A Spectral Theory Of Noncommuting Operators [2024 ed.] 3031516044, 9783031516047

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Rongwei Yang

A Spectral Theory Of Noncommuting Operators

A Spectral Theory Of Noncommuting Operators

Rongwei Yang

A Spectral Theory Of Noncommuting Operators

Rongwei Yang Department of Mathematics & Statistics University at Albany, the State University of New York Albany, New York, USA

ISBN 978-3-031-51604-7 ISBN 978-3-031-51605-4 https://doi.org/10.1007/978-3-031-51605-4

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Mathematics Subject Classification: 46L05, 47A10, 37F10, 14F40, 15A22, 20C07, 32A10 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland Paper in this product is recyclable.

Preface

Mathematics is like an ever-growing tree. Now, 2300 years after Euclid of Alexandria, it has become too big for a student to even picture a small branch, not mentioning to see its trunks or roots. While attending to branches and leaves is an indispensable and often delightful experience for a student, the mathematical community needs to offer vantage points for those who aspire to catch a bigger picture. This book takes a step in this direction. Centered around a spectral theory of non-commuting linear operators, its content involves some fundamental subjects in algebra, analysis, dynamics, geometry, group representations, matrix theory, and topology. It can serve as a survey of the theory as well as a textbook for graduate students in the area of functional analysis and representation theory. Joint spectra provide an important mechanism for the study of several interacting operators. Given linear operators .A1 , . . . , An on a Hilbert space .H, their projective spectrum .P (A) is the collection of complex vectors .z = (z1 , . . . , zn ) ∈ Cn such that the linear pencil .A(z) = z1 A1 + · · · + zn An is not invertible. Unlike most other notions of joint spectra which use .A1 − z1 I, . . . , An − zn I in their definitions, projective spectrum places no emphasis on the identity map I , and it treats different linear operators in an equal footing. If these operators represent input (or perceptions) from various observers, the underlying philosophy of this definition is to take an equal account of them. In the technical aspect, the symmetry and simplicity of this definition makes it possible to compute a large amount of examples in various fields of mathematics and thereby provides guiding intuitions for the growth of the theory. This book has two parts that can be read independently. The first three chapters study characteristic polynomials in several variables. The characteristic polynomial of a square matrix A is defined as .det(zI −A), z ∈ C, and it is a basic subject in linear algebra. However, defining an analogous notion for several possibly non-commuting matrices has been a difficult challenge. Chapter 1 addresses this issue. Chapters 2 and 3 apply the idea to representations of finite groups and finite dimensional complex Lie algebras. The three chapters are suitable for an advanced course in linear algebra.

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The remaining chapters of the book carry the idea to various infinite dimensional settings in which the discussion requires graduate level functional analysis, complex analysis, group theory, and differential geometry. Chapter 4 defines the notion of projective spectrum .P (A) for elements .A1 , . . . , An in a Banach algebra. It is computable in some well-known examples of .C ∗ -algebras. Chapter 5 makes an indepth study of the projective spectrum associated with the infinite dihedral group .D∞ . Chapters 6 and 7 discuss the topological and geometric properties of the projective resolvent set .Cn \ P (A). A wild range of topics are discussed here, including Chern–Simon forms, hyperplane arrangement, cyclic cohomology, Kähler metric, unilateral shift, and quasi-nilpotent operators. Chapter 8 takes a closer look at the projective spectrum of compact operators, in which case there is a natural holomorphic line bundle over .P (A). Chapters 9 and 10 describe applications of projective spectrum to some important topics in the representation theory of infinite groups. Topics include weak containment, amenability, and self-similarity. It shows that projective spectrum naturally connects self-similarity with Julia sets. These seven chapters can provide useful material for graduate topic courses in functional analysis, spectral theory, representation theory, and complex dynamics. Although the book has made an earnest attempt to be self-contained, the goal is not always achieved due to the wide range of fields it covers. But a good amount of reference is provided for those who wish to explore further. A unique feature of this book is worth mentioning. In addition to a small amount of exercises at the end of most sections, it includes a few projects designed for undergraduate research and graduate theses. As one often heard, experience is the best teacher. The theses of my students Patrick Cade, Bryan Goldberg, Mai Tran, Fatemeh Azari Key, Michael Muller, and Kate Howell have contributed to part of the content. It has been a great pleasure working with them. A part of the material has been used for topic courses at the University at Albany. Students’ feedback provided valuable input. This book takes its shape from the lecture notes delivered to students at Shandong University and Dalian University of Technology. The author thanks Penghui Wang and Yixin Yang for organizing the semester-long seminar in spring 2022. Finally, I would like to dedicate this book to my family, teachers, and friends. Albany, NY, USA

