (and it must be a rotation h(z) = eiθ z if either |h0 (0)| = 1 or |h(z)| = |z| for some z 6= 0). Without the assumption of h(0) = 0, the SchwarzPick version states that |h0 (z)| ≤

1−|h(z)|2 . 1−|z|2

This can also be extended

to bounded symmetric domains using the Hahn-Banach theorem. Proposition 3.5.19. Let Dj be the open unit ball of a JB*-triple Vj , for j = 1, 2. Given a holomorphic map f : D1 −→ D2 , we have kf 0 (z)k ≤

kB(f (z), f (z))1/2 k 1 − kzk2

(z ∈ D1 )

where B(·, ·) denotes the Bergman operator. In particular, if V2 = C, we have kf 0 (z)k ≤

1−|f (z)|2 1−kzk2

for all z ∈ D1 ,

0 and √ if V2 2 is a Hilbert space of dimension at least 2, then kf (z)k ≤ 1−kf (z)k . 1−kzk2

Proof. As before, we denote by g. the M¨obius transformation. First, assume f (0) = 0. Let z ∈ D1 \{0} and as in the proof of Lemma 3.5.18, define a holomorphic map ζ : D −→ D by ζ(α) = αz/kzk. For each ϕ ∈ V2 with kϕk ≤ 1, apply the Schwarz lemma to the holomorphic function ϕ ◦ f ◦ ζ on D, we have |(ϕ ◦ f ◦ ζ)0 (0)| ≤ 1 which gives |ϕ(f 0 (0)z)| ≤ kzk and it follows that kf 0 (0)k ≤ 1.

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Now, without the assumption of f (0) = 0, but if f (w) = 0 for some w ∈ D1 , then we have f ◦ gw (0) = 0 and the preceding arguments imply 0 kf 0 (w)k = k(f ◦ gw ◦ g−w )0 (w)k = k(f ◦ gw )0 (0)g−w (w)k 1 . = k(f ◦ gw )0 (0)B(w, w)−1/2 k ≤ 1 − kwk2

Finally, for each z ∈ D1 , we have (g−f (z) ◦ f )(z) = 0 and therefore, as in the previous case, we deduce k(g−f (z) ◦ f )0 (z)k ≤

1 . 1 − kzk2

Observe that 0 0 −1/2 0 (g−f (z) ◦ f )0 (z) = (g−f f (z). (z) (f (z))f (z) = B(f (z), f (z))

Hence we have kf 0 (z)k = kB(f (z), f (z))1/2 (g−f (z) ◦ f )0 (z)k ≤

kB(f (z), f (z))1/2 k . 1 − kzk2

The last assertion follows from Examples 3.2.17 and 3.4.9. We consider another variant of the Schwarz lemma, which is given by the classical theorem of Wolff [177]. One can view the Schwarz lemma as an invariant theorem in the following sense. Given a holomorphic self-map h on D with h(0) = 0, the Schwarz lemma asserts that every open Euclidean disc D(0, r) := {z ∈ D : |z| < r} is invariant under h. More generally, if h fixes some z ∈ D instead of 0, then we have h(∆(z, r)) ⊂ ∆(z, r)

(0 < r < 1)

for each Poincar´e disc ∆(z, r) := {x ∈ D : ρ(x, z) < tanh−1 r}

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(with radius tanh−1 r) by the Schwarz-Pick lemma. If h has no fixedpoint in D, then we have h(∆(z, r)) ⊂ ∆(h(z), r). By considering the Poincar´e discs ∆(z, r) for which z tends to a boundary point ξ ∈ ∂D and r → 1, one can show that the ‘limit’ of these discs is invariant under h, and is in fact a horodisc with horocentre at ξ. This can be viewed as a boundary analogue of the Schwarz lemma. Actually, it can be shown that ξ is a boundary fixed-point of h, meaning ξ = limr→1− h(rξ). We now make precise Wolff’s theorem and extend it to bounded symmetric domains. Given a holomorphic self-map f on D, without fixed-point, Wolff has shown in [177] that there is a (unique) boundary point ξ ∈ ∂D so that every Euclidean disc internally tangent to D at ξ is invariant under f . Such a disc is called a horodisc with horocentre at ξ, and is of the form H(ξ, λ) =

1 λ ξ+ D 1+λ 1+λ

where λ > 0 and the radius of the disc H(ξ, λ) is written 1/(1 + λ) so that the horodiscs can be parametrised by positive real numbers. To show that H(ξ, λ) as the ‘limit’ of a sequence ∆(zk , rk ) of Poincar´e discs as mentioned earlier, where zk → ξ and rk → 1, we first need to choose the sequence (zk ). For this, we pick a sequence αk ∈ (0, 1) with αk ↑ 1. By comparing the identity function idk on the closed disc D(0, αk ) ⊂ D with the function idk − αk f , one finds a point zk ∈ D(0, αk ) satisfying αk f (zk ) = zk , using Rouch´e’s theorem. Further, choosing a subsequence if necessary, we may assume (zk ) converges to some point ξ ∈ D. Since f has no fixed-point in D, we must have ξ ∈ ∂D. Given λ > 0, pick rk ∈ (0, 1) so that 1 − rk2 = λ(1 − |zk |2 ), for sufficiently large k. The Poincar´e disc Pk (λ) := ∆(zk , rk ) = {z ∈ D : ρ(z, zk ) < tanh−1 rk }

(3.38)

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227

is the image gzk (D(0, rk )) of the Euclidean disc D(0, rk ) under the M¨ obius transformation gzk (z) =

z + zk 1 + zk z

(z ∈ D).

We define the limit of the Poincar´e discs Pk (λ) to be the set S(ξ, λ) := {z ∈ D : z = lim xk , xk ∈ Pk (λ)}. k

Then the horodisc H(ξ, λ) is exactly the interior S0 (ξ, λ) of S(ξ, λ). In fact, we have H(ξ, λ) = S0 (ξ, λ) =

  |1 − zξ|2 1 λ 1 z∈D: < = ξ+ D. 1 − |z|2 λ 1+λ 1+λ (3.39)

Further, the f -invariance of these horodiscs implies that ξ is a boundary fixed-point of f . Indeed, given ε > 0, the intersection D(ξ, ε) ∩ D contains a horodisc H(ξ, λ) of sufficiently small radius 1/(1 + λ) (Draw a picture!). Hence for λ/(1 + λ) < r < 1, we have rξ ∈ H(ξ, λ) and f (rξ) ∈ H(ξ, λ) ⊂ D(ξ, ε). One may observe that our presentation of Wolff’s theorem here bears little resemblance to the original work in [177]. We formulate it as such so that the preceding construction can be extended naturally to bounded symmetric domains, which is what we are going to do presently. There is another more direct approach to the construction of H(ξ, λ), however, also applicable to bounded symmetric domains. We discuss this first. This approach depends on the crucial observation that the horodisc H(ξ, λ) is identical with the following sets:   |1 − zz k |2 1 H(ξ, λ) = z ∈ D : lim < k 1 − |z|2 λ   2 1 − |zk | 1 = z ∈ D : lim < k 1 − |g−zk (z)|2 λ

(3.40)

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which can be formulated for bounded symmetric domains. Hinted by this, we proved the following lemma. Lemma 3.5.20. Let D be a bounded symmetric domain realised as the open unit ball of a JB*-triple V . Let (zk ) be a sequence in D (norm) converging to a boundary point ξ ∈ ∂D. The function F : D → [0, ∞) given by F (x) = lim sup k→∞

1 − kzk k2 1 − kg−zk (x)k2

(x ∈ D)

is well-defined and continuous. Proof. For each x ∈ D, we have, by Lemma 3.2.28 and (3.27), 1 − kzk k2 = kB(x, x)−1/2 B(x, zk )B(zk , zk )−1/2 k(1 − kzk k2 ) 1 − kg−zk (x)k2 ≤ kB(x, x)−1/2 B(x, zk )kkB(zk , zk )−1/2 k(1 − kzk k2 ) (1 + kxkkzk k)2 1 + kxk = kB(x, x)−1/2 B(x, zk )k ≤ ≤ 2 1 − kxk 1 − kxk and hence the defining sequence for F (x) is bounded. Therefore F is well-defined. For continuity, let x, y ∈ D and write xk = yk =

1 − kzk k2 . Then we have 1 − kg−zk (y)k2

1 − kzk k2 , also 1 − kg−zk (x)k2

|xk − yk | ≤ kB(x, x)−1/2 B(x, zk ) − B(y, y)−1/2 B(y, zk )k which gives |F (x) − F (y)| = | lim sup xk − lim sup yk | k −1/2

≤ lim sup kB(x, x)

k

B(x, zk ) − B(y, y)−1/2 B(y, zk )k

k

= kB(x, x)−1/2 B(x, ξ) − B(y, y)−1/2 B(y, ξ)k since (zk ) norm converges to ξ. Now continuity of F follows from that of the function h(x) = kB(x, x)−1/2 B(x, ξ)k.

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A computation similar to the previous proof yields the following result. Lemma 3.5.21. Let (xk ) be a sequence in D norm converging to x ∈ D. Then we have lim sup k→∞

1 − kzk k2 1 − kzk k2 = lim sup . 2 1 − kg−zk (xk )k2 k→∞ 1 − kg−zk (x)k

The sequence (zk ) in Wolff’s theorem for D has a limit point ξ by relative compactness of D. An infinite dimensional domain D need not be relatively compact and the existence of limit points is not guaranteed. For this reason, we consider compact maps f : D → D, which are the ones having relatively compact image f (D), that is, the norm closure f (D) is compact in V . All continuous self-maps on a finite dimensional bounded domain are necessarily compact. Now let f : D −→ D be a compact holomorphic map without fixed-point. To prove an analogue of Wolff’s theorem, the first step is to construct a sequence (zk ) in D converging to the boundary, as in the case of D. As before, pick a sequence αk ∈ (0, 1) with αk ↑ 1, and consider the function αk f : D −→ D. To find a fixed-point for αk f , one can make use of the fixed-point theorem of Earle and Hamilton [59], instead of Rouch´e’s theorem, which states that a holomorphic selfmap h on a bounded domain D in a complex Banach space must have a fixed-point if the image h(D) is strictly contained in D. Using this result, we obtain αk f (zk ) = zk for some zk ∈ D and for each k. By compactness of f , we may assume (zk ) norm converges to some point ξ ∈ D, via a subsequence if necessary. Since f has no fixed-point in D, we must have ξ ∈ ∂D. We are now ready to extend Wolff’s theorem to all bounded symmetric domains, following the observation in (3.40).

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Theorem 3.5.22. Let f be a fixed-point free compact holomorphic selfmap on a bounded symmetric domain D. Then there is a sequence (zk ) in D converging to a boundary point ξ ∈ D such that, for each λ > 0, the set  H(ξ, λ) =

x ∈ D : lim sup k→∞

1 − kzk k2 1 < 1 − kg−zk (x)k2 λ



is a convex domain and f -invariant, that is, f (H(ξ, λ)) ⊂ H(ξ, λ). S T Moreover, we have D = λ>0 H(ξ, λ) and 0 ∈ λ0 in Theorem 3.5.22 is decreasing in the sense that λ > λ0 implies H(ξ, λ) ⊂ H(ξ, λ0 ). Hence, given H(ξ, λ) 6= ∅, then H(ξ, λ0 ) 6= ∅ for all λ0 ≤ λ. In fact, we have H(ξ, λ) 6= ∅ for all λ > 0, by Theorem 3.5.27 below. In view of the preceding discussion, we shall call H(ξ, λ) a horoball at ξ (of ‘radius’

1 1+λ )

in higher dimensional bounded symmetric do-

mains. Example 3.5.25. Let D be the open unit ball of the JB*-triple C(Ω) of complex continuous functions on a compact Hausdorff space Ω, with Jordan triple product {x, y, z} = xyz where y denotes the complex

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conjugate of the function y ∈ C(Ω). For a, b ∈ D, the Bergmann operator B(a, b) is given by a product of functions: B(a, b)(z) = (1 − ab)2 z

(z ∈ C(Ω))

where 1 denotes the constant function with value 1 and we have kB(a, b)k = k(1 − ab)2 k = sup{|1 − a(ω)b(ω)|2 : ω ∈ Ω}. Let (zk ) be a sequence in D converging to ξ ∈ ∂D as in Theorem 3.5.22. We may assume 0 ∈ / zk (Ω) by omitting the first few terms of the sequence. Then kB(x, x)−1/2 B(x, zk )B(zk , zk )−1/2 k(1 − kzk k2 )

(1 − xz k )2 (1 − kzk k2 )

. = (1 − |x|2 )(1 − |zk |2 )

2

kk ≤ 1 and the sequence (1 − xz k ) converges to 1 − xξ in Since 1−kz 2 1−|zk | C(Ω), we have lim sup kB(x, x)−1/2 B(x, zk )B(zk , zk )−1/2 k(1 − kzk k2 ) k

(1 − xξ)2 1 − kzk k2

= lim sup 1 − |x|2 1 − |zk |2 k and for λ > 0,  H(ξ, λ) =

  

(1 − xξ)2 1 − kzk k2 1

x ∈ D : lim sup < 1 − |x|2 1 − |zk |2 λ k

which reduces to the horodisc in (3.39) if Ω is a singleton in which case the function

1−kzk k2 1−|zk |2

= 1.

Another construction of the horodisc H(ξ, λ) in D is by taking the limit of a sequence of Poincar´e discs Pk (λ), as noted in (3.39). We now extend this construction to bounded symmetric domains.

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Naturally, one needs to replace the Poincar´e metric by another invariant distance on a bounded symmetric domain D. We choose the Kobayashi distance kD . Analogous to (3.38), we define a Kobayashi ball Dz (r), centred at z ∈ D with radius tanh−1 r > 0, to be the set Dz (r) := {x ∈ D : kD (x, z) < tanh−1 r} = g−z (D(0, r)) where g−z is the M¨ obius transformation and, to avoid confusion with the notation for Bergman operators, D(0, r) denotes the (norm) open ball centred at 0 of radius r > 0 throughout this section. We note that the Kobayashi ball Dz (r) is convex by Corollary 3.5.10. We define the limit of a sequence (Dzk (rk )) of Kobayashi balls to be the set lim Dzk (rk ) := {x ∈ D : x = lim xk , xk ∈ Dzk (rk )}. k

k

(3.41)

Now, given a fixed-point free compact holomorphic map f : D −→ D, we can find a sequence (zk ) in D converging to a boundary point ξ ∈ ∂D, as before. For each λ > 0, one wishes to find a suitable sequence of Kobayashi balls ‘converging’ to the horoball H(ξ, λ), that is, in view of (3.39), H(ξ, λ) is the interior of the limit of these balls. Mimicking the one-dimensional construction again, given λ > 0, we can choose rk ∈ (0, 1) satisfying λ(1 − kzk k2 ) = 1 − rk2 from some k onwards. To simplify notation, we write Dk (λ) := Dzk (rk ) for the Kobayashi ball centred at zk with radius tanh−1 rk . We denote the limit of the sequence (Dk (λ)) by S(ξ, λ) := lim Dk (λ) k

(3.42)

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and the interior by S0 (ξ, λ). We note that S(ξ, λ) 6= ∅ since ξ = limk zk ∈ S(ξ, λ). For D = D, both H(ξ, λ) and S0 (ξ, λ) just defined are identical to the ones in (3.39). What is an appropriate generalisation of the disc λ 1 ξ+ D 1+λ 1+λ in (3.39) to higher dimensions? A simple computation reveals that this disc can be written as r

λ ξ+B 1+λ where B

q

λ 1+λ ξ,

q

λ 1+λ ξ



λ ξ, 1+λ

r

λ ξ 1+λ

!1/2 (D)

(3.43)

is the Bergman operator on D. The formu-

lation in (3.43) can be carried over to all bounded symmetric domains. Question: with the notions in (3.42) and (3.43), do we have, as in (3.39), that for all bounded symmetric domains D, λ H(ξ, λ) = S0 (ξ, λ) = ξ+B 1+λ

r

λ ξ, 1+λ

r

λ ξ 1+λ

!1/2 (D)

with perhaps some appropriate modification of the last set? The answer is affirmative for finite-rank domains. First, we always have the inclusion H(ξ, λ) ⊂ S0 (ξ, λ). Proposition 3.5.26. For a fixed-point free compact holomorphic selfmap f on any bounded symmetric domain D and for λ > 0, we have H(ξ, λ) ⊂ S0 (ξ, λ)

and

S(ξ, λ) ∩ D ⊂ {x ∈ D : F (x) ≤ 1/λ}

where F is the function in Lemma 3.5.20 and S(ξ, λ) ∩ D is also an f -invariant domain.

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Proof. Let x ∈ H(ξ, λ). Then we have, from some k onwards, 1 1 − kzk k2 1 − kzk k2 < = 1 − kg−zk (x)k2 λ 1 − rk2 which implies kg−zk (x)k < rk , that is, x ∈ Dk (λ). Hence x ∈ S(ξ, λ). We have shown H(ξ, λ) ⊂ S(ξ, λ) and therefore H(ξ, λ) ⊂ S0 (ξ, λ) since H(ξ, λ) is open. For the second assertion, let x ∈ S(ξ, λ) ∩ D with x = limk xk and xk ∈ Dk (λ). Since kg−zk (xk )k < rk for each k, we have from Lemma 3.5.21 that lim sup k→∞

1 − kzk k2 1 − kg−zk (x)k2

1 − kzk k2 2 k→∞ 1 − kg−zk (xk )k 1 1 − kzk k2 = . ≤ lim sup 2 λ 1 − rk k→∞ = lim sup

This proves S(ξ, λ) ∩ D ⊂ {x : F (x) ≤ 1/λ}. To see that S(ξ, λ) ∩ D is f -invariant, write fk = αk f where as before, αk ∈ (0, 1) is chosen so that fk (zk ) = zk . Then for each x ∈ S(ξ, y) ∩ D with x = limk xk and xk ∈ Dk (λ), the Schwarz-Pick lemma yields kg−zk (fk (xk ))k = kg−fk (zk ) (fk (xk ))k ≤ kg−zk (xk )k ≤ kg−zk (y)k and hence fk (xk ) ∈ Dk (λ). It follows that f (x) = lim f (xk ) = lim fk (xk ) ∈ S(ξ, y) ∩ D. k

k

Although the question raised before Proposition 3.5.26 remains open for all bounded symmetric domains, a complete answer can be given for all finite-rank bounded symmetric domains, and in particular, for finite dimensional ones. This has been proved in [50, Theorem 5.13, Theorem 5.14]. We state the result below, but suppress the lengthy proof which relies substantially on Jordan theory.

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Theorem 3.5.27. Let f be a fixed-point free compact holomorphic selfmap on a bounded symmetric domain D of finite rank p. Then there is a sequence (zk ) in D converging to a boundary point ξ=

m X

(αj > 0, m ∈ {1, . . . , p})

αj ej

j=1

where e1 , . . . , em are orthogonal minimal tripotents in ∂D, such that for each λ > 0, the f -invariant horoball H(ξ, λ) is given by H(ξ, λ) = S0 (ξ, λ) =

m X j=1

 1/2 s s m m X X σj λ σj λ σj λ ej + B  ej , ej  (D) 1 + σj λ 1 + σj λ 1 + σj λ j=1

j=1

which is affinely homeomorphic to D, where σj ≥ 0 and max{σj : j = 1, . . . , m} = 1. Remark 3.5.28. For p = 1, that is, D is a Hilbert ball, in the above theorem, we have λ S0 (ξ, λ) = ξ+B 1+λ

r

λ ξ, 1+λ

r

λ ξ 1+λ

!1/2 (D)

and in one dimension, D = D, it reduces to Wolff’s horodisc in (3.39) as noted earlier: S0 (ξ, λ) =

λ 1 ξ+ D. 1+λ 1+λ

We note that both Theorems 3.5.22 and 3.5.27 assume the compactness property of the holomorphic self-map f in infinite dimension. The question remains open whether this condition can be removed. For Hilbert balls, however, the answer is affirmative for the following reason. Let f be a fixed-point free holomorphic self-map on a bounded symmetric domain D. We observe that, in the previous construction

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of the horoballs H(ξ, λ), the existence of the sequence (zk ) in D does not depend on the compactness of f , but the existence of the limit ξ ∈ ∂D does require the compactness assumption. In the case of a Hilbert ball D, one can circumvent this requirement. To see this, let (zk ) be chosen, via the Earle-Hamilton fixed-point theorem as before, satisfying αk f (zk ) = zk with 0 < αk ↑ 1. By relatively weak compactness of the Hilbert ball D, we may assume, by choosing a subsequence if necessary, that (zk ) converges weakly to some point ξ ∈ D. We show ξ ∈ ∂D, otherwise we have kg−f (ξ) (f (zk ))k ≤ kg−ξ (zk )k. Using (3.28) and substituting f (zk ) = zk /αk , one deduces 1 − αk−2 kzk k2 |1 − hzk , ξi|2 |1 − hαk−1 zk , f (ξ)i|2 ≥ . 1 − kzk k2 1 − kξk2 1 − kf (ξ)k2 Since

2 1−α−2 k kzk k 1−kzk k2

≤ 1, we have 1≥

1 1 − kg−f (ξ) (ξ))k2

by letting k → ∞. This gives kg−f (ξ) (ξ)k = 0 and f (ξ) = ξ, contradicting the non-existence of a fixed point in D. It follows that (zk ) actually norm converges to ξ since lim supk→∞ kzk k ≤ 1 = kξk ≤ lim inf k→∞ kzk k. Now one can follow the same proofs for Theorems 3.5.22 and 3.5.27, but without the compactness assumption of f in which case, the horoball H(ξ, λ) is indeed ‘internally tangent to’ D. Theorem 3.5.29. Let f : D −→ D be a fixed-point free holomorphic map on a Hilbert ball D in Theorem 3.5.27. Then for each λ > 0, we have H(ξ, λ) ∩ ∂D = {ξ}.

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Proof. We have H(ξ, λ) = S0 (ξ, λ) = S(ξ, λ). Let p ∈ H(ξ, λ) ∩ ∂D. Retaining the previous notation, we have from (3.42), p = limk pk for some pk ∈ Dk (λ), where kg−zk (pk )k < rk and λ(1 − kzk k2 ) = 1 − rk2 . Using (3.28), we deduce 2

|1 − hpk , zk i|

 1 − kzk k2 ≤ (1 − kpk k ) 1 − kg−zk (pk )k2   1 − kzk k2 1 − kpk k2 2 < (1 − kpk k ) →0 = λ 1 − rk2 2



as k → ∞, which gives hp, ξi = 1 and p = ξ. To conclude this section, we discuss briefly the dynamics of a holomorphic self-map f on a bounded symmetric domain D, which concerns the asymptotic behaviour of the iterates (f n ) of f , where n-times

z }| { fn = f ◦ · · · ◦ f . In other words, one considers (f, D) as a discrete-time dynamical system and studies its limit sets, which can be described as the images of subsequential limits of the iterates (f n ). We will call a subsequential limit h = lim f nk k→∞

in the topology of locally uniform convergence, a limit function of (f n ). The case of the disc D is well understood. Let f : D −→ D be a holomorphic map. Then either f has a fixed-point in D or has none. In the former case, let ga : D −→ D be the M¨ obius transformation induced by the fixed-point a ∈ D. Then g−a ◦ f ◦ ga (0) = 0. In particular, if f is biholomorphic, then it is conjugate to a linear isometry ϕ : D −→ D (cf. Lemma 1.1.8) and the dynamics of f is essentially the same as ϕ. For D = D, the isometry ϕ is a rotation: ϕ(z) = eiθ z for some θ ∈ [0, 2π). It follows that the

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orbit {f n (z)} is finite if θ is a rational multiple of π, in which case f p is the identity map for some p ∈ N. If θ is an irrational multiple of π, then {einθ : n = 1, 2, . . .} is dense in ∂D and from this, one can infer the dynamics of f . Remark 3.5.30. For D = D and a biholomorphic map f = αgc with a fixed-point in D that is not the identity map, where c ∈ D and α ∈ ∂D, it has been proved by Rigby in [148] that f p is the identity map exactly when p−1 X r=0

min{r, p−1−r}

α

r

X s=0

   r p−1−r |c| = 0. s s 2s

For a Hilbert ball D and c ∈ D\{0}, it has also been shown in [148] that αgc has a fixed-point in D if and only if |1 − α| > 2kck. Hence there are two remaining cases to be discussed, namely, (I) f has a fixed-point in D and is not biholomorphic, (II) f has no fixed-point in D. Case (I). We begin with the unit disc D. Again, Schwarz lemma provides a solution. The following result is well-known. Theorem 3.5.31. Let f : D −→ D be holomorphic with a fixed-point a ∈ D. If f is not biholomorphic, then the sequence (f n ) of iterates converge locally uniformly to a constant function with value a. Proof. Conjugating with a M¨obius transformation as before, we may assume a = 0. Since f is not biholomorphic, Schwarz lemma says that we must have |f (z)| < |z| for all z 6= 0. Let 0 < r < 1 and let M (r) = sup{|f (z)| : |z| ≤ r}. Then we have M (r) < r and we may assume M (r) > 0. Define a holomorphic map h : D −→ D by h(z) =

f (rz) . M (r)

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Then apply the Schwarz lemma once more, we obtain |f (z)| ≤

M (r) |z| r

(|z| ≤ r).

Write c := M (r)/r ∈ (0, 1). Iterating the above inequality gives |f n (z)| ≤ cn |z| ≤ cn r for all z in the closed disc D(0, r) ⊂ D. This implies (f n ) converges uniformly on D(0, r) to 0, and hence locally uniformly on D. Remark 3.5.32. An interesting consequence of the preceding theorem is that a holomorphic map f : D −→ D, which is not biholomorphic, can have at most one fixed-point in D. It is natural to ask if the preceding result and proof can be extended to bounded symmetric domains D. For this, one would need the crucial inequality kf (z)k < kzk in the case of a holomorphic self-map f on D, which has a fixed-point and is not biholomorphic. Unfortunately, it is unavailable even for the 2-dimensional Euclidean ball B = {(z, w) ∈ C2 : |z|2 + |w|2 < 1}. The holomorphic map f : (z, w) ∈ B 7→ (z, 0) ∈ B is an example, where f (z, 0) = (z, 0). One may try an alternative approach to Theorem 3.5.31 for D using instead the derivative f 0 (a) at the fixed-point a and the Schwarz lemma. Indeed, under the assumption of the theorem with a = 0, we would have |f 0 (0)| < 1 by the Schwarz lemma, since f is not biholomorphic. It follows from continuity that |f 0 (z)| < 1 on a closed disc D(0, r) with 0 < r < 1. Hence we have |f (z) − f (w)| ≤ c|z − w| for some c ∈ (0, 1) and all z, w ∈ D(0, r). From this, one concludes that (f n ) converges locally uniformly to 0, using the well-known contraction mapping theorem.

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Although the above example of f (z, w) = (z, 0) also shows that the inequality kf 0 (0)k < 1 is unavailable for a holomorphic self-map f on the ball B, which fixes 0 and is not biholomorphic, nevertheless, Cauchy inequality implies that, if (f n ) converges to 0 locally uniformly, then we have the weaker inequality ρ(f 0 (0)) = lim k(f n )0 (0)k1/n = lim k(f 0 )n (0)k1/n < 1 n→∞

where

ρ(f 0 (0))

n→∞

denotes the spectral radius of f 0 (0). This inequality

turns out to be also sufficient for the convergence of the iterates (f n ). The proof of sufficiency and detailed references for the following theorem, due to Vesentini, Khatskevich and Shoikhet, can be found in [146, Proposition 5.3]. Theorem 3.5.33. Let D be a bounded domain in a complex Banach space and let f : D −→ D be a holomorphic map with a fixed-point a ∈ D. The following conditions are equivalent. (i) (f n ) converges locally uniformly to a constant function with value a. (ii) The spectral radius ρ(f 0 (a)) is strictly less than 1. Case (II). We now consider a fixed-point free holomorphic selfmap f on a bounded symmetric domain D. In the case of D = D, we have the following celebrated Denjoy-Wolff theorem [55, 178] which, together with Theorem 3.5.31, determines completely the dynamics of a holomorphic self-map on D that is not biholomorphic (cf. Remark 3.5.32). Theorem 3.5.34. Let f : D −→ D be a fixed-point free holomorphic map. Then there is a unique boundary point ξ ∈ ∂D such that the iterates (f n ) converge locally uniformly to the constant function f (·) = ξ.

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This question has been investigated by many authors, and also for other domains. It is impossible to discuss details of all these works. One can find, for example, substantial literature on this topic in [2, 145, 146]. Some recent literature, by no means complete, is included in [39]. For infinite dimensional bounded symmetric domains, we consider as before compact holomorphic maps. Let us begin the discussion with a useful criterion for the existence of a fixed-point for a compact holomorphic self-map on a domain, stated below, which has been proved in [114]. Lemma 3.5.35. Let B be the open unit ball of a complex Banach space and let f : B −→ B be a compact holomorphic map. The following conditions are equivalent. (i) f is fixed-point free. (ii) There exists a ∈ B such that supk kf nk (a)k = 1 for every subsequence (f nk ) of the iterates of f . An important consequence of this criterion and Lemma 3.4.11 is that, for a fixed-point free compact holomorphic self-map f on a bounded symmetric domain D, realised as the open unit ball of a JB*triple, the image h(D) of a limit function h = limk f nk is entirely contained in a single boundary component of the boundary ∂D. H´erver has shown in [82] that the Denjoy-Wolff theorem can be extended to finite dimensional Euclidean balls (see also [126]), but also shown in [83] that the theorem fails in the case of the bidisc D × D. The iterates of a fixed-point free holomorphic map f : D × D −→ D × D need not converge to a constant function. Instead, we have a phenomenon which can be formulated as follows.