Rongwei Yang

-微塵中

百千世界

Contents

1

Characteristic Polynomial in Several Variables . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Two Simple Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 The Lie Algebra su2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.2 The Infinite Dihedral Group. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 General Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 The Coefficients of Characteristic Polynomial . . . . . . . . . . . . 1.2.2 A Binomial Expansion in Three Variables . . . . . . . . . . . . . . . . 1.2.3 Linear Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.4 Commuting Normal Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.5 Unitary Invariants. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Determinantal Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 Irreducible Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.2 Dickson’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.3 Real-Zero Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 2 3 3 5 5 8 10 12 14 17 17 18 20

2

Finite Dimensional Group Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Basic Elements of Representation Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Group Determinant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 An Old Theorem of Frobenius . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Group Determinant on a Generating Set . . . . . . . . . . . . . . . . . . 2.3 Abelian Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Word Length Metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 The Characteristic Polynomial Determines an Abelian Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 The Coxeter Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 A Complete Invariant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.2 The Tits Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

23 23 26 27 28 31 31

Finite Dimensional Lie Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Characteristic Polynomial for Lie Algebras. . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 Invariance Under the Automorphism Group . . . . . . . . . . . . . .

41 41 43

3

33 36 36 38

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3.1.2 The Irreducible Representations of sl2 . . . . . . . . . . . . . . . . . . . . 3.1.3 Semidirect Sum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Simple Lie Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Root System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Dynkin Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.3 A Classification by Characteristic Polynomials . . . . . . . . . . . Solvable Lie Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Spectral Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Spectral Invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

46 48 51 52 54 55 59 60 62

4

Projective Spectrum in Banach Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Joint Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 The Koszul Complex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.2 Taylor Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.3 Fredholm Tuples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.4 Essential Taylor Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.5 Harte Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Projective Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Connection with Taylor Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Stein Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Two Important C ∗ -Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 The Cuntz Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 The Irrational Rotation Algebra. . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Free Group von Neumann Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 Noncommutative Probability Space . . . . . . . . . . . . . . . . . . . . . . . 4.4.2 Spectral Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

65 65 66 69 70 72 73 75 77 78 82 83 85 87 88 89

5

The .C ∗ -Algebra of the Infinite Dihedral Group .D∞ . . . . . . . . . . . . . . . . . . . 5.1 Projective Spectrum of the Generators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1 Regular Representation via the Bilateral Shift . . . . . . . . . . . . 5.1.2 Connectedness of the Resolvent Set . . . . . . . . . . . . . . . . . . . . . . . 5.2 Two Projections in Generic Position . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Universal Projections in C ∗ (D∞ ) . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2 Two Projection Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Fuglede–Kadison Determinant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 Basic Properties and Harpe–Skandalis Extension. . . . . . . . . 5.3.2 A Two-Variable Jacobi’s Formula . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.3 On the Fundamental Group of the Resolvent Set . . . . . . . . . 5.3.4 The FK Determinant of C ∗ (D∞ ) . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 An Application to Group of Intermediate Growth . . . . . . . . . . . . . . . . . . 5.4.1 The Growth of Solvable and Nilpotent Groups . . . . . . . . . . . 5.4.2 The Grigorchuk Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

93 93 94 96 97 98 101 104 104 106 108 110 113 114 115

6

The Maurer–Cartan Form of Operator Pencils . . . . . . . . . . . . . . . . . . . . . . . . . 119 6.1 Curvature and Chern–Weil Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 6.1.1 Connection and Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

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6.1.2 Invariant Linear Functional . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.3 The Chern Class . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.4 Chern–Simons Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.5 Chern–Simons Forms of Operator Pencils . . . . . . . . . . . . . . . . Trace Formula and Hyperplane Arrangement . . . . . . . . . . . . . . . . . . . . . . 6.2.1 An Example with the Free Groups. . . . . . . . . . . . . . . . . . . . . . . . . 6.2.2 Theorems of Arnold and Brieskorn . . . . . . . . . . . . . . . . . . . . . . . . 6.2.3 Abelian Banach Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Jacobi’s Formula in Higher Orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.1 su2 and the Chern Character . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.2 An Extension to General Matrices . . . . . . . . . . . . . . . . . . . . . . . . . A Note on Cyclic Cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.1 Hochschild Cohomology and Cyclic Cocycle. . . . . . . . . . . . . 6.4.2 A Map into the de Rham Cohomology . . . . . . . . . . . . . . . . . . . . 6.4.3 Cyclic Cocycles on the Irrational Rotation Algebra . . . . . .

123 125 127 128 130 131 133 134 135 136 137 139 139 141 143

7

Hermitian Metrics on the Resolvent Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Hermitian Vector Bundle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.1 The Dolbeault Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.2 Hermitian Metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.3 Kähler Metric and the Ricci Form . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 The Fundamental Form of Operator Pencils . . . . . . . . . . . . . . . . . . . . . . . . 7.2.1 Operator-Valued Differential Forms . . . . . . . . . . . . . . . . . . . . . . . 7.2.2 Kähler Metric on the Resolvent Set. . . . . . . . . . . . . . . . . . . . . . . . 7.3 The Issue of Completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.1 Distance in the Resolvent Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.2 Another Example with D∞ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 The Fundamental Form of a Single Operator . . . . . . . . . . . . . . . . . . . . . . . 7.4.1 Two Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.2 Ricci Curvature and Eigenvector . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.3 Non-Euclidean Circles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5 Extremal Equation and the Unilateral Shift . . . . . . . . . . . . . . . . . . . . . . . . . 7.5.1 Variational Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5.2 Inner Functions and the Extremal Length of Circles. . . . . . 7.6 The Power Set of Quasinilpotent Operators . . . . . . . . . . . . . . . . . . . . . . . . 7.6.1 Gauging the Singularities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6.2 The Volterra Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6.3 A Touch on the Hyper-Invariant Subspace . . . . . . . . . . . . . . . .

145 145 146 148 149 152 153 156 158 159 162 165 166 167 168 172 172 175 180 181 183 185

8

Compact Operators and Kernel Bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 The Fredholm Determinant and Logarithmic Integral . . . . . . . . . . . . . . 8.1.1 The Argument Principle for Operator Functions . . . . . . . . . . 8.1.2 The Trace of Residues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 The Projective Spectrum of Compact Operators. . . . . . . . . . . . . . . . . . . . 8.2.1 Thin Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.2 Normal Compact Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

187 187 188 190 193 193 194

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8.3

Kernel Bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.1 On the Equivalence of Holomorphic Bundles . . . . . . . . . . . . . 8.3.2 Cowen–Douglas Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.3 The Kernel Bundle of Compact Operators . . . . . . . . . . . . . . . . 8.3.4 An Example with Rank-1 Projections . . . . . . . . . . . . . . . . . . . . . 8.3.5 A Criterion for the Unitary Equivalence. . . . . . . . . . . . . . . . . . .

197 198 202 205 207 209

9

Weak Containment and Amenability. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 Locally Compact Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1.1 Convolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1.2 Revisit the Regular Representation . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Weak Containment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.1 Some General Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.2 The Weak Containment of the Trivial Representation . . . . 9.3 Amenability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.1 Invariant Mean . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.2 The Markov–Kakutani Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.3 A Spectral Description of Amenability . . . . . . . . . . . . . . . . . . . . 9.4 Haagerup Groups and Kazhdan’s Property (T) . . . . . . . . . . . . . . . . . . . . . 9.4.1 Definitions and Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.2 A Description in Several Variables . . . . . . . . . . . . . . . . . . . . . . . .

213 213 214 215 217 218 221 223 224 225 226 229 229 231

10

Self-similarity and Julia Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1 Some Basics of Complex Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1.1 Complex Dynamics in Cn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1.2 The Green Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1.3 Complex Dynamics in Pn and the Indeterminacy Sets. . . . 10.2 Self-similar Group Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2.1 Reshuffling the Dyatic Intervals . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2.2 Three Illuminating Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3 Renormalization Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3.1 The Infinite Dihedral Group. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3.2 The Lamplighter Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3.3 The Grigorchuk Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4 The Julia Set of D∞ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4.1 Determining the Indeterminacy Set . . . . . . . . . . . . . . . . . . . . . . . . 10.4.2 Projective Spectrum and the Julia Set . . . . . . . . . . . . . . . . . . . . . 10.4.3 The Limit of Iterations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