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Theorem 3.5.36. Given a fixed-point free holomorphic map f : D × D −→ D × D, there is one single boundary component Γ ⊂ ∂(D × D) such that the image h(D × D) of all subsequential limits h = limk f nk of (f n ) is contained in the closure Γ. The closed boundary components of ∂(D × D) are the singletons {(ξ, η)} and sets of the form {ξ} × D and D × {η}, where |ξ| = |η| = 1. Hence there are three distinct possibilities for the iterates (f n ) on the bidisc. On the other hand, the boundary components of a Hilbert ball are exactly the boundary points. View in these perspectives, a formulation of a possible generalisation of the Denjoy-Wolff theorem emerges. We state it as a conjecture since it has not been completely proved at present. Conjecture. Let D be a bounded symmetric domain in a complex Banach space V and let f : D −→ D be a fixed-point free compact holomorphic map. Then there is one single boundary component Γ in the boundary ∂D such that h(D) ⊂ Γ for all limit functions h of (f n ). This conjecture is true for finite dimensional Euclidean balls as well as the bidisc, as already discussed. In infinite dimension, the conjecture is false without the assumption of compactness of the self-map f . Stachura [160] has given an example of a fixed-point free biholomorphic map f on an infinite dimensional Hilbert ball, with a subsequence (f nk ) satisfying lim supn kf n (0)k = 1 and limk f nk (0) = 0. Nevertheless, the conjecture is true for all Hilbert balls. This has been proved by Chu and Mellon [47]. In fact, compactness of the selfmap f is not a necessary condition for a Denjoy-Wolff theorem for Hilbert balls. We will now prove a more general result without the compactness assumption, which includes the Chu-Mellon result as a

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special case. Actually, we will give necessary and sufficient conditions for the Denjoy-Wolff theorem to hold on a Hilbert ball. We prove two lemmas first. Lemma 3.5.37. Let D be the open unit ball of a Hilbert space with inner product h·, ·i. Given a sequence (ak ) in D norm converging to a ∈ ∂D and a sequence (vk ) in D weakly convergent to some v ∈ D, we have lim kg−ak (vk )k = 1.

(3.44)

k→∞

If there is a sequence (bk ) in D norm converging to b ∈ ∂D and a 6= b, then lim kg−ak (bk )k = 1.

(3.45)

k→∞

Proof. By Example 3.2.29, the Bergman operators B(ak , ak ) norm converges to 0. B(c, c)1/2 (I

For c ∈ D, the M¨obius transformation gc (v) = c +

+ v c)−1 (v) can be written as gc (v) = c +

B(c, c)1/2 (v) 1 + hv, ci

(v ∈ D).

It follows that

1 > kg−ak (vk )k =

−ak + ≥ kak k −

1 1/2 B(ak , ak ) (vk )

1 − hvk , ak i

1 kB(ak , ak )1/2 (vk )k −→ 1, |1 − hvk , ak i|

as

k→∞

where hv, ai = 6 1. This proves (3.44). To show (3.45), observe from Lemma 3.2.28 that 1 − kg−ak (bk )k2 = kB(bk , bk )−1/2 B(bk , ak )B(ak , ak )−1/2 k−1 ≤

kB(bk , bk )1/2 kkB(ak , ak )1/2 k −→ 0 as k → ∞ kB(bk , ak )k

where, by Example 3.2.29, lim B(bk , ak ) = B(b, a) 6= 0 while k

lim B(bk , bk ) = B(b, b) = 0 k

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and B(a, a) = 0. Lemma 3.5.38. Let D be a Hilbert ball and f : D −→ D a holomorphic map such that the iterates f n converge pointwise to a constant map h : D −→ ∂D. Then (f n ) converges locally uniformly to h. Proof. Let h(D) = {ξ}. Let B ⊂ D be an open ball strictly contained in D so that dist(B, ∂D) > 0. Let D(ξ, ε) be an open ball of radius ε > 0 such that D(ξ, ε) ∩ B = ∅. We show that f n (B) ⊂ D(ξ, ε) ∩ D from some n onwards. This would complete the proof. Suppose what we claim to show is false. Then there is a subsequence (f nk ) such that f nk (bk ) ∈ / D(ξ, ε), where bk ∈ B and f nk (bk ) converges weakly to some v ∈ D, by weak compactness of D. Fix a point y ∈ D. We first note that v ∈ ∂D, for otherwise, (3.44) implies kD (y, bk ) ≥ kD (f nk (y), f nk (bk )) = tanh−1 kg−f nk (y) (f nk (bk ))k −→ ∞ which contradicts the fact that sup{kD (y, bk )} = sup{tanh−1 kg−y (bk )k} < ∞ k

k

since bk ∈ B for all k. Hence kvk = 1 and the sequence (f nk (bk )) norm converges to v. Therefore v ∈ / D(ξ, ε). To complete the proof, compare the two sequences (f nk (bk )) and (f nk (y)), having limits in the boundary ∂D. Since kD (f nk (bk ), f nk (y)) ≤ kD (bk , y) ≤ sup{kD (y, bk )} < ∞ k

we must have v = ξ by (3.45), which contradicts v ∈ / D(ξ, ε). By means of the preceding two lemmas, we arrive at the following dichotomy for a fixed-point free holomorphic self-map on a Hilbert ball.

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Proposition 3.5.39. Let f : D −→ D be a fixed-point free holomorphic map on a Hilbert ball D and let a ∈ D. Then either lim inf kf 2n (a)k < 1 n→∞

or (f n ) converges locally uniformly to a constant map taking value at the boundary ∂D. Proof. Given lim inf kf 2n (a)k ≮ 1, we must have lim kf 2n (a)k = 1. n

n→∞

Let a ∈ H(ξ, λ) for some λ > 0, where H(ξ, λ) ∩ ∂D = {ξ} by Theorem 3.5.29. We first show that (f 2n (a)) converges to ξ. Indeed, f 2n (a) ∈ H(ξ, λ) and (3.28) implies lim sup |1 − hf

2n

2

(a), ξi|

= lim sup(1 − kf

n

2n

2



(a)k )

n

≤

1 (1 − kf 2n (a)k2 ) −→ 0 λ

1 − kzk k2 1 − kg−zk (a)k2



as n → ∞

which gives limn hf 2n (a), ξi = 1 and limn f 2n (a) = ξ. We next show limn f 2n+1 (a) = ξ. For this, it suffices to show that ξ is the only weak limit point of the sequence (f 2n+1 (a)). Let (f 2nk +1 (a)) be a subsequence of (f 2n+1 (a)) weakly convergent to ζ ∈ D. If ζ ∈ D, (3.44) implies kD (a, f (a)) ≥ kD (f 2nk (a), f 2nk +1 (a)) = tanh−1 kg−f 2nk (a) (f 2nk +1 (a))k −→ ∞ which is impossible. Hence we have ζ ∈ H(ξ, λ) ∩ ∂D = {ξ} and limn f n (a) = ξ. Using Lemma 3.5.38, we complete the proof by showing that (f n ) converges pointwise to the constant map h(·) = ξ. Let y ∈ D and v ∈ D be any weak limit point of the sequence (f n (y)). Then (f n (y)) admits a

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subsequence (f nk (y)) weakly converging to v and v ∈ H(ξ, µ) for some µ > 0. We show that kvk = 1. Otherwise, v ∈ D implies kD (a, y) ≥ kD (f nk (a), f nk (y)) = tanh−1 kg−f nk (a) (f nk (y))k → ∞ by (3.44), which is a contradiction. Hence we have v ∈ (ξ, µ)∩∂D = {ξ}, by Theorem 3.5.29. This shows that ξ ∈ ∂D is the only weak limit point of the sequence (f n (y)). Therefore limn f n (y) = ξ. As y ∈ D was arbitrary, we have shown that (f n ) converges pointwise to the constant map h(·) = ξ. If a fixed-point free self-map f on a Hilbert ball D has a convergent orbit, then its limit must lie in the boundary and the theorem below now follows immediately from the preceding proposition. Theorem 3.5.40. Let f : D −→ D be a fixed-point free holomorphic map on a Hilbert ball D. The following conditions are equivalent: (i) lim kf 2n (a)k = 1 for some a ∈ D; n→∞

(ii) an orbit (f n (a)) converges for some a ∈ D; (iii) (f n ) converges locally uniformly to a constant map h(·) = ξ ∈ ∂D. Remark 3.5.41. The Denjoy-Wolff theorem proved in [47] for compact holomorphic maps on Hilbert balls D is a special case of the above result. Indeed, given a fixed-point free compact holomorphic self-map f on D, we have, for some a ∈ D, that supk kf nk (a)k = 1 for all subsequences (f nk ) of (f n ) by Lemma 3.5.35. The example in [160] reveals that condition (i) above cannot be weakened to limk kf nk (a)k = 1 for some subsequence (f nk ) since there is a biholomorphic map f [160] on an infinite dimensional Hilbert ball such that limk kf nk (0)k = 1 for some subsequence (f nk ) of (f n ), but failing the Denjoy-Wolff theorem.

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shown by the following result. Proposition 3.5.42. Let f be a fixed-point free holomorphic self-map on a bounded symmetric domain D such that one orbit (f n (a)) converges for some a ∈ D. Then all limit functions of (f n ) take values in one single boundary component Γ in D. Proof. Let h and h1 be limit functions of (f n ). By Lemma 3.4.11, h(D) is contained in some boundary component Γ ⊂ D and h1 (D) is contained in another boundary component, say Γ1 ⊂ D. By assumption, we have h(a) = h1 (a) ∈ Γ ∩ Γ1 and hence Γ = Γ1 . We also note that the existence of a unique limit function of the iterates (f n ) of a compact holomorphic self-map f , with or without fixed-point, would imply locally uniform convergence of (f n ). It may be of interest to compare this fact with Lemma 3.5.38, where compactness of the self-map f is not assumed. Proposition 3.5.43. Let f be a compact holomorphic self-map on a bounded symmetric domain D. If the iterates (f n ) has a unique limit function h, then (f n ) converges locally uniformly to h. Proof. If (f n ) does not converge to h locally uniformly, then there exists ε > 0, and a subsequence (f nk ) of (f n ) such that kf nk − hkB ≥ ε

(k = 1, 2, . . .)

on some closed ball B strictly contained in D, where k · kB denotes the supremum norm on B. By [47, Lemma 1], (f nk ) has a subsequence (f mk ) converging locally uniformly to a holomorphic map on D, which must be h by the uniqueness assumption. This contradicts kf mk − hkB ≥ ε for all mk .

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Example 3.5.44. Let B be the open unit ball of the Hilbert space `2 of square summable complex sequences. Define f : B −→ B by       1 + x1 1 − x1 x1 1 − x1 x2 f (x1 , x2 , . . .) = , , ,... 2 2 2 2 3    1 + x1 1 − x1  x1 x2 = , 0, 0, . . . + 0, , , . . . . 2 2 2 3 Then f is holomorphic and fixed-point free. Moreover, f is compact since it is the sum of two compact maps. The iterates (f n ) converge to h(·) = (1, 0, 0, . . .). Finally, returning to the previous Conjecture for bounded symmetric domains, we prove the following result for domains of finite rank. Theorem 3.5.45. Let D be a bounded symmetric domain of finite rank in a complex Banach space V and let f : D −→ D be a fixed-point free compact holomorphic map. Then there is one single boundary component Γ in the boundary ∂D such that for each limit function h of (f n ), we have h(D) ⊂ Γ whenever h(D) is weakly closed. More precisely, there is a boundary point ξ ∈ ∂D of the form ξ=

m X

αj ej

(αj > 0, m ≤ p = rank D)

j=1

for some orthogonal minimal tripotents e1 , . . . , em ∈ ∂D, such that for each limit function h of (f n ) with weakly closed range, we have h(D) ⊂ Γe , where Γe is the boundary component of e = e1 + · · · + em . Remark 3.5.46. The conjecture would be completely proved if the conditions of D and h(D) being finite-rank and weakly closed respectively are removed from the above theorem. Proof. Let p = rank D and let ξ =

Pm

j=1 αj ej

be the boundary point

obtained in Theorem 3.5.27, where αj > 0, m ≤ p and e1 , . . . , em are

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orthogonal minimal tripotents. Let e = e1 + · · · + em which is a tripotent in ∂D. Let h be a limit function such that h(D) is weakly closed. Since f is a compact map, the remark following Lemma 3.5.35 implies h(D) ⊂ ∂D. By Lemma 3.4.11, h(D) is contained in a boundary component Γu of D for some tripotent u ∈ ∂D. For n = 1, 2, . . ., pick yn in the horoball H(ξ, n) = S0 (ξ, n). By f -invariance, we have h(yn ) ∈ S(ξ, n), which, by Theorem 3.5.27, is of the form h(yn ) =

m X j=1

1/2 m r m r X X σj n σj n σj n ej +B  ej , ej  (wn ) 1 + σj n 1 + σj n 1 + σj n 

j=1

j=1

for some wn ∈ D. Let (wnk ) be a subsequence of (wn ) weakly converging to w ∈ D, say. Then the sequence (h(ynk )) weakly converges to  1/2   m m m m m X X X X X ej + B  ej , ej  (w) = e j + P0  ej  (w) j=1

j=1

j=1

j=1

j=1

⊂ e + P0 (D) = Γe where Γe is the boundary component in ∂D containing the tripotent e (cf. Example 3.4.15). Since h(D) is weakly closed, we have ∅ = 6 h(D) ∩ Γe ⊂ Γu ∩ Γe and Γu meets either Γe or a boundary component of ∂Γe . By Example 3.4.15, the latter is also a boundary component of D. It follows that either Γu = Γe or Γu is a boundary component of ∂Γe , that is, Γu ⊂ Γe which gives h(D) ⊂ Γe . In the special case of a finite product of Hilbert balls, a careful examination reveals that the preceding theorem has been proved in [49, Theorem 3.2], which generalises Herv´e’s result for the bidisc.

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We close the section with an example of limit functions of which the image can be a singleton or a whole boundary component, which is not closed. Example 3.5.47. Let D be a finite-rank bounded symmetric domain of rank p. Pick any non-zero a ∈ D, with spectral decomposition a = α1 e1 + · · · + αp ep , where kak = α1 ≥ · · · ≥ αp ≥ 0. Let ga : D → D be the M¨obius transformation induced by a, which is not a compact map if D is infinite dimensional. Let x = β1 e1 + β2 e2 + · · · + βp ep , where β1 , β2 , . . . , βp ∈ D so that x ∈ D. By orthogonality, we have x a = (β1 e1 + β2 e2 + · · · + βp ep ) (α1 e1 + · · · + αp ep ) = β1 α1 e1 e1 + · · · + βp αp ep ep and (x a)n (x) = β1n+1 α1n e1 +· · ·+βpn+1 αpn ep for n = 1, 2, . . . . It follows that ga (x) = a + B(a, a)1/2 (1 + x a)−1 (x) = a + B(a, a)1/2 (1 − x a + (x a)2 − (x a)3 + · · · )(x) = a + B(a, a)1/2 (β1 e1 + β2 e2 + · · · + βp ep − (β12 α1 e1 + · · · + βp2 αp ep ) + · · · ) = a + B(a, a)1/2 [(1 − β1 α1 + β12 α12 + · · · )β1 e1

= = = =

+ · · · + (1 − βp αp + βp2 αp2 + · · · )βp ep ]   βp ep β1 e1 1/2 + ··· + a + B(a, a) 1 + β 1 α1 1 + βp αp 2 (1 − αp2 )βp ep (1 − α1 )β1 e1 α1 e1 + · · · + αp ep + + ··· + 1 + β1 α1 1 + βp αp αp + βp α1 + β1 e1 + · · · + ep 1 + α1 β 1 1 + αp βp g α1 (β1 )e1 + · · · + g αp (βp )ep

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where g αj is the M¨ obius transformation on the complex disc D, induced by αj for j = 1, . . . , p. If αj = 0, then g αj is the identity map. If αj > 0, then the iterates (g nαj ) converge locally uniformly to the constant map with value αj /|αj | = 1. Hence the iterates gan (x) = g nα1 (β1 )e1 + · · · + g nαp (βp )ep

(n = 2, 3, . . .)

converge to  e1 + γ2 e2 + · · · + γp ep ,

γj =

1 βj

(αj > 0) (αj = 0)

(j = 2, . . . , p).

In particular, if αj > 0 for all j, then the iterates (gan ) converge pointwise to the constant map g(·) = ξ = e1 + · · · + ep , in which case h(D) = {ξ} for every limit function h. On the other hand, if J = {j : αj > 0} is a proper subset of {1, . . . , p}, then lim gan (x) = n

where e =

P

j∈J

X

ej +

j∈J

X

β j e j ∈ e + De

j ∈J /

ej is a tripotent in ∂D and De = V0 (e) ∩ D. It follows

that, in this case, the image of every limit function h of (gan ) is the whole boundary component e + De since for any e + z ∈ e + De with P z ∈ De and spectral decomposition z = j ∈J / βj uj , we have h(

X j∈J

αj ej +

X j ∈J /

βj uj ) = e +

X

βj uj .

j ∈J /

Notes. The Carath´eodory distance was introduced by Carath´eodory [33], and the Kobayashi distance was first introduced in [107] for finite dimensional complex manifolds. In Proposition 3.5.17, the completeness of the Carath´eodory distance on a bounded homogeneous domain has been proved in [172, p. 279]. The domination of the Bergman metric

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over the Carath´eodory metric stated in Example 3.5.13 has been proved essentially in the paper [122]. For more details of invariant metrics on infinite dimensional domains, we refer to the book [63]. A succinct exposition of iteration of a holomorphic self-map on the disc D and historical remarks has been given in [29]. For Euclidean balls in Cn , this topic has been treated thoroughly in [2, Chapter 2.2]. Various forms of generalisation of Wolff’s theorem and the Denjoy-Wolff theorem to other domains in higher dimensions have been shown by many authors (see, for example, references listed in [2, 38, 39, 50, 145, 146]). The invariant domains for finite dimensional bounded symmetric domains obtained in [129] have a similar form to the one in Theorem 3.5.27. Ellipsoids as invariant domains in Hilbert balls has been shown in [67]. The unified treatment for all bounded symmetric domains in Theorem 3.5.22 was given in [50]. Horospheres have also been used to extend the Denjoy-Wolff theorem to other domains. For instance, a notion of horospheres has been used in [26] and [4] to prove a DenjoyWolff theorem for bounded strictly convex domains and weakly convex domains in Cn , respectively. The only strictly convex bounded symmetric domains are the Hilbert balls. Nevertheless, the result in [4] makes use of the concept of a sequence horosphere. For polydiscs, the sequence horospheres with pole at the origin [4, p. 1516] are the same as our horoballs (cf. [49, Proposition 2.4]). Equivalent conditions in Theorem 3.5.40 for the Denjoy-Wolff theorem have been established in [49], where one only assumes that the self-maps contract the Kobayashi distance, but need not be holomorphic. Example 3.5.44 is taken from [47] and, Example 3.5.47 as well as Theorem 3.5.45 have been shown in [50]. Needless to say, the study of holomorphic dynamics on bounded symmetric domains is incomplete and the closely related topic of angular derivatives in infinite dimension has yet to be explored further.

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3.6

Siegel domains

The holomorphic equivalence of the open unit disc and the upper halfplane in C, via the Cayley transform, is fundamental in classical complex analysis. Siegel domains generalise the notion of the upper halfplane and play an important role in several complex variables. Indeed, a seminal result of Vinberg, Gindikin and Piatetski-Shapiro [171] asserts that every bounded homogeneous domain in Cd is biholomorphically equivalent to a homogeneous Siegel domain. In this section, we discuss bounded symmetric domains which are biholomorphic to a Siegel domain, from a Jordan perspective. Given a real Banach space V , one can equip its complexification Vc = V ⊕ iV with a norm k · kc so that (Vc , k · kc ) is a complex Banach space and the isometric embedding v ∈ V 7→ (v, 0) ∈ V ⊕ iV identifies V as a real closed subspace of Vc . Although there are many choices of the norm k · kc , they are all equivalent to the `∞ -norm k · kmax if we require ku + ivkc ≥ ku + ivkmax := max(kuk, kvk)

(u, v ∈ V )

by the open mapping theorem. We will always assume Vc is equipped with such a norm and by a slight abuse of language, call (Vc , k · kc ) the complexification of V . We denote the conjugation in Vc by u + iv := u − iv so that the imaginary part Im z of an element z ∈ Vc is given by Im z = 1 2 (z

− z).

Definition 3.6.1. Let V be a real Banach space with complexification Vc and let Ω ⊂ V be a (non-empty) open cone. Let W be a complex Banach space and F : W × W −→ Vc a continuous mapping, which is

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conjugate linear in the first variable, linear in the second variable and satisfies F (w1 , w2 ) = F (w2 , w1 ). The set D(Ω, F ) := {(z, w) ∈ Vc ⊕ W : Im z − F (w, w) ∈ Ω} is called a Siegel domain (of the second kind). If W = {0}, then D(Ω, F ) reduces to D(Ω) := {z ∈ Vc : Im z ∈ Ω} = V ⊕ iΩ which is called a tube domain over the cone Ω (or a Siegel domain of the first kind). Example 3.6.2. The motivating example of Siegel domains is of course the complex upper half-plane which is a tube domain over (0, ∞). More generally, let V be the space of n×n real symmetric matrices and Ω ⊂ V the cone of positive definite matrices. The tube domain D(Ω) over Ω is also called the Siegel upper half-plane (of degree n). To discuss Siegel domains, we begin with cones and partial ordering. Let Ω be an open cone in a real Banach space V with norm k · k. Then we have int Ω = Ω by the following lemma. Lemma 3.6.3. Let C be an open convex set in a real topological vector space V . Then int C = C. Proof. There is nothing to prove if C is empty. Pick any q ∈ C. Let p ∈ int C. Then p is an internal point of C, that is, every line through p meets C in a set containing an interval around p (cf. [58, p. 410, 413]). In particular, for the line joining p and q, there exists δ ∈ (0, 1) such that p ± δ(q − p) ∈ C. Since C is open and q is an interior point of C,

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we have λ(p − δ(q − p)) + (1 − λ)q ∈ C for 0 < λ < 1 (cf. [58, p. 413]). Hence p=

δ 1 (p − δ(q − p)) + q ∈ C. 1+δ 1+δ

Trivially, V is a cone in itself. In the sequel, we shall exclude this case since Siegel domains of interest to us are the ones biholomorphic to bounded domains, but D(V ) = V ⊕ iV is not biholomorphic to a bounded domain! If Ω is an open cone properly contained in V , then we must have 0∈ / Ω although the closure Ω contains 0. The closure Ω of Ω is also a cone, which induces a partial ordering ≤ in V so that x ≤ y ⇔ y − x ∈ Ω. We also write y ≥ x for x ≤ y. As usual, a continuous linear functional f ∈ V ∗ is called positive if f (Ω) ⊂ [0, ∞). By the Hahn-Banach separation theorem, we have Ω = {v ∈ V : f (v) ≥ 0 for each f ∈ V ∗ with f (Ω) ⊂ [0, ∞)}. We note that each element e ∈ Ω is an order unit, that is, for each v ∈ V , we have −λv ≤ v ≤ λe for some λ > 0. Indeed, since Ω is open, e − Ω is a neighbourhood of 0 ∈ V and therefore one can find r > 0 such that B(0, r) = {x ∈ V : kxk < r} ⊂ e − Ω. For v 6= 0, we have ±(r/2kvk)v ∈ B(0, r) which implies −

2kvk 2kvk e≤v≤ e. r r

(3.46)

The preceding argument also implies V = Ω − Ω.

(3.47)

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An order unit e ∈ Ω induces a semi-norm k · ke on V , defined by kxke = inf{λ > 0 : −λe ≤ x ≤ λe}

(x ∈ V )

which satisfies −kxke e ≤ x ≤ kxke e

(3.48)

{x ∈ V : kxke ≤ 1} = {x ∈ V : −e ≤ x ≤ e}.

(3.49)

and

Since {x ∈ V : kxke = 0} = Ω ∩ −Ω, the semi-norm k · ke is a norm if and only if Ω ∩ −Ω = {0}. Definition 3.6.4. An open cone Ω in a real Banach space V is called regular if Ω ∩ −Ω = {0} (in which case, Ω is properly contained in V ). If Ω is a regular cone, then as noted above, k · ke is a norm, called the order-unit norm induced by e. All order-unit norms induced by elements in Ω are mutually equivalent. Henceforth, let Ω be a regular open cone in V . It follows from (3.49) that every linear map ψ : V −→ V which is positive, meaning ψ(Ω) ⊂ Ω, is continuous with respect to the order-unit norm k · ke and moreover, kψke = kψ(e)ke , where the former denotes the norm of ψ with respect to k · ke . In particular, if ψ : V −→ R is a positive linear functional, then kψke = ψ(e). A positive linear map ψ : V −→ V is an isometry if and only if ψ(e) = e [40, Proposition 2.3]. Lemma 3.6.5. Let Ω be a regular open cone in a real Banach space (V, k · k), partially ordered by Ω. Then for each e ∈ Ω, the order-unit norm k · ke satisfies k · ke ≤ ck · k. for some c > 0.

(3.50)

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Proof. Let v ∈ V \{0}. By (3.46), we have kvke ≤ (2/r)kvk for some r > 0. Let (V, k · ke ) denote the vector space V equipped with the orderunit norm k·ke , and (V, k·ke )∗ its dual space. It follows from (3.50) that every k · ke -continuous linear functional on V is also k · k-continuous. On the other hand, given f ∈ V ∗ satisfying f (e) = 1 = kf ke , then f is positive and hence continuous with respect to the norm k · ke . Denote the state space (with respect to the order unit e) by S = {f ∈ (V , k · ke )∗ : f (e) = 1 = kf ke } = {f ∈ V ∗ : f (e) = 1, f is positive}

(3.51)

which is a weak* compact convex set in the dual V ∗ and we have kxke = sup{|f (v)| : f ∈ S}

(x ∈ V )

(cf. [77, Lemma 1.2.5]). Lemma 3.6.6. Let Ω be a regular open cone in a real Banach space V and let e ∈ Ω, which induces an order-unit norm k · ke on V . Then we have Ω=

\

f −1 (0, ∞).

f ∈S

Proof. Given that V is partially ordered by the closure Ω, we have \ Ω= f −1 [0, ∞) (3.52) f ∈S

since f /f (e) ∈ S for each non-zero positive linear functional f ∈ V ∗ . Let a ∈ Ω. Then for each f ∈ S, we have f (a) > 0 since a is an order unit, which implies e ≤ λa for some constant λ > 0 and hence 1 ≤ λf (a). This proves Ω⊂

\ f ∈S

f −1 (0, ∞).

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Conversely, let a ∈ V and f (a) > 0 for all f ∈ S. Then a ∈ Ω and by weak* compactness of S, one can find some δ > 0 such that f (a) ≥ δ for all f ∈ S. Let     δ δ ⊂ x ∈ V : kx − ake < N = x ∈ V : kx − ak < 2c 2 where c > 0 is given in (3.50). Then N is an open neighbourhood of a and, N ⊂ Ω since δ δ δ x ∈ N ⇒ − e ≤ x − a ⇒ a − e ≤ x ⇒ ≤ f (x) 2 2 2 0

for all f ∈ S. Hence a belongs to the interior Ω of Ω and, as Ω is open 0

and convex, we have a ∈ Ω = Ω . We see from Lemma 3.6.5 that if Ω is a regular open cone in a finite dimensional Banach space V , then the order-unit norm k · ke induced by e ∈ Ω is equivalent to the norm of V , by the open mapping theorem. In fact, the equivalence of the two norms is related to the basic concept of a normal cone in the theory of partially ordered topological vector spaces. Lemma 3.6.7. Let Ω be a regular open cone in a real Banach space V with norm k · k. Then the order-unit norm k · ke induced by e ∈ Ω is equivalent to k · k if and only if Ω is a normal cone in V , that is, there is a constant γ > 0 such that 0 ≤ x ≤ y implies kxk ≤ γkyk for all x, y ∈ V . In particular, (V, k · ke ) is a Banach space if Ω is a normal cone. Proof. By the definition of the order-unit norm k·ke , we have 0 ≤ x ≤ y in V implies kxke ≤ kyke . Hence Ω is normal in (V, k · ke ). If k · k is equivalent to k · ke , then evidently Ω is also normal in (V, k · k). Conversely, let Ω be normal in (V, k · k). We have already noted in (3.50) that k · ke ≤ ck · k for some constant c > 0. By (3.49) and

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normality of Ω, there is a constant γ > 0 such that kxke ≤ 1 ⇔ −e ≤ x ≤ e ⇒ 0 ≤ x + e ≤ 2e ⇒ kx + ek ≤ 2γkek ⇒ kxk < 2(γ + 1)kek which implies k · k ≤ 2(γ + 1)kekk · ke and the equivalence of k · k and k · ke . We note that a self-dual cone Ω in a Hilbert space H is regular, and also normal since it has been shown in [40, Lemma 2.6] that the order-unit norms induced by elements in Ω are all equivalent to the norm of H. The partially ordered Banach spaces related to symmetric Siegel domains are the JB-algebras. The partial ordering ≤ in a JB-algebra A is defined by the closed cone {a2 : a ∈ A} and if A is unital, then the identity e is an order-unit in the interior of the cone, which is regular, and the induced order-unit norm coincides with the original norm of A [77, Proposition 3.3.10]. We now discuss Siegel domains which are biholomorphic to bounded symmetric domains in Banach spaces. Theorem 3.6.8. Let Ω be a regular open cone in a real Banach space V . The following conditions are equivalent. (i) The tube domain V ⊕ iΩ is biholomorphic to a bounded symmetric domain. (ii) Vc is a unital JB*-algebra with an equivalent norm and Ω = {x2 : x ∈ V }.

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(iii) V is a unital JB-algebra with an order-unit norm and Ω = {x2 : x ∈ V }. In this case, the tube domain V ⊕ iΩ is biholomorphic to the open unit ball D of the JB*-algebra Vc via the Cayley transform z ∈ V ⊕ iΩ 7→ (z − ie)(z + ie)−1 ∈ D where e is the identity of Vc , and the symmetry at ie ∈ V ⊕ iΩ is given by the map z 7→ −z −1 . Proof. (i) ⇒ (ii). We make use of the results in [103]. By Theorem 3.2.18, there is a Jordan triple product on Vc , induced by a symmetry of V ⊕ iΩ, such that Vc becomes a JB*-triple in an equivalent norm k · ksp . Let e ∈ Ω. Using the symmetry sie at ie ∈ V ⊕ iΩ, it has been shown in [103, Theorem 2.5] that Vc is the Jordan triple associated to the symmetric domain V ⊕ iΩ, where the associated Jordan triple product {·, ·, ·} in Vc satisfies {e, a, e} = a for all

a ∈ V.

(3.53)

By Remark 3.2.6, this Jordan triple product coincides with the triple product of the JB*-triple (Vc , k · ksp ). It follows from (3.53) that e is a tripotent in Vc and P2 (e)(z) = z for all z ∈ Vc . Hence, by Example 2.4.18, Vc = P2 (e)(Vc ) is a JB*algebra with identity e, Jordan product z ◦ w = {z, e, w} and involution z ∗ = {e, z, e}. Moreover, it has been shown in [103, (4.1), (4.6)] that Ω = {x2 : x ∈ V } and the symmetry at ie is the inverse map z ∈ V ⊕ iΩ 7→ −z −1 ∈ V ⊕ iΩ. (ii) ⇔ (iii). This has been proved in [179]. (iii) ⇒ (i). By assumption, the identity e of the JB-algebra V is an order-unit in the interior int{x2 : x ∈ V } = int Ω = Ω. By Lemma

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3.6.5, the complete order-unit norm k · ke is equivalent to the original norm of V . Hence the complexification Vc of (V, k · ke ) is a JB*-algebra in a norm k · k equivalent to the norm k · kc of Vc . By Theorem 3.2.20, the open unit ball D = {z ∈ Vc : kzk < 1} is a bounded symmetric domain. To complete the proof, we show that the tube domain V ⊕ iΩ is biholomorphic to D via the Cayley transform z ∈ V ⊕ iΩ 7→ (z − ie)(z + ie)−1 ∈ D.