235 235 238 239 242 243 243 245 246 247 247 248 250 251 252 255

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269

Chapter 1

Characteristic Polynomial in Several Variables

For a square matrix .A ∈ Mk (C), its characteristic polynomial .QA (z) = det(z − A) is a basic subject of study in linear algebra. The idea of connecting a matrix with a polynomial has far-reaching impact in many ways. However, studies in mathematics, science, technology, and engineering often concern with several matrices .A1 , . . . , An with algebraic connections. Does there exist a proper notion of joint characteristic polynomial for these matrices? This problem is highly nontrivial even for a pair of matrices A and B. An immediate approach is to consider .det(A − zB), where .A − zB is called a linear pencil of the two matrices. However, in the case A is invertible, one has .det(A − zB) = det A det(I − zA−1 B), and .A−1 B hardly reveals how A and B interact or how they behave jointly as a pair. Ideally, a definition of joint characteristic polynomial should have the following properties. (1) It is a natural extension of the one variable characteristic polynomial. (2) It is easy to compute in a variety of examples. (3) It is able to reveal the algebraic connections among the matrices, if there are any. (4) It could reflect the joint behavior of the matrices as a system. An old idea in group representation theory and determinantal representation of polynomials offered a more promising approach, albeit with different purposes. In the late 1890s, Dedekind and Frobenius studied the determinant of the multiparameter linear pencil A(z) := z0 I + z1 A1 + · · · + zn An , z = (z0 , . . . , zn ) ∈ Cn+1 ,

.

where the set .{I, A1 , . . . , An } is a finite group of matrices. They discovered that its factorization displays a mysterious and yet remarkable pattern. Indeed, this discovery is the starting point of group representation theory. On another front, inspired by some earlier work, Dickson [65] determined the type of homogeneous

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 R. Yang, A Spectral Theory Of Noncommuting Operators, https://doi.org/10.1007/978-3-031-51605-4_1

1

2

1 Characteristic Polynomial in Several Variables

polynomials that can be represented as the determinant of a linear pencil of matrices. This work in part motivated the development of algebraic geometry. Indeed, linear pencils of matrices have already played an important role in a wide range of fields in mathematics, science, technology, and engineering. In light of these facts, the following definition makes a perfect sense. Definition 1.1 Given square matrices .A1 , . . . , An of equal size, their joint characteristic polynomial is defined as .QA (z) = det A(z), z ∈ Cn+1 . Here, the letter “A” in .QA (z) refers to the pencil .A(z) or the tuple .(A1 , . . . , An ). The zero set of .QA , which we denote by .P (A), and its complement .P c (A) = Cn+1 \ P (A) shall be called the eigensurface and the joint resolvent set of the matrices, respectively. Since .A(z) is homogeneous, the sets .P (A) and .P c (A) can be defined in the n-dimensional complex projective space .Pn . Let .φ : Cn+1 → Pn be the canonical projection. We denote .φ(P (A)) by .p(A) and .φ(P c (A)) by .pc (A), respectively. In particular, .p(A) is a projective hypersurface in .Pn . Remark 1.2 The following aspects of the definition address the four expectations in the beginning. (1) When .n = 1, one has .QA (z) = det(z0 + z1 A1 ) which differs from the classical characteristic polynomial by a change of variables. (2) The computation is straightforward, and it is fun to do in many examples. (3) When .n > 1, the coefficients of .zjs in .QA reveal information about .Aj . And coefficients of the terms .zis zjt tell how .Ai and .Aj interact. (4) The factorization of .QA reflects inherent algebraic properties of .A1 , . . . , An . This was already demonstrated in the work of Dedekind and Frobenius. A Note on Notations Throughout this book, the algebra of all .k × k complex matrices is denoted by .Mk (C). The symbols .GLk , .SLk , .Uk , and .SUk denote the general linear group, the special linear group, the unitary group, and the special unitary group of .k × k matrices, respectively. Their corresponding Lie algebras are denoted, respectively, by .glk , slk , uk , and .suk .

1.1 Two Simple Examples Definition 1.1 is straightforward and simple. This feature makes it possible to compute a great amount of examples across various fields, offering necessary intuitions and insights to guide our exploration. We will get acquainted with many of them as the book progresses. To convey this message more clearly, we start with two simple examples. They will be revisited quite a few times later.

1.1 Two Simple Examples

3

1.1.1 The Lie Algebra su2 The 3-dimensional simple Lie algebra .su2 plays an important role in mathematics and physics. It is spanned by the Pauli matrices ⎛ ⎞ ⎛ ⎛ ⎞ ⎞ 01 0 −i 1 0 .σ1 = , σ2 = , σ3 = . 10 i 0 0 −1 Their commutators satisfy the relation: [σ1 , σ2 ] = 2iσ3 , [σ2 , σ3 ] = 2iσ1 , [σ3 , σ1 ] = 2iσ2 .

.

(1.1)

If one sets .σ (z) = z0 I + z1 σ1 + z2 σ2 + z3 σ3 , then the characteristic polynomial Qσ (z) = det σ (z) = z02 − z12 − z22 − z32 . Since .{I, σ1 , σ2 , σ3 } is a basis for the matrix algebra .M2 (C), the linear map .σ is a bijection from the resolvent set .P c (σ ) to the general linear group .GL2 . Therefore, it is possible to define a multiplication in .P c (σ ) to turn it into a group and make .σ a group isomorphism (exercise). On the other hand, if .ad is the adjoint representation of .su2 , namely

.

.

ad x(y) = [x, y], x, y ∈ su(2),

and .Ti = ad σi , i = 1, 2, 3, then with respect to the basis .{σ1 , σ2 , σ3 } we have the matrix representations ⎛

⎞ ⎛ ⎞ ⎛ ⎞ 0 0 0 0 0 2i 0 −2i 0 .T1 = ⎝0 0 −2i ⎠ , T2 = ⎝ 0 0 0 ⎠ , T3 = ⎝2i 0 0⎠ . 0 2i 0 −2i 0 0 0 0 0 The characteristic polynomial in this case is ⎛ ⎞ QT (z) = z0 z02 − 4(z12 + z22 + z32 ) .

.

(1.2)

Observe that .QT (z) has a factor .z0 . This is not a coincidence.

1.1.2 The Infinite Dihedral Group The infinite dihedral group D∞ =

.

is probably the simplest nonabelian infinite group. It serves as a good example as well as a building block for many mathematical theories. The characteristic

4

1 Characteristic Polynomial in Several Variables

polynomial associated with its representations plays an important role later in the book. Fix a .θ ∈ [0, 2π ) and consider the unitary representation .ρθ : D∞ → U2 defined by ⎛ ρθ (a) =

.

⎞ ⎛ ⎞ 0 eiθ 01 , ρ (t) = . θ e−iθ 0 10

(1.3)

Then the linear pencil .Aθ (z) = z0 I + z1 ρθ (a) + z2 ρθ (t) has characteristic polynomial .QAθ (z) = z02 − z12 − z22 − 2z1 z2 cos θ . Some simple calculations verify the following properties. (1) .QAθ (z) is reducible if and only if .θ = 0 or .π , and this happens if and only if .ρθ (a) commutes with .ρθ (t). In this case .ρθ a direct sum of two 1-dimensional representations and thus is reducible. (2) When .θ /= 0 or .π , the complex algebra generated by .I, ρθ (a) and .ρθ (t) is equal to .M2 (C). In other words, the map .ρθ extends to an isomorphism between the group algebra .C[D∞ ] and .M2 (C). We say that .ρθ is an irreducible representation in this case. (3) Two such representations .ρα and .ρβ are said to be unitarily equivalent if there exists a unitary .V ∈ U2 such that .V ∗ ρα (g)V = ρβ (g), g ∈ D∞ . This occurs if and only if .QAα (z) = QAβ (z). In fact, we have the following theorem due to Halmos [115]. Theorem 1.3 If .ρ is an irreducible representation of .D∞ , then either .ρ is 1dimensional or, up to unitary equivalence, it is of the form .ρθ with .0 < θ < π. Therefore, the polynomial .QAθ is a complete invariant for irreducible representations of .D∞ . It is an interesting question whether the same holds for other groups. Exercise 1.4 1. Regarding the Pauli matrices in Sect. 1.1.1, define a multiplication in .P c (σ ) to make the linear map .σ a group isomorphism between .P c (σ ) and .GL2 . 2. Consider the three-dimensional simple Lie algebra .sl2 with basis .{H, X, Y } such that [H, X] = 2X, [H, Y ] = −2Y, [X, Y ] = H.

.

Compute the characteristic polynomial associated with the adjoint representation of .sl2 and show that it differs from the characteristic polynomial (1.2) by a linear change of variables. 3. Verify observations (1), (2), and (3) in Sect. 1.1.2. 4. Find several matrices of your interest and compute their joint characteristic polynomial. Is the polynomial irreducible?

1.2 General Properties

5

1.2 General Properties The fundamental theorem of algebra asserts that every one-variable complex polynomial is a product of linear factors. And this implies that a two-variable homogeneous polynomial has similar factorizations, namely, for any complex coefficients .a1 , . . . , ak , we can write k Π

z1k + a1 z1k−1 z2 + · · · + ak z2k =

.

(z1 + bj z2 )

j =1

for some .b1 , . . . , bk ∈ C. This is no longer true with a polynomial in three or more variables. For example, one checks that .z02 − z12 − z22 is not a product of two linear factors. In this case, determining whether a given polynomial is irreducible is usually a rather challenging task. This section describes some general properties of characteristic polynomials in several variables. Many more remain to be discovered.