(3.54)

By Corollary 2.4.20, z + ie ∈ V ⊕ iΩ is invertible. We need to show k(z − ie)(z + ie)−1 k < 1. Let z = v + iω ∈ V ⊕ iΩ, where ω is a positive invertible element in the JB-algebra V . Let B be the closed ∗-subalgebra of Vc generated by the self-adjoint elements v, ω, e. Then B is isometrically ∗-isomorphic to a closed ∗-subalgebra of the JB*-algebra L(H) of bounded linear operators on some Hilbert space H, with inner product h·, ·i, (see, for example, [179, Corollary 2.2]). Hence we can identify v and ω as selfadjoint operators on H. Let ξ ∈ H and η = (z + ie)−1 ξ ∈ H. Then we have k(z + ie)ηk2 − k(z − ie)ηk2 = h(v − i(e + ω))(v − i(e + ω))η, ηi − h(v + i(e − ω))(v − i(e − ω))η, ηi =

4hωη, ηi

which gives kξk2 − k(z − ie)(z + ie)−1 ξk2 = 4hωη, ηi = 4h(z ∗ − ie)−1 ω(z + ie)−1 ξ, ξi where hωη, ηi > 0 by positivity and invertibility of ω. Therefore we have proved IH − ((z − ie)(z + ie)−1 )∗ (z − ie)(z + ie)−1 = 4(z ∗ − ie)−1 ω(z + ie)−1 ,

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where IH ∈ L(H) is the identity, and it follows that k(z−ie)(z+ie)−1 k < 1. Hence the Cayley transform in (3.54) is well-defined and holomorphic. In fact, it is biholomorphic since the holomorphic map γ : z ∈ D 7→ i(e + z)(e − z)−1 = −ie + 2i(e − z)−1 ∈ V ⊕ iΩ is its inverse. Indeed, the inverse (e − z)−1 exists since kzk < 1. Also, 0 ≤ z ∗ z ≤ e implies i(e + z)(e − z)−1 − (i(e + z)(e − z)−1 )∗ = 2i(e − z ∗ )−1 (e − z ∗ z)(e − z)−1 and i(e + z)(e − z)−1 ∈ V ⊕ iΩ. So γ is well-defined and γ((z − ie)(z + ie)−1 ) = −ie + 2i(e − (z − ie)(z + ie)−1 )−1 = −ie + 2i(2i(z + ie)−1 )−1 = z for z ∈ V ⊕ iΩ. Remark 3.6.9. In [144], Siegel domains are defined over regular open cones and are finite dimensional. They are biholomorphic to bounded domains (see also [108, Chapter II, Sec. 5]) but this need not be true for infinite dimensional Siegel domains (cf. [72]). In fact, condition (i) in the preceding theorem is equivalent to saying that V ⊕ iΩ is biholomorphic to a bounded domain and there is a symmetry at one point in V ⊕ iΩ, for the latter condition already implies that the domain is homogeneous (and hence symmetric), which has been shown in [103, Theorem 2.5]. In this case, Ω is actually linearly homogeneous (cf. [24, (2.1)]). Remark 3.6.10. Considering Vc as a JB*-triple in the proof of Theorem 3.6.8, the Cayley transform can be derived from the vector field ∂ (e − {z, e, z}) ∂z on the tube domain, as shown in [103]. In fact, the ∂ exponential exp(− πi 4 (e − {z, e, z}) ∂z ) is the biholomorphic map z 7→

i(z − ie)(z + ie)−1 on V ⊕ iΩ.

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Bounded symmetric domains We have seen in Theorem 3.6.8 that the open unit ball of a JB*-

algebra is biholomorphic to a tube domain if and only if the algebra has an identity. A natural question is: can the open unit ball of a JB*-triple be realised as a Siegel domain? This question has been answered in [103] to which we refer the interested reader for a proof of the following theorem. Theorem 3.6.11. Let Z be a JB*-triple. Then the open unit ball of Z is biholomorphic to a Siegel domain (of the second kind) if and only if Z has a maximal tripotent e, in which case the Siegel domain can be constructed from the Peirce decomposition Z = Z2 (e) ⊕ Z1 (e): D = {(z, w) ∈ Z2 (e)sa ⊕Z1 (e) : Im z−F (w, w) ∈ {{x, e, x} : x ∈ Z2 (e)}} where Z2 (e)sa = {z ∈ Z2 (e) : z = {e, z, e}} and F (w1 , w2 ) = 2{w1 , w2 , e}. If we call a Siegel domain symmetric whenever it is biholomorphic to a bounded symmetric domain, then one can summarise the preceding two theorems by saying that symmetric Siegel domains of the first kind are exactly (via biholomorphism) the open unit balls of unital JB*algebras, those of the second kind are the open unit balls of JB*-triples containing a maximal tripotent. Tube domains biholomorphic to bounded symmetric domains in Hilbert spaces are exactly the ones over linearly homogeneous self-dual cones. By Theorem 2.4.14, these cones are of the form Ω = int {x2 : x ∈ H} for some unital JH-algebra H. In fact, unital JH-algebras also carry the structure of a JB-algebra, and can be classified. Lemma 3.6.12. Let A be a unital JH-algebra. Then A decomposes into a finite direct sum A = A1 ⊕ · · · ⊕ Ad

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of unital JB-algebras A1 , . . . , Ad , where each Aj is either a finite dimensional JB-algebra or a real spin factor, for j = 1, . . . , d. Proof. Let e ∈ A be the identity. Then e is an order-unit in the linearly homogeneous self-dual cone int {a2 : a ∈ A} and the order-unit norm k · ke is equivalent to the Hilbert space norm k · kh . We show that A, when equipped with the order-unit norm, is a unital JB-algebra. By [77, Proposition 3.1.6], it suffices to show that −e ≤ a ≤ e implies 0 ≤ a2 ≤ e for each a ∈ A. We first observe that, for each projection p ∈ A, we have p, e − p ≤ e. Indeed, we have p, e − p ∈ C since p = p2 and e − p = (e − p)2 . Let −e ≤ a ≤ e. Then by [37, p.108], there are mutually orthogonal (nonP zero) projections {pk }k in A such that a = ∞ k=1 λk pk . By self-duality, 0 ≤ e − a implies 0 ≤ he − a, pk i and λk hpk , pk i = ha, pk i ≤ he, pk i = hpk , pk i for all k. Likewise 0 ≤ e + a implies −hpk , pk i ≤ λk hpk , pk i for all k. Hence we have −1 ≤ λk ≤ 1 and λ2k ≤ 1. It follows that a2 =

∞ X k=1

λ2k pk ≤

∞ X

pk ≤ e.

k=1

This proves that (A, k · ke ) is a unital JB-algebra. Moreover, it is a reflexive Banach space since it is isomorphic to the Hilbert space (A, k · kh ). By Corollary 3.3.6, (A, k · ke ) must be a finite `∞ -sum of unital JB-algebras, each of which is either a finite dimensional algebra or a real spin factor. The classification of finite dimensional unital JB-algebras has been listed in (2.9).

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Corollary 3.6.13. Let Ω be a regular open cone in a real Banach space V . The following conditions are equivalent. (i) V ⊕ iΩ is biholomorphic to a bounded symmetric domain in a Hilbert space. (ii) V is a reflexive JB-algebra with an equivalent norm and Ω = {x2 : x ∈ V }. (iii) V is a unital JH-algebra with an equivalent norm and Ω = {x2 : x ∈ V }. (iv) V is a Hilbert space in an equivalent norm, where Ω is a linearly homogeneous self-dual cone. Proof. (i) ⇒ (ii). By Theorem 3.6.8 and condition (i), V carries the structure of a reflexive (unital) JB-algebra in an equivalent norm and Ω = {x2 : x ∈ V }. (ii) ⇒ (iii). By the proof of the preceding lemma, V is a finite `∞ L`∞ sum j Aj , where Aj is either a real spin factor or a finite dimensional JB-algebra. Each Aj ia a unital JH-algebra (in an equivalent norm) and L hence the `2 -sum V = `j2 Aj is a unital JH-algebra. (iii) ⇒ (i). This follows from the fact that a unital JH-algebra is also a JB-algebra in an equivalent norm, shown in Lemma 3.6.12. (iii) ⇔ (iv). See Theorem 2.4.14. We have noted after Theorem 2.4.14 that a linearly homogeneous self-dual cone in a real Hilbert space carries the structure of a Riemannian symmetric space. In fact, if a tube domain V ⊕iΩ is biholomorphic to a bounded symmetric domain, we have seen that V carries the structure of a unital JB-algebra and Ω is the regular open cone of squares in V . By Remark 3.6.9, Ω is linearly homogeneous. It is also a normal

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cone since the norm of V is the order-unit norm k · ke induced by the identity e ∈ Ω. Moreover, we can equip Ω with a compatible tangent norm b : T Ω −→ [0, ∞), induced by k · ke . With respect to this tangent norm, Ω is a real symmetric Banach manifold, where the symmetry at e is the inverse map x ∈ Ω 7→ x−1 ∈ Ω. To see this, we first define the tangent norm b, following [168, 12.31]. Let L(V ) be the real Banach Lie algebra of bounded linear operators on V , in the usual Lie brackets [S, T ] = ST − T S

(S, T ∈ L(V )).

The open subgroup GL(V ) of L(V ), consisting of invertible elements, is a real Banach Lie group with Lie algebra L(V ). The linear maps g ∈ GL(V ) satisfying g(Ω) = Ω form a subgroup of GL(V ), denoted by G(Ω) = {g ∈ GL(V ) : g(Ω) = Ω}.

(3.55)

We shall call G(Ω) the linear automorphism group of Ω so that linear homogeneity of Ω is saying that G(Ω) acts transitively on Ω. An element g ∈ GL(V ) belongs to G(Ω) if and only if g(Ω) = Ω. Hence G(Ω) is a closed subgroup of GL(V ) and can be topologised to a real Banach Lie group with Lie algebra g(Ω) = {X ∈ L(V ) : exp tX ∈ G(Ω), ∀t ∈ R}

(3.56)

(cf. [168, p. 387]). By a remark before Lemma 3.6.5, a linear isomorphism g ∈ G(Ω) satisfying g(e) = e is an isometry with respect to the order unit norm k · ke and hence one can define b(p, v) = kh(v)ke

((p, v) ∈ T Ω)

(3.57)

for any h ∈ G(Ω) satisfying h(p) = e. The tangent norm b is G(Ω)invariant, in other words, each g ∈ G(Ω) is a b-isometry. Indeed, given

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g ∈ G(Ω), and (p, v) ∈ T Ω with b(p, v) = kh(v)ke for some h(p) = e, pick h1 ∈ G(Ω) such that h1 (g(p)) = e. Then h1 gh−1 (e) = e and hence h1 gh is an isometry with respect to k · ke , which gives b(g(p), g 0 (p)(v)) = b(g(p), g(v)) = kh1 (g(v))ke = kh1 gh−1 (h(v))ke = kh(v)ke . Now, in the case where V ⊕ iΩ is biholomorphic to a bounded symmetric domain, the symmetry z ∈ V ⊕ iΩ 7→ −z −1 ∈ V ⊕ iΩ at ie restricts to the inverse map z ∈ iΩ 7→ −z −1 ∈ iΩ since z −1 = Q−1 z (z) ∈ iV ∩ (V ⊕ iΩ) = iΩ, where Qz = {z, ·, z} is the quadratic operator on the Jordan algebra V ⊕ iV and Qz (iV ) = iV for z ∈ iΩ, by invertibility. It follows that the inverse map se : x ∈ Ω 7→ x−1 = Q−1 x (x) ∈ Ω is a symmetry at e for the G(Ω)-invariant tangent norm b since s0e (x) = −Q−1 and Q−1 ∈ G(Ω) (cf. [77, 3.2.11, 3.3.6]). We have therefore x x proved, in view of Remark 3.6.9, the following corollary. Corollary 3.6.14. Let Ω be a regular open cone in a real Banach space V . For the two conditions below, we have (i) ⇒ (ii). (i) V ⊕ iΩ is biholomorphic to a bounded symmetric domain. (ii) Ω is a normal linearly homogeneous cone and a symmetric Banach manifold in a G(Ω)-invariant tangent norm. Problem 3.6.15. The formulation of the preceding corollary hints at the question of the converse (ii) ⇒ (i). Although there is tangible evidence to suggest the validity of this implication, it remains unresolved. For a regular open cone Ω in a finite dimensional Euclidean space V , it has been shown in [158] and [166] that if Ω is linearly homogeneous and

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also a symmetric space in the canonical Riemannian metric defined in (2.52), then it is self-dual and hence V ⊕ iΩ is indeed biholomorphic to a bounded symmetric domain by Corollary 3.6.13. However, it seems that the implication (ii) ⇒ (i) could be false without the homogeneity condition on Ω. We note that, besides the Riemannian metric, one can ask the question of symmetry with respect to other compatible tangent norms on a linearly homogeneous regular open cone Ω in a real Banach space V. By Lemma 3.6.7, the order-unit norms induced by the order units in Ω are all equivalent to the norm of V and we can define a tangent norm τ : T Ω −→ [0, ∞) by τ (p, v) = kvkp

((p, v) ∈ Ω × V )

(3.58)

where k · kp denotes the order-unit norm induced by the order unit p ∈ Ω. To see that τ is continuous, let (pn ) converge to p in Ω and (vn ) converge to v in V . Given 1 > ε > 0, kpn − pkp → 0 implies −εp ≤ pn − p ≤ εp and (1 − ε)p ≤ pn ≤ (1 + ε)p from some n onwards, which gives −(1 + ε)kvn kpn p ≤ −kvn kpn pn ≤ vn ≤ kvn kpn pn ≤ (1 + ε)kvn kpn p and hence kvn kp ≤ (1 + ε)kvn kpn . Likewise p ≤ kvn kp 1−ε

pn 1−ε

implies kvn kpn ≤

and therefore 1−ε≤

kvn kp ≤ 1 + ε. kvn kpn

Since kvn kp → kvkp , we conclude kvn kpn → kvkp . In fact, the tangent norm τ coincides with b : T Ω −→ [0, ∞) defined in (3.57) (in the setting of unital JB-algebras). This follows from the

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fact that τ is G(Ω)-invariant, which implies τ = b. For if h ∈ G(Ω), then we have, for v ∈ Tp Ω = V , τ (h(p), h0 (p)(v)) = τ (h(p), h(v)) = kh(v)kh(p) = kvkp = τ (v, p) where the third identity follows from the equivalent conditions −λh(p) ≤ h(v) ≤ λh(p) ⇔ λp ≤ v ≤ λp

(λ > 0).

By [138, Lemma 1.3, Theorem 1.1], the integrated distance dτ of τ on Ω coincides with Thompson’s metric (x, y ∈ Ω)

dτ (x, y) = max{log M (x/y), log M (y/x)} where M (a/b) := inf{β > 0 : βa ≥ b}

(a, b ∈ Ω).

It is interesting that the distance dτ is related to the Carath´eodory distance on the corresponding tube domain V ⊕ iΩ. Let V ⊕ iΩ be biholomorphic to a bounded domain and let V+∗ = {f ∈ V ∗ : f (Ω) ⊂ (0, ∞)} be the cone of ‘strictly positive’ continuous linear functionals on V . Following Vessentini [169], we define a Carath´eodory-type distance δ : Ω × Ω → [0, ∞) by   f (x) ∗ δ(x, y) = sup log : f ∈ V+ f (y)

(x, y ∈ Ω).

By regularity of Ω, it can be shown that δ is a distance on Ω, which is invariant under all affine automorphisms of Ω [169, Proposition 4.3]. Further, δ actually coincides with the restriction of the Carath´eodory distance c on V ⊕ iΩ to iΩ: δ(x, y) = c(ix, iy)

(x, y ∈ Ω)

(cf. [169, Theorems I, II]), as well as the distance dτ which is shown below.

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Lemma 3.6.16. Let Ω be a linearly homogeneous regular open cone in a real Banach space V . Then the Carath´eodory-type distance δ on Ω coincides with dτ . Proof. For each u ∈ Ω, we denote the state space with respect to the order-unit u, with order-unit norm k · ku , by Su = {f ∈ V ∗ : f (u) = kf ku = 1} ⊂ V+∗ where kf ku = sup{|f (x)| : kxku ≤ 1}. For each f ∈ V+∗ , we have kf ku = f (u) and hence f /kf ku ∈ Su . We make use of the identity   f (x) ∗ δ(x, y) = sup log : f ∈ V+ f (y)

(x, y ∈ Ω)

where f (x)/kf ky f (x) = = f (x)/kf ky f (y) f (y)/kf ky

and f /kf ky ∈ Sy .

Hence δ can be written as δ(x, y) = sup{| log f (x)| : f ∈ Sy }     1 : f ∈ Sy = sup max log f (x), log f (x)     1 = log max sup{f (x) : f ∈ Sy }, sup : f ∈ Sy . f (x) We have sup{f (x) : f ∈ Sy } = kxky = inf{β > 0 : x ≤ βy} = M (y/x). We complete the proof by showing   1 M (x/y) = sup : f ∈ Sy . f (x) Indeed, y ≤ βx implies 1 ≤ βf (x) for each β > 0 and f ∈ Sy , which gives sup {1/f (x) : f ∈ Sy } ≤ M (x/y).

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To reverse the inequality, let µ = inf{f (x) : f ∈ Sy }. Since f (x) = kf kx > 0 for all f ∈ Sy , we have µ > 0 by weak* compactness of Sy . Observe that 1/µ = sup {1/f (x) : f ∈ Sy } and µy ≤ x since f (x) ≥ µ = µf (y) for all f ∈ Sy . It follows that 1/µ ≥ M (x/y) as desired. We conclude this section with a final remark relating to Problem 3.6.15, by the following proposition. Proposition 3.6.17. Let Ω be a regular open cone in a real Banach space V such that V ⊕ iΩ is biholomorphic to a bounded domain. Then the following conditions are equivalent. (i) V ⊕ iΩ is biholomorphic to a bounded symmetric domain. (ii) Ω admits a symmetry, with respect to a compatible tangent norm, which extends to a symmetry of the domain Ω ⊕ iV . If V is a Hilbert space, then these conditions are equivalent to Ω being a linearly homogeneous self-dual cone. Proof. We note that Ω ⊕ iV is biholomorphic to the tube domain D(Ω) = V ⊕ iΩ. If a symmetry s : Ω → Ω at e ∈ Ω extends to a symmetry se : Ω ⊕ iV → Ω ⊕ iV , then D(Ω) admits a symmetry and by Remark 3.6.9, D(Ω) is already biholomorphic to a bounded symmetric domain. Conversely, if V ⊕ iΩ is biholomorphic to a bounded symmetric domain, then V is a JB-algebra with identity e ∈ Ω, and the inverse map z ∈ Ω⊕iV 7→ z −1 ∈ Ω⊕iV restricts to the symmetry x ∈ Ω 7→ x−1 ∈ Ω, with respect to the tangent norm b defined in (3.57).

Notes. Theorem 3.6.8 was proved in [24] prior to the appearance of [99], without using the underlying JB*-triple structure of a bounded symmetric domain. This theorem can be viewed as an infinite dimensional extension of Koecher’s seminal work. Indeed, in his

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‘elementary approach to bounded symmetric domains’ (in finite dimensions) [111], Koecher showed that the tube domain over the open cone of squares of a finite dimensional JB-algebra, that is, a formally real Jordan algebra, is biholomorphic to a bounded symmetric domain obtained from the TKK construction. Finite dimensional linearly homogeneous self-dual cones are also discussed in [5] in connection with compactification of locally symmetric varieties. (Note added in proofs. The author has recently given an affirmative answer to Problem 3.6.15 in a preprint (arXiv:2006.06449) entitled ‘Siegel domains over Finsler symmetric cones’.)

3.7

Holomorphic homogeneous regular domains

The notion of a holomorphic homogeneous regular (HHR) complex manifold M (of finite dimension) has been introduced by Liu, Sun and Yau [119, Definition 7.2] in connection with the estimation of several canonical metrics on the moduli and Teichm¨ uller spaces of Riemann surfaces. It can be described by saying that a particular function σ : M → (0, 1] called the squeezing function, has a strictly positive lower bound (cf. [53]). These manifolds possess many important geometric properties (e.g. all classical metrics on them are equivalent) [119, 120] and have also been studied by several authors (see, for example, [53, 54, 62, 104, 182]) in the case of complex domains. In particular, it has been shown in [182] that a holomorphic homogeneous regular bounded domain D in Cn must be pseudoconvex and all bounded strongly convex domains in Cn are holomorphic homogeneous regular. It has been shown recently in [104] that actually all bounded convex domains in Cn are holomorphic homogeneous regular. The squeezing function on a bounded homogeneous domain in Cn is constant, by its holomorphic invariance, and has been computed ex-

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plicitly for the four series of classical Cartan domains in [115]. In view of these interesting works, it is natural to ask if they can be extended to the setting of infinite dimensional domains. In this section, we do just that. We extend the concept of a holomorphic homogeneous regular domain and generalise the aforementioned results to the infinite dimensional setting. In particular, we determine completely which bounded symmetric domains are HHR and compute the squeezing functions for all these domains, including the two exceptional domains, which were left untreated in [115]. The concept of the squeezing function for finite dimensional complex manifolds involves comparing a given manifold with various Euclidean balls via holomorphic embeddings. For infinite dimensional domains, we consider their holomorphic embeddings in Hilbert balls, as a natural infinite dimensional generalisation. Definition 3.7.1. A map f : D1 → D2 between two domains is called a holomorphic embedding of D1 into D2 if f (D1 ) is a domain in D2 and f is biholomorphic onto f (D1 ). The set of all holomorphic embeddings of D1 into D2 is denoted by Hemb (D1 , D2 ). Let D be a domain in a complex Banach space V . In this section, we denote by BH = {x ∈ H : kxk < 1} the open unit ball of a Hilbert space H. The set Hemb (D, BH ) of holomorphic embeddings may be empty. For instance, if D is the open unit ball of the Banach space `∞ of bounded complex sequences, then Hemb (D, BH ) = ∅ for any Hilbert ball BH . In fact, Hemb (D, BH ) 6= ∅ if and only if the ambient Banach space V of D is linearly homeomorphic to H. Indeed, if there is a holomorphic embedding f : D → BH , then V , as the tangent space at a point p ∈ D,

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must be linearly homeomorphic to H, which is the tangent space of f (D) at f (p). Conversely, if ϕ : V → H is a linear homeomorphism, then we have ϕ(D) ⊂ RBH for some R > 0, and for each p ∈ D, the map f : z ∈ D 7→ ϕ(z − p)/2R ∈ BH is a biholomorphic map onto the domain f (D) in BH , with f (p) = 0 and rBH ⊂ f (D) ⊂ BH for some r > 0. Given Hemb (D, BH ) 6= ∅, then for each p ∈ D, the set F(p, D) = {f ∈ Hemb (D, BH ) : f (p) = 0} is non-empty, as just noted. Hence we can define the squeezing function σD : D → (0, 1] by σD (p) =

sup {r > 0 : rBH ⊂ f (D)}. f ∈F (p,D)

The squeezing constant σ ˆD for D is defined to be σ ˆD = inf σD (p). p∈D

Both the squeezing function and squeezing constant are biholomorphic invariants. Remark 3.7.2. We note that, if Hemb (D, BH ) 6= ∅, then the definition of the squeezing function for a domain D ⊂ V does not depend on the chosen Hilbert ball BH . Indeed, if there is a holomorphic embedding of D into another Hilbert ball BK of a Hilbert space K, then the previous remarks imply that there is a continuous linear isomorphism T : H → K. Denote the inner products of H and K by h·, ·iH and h·, ·iK respectively. Let α : H ∗ → H and β : K → K ∗ be the canonical isometries. Then the linear isomorphism αT ∗ βT : H → H satisfies hαT ∗ βT x, yiH = hT x, T yiK

(x, y ∈ H)

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and the linear isomorphism T (αT ∗ βT )−1/2 : H → K is an isometry. It follows that the squeezing functions σD defined in terms BH and BK respectively are identical. We now extend the concept of a finite dimensional HHR manifold introduced in [119, 120] to infinite dimensional domains. A finite dimensional HHR domain is also called a domain with uniform squeezing property in [182]. Definition 3.7.3. A domain D in a complex Banach space V is called holomorphic homogeneous regular (HHR) if D admits a holomorphic embedding into some Hilbert ball BH and its squeezing function σD : D → (0, 1] has a strictly positive lower bound, that is, σ ˆD > 0. Remark 3.7.4. If D is an HHR domain in a Banach space V , then as noted previously, V must be linearly homeomorphic to a Hilbert space. We call V an isomorph of a Hilbert space. The class of these Banach spaces has been characterised by many authors, for instance, it has been shown in [116] that a Banach space is an isomorph of a Hilbert space if and only if it is of type 2 and cotype 2. We refer to [141, Chapter IV] for more details. We begin our discussion of infinite dimensional HHR domains by showing some properties of the squeezing function. We will make use of the Carath´eodory pseudo-distance cD on a domain D. We first show that the squeezing function is continuous. Proposition 3.7.5. Let D be a domain in a Banach space V linearly homeomorphic to a Hilbert space H. Then the squeezing function σD : D → (0, 1] is continuous. Proof. Let (zk ) be a sequence converging to a ∈ D. We show lim inf σD (zk ) ≥ σD (a) ≥ lim sup σD (zk ).

k→∞

k→∞

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Let 0 < 2ε < σD (a) and pick σD (a) ≥ ρ > σD (a) − ε such that there is a holomorphic embedding f : D → BH satisfying f (a) = 0 and ρBH ⊂ f (D). By continuity, we have kf (zk )k < ε for k > K, for some K > 0. Consider the holomorphic embedding fk : D → BH given by fk (ω) =

f (ω) − f (zk ) 1+ε

(ω ∈ D)

which satisfies fk (zk ) = 0 and ρ−ε BH ⊂ fk (D). 1+ε This gives σD (zk ) ≥

σD (a) − 2ε ρ−ε > 1+ε 1+ε

for k > K and hence limk→∞ inf σD (zk ) ≥ σD (a) since ε > 0 was arbitrary. For the upper limit, let 0 < 2ε < limk inf σD (zk ) and let fk : D → BH be a holomorphic embedding satisfying fk (zk ) = 0 and ρk BH ⊂ fk (D) for some σD (zk ) ≥ ρk > σD (zk ) − ε. Since the Carath´eodory pseudo-distance cD (zk , a) → 0 as k → ∞, we have tanh−1 kfk (a)k = cBH (0, fk (a)) ≤ cD (zk , a) → 0 and hence there exists some M > 0 such that kfk (a)k < ε for k > M . By analogous arguments as before, one obtains σD (a) ≥

σD (zk ) − 2ε ρk − ε > 1+ε 1+ε

for k > M , which gives σD (a) ≥ limk→∞ sup σD (zk ).

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that if there is a sequence (pk ) in a finite dimensional bounded domain D with limk σD (pk ) = 0, then the sequence admits a subsequence (pj ) converging to a boundary point p ∈ ∂D, this is not immediately clear for infinite dimensional domains. Nevertheless, one can still show, in infinite dimension, (pk ) has a subsequence (pj ) for which the distance d(pj , ∂D) to the boundary tends to 0. We prove a lemma first. Lemma 3.7.6. Let Ω be a bounded domain in an isomorph V of a Hilbert space H and ϕ : V → H a linear homeomorphism. Then there is a constant m > 0 such that for each q ∈ Ω satisfying BV (q, s) ⊂ Ω for some s > 0, we have σΩ (q) ≥

s m2 kϕkkϕ−1 k

.

Proof. By a translation, we may assume q = 0. Since Ω is bounded, we have Ω ⊂ BV (0, m) for some m > 0 and 1 BH (0, m) ⊂ ϕ(BV (0, m)) ⊂ BH (0, mkϕk) = mkϕkBH . (3.59) kϕ−1 k The restriction of ϕ to Ω, still denoted by ϕ, is a holomorphic embedding of Ω into mkϕkBH satisfying ϕ(q) = 0. It follows from (3.59) that s BH (0, m) ⊂ ϕ(BV (0, s)) ⊂ ϕ(Ω) ⊂ ϕ(BV (0, m)) ⊂ mkϕkBH . mkϕ−1 k Hence we have σΩ (q) ≥

s m2 kϕkkϕ−1 k

.

Lemma 3.7.7. Let (pk ) be a sequence in a bounded domain Ω in an isomorph V of a Hilbert space such that limk→∞ σΩ (pk ) = 0. Then there is a subsequence (pj ) of (pk ) such that lim d(pj , ∂Ω) = 0

j→∞

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and for each j, there exists a boundary point qj ∈ ∂Ω with kpj − qj k = d(pj , ∂Ω). Proof. Let (pk ) be the given sequence satisfying lim σΩ (pk ) = 0.

(3.60)

k→∞

Since the bounded domain Ω is relatively weakly compact in V , there is a subsequence (pj ) in Ω converging weakly to some point p ∈ Ω. (We do not know if the squeezing function σΩ is weakly continuous on Ω.) Let rj = d(pj , ∂Ω) denote the distance from pj to the boundary ∂Ω. We first show that limj→∞ rj = 0. Otherwise, we may assume (by choosing a subsequence) rj ≥ s,

for some

s>0

for all j. For all z ∈ ∂Ω, we have kz − pj k ≥ rj . Observe that BV (pj , rj ) ⊂ Ω, for if there exists some ω ∈ BV (pj , rj )\Ω, then we must have ω ∈ / Ω. Therefore the (real) line joining pj and ω must intersect ∂Ω at a point z0 say, which gives a contradiction that rj ≤ kz0 − pj k ≤ kω − pj k < rj . By Lemma 3.7.6, there exists m > 0 such that σΩ (pj ) ≥

rj 2 m kϕkkϕ−1 k

≥

s m2 kϕkkϕ−1 k

> 0,

contradicting limj σΩ (pj ) = 0. Therefore we have established rj = d(pj , ∂Ω) → 0

as

j → ∞.