1.2.1 The Coefficients of Characteristic Polynomial Given a .k × k matrix A, we define .c0 (A) = 1 and ⎛

tr A tr A2 .. .

⎜ ⎜ 1 ⎜ .cm (A) = det ⎜ ⎜ m! ⎝tr Am−1 tr Am

m−1 0 tr A m − 2 .. .. . . tr Am−2 · · · tr Am−1 · · ·

··· ··· .. .

0 0 .. .

⎞

⎟ ⎟ ⎟ ⎟ , 1 ≤ m ≤ k. ⎟ ··· 1 ⎠ · · · tr A

(1.4)

Then the Plemelj-Smithies formula [100] expresses the characteristic polynomial of A in the form

.

det(λ + A) =

k ∑

λk−m cm (A).

(1.5)

m=0

Hence, for matrices .A1 , . . . , An ∈ Mk (C), if we set .A∗ (z' ) = z1 A1 + · · · + zn An , where .z' stands for the vector .(z1 , . . . , zn ), the function .QA (z0 , z' ) is the classical characteristic polynomial of the matrix .A∗ (z' ). The following fact is thus immediate. Proposition 1.5 If .Aj , Bj , j = 1, . . . , n are square matrices of equal size and QA = QB , then for each .z' ∈ Cn the matrices .A∗ (z' ) and .B∗ (z' ) have the same set of eigenvalues counting multiplicity. In particular, the classical spectra .σ (Aj ) = σ (Bj ) for each j .

.

6

1 Characteristic Polynomial in Several Variables

This means that the characteristic polynomial is a good invariant for matrix tuples. For a single square matrix A, its eigenspace is invariant under the action of A. Hence A’s structure is reflected through the factorization of its characteristic polynomial. For several matrices .A1 , . . . , An ∈ Mk (C), it is usually a tedious job to determine whether they share a common invariant subspace. If such space, say N , does exist, then with respect to the orthogonal decomposition .Ck = N ⊕ N ⊥ , the matrices have the block-matrix forms ⎛ 1 1 ⎞ ⎛ n n ⎞ A11 A12 A11 A12 , . . . , An = . .A1 = 1 0 A22 0 An22 Thus .QA (z) = QA11 (z)QA22 (z), where .QAii is the characteristic polynomial of the submatrices .A1ii , . . . , Anii , .i = 1, 2. This simple observation reveals the following fact. Proposition 1.6 For .A1 , . . . , An ∈ Mk (C), if their characteristic polynomial is irreducible, then they have no nontrivial common invariant subspace. However, the converse of Proposition 1.6 is not true. Example 1.7 Consider the following matrices ⎛

⎞ 0 0 0 .A1 = ⎝ 1 0 0 ⎠ , 0 −1 0

⎛

⎞ 010 A2 = ⎝ 0 0 1 ⎠ . 000

(1.6)

Both matrices are nilpotent, and their characteristic polynomial .QA (z) = z03 . But .A1 and .A2 have no nontrivial common invariant subspace. In fact, the algebra generated by .I, A1 , and .A2 is equal to .M3 (C). Using the Plemelj-Smithies formula, a general joint characteristic polynomial can be expressed as QA (z) = det(z0 + A∗ (z' )) =

k ∑

.

z0k−m cm (A∗ ), (z0 , z' ) ∈ Cn+1 .

(1.7)

m=0

For simplicity, we shall denote .cm (A∗ ) by .qm and write QA (z) =

k ∑

.

z0k−m qm (z' ).

(1.8)

m=0

Clearly, .qm is a degree-m homogeneous polynomial in .z' . The following fact can be verified by direct computation. Proposition 1.8 For any .k × k matrices .A1 , . . . , An , we have

1.2 General Properties

(a) .q1 (z' ) = (b) .q2 (z' ) =

7

∑n j =1 zj tr Aj .( ) 1 ∑n i,j =1 zi zj tr Ai tr Aj − tr (Ai Aj ) . 2

It is possible to express .qm in a similar way for .m ≥ 3, but the amount of computation would be challenging. A more elegant approach can be given by the cofactor matrix. For a matrix .A = (aij ) ∈ Mk (C), we let .C = (Cij ) be its cofactor matrix, e.g., .Cij = (−1)i+j det(Aij ), where .Aij is the .(i, j )th minor of A. The cofactor theorem states that

.

det A =

k ∑

aim Cim , 1 ≤ i ≤ k.

(1.9)

m=1

Treating A as a multivariable function in its entries .aij , 1 ≤ i, j ≤ k, we obtain

.

⎞ k k ⎛ ∑ ∑ ∂aim ∂ det A ∂Cim ∂Cim = Cij + Cim + aim aim . = ∂aij ∂aij ∂aij ∂aij m=1

m=1

Since .Cim is independent of the ith row of A, we have . follows that .

∂Cim = 0 for every m. It ∂aij

∂ det A = Cij , 1 ≤ i, j ≤ k. Therefore, ∂aij d det(A) =

n ∑

.

Cij daij = tr (C T dA).

(1.10)

i,j =1

In the case A is invertible, dividing (1.10) by .det A, we have the Jacobi’s formula ⎛ ⎞ d log det A = tr A−1 dA .

.

The following consequence is instrumental for several later chapters. Lemma 1.9 If .Ω is a complex domain in .Cn and .f : Ω → GLk is differentiable, then ⎛ ⎞ −1 .d log det f (z) = tr f (z)df (z) , z ∈ Ω. Interestingly, the polynomials .q1 , . . . , qk in expansion (1.8) have a generating function in the following sense. Proposition 1.10 For .1 ≤ m ≤ k − 1, we have qk−m (z' ) =

.

⎛ ⎞| | 1 ∂ m−1 | tr C (z) , A | m! ∂z0m−1 z0 =0

8

1 Characteristic Polynomial in Several Variables

where .CA (z) is the cofactor matrix of the linear pencil .A(z). Proof In (1.10), we substitute the general matrix A by the linear pencil .A(z) to obtain ⎛ ⎞ T .dQA (z) = tr CA (z)(dz0 + A1 dz1 + · · · + An dzn ) . It follows that ∂QA (z) = tr CAT (z) = tr CA (z). (1.11) ∂z0 | Setting .z0 = 0, we obtain .qk−1 (z' ) = tr CA (z)|z =0 . In view of expansion (1.8), one 0 obtains the formula for .qk−m (z' ) by taking the .(m − 1)th partial derivative of (1.11) ⨆ ⨅ with respect to .z0 and then setting .z0 = 0. .

1.2.2 A Binomial Expansion in Three Variables It is useful to give a general formula for the coefficients of the characteristic polynomial of two matrices. The formula for the case of three or more matrices can be derived similarly. First, we recall the notion of minor matrix. Let .A = (aij ) be an .k × k matrix. If the rows and columns chosen are given by subscripts .1 ≤ i1 < · · · < ip ≤ k, 1 ≤ j1 < · · · < jp ≤ k, respectively, then the corresponding .p × p minor of A is denoted by ⎛ A

.

I' J'

⎞

⎛ =A

i1 . . . ip j1 . . . jp

⎞ p

:= det(aim jl )m,l=1 ,

' ' with multi-index .I = (i1 . . . ip ) and .J = (j1 . . . jp ). ⎞ ⎛ notation

Let .As = aijs , s = 1, 2, be two .k × k matrices and set .B = z1 A1 + z2 A2 . Then .det B is a degree-k homogeneous polynomial bk z1k + bk−1 z1k−1 z2 + · · · + b1 z1 z2k−1 + b0 z2k .

.

By setting .z1 = 0 or .z2 = 0, we see that .bk = det A1 and .b0 = det A2 . The value of bp for .0 < p < k can be expressed by the minors of .A1 and .A2 . To this end, we let .Λp be the set of ordered partitions .

{1, 2, . . . , k} = {i1 < · · · < ip } ∪ {ip+1 < · · · < ik } := I,

.

(1.12)

1.2 General Properties

9

and we write .I = I ' ∪ I '' correspondingly. Note that .I ' determines .I '' . For each .I ∈ Λp , we let .σI be the permutation .m → im , 1 ≤ m ≤ k, and let .sgn(σ ) denote the signature of the permutation .σ on .(1, 2, . . . , k). ⎛ ⎞ Proposition 1.11 Let .As = aijs , s = 1, 2 be two .k × k matrices. Then, det(z1 A1 + z2 A2 ) = bk z1k + bk−1 z1k−1 z2 + · · · + b1 z1 z2k−1 + b0 z2k , where

.

⎛

∑

bp =

sgn(σI σJ )A1

.

I,J ∈Λp

I' J'

⎞

⎛ A2

I '' J ''

⎞ , 0 < p < k.

However, its proof is somewhat tedious. We refer the readers to [132] for details. ⎛ '⎞ I = 0 for Example 1.12 In the case .A1 is a diagonal matrix, we have .A1 J' .I /= J . Since .sgn(σI σI ) = 1, we have bp =

∑

.

⎛ A1

I ∈Λp

I' I'

⎞

⎛ A2

I '' I ''

⎞ .