We next show that there exists qj ∈ ∂Ω such that kpj − qj k = rj . Indeed, there is a sequence (qn ) in ∂Ω satisfying limn kqn −pj k = rj . By weak compactness of the boundary ∂Ω, we may assume, by choosing a

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subsequence, that the sequence (qn ) weakly converges to some qj ∈ ∂Ω. Then we have rj = lim inf kqn − pj k ≥ kqj − pj k ≥ rj . n

It has been shown in [119, Theorem 7.2] that the Bergman, Carath´eodory and Kobayashi metrics are equivalent on HHR manifolds, where two differential metrics ω1 and ω2 are said to be equivalent if they are quasi-isometric in the sense that C −1 ω1 ≤ ω2 ≤ Cω2 for some constant C > 0. Likewise, the Carath´eodory and Kobayashi pseudo-metrics are equivalent on HHR domains in Banach spaces. Proposition 3.7.8. Let D be an HHR domain in an isomorph V of a Hilbert space H, with the squeezing constant σ ˆD . Then we have σ ˆD KD (x, v) ≤ CD (x, v) ≤ KD (x, v) for all x ∈ D and v ∈ Tx D. Proof. We need only show the first inequality. Let x ∈ D and v ∈ V , where we identify the tangent space Tx D with V . Let fx : D → BH be a holomorphic embedding such that fx (x) = 0 and rBH ⊂ fx (D) ⊂ BH for some r > 0. For each ϕ ∈ H ∗ with kϕk ≤ 1, the composite map ϕfx : D → D satisfies ϕfx (x) = 0 and hence |ϕfx0 (x)(v)| = |(ϕfx )0 (x)(v)| ≤ CD (x, v). It follows that kfx0 (x)(v)k ≤ CD (x, v).

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Define a holomorphic map γ : D → rBH by γ(α) =

αrfx0 (x)(v) kfx0 (x)(v)k

(α ∈ D).

The restriction of the inverse fx−1 : fx (D) → D to rBH ⊂ fx (D) is well-defined and its composite with γ gives a holomorphic map fx−1 γ : D → D satisfying fx−1 γ(0) = x and   0   0 kfx (x)(v)k kfx (x)(v)k = (fx−1 )0 (0)γ 0 (0) (fx−1 γ)0 (0) r r = fx0 (x)−1 (fx0 (x)(v)) = v. This gives K(x, v) ≤

kfx0 (x)(v)k r

≤ CD (x, v)/r and hence

σ ˆD KD (x, v) ≤ CD (x, v) by the definition of the squeezing constant. In finite dimensions, bounded HHR domains are pseudoconvex, which has been proved in [182, Lemma 2]. To extend this result to infinite dimension, let us recall the definition of pseudoconvexity. For this, we first introduce the concept of a plurisubharmonic function. Loosely speaking, these functions are ‘subharmonic on complex lines’. Let D be a domain in a complex Banach space V . A function f : D −→ R ∪ {−∞} is called plurisubharmonic if it satisfies the following conditions: (i) f is upper semi-continuous, (ii) f restricts to a subharmonic function on every complex line in D, that is, on the open set U = {α ∈ C : a+αb ∈ D}, where a, b ∈ V , the function α ∈ U 7→ f (a + αb) is subharmonic. Definition 3.7.9. A domain D in a complex Banach space V is called pseudoconvex if the function z ∈ D 7→ − log d(z, ∂D) ∈ R ∪ {−∞} is plurisubharmonic, where d(z, ∂D) is the distance from z to ∂D.

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Bounded symmetric domains In finite dimensions, a celebrated result of Oka [139] reveals that

pseudoconvex domains are exactly the domains of holomorphy, which solves the classical Levi problem (see also [25, 136]). In all Banach spaces, domains of holomorphy are pseudoconvex [133, 11.4, 37.7]. The converse is true in separable Banach spaces with the bounded approximation property (cf. [137] and [133, 5.8]), but false in general [95]. We show that bounded HHR domains are domains of holomorphy, which extends the result of [182, Lemma 2]. Theorem 3.7.10. Let D be a bounded HHR domain in a complex Banach space V . Then D is a domain of holomorphy. Proof. In view of Proposition 3.5.16, we need only show that the Carath´eodory distance in D is complete. By the hypothesis, the squeezing constant σ ˆD takes the value, say, r ∈ (0, 1]. Let (xn ) be a cD -Cauchy sequence in D. We show that (xn ) cD -converges. Let ε = tanh−1 2r . Then there is a number N > 0 such that cD (xn , xN ) < ε for n > N. Let f : D → BH be a holomorphic embedding into a Hilbert ball BH with f (xN ) = 0 and BH (0, 3r 4 ) ⊂ f (D). Then the inverse holomorphic map g := f −1 : f (D) → D is well-defined on the ball BH (0, 3r 4 ). We have, for n > N , cBH (0, f (xn )) = cBH (f (xN ), f (xn )) ≤ cD (xN , xn ) < ε = tanh−1

r 2

as well as lim cBH (f (xm ), f (xn )) ≤

n,m→∞

lim cD (xm , xn ) = 0.

n,m→∞

Since BH is complete in the Carath´eodory distance, there is a subsequence (xnk ) of (xn ) such that f (xnk ) converges to some y0 ∈ BH with respect to cBH , and cBH (0, y0 ) ≤ ε. Hence, as noted previously, we have

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y0 ∈ B H (0, 2r ) ⊂ BH (0, 3r 4 ) ⊂ f (D) and also, lim cD (xnk , g(y0 )) ≤ lim cD (g(ynk ), g(y0 )) k→∞

k→∞

4 cB (f (g(ynk )), f (g(y0 ))) 3r H 4 = lim cBH (ynk , y0 ) = 0. k→∞ 3r ≤ lim

k→∞

It follows that the sequence (xn ) converges to g(y0 ) in D with respect to cD and the proof is complete. We now consider bounded symmetric domains. In finite dimensions, it is well-known that a bounded symmetric domain of rank ` contains a polydisc of dimension ` as a totally geodesic submanifold [108, p. 41]. To see that this is also the case for infinite dimensional bounded symmetric domains of finite rank, we only need to consider the irreducible ones. There are only two classes of such domains, namely, the Type IV domains realisable as the Lie balls, which are of rank 2, and the Type I domains of rank `, which can be realised as the open unit ball of the JB*-triple L(C` , K) of bounded linear operators between Hilbert spaces C` and K, with ` ≤ dim K ≤ ∞ and ` < ∞. Given ` < ∞, every operator T ∈ L(C` , K) is a Hilbert-Schmidt operator in the Hilbert-Schmidt norm ` X kT k2 = ( kT ek k2 )1/2 k=1

satisfying kT k ≤ kT k2 ≤ orthonormal basis in

√

`kT k, where {e1 , . . . , e` } is the standard

C` .

Let D be the closure of D = {z ∈ C : |z| < 1} and D the closure of the open unit ball D = {T ∈ L(C` , K) : kT k < 1}.

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Fix orthonormal basis vectors uα1 , . . . , uα` from an orthonormal basis {uα } in K. Then the continuous map ϕ : D × · · · × D → D, defined by ϕ(z1 , . . . , z` ) =

` X

zk (ek ⊗ uαk )

`

(z1 , . . . , z` ) ∈ D ,

(3.61)

k=1

restricts to an injective holomorphic map ϕ : D × ··· × D → D with ϕ(0, . . . , 0) = 0, where ek ⊗ uαk : C` → K is the rank-one operator ek ⊗ uαk (h) = hh, ek iuαk

(h ∈ C` )

with kek ⊗ uαk k = kek ⊗ uαk k2 = 1. This also implies that ϕ maps the boundary ∂D` of D` into the boundary ∂D = {T ∈ L(C` , K) : kT k = 1}. Let D be the open unit ball of a spin factor V , which is of rank 2. Let e1 and e2 be two mutually (triple) orthogonal minimal tripotents in V . Then we have kλe1 +µe2 k = max{|λ|, |µ|} for λ, µ ∈ C, by Corollary 3.2.24. Hence one can define a continuous map 2

ϕ : (z1 , z2 ) ∈ D 7→ z1 e1 + z2 e2 ∈ D

(3.62)

which restricts to an injective holomorphic map from D2 to D satisfying ϕ(0) = 0 and ϕ(∂D2 ) ⊂ ∂D. Given a Hilbert space H, a holomorphic map f : Dn → H admits a power series representation in terms of homogeneous polynomials from Cn to H. We recall that a homogeneous polynomial p of degree d from Cn to H is given by p(z1 , . . . , zn ) = P ( (z1 , . . . , zn ), . . . , (z1 , . . . , zn ) ) ∈ H, (z1 , . . . , zn ) ∈ Cn where P : |Cn × ·{z · · × Cn} → H is a d-linear map. Let {eα } be an d-times orthonormal basis in H. We can write X p(z1 , . . . , zn ) = qα (z1 , . . . , zn )eα α

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285

where qα (z1 , . . . , zn ) is a homogeneous polynomial of degree d in n complex variables z1 , . . . , zn and has the form cα; j1 ,...,jn z1j1 · · · znjn

X

qα (z1 , . . . , zn ) =

(cα; j1 ,...,jn ∈ C).

j1 +···+jn =d

Hence a holomorphic map f : Dn → H has a representation f (z1 , . . . , zn ) = f (0) +

∞ X

pd (z1 , . . . , zn ),

(z1 , . . . , zn ) ∈ Dn

d=1

where pd is a homogeneous polynomial of degree d from Cn to H and has the from pd (z1 , . . . , zn ) =

X

X

cdα; j1 ,...,jn z1j1 · · · znjn eα

(cdα; j1 ,...,jn ∈ C).

α j1 +···+jn =d

(3.63) Let h : D →

D0

be a biholomorphic map between two open unit

balls D, D0 of Banach spaces V and V 0 respectively. If h(0) = 0, then by Cartan’s uniqueness theorem, h is the restriction of the derivative h0 (0) : V → V 0 , which is a linear isometry. In particular, h extends to ¯ :D ¯ → D0 between the closures D and D0 , where a continuous map h ¯ = h0 (0)| ¯ . Moreover, h(∂D) ¯ h = ∂D0 . D Let D be a bounded symmetric domain, realised as the open unit ball of a JB*-triple V . Given a holomorphic embedding f : D → BH of D into a Hilbert ball BH , the image f (D) is a bounded symmetric domain and hence there is an equivalent norm k · ksp on H such that (H, k·ksp ) is a JB*-triple and f (D) identifies (via a biholomorphic map) as the open unit ball of (H, k·ksp ). If f (0) = 0, then the previous remark implies that f extends to a continuous map f¯, which maps ∂D onto the boundary ∂f (D) of the domain f (D). The following lemma is a simple infinite dimensional extension of Alexander’s result in [10, Proposition 1] (see also [115, Lemma 1]).

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Lemma 3.7.11. Let D be a bounded domain with boundary ∂D and B a Hilbert ball such that the following two continuous maps

D

`

ϕ − →

D

f − →

B

on the closures restrict to holomorphic maps

D`

ϕ − →

D

f − →

B

with open image f (D), satisfying ϕ(∂D` ) ⊂ ∂D and f (∂D) ⊂ ∂f (D). If ρB ⊂ f (D) for some ρ > 0, then `ρ2 ≤ 1.

Proof. Let {eα } be an orthonormal basis in the Hilbert space containing the ball B. By (3.63), the holomorphic map f ◦ ϕ on D` has a power series representation

f ◦ ϕ(z1 , . . . , z` ) =

∞ X

pd (z1 , . . . , z` )

d=1

where pd (z1 , . . . , z` ) is a d-homogeneous polynomial of the form

pd (z1 , . . . , z` ) =

X

X

cdα; j1 ,...,j` z1j1 · · · z`j` eα

(cdα; j1 ,...,j` ∈ C).

α j1 +···+j` =d

Since ρB ⊂ f (D), we have kf (w)k ≥ ρ for each w ∈ ∂D. Noting

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that f ◦ ϕ(∂D` ) ⊂ ∂f (D), we deduce Z 2π 1 2 kf ◦ ϕ(0, . . . , eiθj , 0, . . . , 0)k2 dθj ρ ≤ 2π 0 Z 2π 1 = lim kf ◦ ϕ(0, . . . , reiθj , 0, . . . , 0)k2 dθj 2π r→1 0 2 Z 2π X X 1 = lim cdα; 0,...,0,d,0,...,0 rd eidθj dθj 2π r→1 0 α d 2 X Z 2π X 1 lim cdα; 0,...,0,d,0,...,0 rd eidθj dθj = 2π r→1 α 0 d XX 2 = lim cdα; 0,...,0,d,0,...,0 r2d r→1

α

d

2 X X d = c α; 0,...,0,d,0,...,0 . α

d

It follows that  ` Z 2π Z 2π 1 1 ≥ lim kf ◦ ϕ(reiθ1 , . . . , reiθ` )k2 dθ1 · · · dθ` ··· r→1 2π 0 0 2 XX X d = lim cα; ν1 ,...,ν` r2d r→1

=

α

XX α

d ν1 +···+ν` =d

X

2 d c α; ν1 ,...,ν`

d ν1 +···+ν` =d

2 2 X X X X ≥ cdα; d,0,...,0 + · · · + cdα; 0,...,0,d ≥ `ρ2 . α

d

α

d

In finite dimensions, the squeezing constant of the classical Cartan domains has been computed by Kubota in [115]. We will now compute the squeezing constants of the remaining finite rank bounded symmetric domains of all dimensions. We begin with the two exceptional domains which are realised as the open unit balls of the JB*-triples M1,2 (O) and H3 (O) respectively,

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where dim M1,2 (O) = 16 and dim H3 (O) = 27. Both JB*-triples carry a Hilbert space structure with the trace form and one can define an inner product by hx, yi =

1 Trace (x y) 9

(x, y ∈ H3 (O)).

(3.64)

so that he, ei = 1 for each minimal tripotent e ∈ H3 (O) (cf. [150, Proposition 2.8, Corollary 2.14]). If e and u are two mutually triple orthogonal tripotents in H3 (O), then he, ui = 0. The 27-dimensional domain D27 ⊂ H3 (O) has rank 3 whereas the 16-dimensional domain D16 ⊂ M1,2 (O) has rank 2. The following two propositions, together with Kubota’s results in [115], give a complete list of squeezing constants of all finite dimensional irreducible bounded symmetric domains. Proposition 3.7.12. The squeezing constant of the exceptional domain √ D27 is given by σ ˆD27 = 1/ 3. Proof. We compute σD27 (0) = σ ˆD27 . We have D27 = {z ∈ H3 (O) : kzk < 1}, where k · k is the norm of the JB*-triple H3 (O). Given z ∈ H3 (O), we have the spectral decomposition z = α1 e 1 + α2 e 2 + α3 e 3

(α1 ≥ α2 ≥ α3 ≥ 0),

and kzk = α1 , where the minimal tripotents e1 , e2 , e3 are mutually orthogonal with respect to the inner product in (3.64). The Hilbert space norm kzk2 of z is given by kzk22 = hz, zi = α12 + α22 + α32 . It follows that kzk ≤ kzk2 ≤

√

3kzk

for all z ∈ H3 (O). This implies B27 ⊂ D27 ⊂

√ 3B27

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where B27 = {z ∈ H3 (O) : kzk2 < 1} is the Hilbert ball in H3 (O). √ Hence we have σ ˆD27 ≥ 1/ 3. To show the reverse inequality, we define 3

a continuous map ϕ : D → D27 by   z1 0 0 ϕ(z1 , z2 , z3 ) =  0 z2 0  = z1 e11 + z2 e22 + z3 e33 0 0 z3 where ejj is the diagonal matrix in H3 (O) with 1 in the jj-entry and 0 elsewhere. Since e11 , e22 , e33 are mutually triple orthogonal minimal tripotents in H3 (O), we see that ϕ restricts to an injective holomorphic map from D3 into D27 with ϕ(0) = 0 and ϕ(∂D3 ) ⊂ ∂D27 . By Lemma 3.7.11 and the remarks before it, for each holomorphic embedding f : D27 → B27 with f (0) = 0 and ρB27 ⊂ f (D27 ), we must have 3ρ2 ≤ 1. This proves the reverse inequality. Proposition 3.7.13. The squeezing constant of the exceptional domain √ D16 is given by σ ˆD16 = 1/ 2. Proof. The arguments are similar to those in the proof of Proposition 3.7.12, we recapitulate for completeness. We consider M1,2 (O) as a √ subtriple of H3 (O). It suffices to show σD16 (0) = 1/ 2. We have D16 = {z ∈ M1,2 (O) : kzk < 1}, where k · k is the norm of the JB*triple M1,2 (O). Each z ∈ M1,2 (O) has a spectral decomposition z = α1 e 1 + α2 e 2

(α1 ≥ α2 ≥ 0),

and kzk = α1 , where the minimal tripotents e1 , e2 are mutually orthogonal with respect to the inner product in (3.64). The Hilbert space norm kzk2 of z is given by kzk22 = hz, zi = α12 + α22 and kzk ≤ kzk2 ≤

√

2kzk

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for all z ∈ M1,2 (O). This implies B16 ⊂ D16 ⊂

√

2B16 ,

where B16 = {z ∈ M1,2 (O) : kzk2 < 1} is the Hilbert ball in M1,2 (O). √ Hence σ ˆD16 ≥ 1/ 2. For the reverse inequality, one defines a continuous 2

map ϕ : D → D16 by ϕ(z1 , z2 ) = z1 e11 + z2 e22 where e11 = (1, 0) and e22 = (0, 1) are mutually triple orthogonal minimal tripotents in M1,2 (O), and ϕ restricts to an injective holomorphic map from D2 into D16 with ϕ(0) = 0 and ϕ(∂D2 ) ⊂ ∂D16 . As before, for each holomorphic embedding f : D16 → B16 satisfying f (0) = 0 and ρB16 ⊂ f (D16 ), we must have 2ρ2 ≤ 1. This proves the reverse inequality. The proof of the two preceding propositions is analogous to Kobota’s in [115] for the classical Cartan domains. We will not repeat Kubota’s computation for the classical domains in the following corollary. Corollary 3.7.14. Let D be a finite dimensional irreducible bounded symmetric domain of rank p. Then its squeezing constant is given by √ σ ˆD = 1/ p. The following result, which characterises HHR bounded symmetric domains, reveals the connection between the rank of a symmetric domain and the extent to which a Hilbert ball can be squeezed inside it. Theorem 3.7.15. Let D be a bounded symmetric domain in a complex Banach space. Then D is HHR if and only if it is of finite rank. In this

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case, D is biholomorphic to a finite product D1 × · · · × Dk of irreducible bounded symmetric domains and we have !−1/2 1 1 σ ˆD = . 2 + ··· + σ 2 σ ˆD ˆD 1 k

(3.65)

If dim Dj < ∞, then Dj is a classical Cartan domain or an excep√ tional domain, and σ ˆDj = 1/ pj where pj is the rank of Dj . If dim Dj = ∞, then Dj is either a Lie ball or a Type I domain of √ finite rank pj . For a Lie ball Dj , we have σ ˆDj = 1/ 2. For a rank pj √ Type I domain Dj , we have σ ˆDj = 1/ pj . Proof. Let D be HHR, realised as the open unit ball of a JB*-triple V . Then V is linearly homeomorphic to some Hilbert space H. In particular, V is reflexive and hence D is of finite rank. Conversely, by Theorem 3.3.5, a finite-rank bounded symmetric domain D decomposes into a finite Cartesian product D = D1 ×· · ·×Dk of irreducible bounded symmetric domains, where each Dj is of finite rank pj and realised as the open unit ball of a Cartan factor Vj for j = 1, . . . , k. To complete the proof, we show that each domain Dj of rank pj √ has squeezing constant σ ˆDj = 1/ pj and σ ˆD = (p1 + · · · + pk )−1/2 . √ By Corollary 3.7.14, we have σ ˆDj = 1/ pj if dim Vj < ∞. In fact, this is also the case even if Vj is infinite dimensional, in which case Vj is either a spin factor or the Type I Cartan factor L(C` , K) with dim K = ∞ > `. We now compute the squeezing constant in these two cases. First, let Dj be a Lie ball, that is, the open unit ball of a spin factor (V, k · k), which has rank 2. In this case, V is a Hilbert space with norm k · kh satisfying k · k ≤ k · kh ≤

√

2k · k

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√ (cf. (2.55) and (2.58)). This gives σ ˆDj ≥ 1/ 2. Making use of the map ϕ in (3.62) and analogous arguments in the proof of Proposition 3.7.13, √ one concludes that σ ˆDj = σ ˆDj (0) = 1/ 2. Next, let Dj be a Type I domain of rank pj , realised as the open unit ball Dj = {T ∈ L(Cpj , K) : kT k < 1} of L(Cpj , K) with dim K = ∞. Equipped with the Hilbert-Schmidt norm k · k2 , the vector space L(Cpj , K) is a Hilbert space. Let B = {T ∈ L(Cpj , K) : kT k2 < 1} be its open unit ball. Since k · k ≤ k · k2 ≤ √ √ √ pj k · k, we have B ⊂ Dj ⊂ pj B and therefore σ ˆDj (0) ≥ 1/ pj . As before, using the map ϕ in (3.61) and similar arguments, we deduce √ σ ˆDj = σ ˆDj (0) = 1/ pj . It remains to establish (3.65). The domain D = D1 × · · · × Dk is the open unit ball of the `∞ -sum V1 ⊕ · · · ⊕ Vk of Cartan factors, where Dj is the open unit ball of Vj of rank pj for j = 1, . . . , k. We observe from the previous arguments that for each pj

domain Dj , one can construct a continuous map ϕj : D restricts to a holomorphic map from

Dp j

→ Dj which

to Dj satisfying ϕj (0) = 0 and

ϕj (∂Dpj ) ⊂ ∂Dj . Hence the product map ϕ := ϕ1 × · · · × ϕk : D

p1

pk

× ··· × D

→ D1 × · · · × Dk = D

is continuous, which restricts to a holomorphic map from Dp1 × · · · × Dpk to D1 × · · · × Dk satisfying ϕ(0, . . . , 0) = (0, . . . , 0) and maps the boundary of Dp1 × · · · × Dpk into the boundary of D1 × · · · × Dk = D. Applying Lemma 3.7.11 again, we deduce that σ ˆD ≤ √

1 1 =q . −2 −2 p1 + · · · + pk σ ˆD1 + · · · + σ ˆD k

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For each j = 1, . . . , k, the previous arguments reveal that there is a Hilbert space Hj with open unit ball Bj such that Bj ⊂ Dj ⊂

√

pj Bj .

Let B be the open unit ball of the Hilbert space direct sum H1 ⊕2 · · · ⊕2 Hk . Then we have B ⊂ D1 × · · · × Dk ⊂ This implies that σ ˆD ≥ √

√

p1 + · · · + pk B.

1 p1 + · · · + pk

which completes the proof. We have noted before that finite dimensional bounded convex domains are HHR. This is of course false in infinite dimension, for instance, the open unit ball of `∞ is not HHR. However, an appropriate extension of this result would be the assertion that bounded convex domains in isomorphs of Hilbert spaces are HHR. It is unclear at present if this is true. We conclude this section by introducing a large class of bounded convex domains including the strongly convex domains, which we call uniformly elliptic domains and show that these domains are HHR in Hilbert spaces. Recall that a finite dimensional bounded domain D ⊂ Cn with a C 2 boundary ∂D is called strongly convex if all normal curvatures of ∂D are positive (cf. [2, p.108]). Such a domain is a manifold with curvature pinched which entails the existence of two positive constants R > r > 0 such that for each q ∈ ∂D, there are two points q 0 , q 00 in D with the property that q is a common boundary point of the Euclidean balls BCn (q 0 , r) and BCn (q 00 , R) satisfying BCn (q 0 , r) ⊂ D ⊂ BCn (q 00 , R). For fixed r and R, it can be seen that q 0 and q 00 are unique and colinear with q. For instance, an ellipsoid is strongly convex and has this property.

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Bounded symmetric domains In view of the fact that Hilbert balls are the only bounded symmet-

ric domains with a C 2 boundary, we generalise the concept of strong convexity to infinite dimension without the assumption of a smooth boundary, to cover a wider class of domains, as follows. Definition 3.7.16. A bounded convex domain Ω in a complex Banach space V is called uniformly elliptic if there exist universal constants r, R with 0 < r < R such that to each q ∈ ∂Ω, there correspond two unique points q 0 , q 00 ∈ Ω, colinear to q, satisfying (3.7.16.1) BV (q 0 , r) ⊂ Ω ⊂ BV (q 00 , R); (3.7.16.2) q ∈ ∂BV (q 0 , r) ∩ ∂BV (q 00 , r), that is, q is a common boundary point of BV (q 0 , r), BV (q 00 , R) and Ω. Evidently, the definition of uniform ellipticity depends on the norm of the ambient Banach space. By the previous remarks, strongly convex domains are uniformly elliptic, but the converse is false. In fact, all open balls in Banach spaces are uniformly elliptic. Indeed, if say, Ω = BV is the open unit ball of a Banach space V , then for each boundary point q ∈ ∂Ω, we have kqk = 1 and BV (q/2, 1/2) ⊂ Ω = BV (0, 1) and q ∈ ∂BV ( 2q , 12 ) ∩ ∂Ω ∩ ∂BV (0, 1). For R = 1 and r = 1/2, the points q 0 = q/2 and q 00 = 0 are unique and colinear to q. By definition, each point p in a uniformly elliptic domain Ω in a Banach space V lies in the ball BV (q 00 , R) for all q ∈ ∂Ω, as in (3.7.16.1) above, although p need not be colinear with q and q 00 . We consider the question of colinearity below. Lemma 3.7.17. Let Ω be a uniformly elliptic domain in a Banach space V and for each q ∈ ∂Ω, let BV (q 0 , r) ⊂ Ω ⊂ BV (q 00 , R),

q ∈ ∂BV (q 0 , r) ∩ ∂BV (q 00 , r)

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be as in the definition of uniform ellipticity. Then for each p ∈ Ω and q ∈ ∂Ω with kp − qk = d(p, ∂Ω), either p is colinear with q and q 00 or, there exists q1 ∈ ∂Ω such that p is colinear with q1 and q100 = q 00 satisfying kp − q1 k = kp − qk. Proof. Let q ∈ ∂Ω ∩ ∂BV (q 00 , R) satisfy kp − qk = d(p, ∂Ω). Suppose p is not colinear with q and q 00 . We show the existence of q1 in the lemma. Consider p ∈ Ω ⊂ BV (q 00 , R). Extend the (real) line through q 00 and p to a point q1 ∈ ∂BV (q 00 , R). Then we have kp − q1 k = d(p, ∂BV (q 00 , R)) ≤ kp − qk. We show that q1 ∈ ∂Ω, which would imply kp − q1 k ≥ kp − qk and complete the proof by uniqueness of q10 and q100 . If q1 ∈ / ∂Ω, we deduce a contradiction. Since q1 ∈ / Ω and p ∈ Ω, the line joining p and q1 must intersect ∂Ω at some point ω, say. Now we have the contradiction kp − qk ≥ kp − q1 k > kp − ωk ≥ d(p, ∂Ω) = kp − qk.

Theorem 3.7.18. Let Ω be a uniformly elliptic domain in a Hilbert space H. Then Ω is HHR. Proof. We need to show that the squeezing function σΩ of Ω has a strictly positive lower bound. Suppose, to the contrary, that there is a sequence (pν ) in Ω such that lim σΩ (pν ) = 0.

ν→∞

(3.66)

We deduce a contradiction. By Lemma 3.7.7, we may assume, by choosing a subsequence, that d(pν , ∂Ω) converges to 0 as ν → ∞ and one can find a boundary point qν ∈ ∂Ω such that kqν − pν k = d(pν , ∂Ω) > 0.

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Bounded symmetric domains Write λν = d(pν , ∂Ω) and let BV (qν0 , r) ⊂ Ω ⊂ BV (qν00 , R),

qν ∈ ∂BV (qν0 , r) ∩ ∂BV (qν00 , R)

be as in the definition of uniformly ellipticity of Ω where, by Lemma 3.7.17, qν can be chosen so that pν lies on the line through qν and qν00 . We complete the proof by a contradiction that there is a subsequence (pν 0 ) of (pν ) and a constant δ > 0 satisfying σΩ (pν 0 ) > δ

for all ν 0 .

In fact, δ depends only on r and R. For each ν, we define a holomorphic embedding Φ ◦ Lν : Ω → H as follows. Let e1 be the unit vector e1 :=

qν00 − qν . kqν00 − qν k

We have (S1) qν00 = Re1 + qν , qν0 = re1 + qν , (S2) pν = λν e1 + qν (λν → 0 as ν → ∞). Since σΩ (pν ) = σΩ−qν (pν − qν ), taking a translation, we may assume qν = 0. Then we have (S10 ) qν00 = Re1 , ϕqν0 = re1 , (S20 ) pν = λν e1 . We now have pν = λν e1 ∈ BH (re1 , r) ⊂ Ω ⊂ BH (Re1 , R)

(3.67)

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where qν0 = re1 ,

qν00 = Re1 .

Extend {e1 } to an orthonormal basis {eγ }γ∈Γ in H. For each z ∈ H, we will write z=

X

zγ eγ = z1 e1 +

γ∈Γ

X

zγ eγ

γ6=1

with zγ ∈ C. We have z ∈ BH (re1 , r) ⇔ kz − re1 k < r where kz −re1 k2 = |z1 −r|2 +

X

|zγ |2 = |z1 |2 −2rRe z1 +r2 +

γ6=1

X

|zγ |2 . (3.68)

γ6=1

We definite a dilation Lν : H → H by Lν (z) =

z1 1 1 X zγ eγ , e +√ λν λν γ6=1

z=

X

zγ eγ

γ∈Γ

which satisfies Lν (pν ) = e1 . The map Lν is a linear homeomorphism of H, with inverse 1 L−1 ν (z) = λν z1 e +

p X λν zγ eγ . γ6=1

Define a Cayley transform Φ : {z ∈ H : Re z1 > 0} → H by √ X z1 − 1 1 X 2zγ γ e + e , z= zγ eγ Φ(z) := z1 + 1 z1 + 1 γ γ6=1

and the holomorphic embedding Φ ◦ Lν : Ω → H

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where Φ(Lν (pν )) = 0. Although Φ depends on ν, we omit the subscript ν indicating this, to simplify notation, since confusion is unlikely. We will show that   r √ r ⊂ Φ(Lν (Ω)) ⊂ BH (0, 1 + R) BH 0, 2 + 2r for sufficiently large ν. Substituting R for r in (3.68), we see that BH (Re1 , R) = {z ∈ H : kz − Re1 k2 < R2 } X = {z ∈ H : |zγ |2 < 2R Re z1 }. γ∈Γ

Given ζ =

γ γ ζγ e

P

Φ

∈ BH , we have −1

√ 1 + ζ1 1 X 2ζγ γ e + e . (ζ) = 1 − ζ1 1 − ζ1 γ6=1

Hence ζ ∈ ΦLν (BH (Re1 , R)) ⇔ L−1 Φ−1 ζ ∈ BH (Re1 , R) √ ν   X 2λν 1 + ζ1 e1 + eγ ∈ BH (Re1 , R) ⇔ λν 1 − ζ1 1 − ζ1 γ6=1 X ⇔ αλ2ν |1 + ζ1 |2 + 2λν |ζγ |2 < 2Rλν (1 − |ζ1 |2 ) γ6=1

⇒

X

|ζγ |2 < R(1 − |ζ1 |2 )

γ6=1

⇒ kζk2 = |ζ1 |2 +

X

|ζγ |2 < 1 + R

γ6=1

and therefore we have, by (3.67), √ ΦLν (Ω) ⊂ ΦLν (BH (Re1 , R)) ⊂ BH (0, 1 + R). (3.69) q r We now show that BH (0, 2+2r ) ⊂ ΦLν (Ω) for sufficiently large ν. For this, we will make use of the inclusion BH (ru, r) ⊂ Ω.

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−1 −1 −1 We have L−1 ν Φ (ζ) ∈ BH (ru, r) if and only if kLν Φ (ζ)−ruk
r and |1+ζ1 |2 ≤ r 2  r . Since λν → 0 as ν → ∞, there exists ν0 such that 1+ 2 + 2r ν ≥ ν0 implies r λν <  2 q r 2 1 + 2+2r and hence λν |1 + ζ1 |2 − 2r(1 − |ζ1 |2 ) + 2

X

|ζγ |2

γ6=1 2

≤ λν |1 + ζ1 | − 2r + 2r|ζ1 | + 2kζk2 r 2 <  2 |1 + ζ1 | − r < −r/2 q r 2 1 + 2+2r which gives λν (λν |1 + ζ1 |2 − 2r(1 − |ζ1 |2 ) + 2

2

P

γ6=1 |ζγ |

2)

< −rλν /2 < 0

and by (3.70), −1 2 2 kL−1 ν Φ (ζ) − ruk < r .

We have therefore shown that, for ν ≥ ν0 , the inclusions BH (0, are satisfied.

p r/(2 + 2r ) ⊂ ΦLν (BH (re1 , r)) ⊂ ΦLν (Ω)

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Bounded symmetric domains Now it follows from this and (3.69) that r r σΩ (pν ) ≥ >0 2(1 + r)(1 + R)

for all ν ≥ ν0 , which contradicts limν→∞ σΩ (pν ) = 0 and completes the proof.

Notes. HHR manifolds can be regarded as a generalisation of Teichm¨ uller spaces T of Riemann surfaces. By Bers embedding theorem, given p ∈ T , there is an embedding fp : T −→ Cd so that fp (p) = 0 and fp (T ) is sandwiched between two Euclidean balls of positive radii, which provides the uniform squeezing property (see the proof of [119, Theorem 7.1]). We refer to Siu’s lecture [159] for a useful survey of the classical Levi problem and its generalisations. Proposition 3.7.5 and its proof are direct generalisation of the finite dimensional result and arguments in [53, Theorem 3.1]. Theorem 3.7.18 generalises the finite dimensional result for strongly convex domains in [182, Proposition 1]. Apart from Proposition 3.7.8, all results in this section have been proven in [46].

3.8

Classification

We discuss a Jordan approach to the classification of bounded symmetric domains, up to biholomorphisms. We have already formulated ´ Cartan’s classification of finite dimensional bounded in Section 3.3 E. symmetric domains in terms of Cartan factors, which has been extended in Corollary 3.3.7 to the classification of finite-rank bounded symmetric domains. However, the formulation in Section 3.3 depends on some results to be proved in this section. We will now see in detail that Cartan’s classification can be viewed as a special case of the Jordan classification developed in what follows.

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Our approach is based on the realisation of bounded symmetric domains as open unit balls of JB*-triples and the following classification theorem, already hinted by the remarks before Theorem 3.2.22. Theorem 3.8.1. Let V and W be JB*-triples with open unit balls BV and BW respectively. Then BV is biholomorphic to BW if and only if V is isometrically triple isomorphic to W . Proof. This follows from Theorem 3.2.22 and the remarks preceding it. Classifying bounded symmetric domains, therefore, amounts to classifying JB*-triples. This is a formidable task in infinite dimension. Indeed, classifying the subclass of C*-algebras is already a vast enterprise. Nevertheless, analogous to Murray and von Neumann’s classification of von Neumann algebras, one can still classify (to some extent) a large class of JB*-triples, namely, the JBW*-triples. This enables us to classify bounded symmetric domains in dual Banach spaces. Lemma 3.8.2. A bounded symmetric domain in a dual Banach space is biholomorphic to the open unit ball of a JBW*-triple. Proof. We recall that a JBW*-triple is a JB*-triple which is a dual Banach space. Let D be a bounded symmetric domain in a dual Banach space V . Then V can be equipped with an equivalent norm k · k so that (V, k · k) is a JB*-triple and D is biholomorphic to the open unit ball of (V, k · k). Since V is a dual Banach space, it is a JBW*-triple. Definition 3.8.3. A JBW*-triple W is called a JBW*-factor or simply, a factor, if it does not contain any weak* closed triple ideal other than {0} and W . Cartan factors are indeed JBW*-factors. We note that, however, a factor can contain non-trivial norm closed triple ideal. For example,

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the factor L(H) of bounded operators on a Hilbert space H contains the norm closed ideal K(H) of compact operators on H. A JB*-triple V is called simple if does not contain any norm closed triple ideal other than {0} and V . The preceding remark says that the JBW*-factor L(H) is not a simple JBW*-triple. To verify if a norm closed subspace J of a JB*-triple V is a triple ideal, it suffices to verify the inclusion {V, J, J} ⊂ J or {V, J, V } ⊂ J, by [28, Proposition 1.3]. It follows that the weak* closure of a triple ideal in a JBW*-triple is also a triple ideal since the triple product is separately weak* continuous in JBW*-triples (cf. [37, Theorem 3.3.9]). Definition 3.8.4. Given an element u in a JBW*-triple V , we denote by J(u) the smallest weak* closed triple ideal of V containing u. A tripotent u ∈ V is called abelian if the Peirce 2-space V2 (u) is an abelian Jordan triple system as defined in Definition 2.2.13. Remark 3.8.5. A minimal tripotent u in a JBW*-triple is clearly abelian. We take this opportunity of correcting a misprint in the definition of abelian tripotents in [37, p. 211] where the triple ideal J(u) should be replaced by the Peirce 2-space V2 (u). Lemma 3.8.6. Let A be a von Neumann algebra. If A is an abelian JBW*-triple, then it is an abelian, that is, a commutative, algebra. Proof. It suffices to show pq = qp for two projections p and q in A, by Example 2.4.7. Let e ∈ A be the identity. By assumption, we have {p, q, {p, e, e}} = {p.{q, p, e}, e}

(3.71)

which simplifies to pqp = 12 (pq + qp) and therefore pqp = qp. Changing the role of p and q in (3.71), we get qpq = pq and hence pq = qp. Definition 3.8.7. A JBW*-triple W is called discrete if it admits an abelian tripotent u such that W = J(u).

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Remark 3.8.8. A discrete JBW*-triple is also known as a type I JBW*triple [86]. To avoid confusion with the notion of a Type I Cartan factor, we will not use this terminology. Our nomenclature, however, is consistent with that in operator algebras where type I von Neumann algebras are also called discrete von Neumann algebras (cf. [154, 2.2.9]). Following a common approach in classification theory, we decompose a JBW*-triple into simpler ones to classify. We first decompose a JBW*-triple into two parts, one atomic and the other non-atomic. For this, we need two lemmas. Lemma 3.8.9. Let J be a weak* closed triple ideal in a JBW*-triple V . Then there is a weak* closed triple ideal J {0} and V = J ⊕ J

in V such that J

J=

.

Proof. We have noted after Theorem 3.4.8 that JBW*-triples contain a maximal tripotent. Let u be a maximal tripotent in J. Then u is a tripotent in V . Let Pj (u) : V −→ V be the Peirce projections, j = 0, 1, 2. For x = x1 +x2 ∈ P1 (u)V +P2 (u)V , we have jxj = 2{u, u, x} ∈ J and hence J = P1 (u)V + P2 (u)V. Let J J

= P0 (u)V which is a weak* closed subtriple of V . We have P2 (u)V = {0}. For y ∈ J

P0 (u)V ∩ J = {0}. Hence J {V, J that is, J

,J

and z ∈ P1 (u)V , we have {z, z, y} ∈ J = {0} and it follows that

} = {J ⊕ J

,J

,J

}⊂J

is a triple ideal in V .

Remark 3.8.10. In the above lemma, it is easy to see that J V : v J = {0}} since the latter set contains J {0} with J.

= {v ∈

and has intersection

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Lemma 3.8.11. Let u be a minimal tripotent in a JBW*-triple V . Then the smallest weak* closed triple ideal J(u) in V containing u is a discrete factor. If v ∈ V is another minimal tripotent, then either J(u) = J(v) or J(u) J(v) = {0}. Proof. If J is a non-zero proper weak* closed triple ideal in J(u), then Lemma 3.8.9 implies J(u) = J ⊕ J closed triple ideal J J ⊕J

with J

for some non-zero proper weak* J = {0}. Hence u = u1 + u2 ∈

and u1 , u2 must be tripotents, contradicting minimality of

u. Therefore J(u) is a discrete factor. The second assertion is an immediate consequence. Lemma 3.8.12. Let W1 and W2 be discrete JBW*-triples. Then their `∞ -sum W1 ⊕ W2 is a discrete JBW*-triple. Proof. Let Wj = J(uj ) for j = 1, 2, where uj is an abelian tripotent in Wj . Then u1 ⊕ u2 is an abelian tripotent in W1 ⊕ W2 . Let J(u1 ⊕ u2 ) be the smallest weak* closed triple ideal in W1 ⊕ W2 containing u1 ⊕ u2 . We show W1 ⊕ W2 = J(u1 ⊕ u2 ), which would complete the proof. First observe that u1 ⊕ 0 = {u1 ⊕ 0, u1 ⊕ u2 , u1 ⊕ u2 } ∈ J(u1 ⊕ u2 ). Let P1 : W1 ⊕ W2 −→ W1 be the natural projection, which is a triple homomorphism. Then the weak* closure P1 (J(u1 ⊕ u2 )) is a weak* closed triple ideal in W1 containing u1 and hence P1 (J(u1 ⊕ u2 )) ⊃ J(u1 ). Let x1 ⊕ x2 ∈ J(u1 ⊕ u2 ) . Then for each y ∈ J(u1 ), there is a net (aα ⊕ bα ) ∈ J(u1 ⊕ u2 ) such that (aα ) weak* converges to y in W1 . Since x1 is triple orthogonal to each aα , it is also triple orthogonal to y. As y ∈ J(u1 ) was arbitrary, we have shown that x1 ∈ J(u1 ) {0}. Likewise, x2 = 0. It follows that J(u1 ⊕ u2 )

=

= {0} and hence

J(u1 ⊕ u2 ) = W1 ⊕ W2 . Let V be a JBW*-triple. It follows from Lemma 3.8.9, Lemma

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3.8.11 and Corollary 3.2.24 that V can be decomposed into an `∞ -sum ! V =

M

J(u)

! ⊕

u

M

J(u)

(3.72)

u

of two weak* closed triple ideals, where the first summand sums over all minimal tripotents u in V and is called the atomic part of V . It is an `∞ -sum of discrete factors. The second summand does not contain any minimal tripotent, called the non-atomic part of V . A JBW*-triple is called atomic if its non-atomic part vanishes and by (3.72), it is an `∞ -sum of discrete JBW*-factors. On the other hand, a JBW*-triple may not contain any minimal tripotent in which case, its atomic part vanishes and it is called non-atomic when this happens. Corollary 3.8.13. A JB*-triple V is triple isomorphic to a closed subtriple of an `∞ -sum of discrete factors. Proof. By (3.72), the second dual V ∗∗ is an `∞ -sum V ∗∗ = Va ⊕ Va of two weak* closed triple ideals in which Va is an `∞ -sum of discrete factors. Let P : V ∗∗ −→ Va be the canonical contractive projection and b : V −→ V ∗∗ the canonical embedding. Both maps are triple homomorphisms and therefore P (Vb ) is a closed subtriple of Va . If P (b v ) = 0, then vb ∈ Va . For each extreme point f in the closed unit ball of V ∗ , there exists a unique minimal tripotent uf ∈ V ∗∗ such that f (uf ) = 1, by [37, Lemma 3.3.15], which implies uf ∈ Va and f (b v ) = f {uf , vb, uf } = 0 (see [37, (3.10)]. Hence vb = 0 and P ◦ b : V −→ P (Vb ) is a triple isomorphism.

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Bounded symmetric domains By a result of Horn and Neher [88], we can decompose further the

non-atomic part of a JBW*-triple V in (3.72) (via a triple isomorphism) into an `∞ -sum as follows. ! V =

M

J(u)

⊕ V1 ⊕ R ⊕ H(A, β)

(3.73)

u

where V1 is a (non-atomic) discrete JBW*-triple, R is a weak* closed right ideal of a continuous von Neumann algebra and H(A, β) = {a ∈ A : β(a) = a} for some continuous von Neumann algebra A, with a linear involutive *-antiautomorphism β : A −→ A. The second and third summand inherit the Jordan triple product from the underlying von Neumann algebra. Of course, any summand in (3.73) may vanish. We recall that a von Neumann algebra A is said to be continuous if it does not contain a non-zero projection p such that the algebra pAp is commutative in which case, viewed as a JBW*-algebra, A does not contain any non-zero abelian tripotent and is therefore non-discrete, by the following lemma. Lemma 3.8.14. Let e be an abelian tripotent in a von Neumann algebra A. Then the algebra ee∗ Aee∗ is commutative. Proof. Write p = ee∗ and q = e∗ e, which are projections in A. We have qe∗ = e∗ and the Peirce 2-space A2 (e) = pAq is an abelian JBW*-triple (cf. Example 2.4.4). It can be seen that pAq is triple isomorphic to the JBW*-triple pAqe∗ = pAe∗ , which is therefore abelian. Since pAp is a JBW*-subtriple of pAe∗ , it is also an abelian JBW*-triple and hence a commutative algebra, by Lemma 3.8.6. In the decomposition (3.73), one can classify the first two summands which are discrete. This has been carried out by Horn in [86, (1.7)], which is stated below.

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Theorem 3.8.15. A JBW*-triple is discrete if and only if it is triple isomorphic to an `∞ -sum M

C(Ωα , Cα )

α

where Ωα is a hyperstonean space, Cα is a discrete factor and C(Ωα , Cα ) is the complex Banach space of Cα -valued continuous functions on Ωα , which is a JBW*-triple in the pointwise triple product {f, g, h}(ω) = {f (ω), g(ω), h(ω)}

(f, g, h ∈ C(Ωα , Cα ), ω ∈ Ωα ).

We recall that a stonean space is a compact (Hausdorff) extremely disconnected topological space, and a stonean space Ω is hyperstonean if and only if C(Ω) is a von Neumann algebra (cf. [154, p. 46]). To classify finite dimensional bounded symmetric domains, it suffices to classify the irreducible ones. For infinite dimensional irreducible bounded symmetric domains in dual Banach spaces, we have the following classification theorem. Theorem 3.8.16. A bounded symmetric domain D in a dual Banach space is irreducible if and only if it is biholomorphic to the open unit ball of a JBW*-factor. Proof. Let D be realised as the open unit ball of a JBW*-triple V . If V admits a non-trivial weak* closed triple ideal J. Then V decomposes into an `∞ -direct sum V = J ⊕∞ J of two non-trivial JBW*-triples J and J . It follows that D is biholomorphic to the product of the open unit balls of J and J , contradicting irreducibility of D. Conversely, let D be the open unit ball of a JBW*-factor. If D is biholomorphic to the product D1 × D2 of two bounded symmetric

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domains D1 and D2 , which are realised as the open unit balls of the JBW*-triples V1 and V2 respectively, then D ≈ D1 × D2 is the open unit ball of the `∞ -sum V1 ⊕∞ V2 , which is not a factor since V1 ⊕ {0} is a non-trivial weak*-closed triple ideal. This contradicts the assumption that D is the open unit ball of a factor. Hence JBW*-factors can be used to classify irreducible bounded symmetric domains in dual Banach spaces, including all finite dimensional irreducible ones. We see from (3.73) and Theorem 3.8.15 that a JBW*-factor must be one of the following forms: (i) C(Ωα , Cα )

(ii) R

(iii) H(A, β)

where the hyperstonean space Ωα reduces to a singleton since for each ω ∈ Ωα , the subspace J = {f ∈ C(Ωα , Cα ) : f (ω) = 0} is a weak* closed triple ideal in C(Ωα , Cα ). Hence C(Ωα , Cα ) identifies with the discrete factor Cα . Examples of continuous von Neumann algebras can be constructed by Murray and von Neumann’s group measure space construction. Let G be an infinite countable discrete group with identity e, in which all conjugacy classes {hgh−1 : h ∈ G}

(g ∈ G\{e})

except {e}, are infinite. For instance, G can be the free group of two or more generators. Let L2 (G) be the Hilbert space of square integrable functions on G with respect to the left Haar measure and let ρ : G −→ L(L2 (G)) be the left regular representation defined by ρ(x)f (y) = f (x−1 y)

(x, y ∈ G, f ∈ L(L2 (G)).

The group von Neumann algebra M(G) is defined to be the von Neumann algebra generated by ρ(G) in the von Neumann algebra L(L2 (G))

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of bounded linear operators on L2 (G). Pick a projection p ∈ M(G). Then R := pM(G) is an example for (ii) above. A JBW*-factor must be discrete if it contains a minimal tripotent, by Lemma 3.8.11. In particular, Cartan factors are discrete. Hence all finite-rank JBW*-triples are discrete since they are finite `∞ -sums of Cartan factors, by Theorem 3.3.5. To conclude the section, we classify the discrete JBW*-factors. It turns out that they are exactly the Cartan factors and in view of Theorem 3.8.16 and Corollary 3.8.18 below, we have extended Cartan’s classification to irreducible bounded symmetric domains in dual Banach spaces. The following result has been proved by Horn in [86, (1.8)]. Theorem 3.8.17. A JBW*-factor is discrete if and only if it is a Cartan factor. Proof. The proof for the necessity is lengthy. We sketch the main steps, following the grid approach in [52]. Let V be a discrete JBW*-factor. Then there is an abelian tripotent e ∈ V such that V = J(e). Since V is a factor, e must be a minimal tripotent, for otherwise we can find two non-zero mutually orthogonal tripotents u and v such that e = u + v and J(u) is a proper weak* closed triple ideal in V , which is impossible. The next step is to show that the rank of the Peirce 1-space V1 (e) is at most 2. Indeed, given a non-zero tripotent v ∈ V1 (e), if it is not already a minimal tripotent in V1 (e), then v is the sum v = v1 + v2 of two mutually orthogonal tripotents v1 , v2 ∈ V1 (e). We need only show that v1 and v2 are minimal tripotents. Take v1 say. We see that e lies outside the Peirce 2-space V2 (v1 ) of v1 for otherwise V2 (e) ⊂ V2 (v1 ) which would imply v2 is orthogonal to

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e — impossible! But e is not in V0 (v1 ) either since this would imply e is orthogonal to v1 = 2{e, e, v1 }. We show e ∈ V1 (v1 ). Consider the Peirce decomposition e = P2 (v1 )e + P1 (v1 )e + P0 (v1 )e and apply to it the Peirce 2-projection of e, we obtain e = P2 (e)e = P2 (e)P2 (v1 )e + P2 (e)P1 (v1 )e + P2 (e)P0 (v1 )e. By the Peirce multiplication rules, we have P2 (v1 )e = {v1 , {v1 , e, v1 }, v1 } ∈ V2 (e) and hence P2 (v1 )e = P2 (e)P2 (v1 )e = λe for some λ ∈ C, as e is a minimal tripotent. But P2 (v1 ) : V −→ V is a contractive projection, we can only have λ = 0 or λ = 1. Since we have already establish e ∈ / V2 (v1 ), we must have λ = 0 and thus e = P2 (e)P1 (v1 )e + P2 (e)P0 (v1 )e. Further, we have P0 (v1 )e = e − 2{v1 , v1 , e} + P2 (v1 )e = e − 2{v1 , v1 , e}, where {v1 , v1 , e} ∈ V2 (e) implies P0 (v1 )e = P2 (e)P0 (v1 )e = e − 2{v1 , v1 , e} = µe

(µ ∈ C)

by minimality of e. Again, we have µ ∈ {0, 1} since P0 (v1 ) is a contractive projection. If µ = 1, the preceding identity implies {v1 , v1 , e} = 0, contradicting e ∈ / V0 (v1 ). It follows that µ = 0 and {v1 , v1 , e} = that is, e ∈ V1 (v1 ). By the Peirce multiplication rules again, we have {e, v1 , e} = {v1 , e, v1 } = 0.

1 2 e,

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Hence {e + v1 , e + v1 , e + v1 } = 2(e + v1 ) and u :=

√1 (e 2

+ v1 ) is a tripotent. By a remark following (3.29), the

Peirce symmetry B(u, 2u) : V −→ V is a triple isomorphism and it follows from B(u, 2u)e = v1 that v1 is a minimal tripotent. Likewise v2 is a minimal tripotent. This proves rank V1 (e) ≤ 2. We now classify V case by case, depending on the rank of V1 (e). Case 0. V1 (e) = {0} in which case, we also have V0 (e) = {0} for otherwise, V2 (e) = Ce would be a proper weak* closed triple ideal in V = V2 (e) ⊕ V0 (e), which is impossible. Hence V ≈ C. Case 1. rank V1 (e) = 1. Let v be a minimal tripotent in V1 (e). Then there are two possibilities: (a) e ∈ V1 (v): it can be shown in this case that V is a Hilbert space, which is a Type I Cartan factor (cf. Example 2.4.9). (b) e ∈ V2 (v): it can be shown that V is a Type III Cartan factor. Case 2. rank V1 (e) = 2. Let v1 and v2 be two mutually orthogonal minimal tripotents in V1 (e). Then we have e ∈ V1 (v1 ) ∩ V1 (v2 ). Let e˜ = {v1 , e, v2 }. Then e˜ is a minimal tripotent orthogonal to e and there are six possibilities: (c) V1 (e + e˜) = {0}: one can show that V is a spin factor. (d) V1 (e + e˜) 6= {0}: in this case, dim V2 (e + e˜) = 4, 6, 8 or 10 and the following possibilities occur: (e) dim V2 (e + e˜) = 4: one shows that V is a Type I Cartan factor.

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Bounded symmetric domains (f ) dim V2 (e + e˜) = 6: one shows that V is a Type II Cartan

factor. (g) dim V2 (e + e˜) = 8: one shows that V is a Type V Cartan factor. (h) dim V2 (e + e˜) = 10: one shows that V is a Type VI Cartan factor. We refer to [52] for a complete proof of the above cases. Finally, we arrive at a Jordan formulation of Cartan’s classification. Corollary 3.8.18. A finite dimensional irreducible bounded symmetric domain is biholomorphic to the open unit ball of a Cartan factor. Proof. Let D be an irreducible bounded symmetric domain in Cn . By Theorem 3.8.16, D is biholomorphic to the open unit ball of a JBW*factor V , which is finite dimensional. By Theorem 2.2.34, V contains a minimal tripotent u. As noted before, V must be discrete since V is a JBW*-factor and we must have V = J(u). By Theorem 3.8.17, V is a Cartan factor. Corollary 3.8.19. Let V be a JB*-triple. Then we have k{x, y, z}k ≤ kxkkykkzk

(x, y, z ∈ V ).

Proof. This is an immediate consequence of the preceding result and Corollary 3.8.13 since the inequality holds in all Cartan factors.

Notes. Corollary 3.8.13 has been proved in [65]. The group von Neumann algebra M(G) constructed before Theorem 3.8.16 is a wellknown example of a type II1 von Neumann algebra with a trivial centre.

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Chapter 4

Function theory 4.1

The class S

The beautiful theory of injective holomorphic functions, alias univalent functions, is central in complex function theory of one variable. By virtue of the Riemann mapping theorem, the study of these functions on simply connected domains can be transported to the open unit disc D, where it is customary to formulate the theory, for instance, the Bieberbach conjecture [15]. Given a univalent function h : D −→ C, we must have h0 (0) 6= 0 and the univalent function f := (h − h(0))/h0 (0) satisfies f (0) = 0 = f 0 (0) − 1, with a Taylor series expansion f (z) = z + a2 z 2 + · · · + an z n + · · ·

(z ∈ D)

which is more convenient to handle. For this reason, it suffices to focus one’s attention on the class of univalent functions f on D satisfying f (0) = 0 and f 0 (0) = 1. Such a function is called a (normalised) shlicht function, or a normalised univalent function, and the class of these functions is usually denoted by S. The Bieberbach conjecture, proved by de Branges [22], says that |an | ≤ n for each f (z) = z +a2 z 2 +· · · ∈ S. A generalisation of the unit disc D to higher dimensions is the Euclidean balls. However, to confine the study of univalent functions in the 313

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setting of Euclidean balls in higher dimensions is of limited generality, which would exclude many interesting domains (e.g. polydiscs) since the Riemann mapping theorem fails in this case. In view of Cartan’s classification of bounded symmetric domains and its infinite dimensional extension, bounded symmetric domains, which are biholomorphic to the open unit balls of JB*-triples, appear to be a more appropriate setting for extending the function theory of D. Let D be a bounded symmetric domain realised as the open unit ball of a JB*-triple V . A holomorphic map f : D −→ V is called normalised if f (0) = 0 and the derivative f 0 (0) : V −→ V is the identity map. By the open mapping theorem, the image f (D) of each schlicht function f ∈ S is a domain in C and f : D −→ C is a holomorphic embedding, as defined in Definition 3.7.1. Generalising the class S, we will denote by S(D) the class of normalised holomorphic embeddings of D in V . In particular, S = S(D). For each f = z + a2 z 2 + · · · ∈ S, a well-known consequence of the inequality |a2 | ≤ 2 is the Koebe distortion theorem: 1 + |z| 1 − |z| ≤ |f 0 (z)| ≤ 3 (1 + |z|) (1 − |z|)3

(f ∈ S)

which implies a growth estimate of f : |z| |z| ≤ |f (z)| ≤ 2 (1 + |z|) (1 − |z|)2

(f ∈ S).

Although the Koebe distortion theorem is not true for all maps f ∈ S(B) on higher dimensional Euclidean balls B, there are numbers of growth and distortion results in literature for convex maps in S(B) and for some other domains (see, for example, [68]). A function f ∈ S(D) is called convex if its image f (D) is a convex domain. We shall only present two distortion theorems in this section for bounded symmetric domains to sample the Jordan connections. Further

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discussion of distortion results related to the Bloch constant will be given in the next section. First, the following growth and distortion results for convex maps have been proved in [75] and [76], respectively. Lemma 4.1.1. Let B be the open unit ball of a complex Banach space V . For each convex map f ∈ S(B), we have kxk kxk ≤ kf (x)k ≤ 1 + kxk 1 − kxk

(x ∈ B).

(4.1)

Lemma 4.1.2. Let B be the open unit ball of a complex Banach space V . For each convex map f ∈ S(B), we have 1 + kxk 1 − kxk CB (x, y) ≤ kf 0 (x)yk ≤ CB (x, y) 1 + kxk 1 − kxk

(4.2)

for each x ∈ B and y ∈ V , where CB (x, y) is the Carath´eodory differential metric on B. We prove the following distortion result for convex maps on bounded symmetric domains. Theorem 4.1.3. Let D be a bounded symmetric domain realised as the open unit ball of a JB*-triple V . For each convex map f ∈ S(D), we have (i)

1 1 ≤ kf 0 (x)k ≤ 2 (1 + kxk) (1 − kxk)2

(ii)

(1 − kxk)kyk kyk ≤ kf 0 (x)yk ≤ 1/2 (1 − kxk)2 (1 + kxk)kB(x, x) k

(x ∈ D). (x ∈ D, y ∈ V )

where B(x, x) is the Bergmann operator on V . Proof. For the lower bound in (i), we apply (4.2) which gives kf 0 (x)yk ≥

1 − kxk 1 − kxk CD (x, y) = kB(x, x)−1/2 yk (x ∈ D, y ∈ V ). 1 + kxk 1 + kxk

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Taking supremum over y ∈ D, we get kf 0 (x)k ≥

1 − kxk 1 − kxk 1 . kB(x, x)−1/2 k = = 1 + kxk (1 + kxk)(1 − kxk2 ) (1 + kxk)2

The upper bound in (i) is a consequence of the right inequality in (ii), which follows immediately from (4.2): kf 0 (x)yk ≤

1 + kxk 1 + kxk kyk CD (x, y) = kB(x, x)−1/2 yk ≤ 1 − kxk 1 − kxk (1 − kxk)2

by Lemmas 3.2.28 and 3.5.11. For the left inequality in (ii), a simple application of (4.2) and Lemma 3.5.11 again gives kf 0 (x)yk ≥

1 − kxk 0 1 − kxk CD (x, y) = kg (x)(y)k, 1 + kxk 1 + kxk −x

where g−x is the M¨ obius transformation on D. Since 0 0 kyk = kB(x, x)1/2 g−x (x)(y)k ≤ kB(x, x)1/2 kkg−x (x)(y)k,

we have kf 0 (x)yk ≥

(1 − kxk)kyk . (1 + kxk)kB(x, x)1/2 k

Given an element x in the open unit ball D of a JB*-triple V , although the Bergman operator B(x, x) : V −→ V need not be a hermitian operator [100, Example 4.5], its square root B(x, x)1/2 is the exponential of a hermitian operator on V [37, (3.3)] and it follows from [20] that the norm of B(x, x)1/2 equals its spectral radius. By Lemma 3.2.10, the spectrum of B(x, x)1/2 is contained in [0, 1] and therefore we have kB(x, x)k ≤ kB(x, x)1/2 k2 ≤ 1

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and the lower bound in Theorem 4.1.3 (ii) is greater than

(1−kxk)kyk 1+kxk .

D is a Hilbert ball and x ∈ D, we have  (1 − kxk2 )2 if D = D 1/2 2 kB(x, x) k = kB(x, x)k = 1 − kxk2 if D 6= D

If

(4.3)

which follows from Example 3.4.9 (cf. [37, Lemma 3.2.8]). We now discuss two-point distortion of convex maps on bounded symmetric domains. We will use the estimates in (4.1) and arguments similar to those in the proof of [69, Theorem 7] to deduce the following result. Theorem 4.1.4. Let D be a bounded symmetric domain realised as the open unit ball of a JB*-triple V . For each convex map f ∈ S(D) and two distinct points a, b ∈ D, we have sinh cD (a, b) max (i) kf (a) − f (b)k ≥ exp cD (a, b)



1 − kak2 1 − kbk2 , kf 0 (a)−1 k kf 0 (b)−1 k

 .

(ii) kf (a) − f (b)k ≤  sinh cD (a, b)ecD (a,b) min kf 0 (a)kkB(a, a)1/2 k, kf 0 (b)kkB(b, b)1/2 k where cD is the Carath´eodory distance. Proof. Using analogous arguments of the proof in [69, Theorem 7], one can show that kf (a) − f (b)k sinh cD (a, b) ≥ max{k(f 0 (a)ga0 (0))−1 k−1 , k(f 0 (b)gb0 (0))−1 k−1 } exp cD (a, b) where ga and gb are M¨ obius transformations. Since kgx0 (0)−1 k = kB(x, x)−1/2 k =

1 1 − kxk2

by Lemmas 3.2.25 and 3.2.28, we have k(f 0 (x)gx0 (0))−1 k ≤ kgx0 (0)−1 kkf 0 (x)−1 k ≤

kf 0 (x)−1 k 1 − kxk2

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for x ∈ D. This proves (i). For the upper bound of kf (a) − f (b)k, define F (x) = ga0 (0)−1 f 0 (a)−1 (f (ga (x)) − f (a))

(x ∈ D).

Then F is a convex map in S(D) and hence (4.1) implies kF (x)k ≤

kxk = sinh cD (x, 0) exp cD (x, 0) 1 − kxk

(x ∈ D)

where 2cD (x, 0) = log 1+kxk 1−kxk . Substituting g−a (b) for x gives kf (b) − f (a)k = kf 0 (a)ga0 (0)F (g−a (b))k ≤ kf 0 (a)ga0 (0)kkF (g−a (b))k ≤ kf 0 (a)kkga0 (0)k sinh cD (g−a (b), 0) exp cD (g−a (b), 0) = kf 0 (a)kkB(a, a)1/2 k sinh cD (a, b) exp cD (a, b). Changing the roles of a and b yields the upper bound in (ii). We will introduce Bloch maps in the following section and prove a distortion theorem for these functions in finite dimensions. Notes.

A detailed treatment of univalent function theory and

generalisations in several complex variables has been given in the book [70]. For a schlicht function f (z) = z + a2 z 2 + · · · ∈ S, the inequality |a2 | ≤ 2 was first proved by Bieberbach [15]. The inequality |an | ≤ n for n = 3, 4, 5, 6 were already proven by 1972, prior to de Branges’s proof for all n. Theorems 4.1.3 and 4.1.4 are taken from [41]. The two-point distortion theorem in [105, Remark] for the unit disc D is a special case of Theorem 4.1.4.

4.2

Bloch constant and Bloch maps

Let H(D) be the class of complex-valued holomorphic functions on the unit disc D and let f ∈ H(D) with f 0 (0) = 1. The celebrated Bloch’s

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theorem states that f maps a subdomain in D biholomorphically onto a disc with radius r(f ) greater than some positive universal constant. The ‘best possible’ constant b for all such functions, that is, b = inf{r(f ) : f ∈ H(D) and f 0 (0) = 1}, is called the Bloch constant. As a direct consequence of his extension of the Schwarz lemma, Ahlfors [6] gave an elementary proof of Bloch’s theorem with an explicit √ lower bound b ≥ 3/4. Using the fact that b = inf{r(f ) : f ∈ H(D), f 0 (0) − 1 = f (0) = 0, sup(1 − |z|2 )|f 0 (z)| ≤ 1} z∈D

(4.4) (cf. [117]) and proving a distortion theorem in [18] for functions f in (4.4), which says √ 1 − 3|z| Re f (z) ≥  3 |z| √ 1− 3 0

  1 , |z| ≤ √ 3

(4.5)

Bonk was able to improve the lower bound for the Bloch constant to √

b>

3 4

+ 10−14 .

A holomorphic function f : D −→ C satisfying |f |B := sup(1 − |z|2 )|f 0 (z)| < ∞ z∈D

is known as a Bloch function, where |f |B is called the Bloch semi-norm. We have therefore seen that one can derive an estimate of the Bloch constant via some distortion result for normalised Bloch functions f on D satisfying |f |B ≤ 1 (equivalently, |f |B = 1 as f is normalised). This will be the theme of our discussion for higher dimensional extension throughout this section, where the following convention is adopted.

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with the Euclidean norm kzk2 =

p |z1 |2 + · · · + |zn |2

(z = (z1 , . . . , zn ) ∈ Cn )

unless stated otherwise. The open Euclidean ball centred at a ∈ Cn with radius r > 0 will be denoted by Bn (a, r) := {z ∈ Cn : kz − ak2 < r}. The Bloch semi-norm |f |B on D can be expressed as |f |B = sup{|(f ◦ g)0 (0)| : g ∈ Aut D}. Indeed, for each g ∈ Aut D, we have |g 0 (0)| = 1 − |g(0)|2 which implies |(f ◦ g)0 (0)| = |f 0 (g(0))|(1 − |g(0)|2 ). On the other hand, for each z ∈ D, and for the M¨ obius transformation g−z induced by −z, we have 0 (1 − |z|2 )|f 0 (z)| = (1 − |z|2 )|(f ◦ gz )0 (0)||g−z (z)| = |(f ◦ gz )0 (0)|.

The above observation leads to a natural generalisation of the concept of a Bloch function to higher dimensions. Definition 4.2.1. Let D be a bounded symmetric domain realised as the open unit ball of a JB*-triple. The Bloch semi-norm of a holomorphic map f : D −→ Cn is defined by |f |B = sup{k(f ◦ g)0 (0)k : g ∈ Aut D} where Cn is equipped with the Euclidean norm. We call f a Bloch map if |f |B < ∞. A Bloch map f : D −→ C is often called a Bloch function.

Remark 4.2.2. Let D be a bounded symmetric domain in a Banach space (V, | · |), and let k · ksp be an equivalent norm so that (V, k · ksp ) is

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a JB*-triple and D identifies with the open unit ball {v ∈ V : kvksp < 1} (cf. Theorem 3.2.18). The Bloch semi-norm |f |B is defined in terms of the norm k·ksp . However, if one uses the norm of (f ◦g)0 (0) : (V, |·|) −→ Cn , given by |(f ◦ g)0 (0)|∗ = sup{k(f ◦ g)0 (0)(v)k2 : |v| ≤ 1}, to define another semi-norm |f |∗B := sup{|(f ◦ g)0 (0)|∗ : g ∈ Aut D}, then we have α|f |∗B ≤ |f |B ≤ β|f |∗B for some 0 < α ≤ β and there is no difference between defining a Bloch map by the condition |f |∗B < ∞, or by |f |B < ∞ although the two semi-norms are generally different. Evidently, the Bloch semi-norm of f : D −→ Cn is invariant under the automorphism group Aut D and as before, the inequality 0 (1 − kzk2 )kf 0 (z)k ≤ (1 − kzk2 )k(f ◦ gz )0 (0)kkg−z (z)k

= (1 − kzk2 )k(f ◦ gz )0 (0)kkB(z, z)−1/2 k

(z ∈ D)

0

= k(f ◦ gz ) (0)k implies sup{(1 − kzk2 )kf 0 (z)k : z ∈ D} ≤ |f |B

(4.6)

although the two sides above need not be equal in higher dimensions, even in the case of a bidisc. We will give an example later. Example 4.2.3. Let f : D −→ Cn be a bounded holomorphic map on the open unit ball D of a JB*-triple V and denote the supremum norm of f by kf k∞ := sup{kf (z)k2 : z ∈ D}. Then f is a Bloch map and |f |B ≤ kf k∞ . Indeed, let kf k∞ < 1. Then for each g ∈ Aut D, the holomorphic map f ◦ g maps D into the Euclidean ball Bn ⊂ Cn . Hence the Schwarz-Pick lemma (cf. Proposition

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3.5.19) gives k(f ◦ g)0 (0)k ≤ kB(f (g(0)), f (g(0)))1/2 k ≤ 1. From this, one deduces |f |B ≤ kf k∞ . On the unit disc D, the holomorphic function ψ(ζ) = tanh−1 (ζ) is a simple example of a unbounded Bloch function with Bloch semi-norm |ψ|B = 1. Given a ∈ D\{0}, let φa ∈ V ∗ be the support functional at a, that is, φa (a) = kak and kφa k = 1. The holomorphic function ψ ◦ φa : D −→ C is clearly unbounded on D, but it is a Bloch function with Bloch semi-norm |ψ ◦ φa |B = |ψ|B since k(ψ ◦ φa )0 (0)k = kψ 0 (φa (0))φ0a (0)k = kψ 0 (0)φa k = 1 and for each g ∈ Aut D, we have k(ψ ◦ φa ◦ g)0 (0)k ≤ |ψ 0 (φa (g(0)))|k(φa ◦ g)0 (0)k |ψ|B (1 − |φa (g(0))|2 ) = |ψ|B ≤ 1 − |φa (g(0))|2 by (4.6) and Proposition 3.5.19. Bloch’s theorem fails in dimension 2 [180]. Nevertheless, one can define the Bloch constant for various families of Bloch maps in higher dimensions. We first extend Bonk’s distortion theorem to locally biholomorphic maps on finite dimensional bounded symmetric domains and will return to the Bloch constant and Block maps afterwards. Definition 4.2.4. Let D be a domain in a complex Banach space V , and W a Banach space. A holomorphic map f : D −→ W is called locally biholomorphic if the derivative f 0 (a) : V −→ W has a continuous inverse for every a ∈ V . We denote by Hloc (D, W ) the class of locally biholomorphic maps f : D −→ W . We begin with extensions of Bonk’s distortion theorem. We observe that Bonk’s theorem concerns normalised Bloch functions f with

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|f |B = 1. In higher dimensional Euclidean balls Bn ⊂ Cn and polydiscs Dn , replacing (f ◦ g)0 (0) in the definition of |f |B by its Jacobian, the Bloch semi-norm decreases to kf k0 := sup{| det(f ◦ g)0 (0)|1/n : g ∈ Aut D} ≤ |f |B

(D = Bn or Dn )

where | det f 0 (0)|1/n ≤ kf k0 and kf k0 = |f |B for n = 1. Using this reduction of the Bloch semi-norm, one can show the following distortion results for Bn and Dn . Theorem 4.2.5. Let f : D −→ Cn be a locally biholomorphic map satisfying det f 0 (0) = 1 = kf k0 , where D = Bn or Dn . Then   −(n + 1)kzk exp 1 − kzk (z ∈ Bn ) | det f 0 (z)| ≥ Re det f 0 (z) ≥ (1 − kzk)n+1 and  exp | det f 0 (z)| ≥ Re det f 0 (z) ≥

−2nkzk 1 − kzk

(1 − kzk)2n

 (z ∈ Dn ).

Both inequalities are sharp. Proof. The first inequality is proved in [121, Theorem 7], and the second one is proved in [174, Theorem 3.2]. The preceding distortion results have also been used to estimate suitably defined Bloch constants on Bn and Dn , which will be discussed later. For now, two natural questions arise: Question 4.2.6. Can we explain the difference of the exponents in the distortion bounds in Theorem 4.2.5? Question 4.2.7. Can we extend Theorem 4.2.5 to other bounded symmetric domains in Cn ?

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orem 4.2.5 to all finite dimensional bounded symmetric domains, and answer both questions affirmatively. Let D be a finite dimensional bounded symmetric domain realised as the open unit ball of a JB*-triple V of dimension dim V = n. Denote by r the rank of D, and by p the genus of D, as defined in Definition 3.3.9. We introduce a constant c(D) :=

pr 2

(4.7)

which satisfies (n + 1)/2 ≤ c(D) ≤ n by (3.33) and will play the role of the constant term in the exponents in Theorem 4.2.5. In fact, we have c(Bn ) = (n + 1)/2 for the Euclidean ball Bn ⊂ Cn as Bn has rank 1, and c(Dn ) = 2n/2 = n for the polydisc Dn ⊂ Cn . Given a holomorphic map f : D −→ Cn , we define n o kf kd = sup (1 − kzk2 )c(D)/n | det f 0 (z)|1/n : z ∈ D which is not a norm, but may be called the quasinorm of f . Note that | det f 0 (0)|1/n ≤ kf kd , |f |B . For D = Bn , we have kf kd = kf k0 (cf. [121, p. 356]), and for D = Dn , we have kf kd ≤ kf k0 =

{(1 − |z1 |2 )1/2 · · · (1 − |zn |2 )1/2 | det f 0 (z)|1/n }

sup z=(z1 ,...,zn

)∈Dn

(cf. [174, p. 654]). We will use the quasinorm kf kd , instead of kf k0 , in the generalisation of Theorem 4.2.5. First, we need two lemmas, one of which is the following version of Julia’s lemma.

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Lemma 4.2.8. (Julia’s lemma) Let ρ, ψ : D∪{1} −→ C be holomorphic functions such that ρ(D) ⊂ D, ρ(1) = 1, ψ(1) = 0 and ψ(D) ⊂ {ζ ∈ C : Re ζ > 0}. Then we have ρ0 (1) > 0 and for each r > 0, ψ(H(1, 1/r)) ⊂ {ζ ∈ C : |ζ − dr| ≤ dr} where d = −ψ 0 (1) > 0 and H(1, 1/r) is the closure of the horodisc   |1 − ζ|2 1 r H(1, 1/r) = ζ ∈ C : 0 and   1−x 0 |ρ(x)| ≥ exp −2ρ (1) 1+x for all x ∈ (−1, 1). Proof. By Julia’s lemma, we have ρ0 (1) > 0. Let ψ = − log ρ. Then we have Re ψ(z) = − log |ρ(z)| > 0 for all z ∈ D and ψ(1) = 0. Moreover, ψ 0 (1) = − log ρ0 (1) < 0. Given −1 < x < 1, let r =

1−x 1+x

and d = −ψ 0 (1). Then x ∈

H(1, 1/r) and by Julia’s lemma again, we have |ψ(x) − dr| ≤ dr and Re ψ(x) ≤ |ψ(x)| ≤ 2dr = −2ψ 0 (1)

1−x 1+x

which gives 

1−x |ρ(x)| ≥ exp −2ρ (1) 1+x 0

 .

Here is a unified distortion result which includes Theorem 4.2.5 as a special case.

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Theorem 4.2.10. Let D be a bounded symmetric domain realised as the open unit ball of a JB∗ -triple V and dim V = n . Let f : D −→ Cn be locally biholomorphic, satisfying det f 0 (0) = 1 = kf kd . Then we have   1 −2c(D)kzk 0 (z ∈ D). (4.8) | det f (z)| ≥ exp 1 − kzk (1 − kzk)2c(D) The estimate in (4.8) is sharp. Proof. Write c = c(D) to simplify notation. Let z ∈ D and u = z/kzk. Define a holomorphic function ρ : D ∪ {1} −→ C by     1 + ζ 2c 0 1−ζ ρ(ζ) = det f u . 2 2 Evidently, ρ(1) = 1. Since f is locally biholomorphic, we have ρ(D) ⊂ D\{0}. Observe from the definition of the quasinorm kf kd = 1 that | det f 0 (w)| ≤

kf knd = (1 + kwk2 + kwk4 + · · · )c (1 − kwk2 )c

(w ∈ D)

where det f 0 (0) = 1. Therefore ρ0 (1) = c and c 2c 2c   1 + ζ 1 + ζ 1 − ζ 1 det f 0 ≤ 0 < |ρ(ζ)| = u 0. Therefore kF kd ≤ 1 and hence kF kd = 1 = det F 0 (0). Since det F 0 (±kzku) = ψ(±kzk) for all z ∈ D, we see that F attains the equality in (4.8). Now we resume the discussion of Bloch maps. As mentioned at the beginning of this section, given a holomorphic map f : D −→ Cn , the Bloch semi-norm |f |B , unlike the one-dimensional case, need not be equal to sup{(1 − kzk2 )kf 0 (z)k : z ∈ D} ≤ |f |B . In spite of this, one can still ask a weaker question whether both seminorms are finite at the same time so that, if the answer is positive, one can also use the ‘simpler’ condition sup{(1−kzk2 )kf 0 (z)k : z ∈ D} < ∞ as an equivalent definition for a Bloch function. Unfortunately, the answer is negative even in the case of a bidisc, but positive for a Hilbert ball!

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Example 4.2.11. Let D = D × D be the bidisc and let f : D −→ C be defined by f (z1 , z2 ) = (1 − z2 ) log

1 , 1 − z1

(z1 , z2 ) ∈ D.

Then we have supz∈D (1 − kzk2 )kf 0 (z)k < ∞, but |f |B = ∞. Indeed, for (x1 , x2 ) ∈ C2 , f 0 (z1 , z2 )(x1 , x2 ) =

(1 − z2 )x1 1 − x2 log . 1 − z1 1 − z1

Observe that 1 − z2 1 kf (z1 , z2 )k ≤ + log 1 − z1 1 − z1 2 |1 − z2 | + + log 2 + π ≤ |1 − z1 | |1 − z1 | 0

where 1/|1 − z1 | ≥ 1/2 implies 2/|1 − z1 | ≥ log(2/|1 − z1 |) ≥ | log(1/|1 − z1 |)| − log 2. Pick z = (z1 , z2 ) ∈ D, where |z1 | > |z2 | or |z1 | ≤ |z2 |. In the former case, we have  |1 − z2 | 2 (1 − kzk )kf (z)k ≤ (1 − |z1 | ) + + log 2 + π |1 − z1 | |1 − z1 | ≤ (1 + |z1 |)(|1 − z2 | + 2 + (log 2 + π)(1 − |z1 |)). 2

0

2



In the case of |z1 | ≤ |z2 |, we have |1 − z1 | ≥ 1 − |z1 | ≥ 1 − |z2 | and hence (1 − kzk2 )kf 0 (z)k = (1 − |z2 |2 )kf 0 (z)k ≤ (1 + |z2 |)(|1 − z2 | + 2 + (log 2 + π)(1 − |z2 |)). We deduce from these inequalities that sup (1 − kzk2 )kf 0 (z)k < ∞. z∈D

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On the other hand, for z = (z1 , 0) ∈ D, we have   2 2 0 f (z1 , 0) 1 − |z1 | , 1 = 1 − |z1 | − log 1 . 2 2(1 − z1 ) 1 − z1 Let gz : D −→ D be the M¨ obius transformation induced by z. We have 0 |f 0 (z)(x1 , x2 )| ≤ k(f ◦ gz )0 (0)kkg−z (z)(x1 , x2 )k

≤ |f |B kB(z, z)−1/2 (x1 , x2 )k.   2 For z = (z1 , 0) and (x1 , x2 ) = 1−|z2 1 | , 1 , we have kB(z, z)−1/2 (x1 , x2 )k = kB(z, z)−1/2 ((1−|z1 |2 )/2, 1)k = k(1/2, 1)k = 1. Hence |f |B

1 − |z1 |2 1 − log ≥ 2(1 − z1 ) 1 − z1 1 1 − |z1 |2 ≥ log − 1 − z1 2|1 − z1 | 1 1 + |z1 | 1 ≥ log ≥ log − −1 1 − z1 2 1 − z1

where the right-hand side is unbounded on D. Example 4.2.12. Let D be the open unit ball of a Hilbert space H. Let f : D −→ C be a holomorphic function satisfying kf kD := sup{(1 − kzk2 )kf 0 (z)k : z ∈ D} < ∞. Then the Bloch semi-norm of f satisfies √ |f |B ≤ (2 + 2 2)(sup (1 − kzk2 )kf 0 (z)k)

(4.9)

(4.10)

z∈D

and hence f is a Bloch function. If H = C, we actually have |f |B = kf kD , already noted before. To show (4.10) for dim H ≥ 2, we first consider the case of dim H = 2, where D = B2 = {(z1 , z2 ) ∈ C2 : |z1 |2 + |z2 |2 < 1}.

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For f : B2 −→ C satisfying (4.9), we derive the inequality √ ∂f p 1 − |x1 |2 ≤ (1 + 2 2 )kf kB ((x1 , 0) ∈ B2 ). (4.11) (x , 0) 1 2 ∂z2 p Given (x1 , 0) ∈ B2 , let r = √12 1 − |x1 |2 so that |x1 |2 + r2 < 1. By Cauchy formula, we have   Z ∂f ∂f 1 (x1 , 0) = ∂z2 ∂z1 2πi |w|=r

∂f ∂z1 (x1 , w) dw. w2

Hence √   ∂f 2πrkf kB2 2 2kf kB2 ∂f 1 ∂z2 ∂z1 (x1 , 0) ≤ 2π r2 (1 − |x1 |2 − r2 ) = (1 − |x |2 )3/2 . 1 Observe that ∂f ∂f (x1 , 0) − (0, 0) = ∂z2 ∂z2

Z 0

1

∂2f (tx1 , 0)x1 dt ∂z2 ∂z1

which gives √ Z 1 ∂f ∂f 2 2kf k B 2 dt + (0, 0) ∂z2 (x1 , 0) ≤ 2 2 3/2 ∂z2 0 (1 − t |x1 | ) √ 2 2kf kB2 kf kB2 ≤ p +p 2 1 − |x1 | 1 − |x1 |2 and therefore √ p ∂f 2 ∂z2 (x1 , 0) 1 − |x1 | ≤ (1 + 2 2 )kf kB2 . Returning to the case of an arbitrary Hilbert ball D and f : D −→ C in (4.9), we note that |f |B = sup{k(f ◦ ga )0 (0)k : a ∈ D} where ga is the M¨ obius transformation induced by a and we have used the fact that each automorphism of D is of the form ϕ ◦ ga for some linear isometry ϕ of D and a ∈ D, by Theorem 3.2.32.

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Let a ∈ D\{0}. Then u = a/kak is an extreme point of the closed ball D and hence a maximal tripotent in H by Theorem 3.4.8. It follows from Example 3.4.9 that H is the orthogonal sum H1 (u) ⊕ H2 (u) of the Peirce spaces and ga0 (0)v

1/2

= B(a, a)

p (v) =

1 − kak2 v (v ∈ H1 (u)) (1 − kak2 )v (v ∈ H2 (u)).

Let v ∈ H1 (u) with kvk ≤ 1. Define a holomorphic function F : B2 −→ C by F (z1 , z2 ) = f (z1 u + z2 v)

for

(z1 , z2 ) ∈ B2 .

Then we have kF 0 (z1 , z2 )k ≤ kf 0 (z1 u + z2 v)k and F satisfies (4.9) since kF kB2 ≤ kf kD < ∞. It follows from (4.11) that p ∂F p 0 2 (kak, 0) 1 − kak2 |f (a)v| 1 − kak = ∂z2 √ √ ≤ (1 + 2 2 )kF kB2 ≤ (1 + 2 2 )kf kD . Finally, for v = v1 ⊕ v2 ∈ H1 (u) ⊕ H2 (u) with kvk ≤ 1, we have p |(f ◦ ga )0 (0)v| = |f 0 (a)ga0 (0)v| = |f 0 (a)( 1 − kak2 v1 + (1 − kak2 )v2 )| √ √ ≤ (1 + 2 2 )kf kD + kf kD = (2 + 2 2 )kf kD which proves (4.10). The previous two examples are somewhat unsettling. In view of the Hilbert ball case, can one not simply use the condition sup{(1 − kzk2 )kf 0 (z)k : z ∈ D} < ∞ for the definition of a Bloch function in higher dimensions? We give a justification that the condition |f |B < ∞ is in fact a more appropriate choice. Indeed, for the unit disc D, another equivalent condition for f : D −→ C to be a Bloch function is that the family Ff = {f ◦ g − f (g(0)) : g ∈ Aut (D)}

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is a normal family (cf. Definition 1.1.10). The following lemma should justify the choice of |f |B < ∞. Lemma 4.2.13. Let D be a bounded symmetric domain in Cn , realised as the open unit ball of norm on Cn equivalent to the Euclidean norm, and f : D −→ Cd a holomorphic function. The following two conditions are equivalent. (i) |f |B < ∞. (ii) Ff = {f ◦ g − f (g(0)) : g ∈ Aut (D)} is a normal family. Proof. (i)⇒(ii). By Lemma 1.1.11, it suffices to show that Ff is uniformly bounded on compact subsets of D, and by Theorem 1.1.1, suffice it to show sup {k(f ◦ g)0 (z)k : g ∈ Aut D} < ∞ z∈K

for each compact subset K of D. This follows from 0 k(f ◦ g)0 (z)k = k(f ◦ (g ◦ gz ))0 (0)kkg−z (z)k ≤ |f |B kkB(z, z)−1/2 k |f |B . = 1 − kzk2

(ii)⇒(i). This follows from the Cauchy inequality since (f ◦ g)0 (0) = (f ◦ g − f (g(0)))0 (0)

(g ∈ Aut D).

We see now that for the holomorphic function f : D × D −→ C in Example 4.2.11, the family Ff is not normal although supz∈D (1 − kzk2 )kf 0 (z)k < ∞. Corollary 4.2.14. Let D be a bounded symmetric domain in Cn , realised as the open unit ball of an equivalent norm in Cn , and let (fk ) be a

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sequence of Cd -valued Bloch maps, with uniformly bounded Bloch seminorm. Then there is a subsequence of (fk − fk (0)) converging locally uniformly to a Bloch map on D. Proof. Let |fk |B ≤ C for all k and some C > 0. Analogous to the proof of the preceding lemma, one shows that, on a compact subset K of D, sup {k(fk − fk (0))0 (z)k ≤ z∈K

C |fk |B ≤ 2 1 − kzk 1 − kzk2

for all k. Hence Theorem 1.1.1 implies that {fk − fk (0)}k is uniformly bounded on compact sets in D and by Lemma 1.1.11, it is a normal family and the conclusion follows. Now we introduce the Bloch constant for a family of Bloch maps and will use the distortion result in Theorem 4.2.10 to estimate its lower bound. We first introduce two more constants besides c(D), attached to a finite dimensional bounded symmetric domain D in Cn , which are related to the squeezing constant σ ˆD discussed in Section 3.7. Since D identifies with the open unit ball with respect to an equivalent norm k · k on the Euclidean space (Cn , k · k2 ), it is sandwiched between two open Euclidean balls. We define s(D) := inf{kzk2 : kzk = 1},

t(D) := sup{kzk2 : kzk = 1}. (4.12)

It can be seen that Bn (0, s) ⊂ D ⊂ Bn (0, t) where s = s(D) > 0 is the radius of the largest Euclidean ball Bn (0, s) contained in D, and t = t(D) is the radius of the smallest Euclidean ball Bn (0, t) containing D. It follows that 0
0, which is the biholomorphic image f (G) of an open set G ⊂ D under f . If n = 1, a schlicht ball in f (D) is usually called a schlicht disc, in which case one can find a holomorphic map h : D −→ D such that f ◦ h : D −→ B1 (f (a), r) is biholomorphic and (f ◦ h)(ζ) = f (a) + rζ

(ζ ∈ D).

For each point a ∈ D, let r(a, f ) denote the radius of the largest schlicht ball of a holomorphic map f : D −→ Cn centered at f (a). We have r(a, f ) = sup{r > 0 : Bn (f (a), r) ⊂ f (D), f −1 ∈ Hemb (Bn (f (a), r), D)}. Let r(f ) = sup{r(a, f ) : a ∈ D}. Lemma 4.2.16. Let D be a bounded symmetric domain in Cn and let f : D −→ Cn be locally biholomorphic, mapping an open subset G ⊂ D biholomorphically onto a schlicht ball Bn (f (a), r(a, f )) for some a ∈ D. Then ∂G ∩ ∂D 6= ∅. In particular, r(a, f ) = kf (a) − bk2 where b = limk f (zk ) ∈ ∂Bn (f (a), r(a, f )) ∩ ∂f (D) for some sequence (zk ) in G, converging to a boundary point in ∂G ∩ ∂D.

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Proof. Assume ∂G ∩ ∂D = ∅. We deduce a contradiction. In this case, there is a sequence (Gk ) of open sets in D satisfying Gk ⊃ Gk+1 and ∞ \

Gk = G.

n=1

By the definition of schlicht balls, one can find two distinct points zk , wk ∈ Gk such that f (zk ) = f (wk ). We may assume without loss of generality that (zk ) converges to some z ∈ ∂G and (wk ) converges to some w ∈ ∂G. It follows that f (z) = f (w) ∈ ∂Bn (f (a), r(a, f )). If z = w, then f is not locally biholomorphic, which is impossible. Hence z 6= w, but the equality f (z) = f (w) would imply that f is not injective on G, which is also impossible. This proves the first assertion, which implies the second one readily. Definition 4.2.17. Let D be a bounded symmetric domain in Cn , realised as the open unit ball of an equivalent norm in Cn , and let B (D, Cn ) denote the subclass of H (D, Cn ) consisting of locally biHloc loc

holomorphic Bloch maps on D. The Bloch constant b(D) for the class B (D, Cn ) is defined to be Hloc B b(D) = inf{r(f ) : f ∈ Hloc (D, Cn ), |f |B = 1 = det f 0 (0)}.

We prove two lemmas before giving a lower estimate of b(D). Lemma 4.2.18. Let D be a bounded symmetric domain in Cn , identified as the open unit ball of a JB*-triple V = (Cn , k · ksp ) in a norm k · ksp equivalent to the Euclidean norm k · k2 in Cn . Given a (non-zero) bounded linear operator T : V −→ Cn , we have kT (v)k2 ≥

s(D) | det T | kT kn−1

where s(D) is defined in (4.12).

(kvksp = 1)

(4.13)

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Proof. We can consider T as a bounded linear operator from the Euclidean space Cn to itself since k · ksp is equivalent to the Euclidean norm. Let T ∗ be the adjoint operator. Then T ∗ T is a positive operator with eigenvalues 0 ≤ λ1 ≤ · · · ≤ λn = kT ∗ T k. By the minimax principle, we have   ∗   ∗ hT T (u), ui hT T (u), ui : u ∈ V \{0} ≤ inf : kuksp = 1 . λ1 = inf kuk22 kuk22 By the definition of s(D) in (4.12), we have kuk2 ≥ s(D) whenever kuksp = 1. In particular, for kvksp = 1, we have | det T |2 = det(T ∗ T ) = λ1 · · · λn ≤ λ1 kT ∗ T kn−1 ≤

hT ∗ T (v), vi ∗ n−1 kT (v)k22 kT k2(n−1) kT T k ≤ s(D)2 kvk22

which proves the desired inequality. For a locally biholomorphic Bloch map f , we have the following lower estimate for the radius of the largest schlicht ball of f centered at f (0). Theorem 4.2.19. Let D be a bounded symmetric domain in Cn . Given B (D, Cn ) with kf k = 1 = det f 0 (0), we have f ∈ Hloc d

Z r(0, f ) ≥ s(D) 0

1

(1 − t2 )n−1 exp (1 − t)2c(D)



−2c(D)t 1−t

 dt ≥

s(D) 1 ≥ 2c(D) 2c(D)

where s(D) is defined in (4.12), and c(D) in (4.7). Proof. Write c = c(D). As before, D identifies with the open unit ball of a norm equivalent to the Euclidean norm of Cn . By Lemma 4.2.16, r(0, f ) is the Euclidean distance from f (0) to a boundary point of f (D). Let Γ : [0, 1] −→ Cn be the line segment from

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f (0) to a point in ∂f (D), which is of Euclidean length `(Γ) = r(0, f ). Define γ : [0, 1) −→ D by γ(s) = f −1 (Γ(s)) where γ(0) = 0 and lims→1 kγ(s)k = 1. By (4.13), we have Z 1

(f ◦ γ)0 (s) ds r(0, f ) = `(Γ) = 2 0   0 Z 1 Z 1

0 γ (s) 0 0 0

= kf (γ(s))γ (s)k2 ds =

f (γ(s)) kγ 0 (s)k kγ (s)kds 0 0 2 Z 1 | det f 0 (γ(s))| ≥ s(D) dkγ(s)k 0 n−1 0 kf (γ(s))k where, by Theorem 4.2.10 (i) and (4.6), Z 1 | det f 0 (γ(s))| dkγ(s)k 0 n−1 0 kf (γ(s))k   Z 1 (1 − kγ(s)k2 )n−1 −2ckγ(s)k ≥ exp dkγ(s)k (1 − kγ(s)k)2c 1 − kγ(s)k 0   Z 1 (1 − t2 )n−1 −2ct = exp dt. (1 − t)2c 1−t 0 Finally, c(D) ≥ (n + 1)/2 by (3.33) and hence we have   Z 1 1 −2c(D)t s(D) exp dt ≥ . r(0, f ) ≥ s(D) 2 1−t 2c(D) 0 (1 − t) This completes the proof. Before making use of Theorem 4.2.19 to estimate the Bloch constant b(D), we derive two lemmas first. Lemma 4.2.20. Let D be a bounded symmetric domain in Cn . For B (D, Cn ) with det f 0 (0) = 1, there is a Bloch function each f ∈ Hloc B (D, Cn ) such that |h| ≤ |f | , det h0 (0) = 1, r(h) ≤ r(f ) and h ∈ Hloc B B

d(h) := sup{| det(h ◦ ϕ)0 (0)| : ϕ ∈ Aut D} = 1.

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Proof. Let d(f ) = sup{| det(f ◦ϕ)0 (0)| : ϕ ∈ Aut D}. We have d(f ) ≥ 1. If d(f ) = 1, there is nothing to prove. Otherwise, we can pick a sequence (ϕk ) ∈ Aut D such that dk := | det(f ◦ ϕk )0 (0)| > 1 and limk dk = d(f ). Define a Bloch function hk : D −→ Cn by hk =

f ◦ ϕk − f ◦ ϕk (0) 1/n

.

dk

Then we have hk (0) = 0, det h0k (0) = 1 and |hk |B ≤ |f |B . Since dk > 1, we have r(hk ) ≤ r(f ). By Lemma 4.2.13, {hk } is a normal family and hence it has a subsequence converging locally uniformly to h, which satisfies |h|B ≤ |f |B , det h0 (0) = 1, r(h) ≤ r(f ) and d(h) = 1. Lemma 4.2.21. Let D be a bounded symmetric domain in Cn . Then B (D, Cn ) satisfying |f | = 1 = d(f ) = one can find a Bloch map f ∈ Hloc B

det f 0 (0) and r(f ) = b(D). B (D, Cn ) Proof. For k = 1, 2, . . ., we can find a Bloch function fk ∈ Hloc

satisfying |fk |B = 1 = det fk0 (0)

1 and r(fk ) ≤ b(D) + . k

B (D, Cn ) such By the preceding lemma, there is a sequence (hk ) in Hloc

that |hk |B ≤ |fk |B , det h0k (0) = 1 = d(hk ) and r(hk ) ≤ r(fk ) ≤ b(D)+ k1 . By Corollary 4.2.14, (hk −hk (0)) contains a subsequence converging B (D, Cn ), which satisfies locally uniformly to a Bloch function h ∈ Hloc

det h0 (0) = 1 = d(h) = |h|B and r(h) = b(D). Corollary 4.2.22. Let D be a bounded symmetric domain in Cn . Then the Bloch constant b(D) satisfies   Z 1 (1 − t2 )n−1 −2c(D)t exp b(D) ≥ s(D) dt. 2c(D) 1−t 0 (1 − t)

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B (D, Cn ) Proof. By Lemma 4.2.21, we can find a Bloch function f ∈ Hloc

satisfying |f |B = 1 = d(f ) = det f 0 (0) and r(f ) = b(D). For each z ∈ D, let gz be the M¨ obius transformation induced by z. Since (1 − kzk2 )c(D)/n | det f 0 (z)|1/n ≤ | det f 0 (z)|1/n = | det(f ◦ gz )0 (0)|1/n ≤ d(f )1/n = 1, we have 1 = | det f 0 (0)|1/n ≤ kf kd ≤ 1 and kf kd = 1. Now b(D) = r(f ) ≥ r(0, f ) and Theorem 4.2.19 applies.

Notes. The lower bound for the Bloch constant has been improved √

in [35] to b >

3 −4 4 +2×10 .

Chern [36] has generalised Bloch’s theorem

to holomorphic maps between two Hermitian manifolds of the same dimension. The proofs of Lemma 4.2.9 as well as Theorem 4.2.5 and Theorem 4.2.10 follow more or less the same idea in [123] for the disc D, where the classical Julia’s lemma plays an important role. Theorem 4.2.10 is a special case of a more general result shown in [42, Theorem 4.1]. The inequality for the Bloch semi-norm in Example 4.10 is an extension of a similar inequality for the Euclidean balls derived in [162, Theorem 4.7]. The notion of a Cn -valued Bloch map on a finite dimensional bounded symmetric domain, under the name of normal mapping of finite order, was first introduced by Hahn [73]. An equivalent, but perhaps less technical, definition of a complex-valued Bloch function on a bounded homogeneous domain was later given by Timoney in [162]. In [121], Cn -valued Bloch maps on Euclidean balls are studied. In [17], complex-valued Bloch functions on Hilbert balls were introduced and studied. The definitions for Bloch maps in these papers are all equivalent to the one we introduce in this section. Theorem 4.2.19 is a particular case of [42, Theorem 5.6] for all

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finite dimensional bounded symmetric domains, which reduces to the results of [121, Theorem 8] and [19, Corollary 3] for Euclidean balls, and to [174, Theorem 3.4] for polydiscs. Our proof is similar to the one given in [121]. A lower bound for the Bloch constant for Type I domains has also been shown in [61]. Lemma 4.2.21 is analogous to [61, Theorem 1.3].

4.3

Banach spaces of Bloch functions

In this section, we focus our attention on complex-valued Bloch functions on bounded symmetric domains of any dimension. These functions form a Banach space in the quotient Bloch semi-norm, where functions of null Bloch semi-norm are identified. We generalise a number of finite dimensional results on Bloch functions to infinite dimensional bounded symmetric domains. In what follows, H(D, B) denotes the class of holomorphic maps between two domains D and B in complex Banach spaces. We begin by showing several equivalent criteria of Bloch functions. Theorem 4.3.1. Let D be a bounded symmetric domain realized as the open unit ball of a JB*-triple V and let f : D −→ C be a holomorphic function. The following conditions are equivalent: (i) f is a Bloch function. (ii) sup{Qf (z) : z ∈ D} < ∞ where  Qf (z) = sup

 |f 0 (z)v| : v ∈ V \ {0} KD (z, v)

and KD (·, ·) is the Kobayashi differential metric. (iii) The radii of the schlicht discs in f (D) are bounded above.

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(iv) f is uniformly continuous as a function from the metric space (D, kD ) to the Euclidean plane C, where kD is the Kobayashi distance. (v) The family Ff (0) := {f ◦ g − (f ◦ g)(0) : g ∈ Aut D} is bounded on rD for 0 < r < 1. (vi) The family {f ◦ h : h ∈ H(D, D)} consists of Bloch functions on D with uniformly bounded Bloch semi-norm. (vii) The family {f ◦ h − (f ◦ h)(0) : h ∈ H(D, D)} is normal. If dim V = n < ∞, these conditions are equivalent to    |f 0 (z)v| : v ∈ V \ {0} 0 such that |f (z1 )−f (z2 )| ≤ 1 whenever z1 , z2 ∈ D satisfy kD (z1 , z2 ) < δ0 . For each s ∈ (0, r), we can find a partition 0 = t1 < t2 < · · · < tn = s such that ρ(tj , tj−1 ) < δ0 , where ρ is the Poincar´e distance on D, and n does not depend on s. Let z ∈ rD \ {0} and s = kzk. Let w = z/kzk ∈ ∂D and define φ : D → D by φ(ζ) = ζw. For every g ∈ Aut D, we have |f (g(z)) − f (g(0))| ≤

n−1 X

|f (g(φ(tj ))) − f (g(φ(tj+1 )))|

j=1

≤ n−1 since kD (g(φ(tj )), g(φ(tj+1 ))) = kD (φ(tj ), φ(tj+1 )) ≤ ρ(tj , tj+1 ) < δ0 for 1 ≤ j ≤ n − 1. (v) ⇒ (i). Fix r ∈ (0, 1). Then there is a constant M > 0 such that |f (g(z)) − f (g(0))| ≤ M for kzk < r and g ∈ Aut D. By the Schwarz lemma, we have k(f ◦ g)0 (0)k ≤

M , r

(g ∈ Aut D)

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which gives |f |B ≤ M/r. (i) ⇒ (iii). Let ∆ = {ζ ∈ C : |ζ − ζ0 | < r} be a schlicht disc in f (D). Then there is a holomorphic map h : D −→ D such that (f ◦ h)(ζ) = ζ0 + rζ

(ζ ∈ D).

Let z = h(0). Since g−z ◦ h : U −→ D is a holomorphic map and g−z ◦ h(0) = 0, we have k(g−z ◦ h)0 (0)k ≤ 1 by Schwarz lemma. Hence r = (f ◦ h)0 (0) = |f 0 (h(0))h0 (0)| = |f 0 (z)gz0 (0)(g−z ◦ h)0 (0)| ≤ kf 0 (z)gz0 (0)k ≤ |f |B . (iii) ⇒ (i). Let the radii of the schlicht discs in f (D) be bounded above by R > 0. Let y ∈ ∂D be arbitrarily fixed and define h : ζ ∈ D 7→ ζy ∈ D. For each g ∈ Aut D, the map f ◦ g ◦ h : D −→ D is holomorphic. If (f ◦ g ◦ h)0 (0) 6= 0, then Bloch’s theorem implies the existence of a schlicht disc in (f ◦ g ◦ h)(D) of radius b|(f ◦ g ◦ h)0 (0)| = b|(f ◦ g)0 (0)y| where b is the Bloch constant. Therefore we have b|(f ◦ g)0 (0)y| ≤ R. We also have this inequality in the case (f ◦ g ◦ h)0 (0) = 0. It follows that |f |B ≤ R/b. (iii) ⇒ (vi). Let the radii of the schlicht discs in f (D) be bounded above by R > 0. Let h ∈ H(D, D). Then f ◦ h is a holomorphic function on D. If (f ◦ h)0 (0) 6= 0, then Bloch’s theorem implies that (f ◦ h)(D) contains a schlicht disc of radius b|(f ◦ h)0 (0)| where b is the Bloch constant. Therefore, we have b|(f ◦ h)0 (0)| ≤ R. We also have this inequality if (f ◦ h)0 (0) = 0. It follows that |f ◦ h|B ≤ R/b. (vi) ⇒ (iii). Let ∆ = {ζ ∈ C : |ζ − ζ0 | < r} be a schlicht disc in f (D). There is a holomorphic map h : D −→ D such that (f ◦ h)(ζ) = ζ0 + rζ

(ζ ∈ D).

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This gives r = |(f ◦ h)0 (0)| ≤ |f ◦ h|B . (vi) ⇒ (vii). By Montel’s theorem, it suffices show that the family {f ◦ h − (f ◦ h)(0) : h ∈ H(D, D)} is uniformly bounded on D(0, r) = {z ∈ C : |z| < r} for 0 < r < 1. Indeed, applying (i) ⇒ (iv) to the Bloch functions f ◦ h, one finds a constant M > 0 such that |f ◦ h(z) − f ◦ h(0)| ≤ M ρ(z, 0) = M tanh−1 |z| ≤ M tanh−1 r for |z| < r. (vii) ⇒ (vi). This can verified by a similar argument for (v) ⇒ (i). An examination of the proof of condition (iv) in Theorem 4.3.1 reveals that a holomorphic function f : (D, kD ) −→ C is a Bloch function if and only if it is a Lipschitz function. Actually, the Lipschitz constant in this case is given by the Bloch seminorm of f . Proposition 4.3.2. Let f be a Bloch function on a bounded symmetric domain D. Then we have   |f (z) − f (w)| |f |B = sup : z, w ∈ D, z 6= w . kD (z, w) Proof. Write |f (z) − f (w)| . kD (z, w) z6=w

lip(f ) = sup

We have shown in Theorem 4.3.1 that |f (z) − f (w)| ≤ |f |B kD (z, w) for all z, w ∈ D. This implies that lip(f ) ≤ |f |B . On the other hand, given g ∈ Aut D and w ∈ D, we have for every t ∈ (0, 1), |f (g(tw)) − f (g(0))| ≤ lip(f )kD (g(tw), g(0)) = lip(f )kD (tw, 0) = lip(f ) tanh−1 (tkwk).

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It follows that |(f ◦ g)0 (0)w| ≤ lip(f )kwk

(w ∈ D)

and hence |f |B ≤ lip(f ). For a finite dimensional bounded symmetric domain D, it has been observed in [11] that h(z, w) ≥ sup{|f (z) − f (w)| : f ∈ H(D, C), βf ≤ 1}

(z, w ∈ D)

where h(·, ·) is the Bergman distance on D. In view of the fact that this inequality becomes an equality for the unit disc D, a natural question has been raised in [11] whether the equality also holds in higher dimensions. We show below that the equality actually holds for all dimensions as soon as the Bergman distance is replaced by the Kobayashi distance. Proposition 4.3.3. Let D be a bounded symmetric domain. Then we have kD (z, w) = sup{|f (z) − f (w)| : f ∈ H(D, C), |f |B ≤ 1}

(4.14)

for z, w ∈ D. Proof. By Proposition 4.3.2, we have sup{|f (z) − f (w)| : f ∈ H(D, C), |f |B ≤ 1} ≤ kD (z, w). To prove the reverse inequality, fix any a ∈ D \ {0} and let f = ψ ◦ φa be the Bloch function on D defined in Example 4.2.3. Then kD (a, 0) = tanh−1 kak = |f (a) − f (0)| ≤ sup{|f (a) − f (0)| : f ∈ H(D, C), |f |B ≤ 1}. By composing with an automorphism of D, we have kD (z, w) ≤ sup{|f (z) − f (w)| : f ∈ H(D, C), |f |B ≤ 1} for any z, w ∈ D. This completes the proof.

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Function theory Let D be a bounded symmetric domain realised as the open unit

ball of a JB*-triple V . Let B(D) denote the complex vector space of all C-valued Bloch functions on D. Usually, one can turn the Bloch semi-norm | · |B into a norm on a quotient of B(D). Instead, we define a norm on B(D) by kf kB := |f |B + |f (0)|

(f ∈ B(D)).

This is indeed a norm by (4.6). We call it the Bloch norm on the space B(D). Theorem 4.3.4. Let D be a bounded symmetric domain realised as the open unit ball of a JB*-triple V . Then B(D) is a Banach space in the Bloch norm k · kB . Proof. Let (fk ) be a Cauchy sequence in B(D). By (4.6), we have kfk0 (z) − fp0 (z)k ≤

kfk − fp kB 1 − kzk2

for all z ∈ D and k, p = 1, 2 . . .. On the other hand, Z 1 kfk0 (tz) − fp0 (tz)kkzkdt |fk (z) − fp (z)| ≤ |fk (0) − fp (0)| + 0

kfk − fp kB ≤ |fk (0) − fp (0)| + 1 − kzk2 for all k, p and z ∈ D. It follows that (fk ) is a Cauchy sequence in H(D, C), in the locally uniform topology. Hence (fk ) converges locally uniformly to some function f ∈ H(D, C). To complete the proof, we show kfk − f kB → 0 as k → ∞. Let ε > 0. There exists k0 ∈ N such that kfk − fp kB < ε for k, p ≥ k0 , which gives |fk (0)−fp (0)|+k(fk0 (g(0))−fp0 (g(0)))g 0 (0)k < ε (g ∈ Aut D, k, p ≥ k0 ).

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For each p ≥ k0 and g ∈ Aut D, the locally uniform convergence of the sequence (fk ) to f implies that fk (0) → f (0) and k(fk0 (g(0)) − fp0 (g(0)))g 0 (0)k → k(f 0 (g(0)) − fp0 (g(0)))g 0 (0)k as k → ∞, and hence |f (0) − fp (0)| + k(f 0 (g(0)) − fp0 (g(0)))g 0 (0)k ≤ ε for p ≥ k0 and g ∈ Aut D. Consequently, kfp − f kB ≤ ε for p ≥ k0 , and f = (f − fp ) + fp ∈ B(D) as well as limp→∞ kfp − f kB = 0. Definition 4.3.5. Given a bounded symmetric domain D, the Banach space (B(D), k · kB ) is called the Bloch space of D. Remark 4.3.6. In literature, the Bloch norm for a finite dimensional bounded symmetric domain D (realised as the open unit ball of a JB*triple) is often defined by the semi-norm βf given in Theorem 4.3.1 (viii), and the Bloch norm β is defined by β(f ) := βf + |f (0)| for a Bloch function f on D. With this norm, B(D) also forms a Banach space. It should be pointed out that the two Banach spaces (B(D), k·kB ) and (B(D), β) are not identical, but linearly homeomorphic. Throughout the chapter, we only consider the Bloch space (B(D), k · kB ). Although the functions in B(D) need not be bounded, it is of inP n terest to note that on the disc D, a Bloch function f (z) = ∞ n=0 an z has bounded Taylor coefficients an . It has been shown in [12, Lemma 2.1] that |an | ≤ 2|f |B . In fact, this bound can be improved slightly in all dimensions.

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Theorem 4.3.7. Let D be the open unit ball of a JB*-triple V . Given f ∈ B(D) with a power series representation f (z) =

∞ X

(z ∈ D)

pn (z)

n=0

by homogeneous polynomials pn of degree n, we have kp1 k ≤ |f |B

kpn k ≤

and

√

e|f |B

(n ≥ 2).

Proof. For each v ∈ V with kvk = 1, we define a Bloch function φv : D −→ C by ∞ X

φv (ζ) = f (ζv) =

pn (v)ζ n

(ζ ∈ D).

n=0

The first inequality follows from the definition of the semi-norm |φv |B . P n Given a Bloch function ψ(ζ) = ∞ n=0 an ζ ∈ B(D) and r ∈ (0, 1), we have an r n =

1 2π

Z

π

ψ(reiθ )e−inθ dθ.

−π

Hence n|an |rn−1 ≤ sup |ψ 0 (ζ)| |ζ|=r

for n ≥ 2, which implies |an | ≤

1 nrn−1 (1

− r2 )

|ψ|B .

Taking r2 = 1 − 1/n, we obtain |an | ≤

 1+

1 n−1

(n−1)/2 |ψ|B ≤

for n ≥ 2. Therefore we have shown |pn (v)| ≤

√

√

e|ψ|B

e |φv |B for n ≥ 2 and

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v ∈ V with kvk = 1. It follows that kpn k =

sup |pn (v)| ≤

√

e|φv |B

kvk=1

= =

√

√

e | sup(1 − |ζ|2 )|φ0v (ζ)| ζ∈D

e | sup(1 − |ζv|2 )|f 0 (ζv)v| ≤

√ e|f |B

ζ∈D

for n ≥ 2. Corollary 4.3.8. The Bloch space B(D) contains an isomorphic copy of the Banach space `∞ of bounded complex sequences. Proof. Let (an ) ∈ `∞ . Then the function f (z) = a0 +

P∞

n=1 an z

2n

is

holomorphic on D and |zf 0 (z)| ≤ k(an )k∞ (2|z|2 + 22 |z|4 + · · · ) = k(an )k∞ (1 − |z|)(2|z|2 + 2|z|3 + (2 + 22 )|z|4 + (2 + 22 )|z|5 + · · · + (2 + 22 + 23 )|z|8 + · · · )   ∞ X X  = k(an )k∞ (1 − |z|) 2k  |z|n n=2

≤ k(an )k∞ (1 − |z|)

∞ X

{k:2k ≤n}

2n|z|n ≤ k(an )k∞ (1 − |z|)

n=2

|z| . (1 − |z|)2

Hence we have (1 − |z|2 )|f 0 (z)| ≤ 2k(an )k∞ and f is a Bloch function on D. One can therefore define a continuous linear monomorphism (an ) ∈ `∞ 7→ f (z) = a0 +

∞ X

n

an z 2 ∈ B(D)

n=0

which is actually a homeomorphism onto a closed subspace of B(D), by Theorem 4.3.7.

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Function theory We infer from the above corollary that the Bloch space B(D) is not

reflexive. In fact, it has been shown in [12] that B(D) is the second dual space of its subspace B0 (D) := {f ∈ B(D) : lim (1 − |z|2 )|f 0 (z)| = 0}. |z|→1

The Banach space B0 (D) is separable and is the | · |B -closure of the polynomials (restricted to D). One can extend the definition of this space to higher dimensions. Polynomials, when restricted to a bounded symmetric domain D, are Bloch functions, by Example 4.2.3. Definition 4.3.9. Let D be a bounded symmetric domain. The little Bloch space B0 (D) is defined to be the k · kB -closure of the polynomials on D. Lemma 4.3.10. Let f ∈ B0 (D). Then we have lim (1−kzk2 )kf 0 (z)k = kzk→1

0. Proof. Let ε > 0 and kf − pkB < ε for some polynomial p ∈ B(D). Then we have (1 − kzk2 )kf 0 (z)k ≤ (1 − kzk2 )k(f − p)0 (z)k + (1 − kzk2 )kp0 (z)k ≤ kf − pkB + (1 − kzk2 )kp0 (z)k < ε + (1 − kzk2 )kp0 (z)k. Since kp0 (·)k is bounded on D, we see that (1 − kzk2 )kf 0 (z)k → 0 as kzk → 1. The converse of the previous lemma is true if D is a Hilbert ball and we actually have the following result. Theorem 4.3.11. Let D be a Hilbert ball and f ∈ B(D). The following conditions are equivalent.

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(i) f ∈ B0 (D). (ii) Qf (z) → 0 uniformly as z → ∂D, that is, for each ε > 0, there exists δ ∈ (0, 1) such that Qf (z) < ε whenever δ < kzk < 1. (iii) lim (1 − kzk2 )kf 0 (z)k = 0. kzk→1

(iv) lim

sup (1 − kzk2 )1/2 (1 − kwk2 )1/2

kzk→1 w∈D\{z}

|f (z) − f (w)| = 0. kz − wk

Proof. (i) ⇒ (ii). Let f ∈ B0 (D). Then |f 0 (z)x| κ(z, x)

=

|f 0 (z)x| kf 0 (z)kkxkkB(z, z)1/2 k ≤ kxk kB(z, z)−1/2 xk

= kf 0 (z)kkB(z, z)1/2 k p which implies Qf (z) ≤ kf 0 (z)k 1 − kzk2 → 0 as kzk → 1 since the null convergence is true for all polynomials. (ii) ⇒ (iii). This follows from the inequality Qf (z) ≥ kf 0 (z)k(1 − kzk2 ) since

|f 0 (z)x| |f 0 (z)x| |f 0 (z)x|(1 − kzk2 ) = ≥ κ(z, x) kxk kB(z, z)−1/2 xk

for x 6= 0. (iii) ⇒ (i). Let f ∈ B(D) satisfy condition (iii). Let ε > 0. There √ exists r ∈ (1/ 2, 1) such that (1 − kzk2 )kf 0 (z)k < ε/2 for r ≤ kzk < 1. Let fk (z) = f ((1 − 1/k)z) for k = 1, 2, . . . . Then fk is holomorphic on D and hence there is a polynomial pk such that sup |fk (z) − pk (z)| ≤ z∈D

1 k

for each k ≥ 1. By Example 4.2.3, fk − pk ∈ B(D) and |fk − pk |B ≤

1 . k

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Function theory By the maximum principle, we have (1 − kzk2 )kfk0 (z)k < ε/2 for

r ≤ kzk < 1 and for all k ≥ 1. Since the sequence (fk ) converges to f uniformly on each closed ball in D, there exists K ≥ 1 such that kf 0 (z) − fk0 (z)k < ε for kzk ≤ r and k ≥ K. This implies (1 − kzk2 )kf 0 (z) − fk0 (z)k < ε for z ∈ D and k ≥ K. By (4.10), we have √ 2+ 2 |f − fk |B ≤ k for k ≥ K. It follows that √ 3+ 2 |f − pk |B ≤ |f − fk |B + |fk − pk |B ≤ k for k ≥ K, proving that f ∈ B0 (D). (i) ⇒ (iv). Let f ∈ B0 (D). For z, w ∈ D, we have Z 1 0 |f (z) − f (w)| = f (tz + (1 − t)w)(z − w)dt 0 Z 1 Z 1 |f |B kz − wk 0 dt ≤ kf (tz + (1 − t)w)kkz − wkdt ≤ 2 0 0 1 − ktz + (1 − t)wk Z 1 dt ≤ |f |B kz − wk 1 − ktz + (1 − t)wk 0 by (4.6). Observe that 1 − ktz + (1 − t)wk ≥ 1 − tkzk − (1 − t)kwk p = t(1 − kzk) + (1 − t)(1 − kwk) ≥ 2 t(1 − t)(1 − kzk)(1 − kwk). Since

R

√

dt t(1−t)

= cos−1 (1 − 2t), we deduce |f (z) − f (w)| π|f |B ≤ p kz − wk 2 (1 − kzk)(1 − kwk)

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and (1 − kzk2 )1/2 (1 − kwk2 )1/2

|f (z) − f (w)| π|f |B ≤ . kz − wk 2

(4.15)

For each t ∈ (0, 1), define ft (z) = f (tz) for z ∈ D. Applying (4.15) twice, we get (1 − kzk2 )1/2 (1 − kwk2 )1/2

|(f − ft )(z) − (f − tt )(w)| π|f − ft |B ≤ kz − wk 2 (4.16)

and |ft (z) − ft (w)| (1 − kzk2 )1/2 (1 − kwk2 )1/2 ktz − twk p p p t 1 − kzk2 1 − kwk2 p |ft (z) − ft (w)| p = p ( 1 − ktzk2 1 − ktwk2 ) 2 2 ktz − twk 1 − ktzk 1 − ktwk πt ≤ (1 − kzk2 )1/2 |f |B . (4.17) 2(1 − t) We have seen in the proof of (i) ⇒ (iii) above that |f − ft |B → 0 as t → 1. If t0 ∈ (0, 1) makes |f − ft0 |B sufficiently small, then lim

kzk→1

πt0 (1 − kzk2 )1/2 |f |B = 0. 2(1 − t0 )

We therefore conclude from (4.16) and (4.17) that lim

sup (1 − kzk2 )1/2 (1 − kwk2 )1/2

kzk→1 w∈D\{z}

|f (z) − f (w)| =0 kz − wk

by the triangle inequality. (iv) ⇒ (iii). For each kwk = 1 and z ∈ D, we have z + tw ∈ D for small t > 0 and p |f (z) − f (z + tw)| 1 − kz + twk2 t↓0 t p p |f (z) − f (w)| ≤ sup 1 − kzk2 1 − kwk2 . kz − wk w∈D\{z}

(1 − kzk2 )|f 0 (z)w| = lim

p

1 − kzk2

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It follows that lim (1 − kzk2 )kf 0 (z)k

kzk→1

≤

lim

sup (1 − kzk2 )1/2 (1 − kwk2 )1/2

kzk→1 w∈D\{z}

|f (z) − f (w)| = 0. kz − wk

Example 4.3.12. Let g = (g1 , . . . , gn ) be a M¨obius transformation of the Euclidean ball Bn . Since each component gj of g extends holomorphically to a neighbourhood of Bn , it belongs to B0 (Bn ) by Theorem 4.3.11.

Notes. In Theorem 4.3.1, condition (viii) was originally used in [162] to define a Bloch function on a finite dimensional bounded homogeneous domain, where the constant βf is defined to be the Bloch semi-norm of f , and was then shown to be equivalent to the other conditions in Theorem 4.3.1, in finite dimensions. Condition (ii) of Theorem 4.3.1 (in terms of Carath´eodory differential metric) has also been used in [53] to define Bloch functions on bounded symmetric domains, which are shown to form a Banach space. A definition of Bloch functions on finite dimensional strongly pseudoconvex domains has been introduced by Krantz and Ma [113]. On a finite dimensional bounded homogeneous domain D, it has been shown in [11, Theorem 3.1] that |f (z) − f (w)| h(z, w) z6=w

βf = sup

for a Bloch function f on D, where z, w ∈ D and h(·, ·) is the Bergman distance. It has been declared in the proof of [17, Proposition 2.7] that the second inequality in Theorem 4.3.7 is true for a Hilbert ball. Corollary 4.3.8 has been proved in [12].

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Theorem 4.3.11 has been proved in [43] and [44]. The proof of (i) ⇔ (iii) in this theorem is analogous to the one for B0 (D) given in [12]. Using similar arguments, the equivalence of (i) and (ii) in this theorem has also been proved in [175, Theorem 3]. The expression of the left-hand side in Theorem 4.3.11 (iv) originates from the paper [85], where it is shown that the Bloch functions on D are exactly the func(w)| < ∞. tions f for which supw∈D\{z} (1 − kzk2 )1/2 (1 − kwk2 )1/2 |f (z)−f kz−wk

This result has been extended to the Euclidean balls by Ren and Tu [147], and to the Hilbert balls by Chu, Hamada, Honda and Kohr [41]. For Euclidean balls, the equivalence (i) ⇔ (iv) in Theorem 4.3.11 has also been proved in [147]. The dimension-free inequality (4.15) in the proof of (i) ⇒ (iv) for Hilbert balls is a modification of a similar one, albeit dimension-dependent, derived in [147]. For a finite dimensional bounded symmetric domain which is not a Euclidean ball, the characterizations of the little Bloch space in Theorem 4.3.11 are no longer valid. It has been shown in [163, Proposition 4.1] that a Bloch function f on such a domain must be constant if it satisfies condition (ii) in Theorem 4.3.11.

4.4

Composition operators

The topic of composition operators is an important one in the study of function spaces and there are many applications. We discuss briefly the composition operators between Bloch spaces of bounded symmetric domains. The question of boundedness, compactness and isometric conditions are our main focus. Throughout this section, we will denote by D and B two bounded symmetric domains realised as the open unit balls of two JB*-triples V and W respectively.

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Function theory Given ϕ ∈ H(D, B), one can form a composition operator Cϕ : f ∈ B(B) 7→ f ◦ ϕ ∈ H(D, C)

which is clearly linear. We call ϕ the symbol of Cϕ . The first natural question is whether f ◦ϕ is a Bloch function on D, and if so, whether Cϕ is a bounded operator between B(B) and B(D). The answer is positive. Theorem 4.4.1. Let D and B be bounded symmetric domains and let ϕ ∈ H(D, B). Then Cϕ (f ) ∈ B(D) for each f ∈ B(B), and Cϕ : B(B) −→ B(D) is a bounded linear operator satisfying   max 1, tanh−1 kϕ(0)k ≤ kCϕ k ≤ max 1, tanh−1 kϕ(0)k + Kϕ where Kϕ = sup sup z∈D v6=0

KB (ϕ(z), ϕ0 (z)v) ≤ 1. KD (z, v)

In particular, kCϕ k = 1 whenever ϕ(0) = 0. Proof. Let f ∈ B(B). By Theorem 4.3.1, |Cϕ (f )|B = |f ◦ ϕ|B = sup{Qf ◦ϕ (z) : z ∈ D}. For z ∈ D and v in the ambient JB*-triple V , we have |(f ◦ ϕ)0 (z)v| KD (z, v)

≤

|f 0 (ϕ(z))ϕ0 (z)v| ≤ Qf (ϕ(z)) KB (ϕ(z), ϕ0 (z)v)

for ϕ0 (z)v 6= 0. If ϕ0 (z)v = 0, then (f ◦ ϕ)0 (z)v = f 0 (ϕ(z))ϕ0 (z)v = 0. Hence Qf ◦ϕ (z) ≤ Qf (ϕ(z)) ≤ |f |B which implies |Cϕ (f )|B ≤ |f |B and Cϕ (f ) ∈ B(D). Observe that Z 1 |f (ϕ(0))| ≤ |f (0)| + |f 0 (tϕ(0))ϕ(0)|dt 0 Z 1 kϕ(0)k ≤ |f (0)| + |f |B dt 2 2 0 1 − t kϕ(0)k

(4.18)

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by (4.6). This gives  Z kCϕ (f )kB ≤ kf kB 1 +

1

0

kϕ(0)k dt 1 − t2 kϕ(0)k2



and therefore Cϕ is a bounded operator. To get the desired upper bound for kCϕ k, let us look at Qf ◦ϕ (z) again. We have |(f ◦ ϕ)0 (z)v| KD (z, v)

|f 0 (ϕ(z))ϕ0 (z)v| KD (ϕ(z), ϕ0 (z)v) KD (ϕ(z), ϕ0 (z)v) KD (z, v) KD (ϕ(z), ϕ0 (z)v) ≤ Qf (z) KD (z, v) =

which gives Qf ◦ϕ (z) = sup v6=0

KD (ϕ(z), ϕ0 (z)v) |(f ◦ ϕ)0 (z)v| ≤ |f |B sup KD (z, v) KD (z, v) v6=0

and therefore |f ◦ ϕ|B ≤ sup sup z∈D v6=0

KD (ϕ(z), ϕ0 (z)v) |f |B = Kϕ |f |B . KD (z, v)

By Theorem 4.3.1, we have |f (ϕ(0))| ≤ |f (0)| + |f (ϕ(0)) − f (0)| ≤ kf kB − |f |B + kD (ϕ(0), 0)|f |B and hence kf ◦ ϕkB ≤ kf kB − |f |B + kD (ϕ(0), 0)|f |B + |f ◦ ϕ|B ≤ kf kB + (−1 + tanh−1 kϕ(0)k + Kϕ )|f |B . If tanh−1 kϕ(0)k + Kϕ − 1 ≤ 0, we have kf ◦ ϕkB ≤ kf kB . On the other hand, if tanh−1 kϕ(0)k + Kϕ − 1 > 0, then kf ◦ ϕkB ≤ (tanh−1 kϕ(0)k + Kϕ )kf kB

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and we have obtained the desired upper bound. Let 1 be the constant function on B with value 1. Then 1 = kCϕ (1)kB = k1kB . This implies kCϕ k ≥ 1. For the lower bound, we note that ϕ(0) = 0 implies kCϕ k ≤ max{1, Kϕ } = 1. Consider the case ϕ(0) 6= 0. Let `ϕ(0) be the support functional at ϕ(0), that is, `ϕ(0) (ϕ(0)) = kϕ(0)k and k`ϕ(0) k = 1. By Example 4.2.3, the function tanh−1 ◦`ϕ(0) is a Bloch function on B with k tanh−1 ◦`ϕ(0) kB = 1. Therefore kCϕ k ≥ k tanh−1 ◦`ϕ(0) ◦ ϕkB ≥ tanh−1 kϕ(0)k. This establishes the lower bound for kCϕ k. Corollary 4.4.2. In the preceding theorem, if Cϕ : B(B) −→ B(D) is an isometry, then we have ϕ(0) = 0 and Kϕ = 1. Proof. Let a = ϕ(0) and g−a the M¨obius transformation induced by −a. We show that ψ(a) = 0 for each continuous functional ψ ∈ W ∗ . Then ψ ◦ g−a is a Bloch function on B and we have |ψ(−a)| + |ψ ◦ g−a |B = kψ ◦ g−a kB = kCϕ (ψ ◦ g−a )kB = |ψ ◦ g−a ◦ ϕ|B ≤ |ψ ◦ g−a |B where the last inequality has been shown in the preceding theorem, which implies ψ(−a) = 0. For the second assertion, we have shown in the preceding theorem that |Cϕ (f )|B ≤ Kϕ |f |B for all f ∈ B(B). Hence we must have Kϕ = 1 as ϕ(0) = 0 and Cϕ is an isometry.

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Example 4.4.3. The converse of Corollary 4.4.2 is false. Let D be the open unit ball of an `∞ -sum V1 ⊕ V2 of two JB*-triples and let ϕ : D −→ D be the projection ϕ(z1 , z2 ) = (z1 , 0). Then ϕ(0, 0) = (0, 0) and Kϕ = 1. Let ψ be the support functional at a point (0, b) ∈ D\{(0, 0)}, that is, ψ(0, b) = kbk and kψk = 1. Then ψ is a Bloch function on D with kψkB ≥ Qψ ((0, 0)) = 1, but kψ ◦ ϕkB = 0. In the remaining section, we write dim B = n to mean that B is a finite dimensional bounded symmetric domain in Cn , realised as the open unit ball of a norm k · k equivalent to the Euclidean norm k · k2 . In this case, a holomorphic map ϕ : D −→ B can be expressed as ϕ(z) = (ϕ1 (z), . . . , ϕn (z))

(z ∈ D)

where D is the open unit ball of a JB*-triple V and the holomorphic functions ϕj : D → C will be called the coordinate components of ϕ. There is a constant c > 0 such that for each y = (y1 , ..., yn ) ∈ Cn ,  1/2 X kyk ≤ ckyk2 = c  |yj |2  . j

A bounded linear map ψ : V −→ Cn can be expressed in the form ψ = (ψ1 , . . . , ψn ) where each ψj is a continuous linear functional on V . In this case, we have

 1/2 X kψk ≤ c  kψj k2  . j

(4.19)

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Function theory Given ϕ = (ϕ1 , . . . , ϕn ) ∈ H(D, B), it is natural to ask if the

composition operator Cϕ sends the little Bloch space B0 (B) into B0 (D). We show below that this is the case for a Hilbert ball D exactly when the coordinate components ϕj of the symbol ϕ are in B0 (D)0 . Proposition 4.4.4. Let D be a Hilbert ball and dim B = n < ∞. Let ϕ = (ϕ1 , . . . , ϕn ) ∈ H(D, B). Then Cϕ maps B0 (B) to B0 (D) if and only if ϕj ∈ B0 (D) for j = 1, . . . , n. Proof. Let Cϕ (B0 (B)) ⊂ B0 (D). Since the coordinate maps pj : (y1 , . . . , yn ) ∈ B 7→ yj ∈ C belong to B0 (B), we have ϕj = pj ◦ϕ = Cϕ (pj ) ∈ B0 (D) for j = 1, . . . , n. Conversely, let ϕj ∈ B0 (D) for j = 1, . . . , n. Let f ∈ B0 (B) and ε > 0. There exists a polynomial P on B such that kf − P kB < ε/2. By (4.6) and (4.19), we have (1 − kzk2 )k(f ◦ ϕ)0 (z)k ≤ (1 − kzk2 )k(P ◦ ϕ)0 (z)k + Q(f −P ) (ϕ(z)) ε ≤ (1 − kzk2 )kP 0 (ϕ(z))kkϕ0 (z)k + 2 n X ε kϕ0k (z)k + . ≤ (1 − kzk2 )kP 0 (ϕ(z))kc 2 k=1

Since ϕj ∈ B0 (D) for j = 1, . . . , n and kP 0 (·)k is bounded on B, Theorem 4.3.11 implies that there exists δ > 0 for which 2

0

(1 − kzk )kP (ϕ(z))kc

n X

kϕ0k (z)k ≤

k=1

whenever kzk > δ. This gives (1 − kzk2 )k(f ◦ ϕ)0 (z)k < ε for kzk > δ, proving Cϕ (f ) ∈ B0 (D).

ε 2

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We now study compactness of the composition operators Cϕ . Proposition 4.4.5. Let D be a bounded symmetric domain realised as the open unit ball of a JB*-triple V . Let ϕ ∈ H(D, B), where dim B = n < ∞. Then Cϕ : B(B) → B(D) is a compact operator if for every ε > 0, there exists a δ > 0 such that KB (ϕ(z), ϕ(z)0 v) δ. Proof. Let ϕ satisfy the given condition. Let (fm ) be a sequence in B(B) such that kfm kB ≤ 1 for m = 1, 2, . . . . We need to show that (fm ◦ ϕ) contains a convergent subsequence in B(B). Since 0 kfm (z)k ≤

1 kfm kB ≤ 2 1 − kzk 1 − kzk2

by (4.6), Montel’s theorem implies the existence of a subsequence (fk ) of (fm ) converging locally uniformly to a holomorphic function f on B with |f |B ≤ 1. Let Fk = fk − f for k = 1, 2, . . . . Then Fk → 0 locally uniformly on B as k → ∞ and kFk kB ≤ 2

(k = 1, 2, . . . ).

(4.21)

Let ε > 0. By assumption, there exists δ ∈ (0, 1) such that (4.20) holds whenever kϕ(z)k > δ which, together with (4.21), implies QFk ◦ϕ (z) ≤ εQFk (ϕ(z)) ≤ 2ε for kϕ(z)k > δ where the first inequality follows from analogous derivation for (4.18). On the other hand, if kϕ(z)k ≤ δ, we have inf{KB (w, y) : kwk ≤ δ, kyk = 1} > 0

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since KB (w, y) = kB(w, w)−1/2 yk > 0 on the compact subset {w ∈ Cn : kwk ≤ δ} × {y ∈ Cn : kyk = 1} of Cn × Cn . Since Fk (w) → 0 uniformly for kwkY ≤ (δ + 1)/2 as k → ∞, there exists K ∈ N such that QFk ◦ϕ (z) ≤ QFk (ϕ(z)) < ε for k > K and kϕ(z)k ≤ δ. It follows that kFk ◦ ϕkB = |Fk (ϕ(0))| + |Fk ◦ ϕ|B → 0 as k → ∞, giving fk ◦ ϕ → f ◦ ϕ in B(D). This proves compactness of Cϕ . In what follows, we consider the special case of Hilbert balls. Proposition 4.4.6. Let D be a Hilbert ball and dim B = n < ∞. Then the composition operator Cϕ : B(B) → B(D) is compact if the symbol ϕ ∈ H(D, B) satisfies the condition that for every ε > 0, there exists δ > 0 such that

1 − kzk2 kϕ0 (z)k < ε 1 − kϕ(z)k2

(4.22)

whenever kϕ(z)k > δ. Proof. Let (fk ) be a sequence in B(B) such that kfk kB ≤ 1 for k = 1, 2, . . . . By (4.6) and Montel’s theorem, we may assume by choosing a subsequence, that (fk ) converges to a function f locally uniformly on B. We have kf kB ≤ 1. Let Fk = fk − f for k = 1, 2, . . . . Then Fk → 0 locally uniformly on B as k → ∞ and kFk kB ≤ 2 for k = 1, 2, . . . .

(4.23)

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Let ε > 0. By assumption, there exists a δ ∈ (0, 1) such that 1 − kzk2 kϕ0 (z)k < ε 1 − kϕ(z)k2

(4.24)

for kϕ(z)kY > δ. Since Fk (w) → 0 uniformly for kwk ≤ (δ + 1)/2 as k → ∞, there exists a K ∈ N such that kFk0 (w)k < ε

(k > K, kwk ≤ δ).

(4.25)

If kϕ(z)k > δ, then we deduce from (4.23), (4.24) and (4.6) that (1 − kzk2 )kFk0 (ϕ(z))ϕ0 (z)k ≤ ε(1 − kϕ(z)k2 )kFk0 (ϕ(z))k ≤ εkFk kB ≤ 2ε. If kϕ(z)k ≤ δ, then by (4.25) and the Schwarz-Pick lemma, we have (1 − kzk2 )kFk0 (ϕ(z))ϕ0 (z)k ≤ kFk0 (ϕ(z))k ≤ ε for k > K. Hence kFk ◦ ϕkB = |Fk (ϕ(0))| + |Fk ◦ ϕ|B → 0 as k → ∞. This proves compactness of Cϕ . Corollary 4.4.7. Let D be a Hilbert ball and dim B = n < ∞. If ϕ ∈ H(D, B) satisfies 1 − kzk2 kϕ0 (z)k = 0, kzk→1 1 − kϕ(z)k2 lim

(4.26)

then Cϕ : B0 (B) −→ B0 (D) is a compact operator between little Bloch spaces. Proof. We note that (4.26) is equivalent to (4.22) and the condition lim (1 − kzk2 )kϕ0 (z)k = 0.

kzk→1

(4.27)

Under condition (4.27), ϕj ∈ B0 (D) for j = 1, 2, . . . , n. Hence Cϕ : B0 (B) −→ B0 (D) is a bounded operator by Proposition 4.4.4. Since Cϕ is compact on B(B) by Proposition 4.4.6, its restriction to the closed subspace B0 (B) is also compact.

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Theorem 4.4.8. Let D be a Hilbert ball and dim B = n < ∞. Let ϕ ∈ H(D, B). Then the composition operator Cϕ : B0 (B) −→ B0 (D) is compact if and only if lim (1 − kzk2 ) sup{k(f ◦ ϕ)0 (z)k : f ∈ B0 (B), kf kB ≤ 1} = 0. (4.28)

kzk→1

Proof. Let Cϕ be compact so that the set E = {Cϕ (f ) : f ∈ B0 (B), kf kB ≤ 1} is relatively compact in B0 (D). Let ε > 0. Then there exist f 1 , . . . , f l ∈ B0 (B) such that E ⊂ Sl

j=1 U (f

j , ε),

where n εo U (f j , ε) = F ∈ B0 (D) : kF − f j ◦ ϕkB < 2

(j = 1, . . . , l).

Since f j ◦ ϕ ∈ B0 (D), there exists r ∈ (0, 1) such that (1 − kzk2 )kD(f i ◦ ϕ)(z)k
r and j = 1, . . . , l by Theorem 4.3.11. Let F ∈ E.

Then F ∈ U (f j0 , ε) for some j0 and hence (1 − kzk2 )kF 0 (z)k ≤ kF − f j0 ◦ ϕkB + (1 − kzk2 )k(f j0 ◦ ϕ)0 (z)k < ε for kzk > r which proves (4.28). Conversely, assume (4.28). Then f ◦ ϕ ∈ B0 (D) for each f ∈ B0 (B) by Theorem 4.3.11. Let (fk ) be a sequence in B0 (B) such that kfk kB ≤ 1 for k = 1, 2, . . . . By Montel’s theorem and choosing a subsequence, we may assume that (fk ) converges locally uniformly to a holomorphic function f on B. We have kf kB ≤ 1 and by (4.28), there exists an r0 ∈ (0, 1) such that (1 − kzk2 )k(fk ◦ ϕ)0 (z)k < ε

(kzk > r0 , k = 1, 2, . . . ).

It follows that (1 − kzk2 )k(f ◦ ϕ)0 (z)k ≤ ε

(kzk > r0 ).

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This implies f ◦ ϕ ∈ B0 (D) and kfk ◦ ϕ − f ◦ ϕkB → 0 as k → ∞, proving compactness of Cϕ .

Notes. The results in this section are taken from [43] and [44]. The upper bound for the composition operator Cϕ in Theorem 4.4.1 is analogous to the one in [11, Theorem 3.2] for finite dimensional bounded homogeneous domains. For Euclidean balls, Theorem 4.4.1 has been shown in [11, Corollary 3.1]. Corollary 4.4.2 has been shown in [11, Theorem 6.1(b)] for composition operators on Bloch spaces of finite dimensional classical bounded symmetric domains and it has been speculated in [11, Remark 6.1] that the result might be true for all finite dimensional bounded symmetric domains. Proposition 4.4.4 is an extension of a result in [157, Theorem 2] for the Euclidean balls. Proposition 4.4.5, together with its converse proved in [43, Proposition 5.3], generalises simultaneously the main theorem in [186] for finite dimensional classical domains, as well as [157, Theorem 3] for finite dimensional bounded symmetric domains and [157, Theorem 4] for the Euclidean balls. For the unit disc D, the result of Proposition 4.4.6 has been shown in [127], and also in [157, Theorem 5] for Euclidean balls. Corollary 4.4.7 has been shown in [127] and [157] for the disc D and Euclidean balls respectively.

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wsbsdbschu

Index A B, 80

atomic JBW*-triple, 305

α(t, z), 25

atomic part, 305

Aut(M, ν), 40

aut M , 25

aut(M, ν), 42

Aut M , 14

abelian Jordan triple system, 80

automorphism group of manifold, 14

abelian Lie algebra, 103 abelian tripotent, 302

automorphism group of Lie algebra, 104

act transitively, 30

automorphism of manifold, 14

Ad, 32 ad, 33, 105 adjoint representation of Lie algebra, 33, 105 adjoint representation of Lie group, 32

B(a, r), open ball, 9 B(D), 347 B0 (D), 350 B(a, r), closed ball, 9 balanced domain, 1

affine boundary component, 206

Banach Lie algebra, 129

analytic (Banach) manifold, 13

Banach Lie group

analytic function, 5

(real or complex), 30

analytic structure, 11

Banach Lie subgroup, 33

analytic vector field, 20

Banach manifold, 11

anisotropic, 87

Bergman kernel, 37

antiautomorphism, 52

Bergman metric, 38

associative inner product, 120

Bergman operator, 82

atlas, 11

Bergman-Shilov boundary, 199 385

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386

page 386

Index

bianalytic function, 5

canonical Jordan triple product, 71

Bieberbach conjecture, 313

canonical part of TKK Lie algebra,

biholomorphic function, 6 biholomorphism, 14 Bloch constant b, 319

109 canonical Riemannian metric on cone, 124

Bloch constant b(D), 335

canonical TKK Lie algebra, 109

Bloch function, 320

Carath´eodory (differential) metric,

Bloch function on D, 319

211

Bloch map, 320

Carath´eodory pseudo-metric, 211

Bloch norm, 346

Carath´eodory tangent norm, 40

Bloch semi-norm, 320

Carath´eodory-type distance, 270

Bloch semi-norm on D, 319

Cartan domain, 193

Bloch space, 347

Cartan factors, 133

Borel embedding, 204

Cartan involution, 108

boundary ∂E, 1

Cartan’s uniqueness theorem, 7

boundary component, 205

Cauchy inequality, 6

boundary fixed-point, 226

Cauchy-Riemann equations, 3

bounded symmetric domain, 47

Cayley algebra, 56

box operator, 53, 80

Cayley transform, 261

brackets of vector fields, 22

centre of a Jordan algebra, 59 centre of Lie algebra, 105

C0 (Ω), 117

characteristic function, 124

Cϕ , composition operator, 356

circular domain, 1

CD (p, v), 211

classical domain, 193

c(D), 324

closed ball, 9

cD , Carath´eodory pseudo-distance, closure: E, 1 211 C*-algebra, 115

commutative Jordan triple system, 80

canonical complex structure, 3

commutator of vector fields, 22

canonical decomposition of g, 146

commutator product of maps, 53

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wsbsdbschu

Index

387

compact dual, 204

differentiable structure, 13

compact map, 229

differential, 16

compact-open topology, 9

dimension of manifold, 12

compatible tangent norm, 34

discrete JBW*-triple, 302

complemented subspace, 16

domain, 1

complete tripotent, 95

domain of holomorphy, 220

complete vector field, 24

dual Banach space, 118

completely positive definite, 27

dual cone, 122

complex extreme point, 202

dual space, 4, 117

complex manifold, 12 complexification, 74, 254 composition operator, 356

exp tX, 25 equivalent differential metrics, 280 Euclidean Jordan algebra, 120

cone, 68 conjugate of a vector space, 69 conjugation of Hilbert space, 134 continuous von Neumann algebra, 306

exceptional domain, 193 exceptional Jordan algebra, 53 exp, 25 exponential map, 32

continuously differentiable, 2

f 0 (a), derivative of f at a, 2

convex map, 314

face, 202

coset space, 34

factor, 301

covariant tensor, 27

finite rank, 187 Finsler function, 35

Da f , derivative of f at a, 2 D, 1 d(A, B), 8

Finsler metric, 35 flat space, 81 formally real Jordan algebra, 65

dfa , derivative of f at a, 2 derivation, 104

G(Ω)-invariant, 267

derivative, 2

G(M ), isometry group of M , 30

diffeomorphism, 14

genus of bounded

differentiable function, 2

symmetric domain, 194

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388

Index

graded Lie algebra, 107 group measure space construction, 308 group von Neumann algebra, 308

holomorphic boundary component, 205 holomorphic embedding, 274 holomorphic extreme point, 202 holomorphic function, 3

H ⊕ R, real spin factor, 58 H(D, B), 340

holomorphic homogeneous regular domains, 276

H(U, W ), 9

homogeneous domain, 48

H(ξ, λ), horoball, 231

homogeneous polynomial, 4

Hn (C), 67

homogeneous space, 34

Hn (F), 58

homomorphism of Lie algebra, 103

Hn (H), 67

homotope of Jordan triple system,

Hn (R), 67

79

Hemb (D, BH ), 274

horoball, 231

Hemb (D1 , D2 ), 274

horocentre, 226

Hloc (D, W ), 322

horodisc, 226

B (D, Cn ), 335 Hloc

hyperbolic domain, 215

H, 57

hyperstonean space, 307

H3 (O), 59 Hartogs theorem, 3

ideal of Jordan algebra, 67

H3 (O), 77

ideal of Lie algebra, 103

hermitian element, 114

idempotent, 59

Hermitian Jordan triple, 69

immersion, 17

Hermitian manifold, 36

indicator function, 137

hermitian matrices over F, 58

inner derivation, 105

hermitian operator, 114

inner ideal, 101

Hermitian symmetric space, 47

integral curve, 24

HHR domains, 276

interior: E 0 or int E, 1

Hilbert ball, 1

invariant distance, 211

holomorphic arc, 205

invariant metric, 211

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Index inverse function theorem, 7 invertible element, 67 involution, 52 involution of Jordan pair, 75 involution of Lie algebra, 104 involutive automorphism, 104 involutive Lie algebra, 107 irreducible cones, 123 isometry group of manifold, 30 isometry of manifold, 29

389 Jordan triple system, 69 k, 1-eigenspace, 107 KD (p, v), Kobayashi pseudo-metric, 217 kD , Kobayashi pseudo-distance, 213 Killing form, 105 Kobayashi (differential) metric, 217 Krein-Milman property, 188 L(H), 132 Ln (V, W ), 4

Jacobi identity, 22, 103

`∞ , 115, 189

JB*-algebra, 124

`∞ -norm, 115

JB*-triple, 114

`∞ -sum, 115

JB-algebra, 120

L(H), 76

JBW*-algebra, 124

L(H, K), 76

JBW*-factor, 301

L(V ), 110

JBW*-triple, 118

left invariant vector field, 30

JBW-algebra, 120

left multiplication, 53

JH-algebra, 120

Lie algebra, 103

joint Peirce decomposition, 99

Lie algebra of Lie group, 31

joint Peirce projection, 101

Lie ball, 134

joint Peirce space, 100

Lie brackets, 103

Jordan algebra, 52

limit function, 238

Jordan C*-algebra, 124

limit of Kobayashi balls, 233

Jordan homomorphism, 52

limit of Poincar´e discs, 227

Jordan isomorphism, 52

linear automorphism group, 267

Jordan pair, 74

linear vector field, 156

Jordan triple, 69

linearly homogeneous cones, 122,

Jordan triple product, 69

267

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390

Index

little Bloch space, 350

non-degenerate trace form, 86

local chart, 11

normal cone, 259

local coordinates, 12

normal family, 9

local flow, 24

normalised holomorphic map, 314

locally biholomorphic map, 322

normalised univalent function, 313

locally uniform convergence, 8

normed Lie algebra, 129

Lorentz cone, 123

numerical range, 114

Mm (C), 76

O, 56

Mm,n (C), 75

O, 56

M¨ obius transform, 48

Octonions, 56

M¨ obius transformation, 169

one-parameter group, 25

M1,2 (O), 76

one-parameter subgroup, 32

main triple identity, 70

open ball, 9

manifold modelled on

open mapping theorem, 27

Banach space, 12

open unit disc, 1

maximal idempotent, 64

operator commute, 59

maximal tripotent, 95

order-unit norm, 257

maximum principle, 11

orthogonal idempotents, 59

mean value theorem, 2

orthogonal tripotents, 90

minimal tripotent, 95 model space for a manifold, 12 Montel’s theorem, 10

p-order cone, 123 P n (V, W ), 4 p, (−1)-eigenspace, 107

ν-isometric, 35

paracompact, 29

ν-isometry, 35

partial isometry, 117

nilpotent, 59

partial ordering, 68

non-degenerate Jordan triple

Peirce k-projection, 94

system, 85

Peirce k-space, 62, 94

non-degenerate TKK Lie algebra, Peirce decomposition, 63, 94 112

Peirce multiplication rules, 97

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Index

391

Peirce symmetry, 183

real extreme point, 202

plurisubharmonic function, 281

real form of complex Lie algebra,

Poincar´e distance, 41

104

Poincar´e metric, 41

real manifold, 12

polarization identity, 74

real spin factor, 58

polynomial vector field, 156

regular open cone, 257

positive definite, 86

Riemannian manifold, 26, 29

positive Jordan triple system, 89

Riemannian metric, 28

positive quaternion, 57

Riemannian symmetric space, 47

power series, 4

right invariant vector field, 30

predual, 118 primitive idempotent, 64

S, class of schlicht functions, 313

principal inner ideal, 102

S(D), 314

ˆD , squeezing constant, 275 principle of analytic continuation, σ s(D), 333 6 projection, 118

s-identity, 59

projection in a JB-algebra, 120

schlicht ball, 334

proper cone, 68

schlicht disc, 334

pseudoconvex domain, 281

schlicht function, 313 Schwarz lemma, 222

quadratic algebra, 57 quadratic operator, 53, 81 quasi-isometric differential metrics, 280 quaternion algebra, 57

Schwarz-Pick lemma, 222 second-order cone, 123 self-adjoint part, 124 self-dual homogeneous cone, 122 semisimple Jordan triple system, 87

ρ, Poincar´e distance, 211

semisimple Lie algebra, 106

rank D, 187

Shilov boundary, 197

rank V , 187

Siegel domain, 255

Radon-Nikodym property, 188

Siegel upper half-plane, 255

rank, 187

simple JB*-triple, 302

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392

Index

simple Jordan algebra, 67

tangent vector, 15

smooth (Banach) manifold, 13

Tits-Kantor-Koecher

smooth curve, 14 smooth function, 2 solvable Lie algebra, 105 special Jordan algebra, 53

Lie algebra, 109 Tits-Kantor-Koecher construction, 102 TKK Lie algebra, 109

spectral decomposition, 93, 137, 191 topology of locally spectrum of a Banach algebra, 197

uniform convergence, 42

spin factor, 134, 193

trace form, 86

squeezing function, 275

transvection, 154

state space, 258

triple homomorphism, 70

stonean space, 307

triple ideal, 101

strictly contained in an open set, 8 triple identity, 70 strictly homogeneous domain, 48

triple isomorphism, 70, 176

submanifold, 19

triple monomorphism, 70

submersion, 17

triple product, 69

subtriple, 69

triple spectrum, 93, 128

symbol of a composition operator, tripotent, 90 356

tube domain, 255

symmetric Banach manifold, 44

Type IV domain, 134

symmetric cone, 122

Type I domain, 134

symmetric space

type I JBW*-triple, 303

of non-compact type, 48

Type II domain, 134

symmetry of a domain, 47

Type III domain, 134

symmetry of a manifold, 43

Type V domain, 134 Type VI domain, 134

tangent bundle, 20 tangent map, 20

U (H), unitary group, 132

tangent norm, 34

u(H), Lie algebra of U (H), 133

tangent space, 15

uniform squeezing property, 276

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Index

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393

unique predual, 118 unit extension of an algebra, 52 unital algebra, 52 unitary tripotent, 95 V ∗ , dual space of V , 4 Vc , complexification of V , 254 vector field, 20 Vitali theorem, 10 weak*-topology of a JBW*-triple, 118 Wolff’s theorem, 225 ∂ , 22 X = h ∂z ∂ X = h ∂z , 22

x3 , the triple product {x, x, x}, 74

page 393