The next fact follows readily. Corollary 1.13 Consider two matrices .A1 , A2 ∈ Mk (C). Then for natural numbers p q p and q satisfying .m := p + q ≤ k, the coefficient of the term .z0k−m z1 z2 in the characteristic polynomial .QA (z) is ∑

∑

1≤s1 = tr (h' h∗ ), h, h' ∈ C[G],

.

defines an inner product on .C[G]. The completion of .C[G] with respect to the norm induced by the inner product shall be denoted by .HG . Then HG =

⎟ ⎞ ⎟ ∑ hg g ⎟⎟ |hg |2 < ∞ ,

⎧∑

.

g∈G

g∈G

and G is an orthonormal basis for .HG . In particular, we have .dim HG = |G|. Given a complex Hilbert space .H, we let .U (H) denote the group of unitary operators on .H and .B(H) denote the set of bounded linear operators on .H. A unitary representation .π of G on .H, often written as .(π, H), is a group homomorphism from G into .U (H). It naturally extends to a .∗-homomorphism from the group algebra .C[G] to the algebra .B(H) such that π(h) = π

⎛∑

.

g∈G

⎞ ∑ hg g := hg π(g), and (π(h))∗ = π(h∗ ). g∈G

A representation .π is said to be reducible if there is a proper subspace .H1 ⊂ H that is reducing for each .π(g), g ∈ G. In this case the homomorphism .g → π ' (g) := π(g)|H1 , g ∈ G, is called a subrepresentation of .π . Equivalently, we say .π ' is contained in .π and write .π ' < π . A representation .π is said to be irreducible if it contains no proper subrepresentation. In this case, the dimension of .π , denoted by .dπ , refers to .dim H. Given a subset .S ⊂ B(H), its commutant is defined as S ' = {A ∈ B(H) | AB = BA, ∀B ∈ S}.

.

Proposition 2.2 Let .(π, H) be a unitary representation of G. Then the following are equivalent. (a) (b) (c) (d)

π is irreducible. π(G) has no nontrivial common reducing subspace. For every nonzero vector .v ∈ H, .span{π(g)v | g ∈ G} is dense in .H. ' .(π(G)) = CI . . .

If G is abelian, then .π(G) ⊂ (π(G))' , and hence the following fact is immediate. Corollary 2.3 Every irreducible representation of an abelian group is onedimensional.

2.1 Basic Elements of Representation Theory

25

Among all representations of a group, there is an inherent one given by the group multiplication. Definition 2.4 Given a countable group G, its (left) regular representation (λG , HG ) is defined by

.

λG (g)h = gh, g ∈ G, h ∈ HG .

.

The reduced .C ∗ -algebra of G, denoted by .Cr∗ (G), is the .C ∗ -algebra generated by the unitary operators .λG (g), g ∈ G, and the group von Neumann algebra ∗ .L(G) is the closure of .Cr (G) with respect to the weak topology on .B(HG ). The representation .λG gives rise to a natural extension of the trace .tr from .C[G] to .L(G) defined by tr F = , F ∈ L(G).

.

(2.2)

We shall call this extension the canonical trace on .L(G). It carries all the properties described in Proposition 2.1. There is a canonical equivalence relation on the set of all unitary representations of G. Suppose .(π, H) and .(ρ, K) are two unitary representations of G. Then .π and .ρ are said to be equivalent, expressed as .π ∼ = ρ, if there is a unitary .V : H → K such that .V π(g)V ∗ = ρ(g), ∀g ∈ G. Example 2.5 Given a countable∑group G, the Hilbert space .ℓ2 (G) is the space of functions .f : G → C such that . g∈G |f (g)|2 < ∞. A standard orthonormal basis for .ℓ2 (G) is .{δg | g ∈ G}, where .δg (g) = 1 and .δg (x) = 0 for all .x /= g. Clearly, the linear map .V : HG → ℓ2 (G) defined by .V (g) = δg , g ∈ G identifies .HG with 2 .ℓ (G), and it is not hard to check that V λG (x)V ∗ δg = δx −1 g , ∀x, g ∈ G.

.

In fact, the left regular representation is traditionally defined on .ℓ2 (G), or on 2 .L (G, μ) when G is locally compact with a Haar measure .μ. The following characterization of irreducibility is due to Burnside [28]. Its proof is a good exercise. Lemma 2.6 A finite dimensional unitary representation .π of G is irreducible if and only if .π(C[G]) = Mdπ (C). ˆ is the set of equivalence classes of irreducible The dual of G, denoted by .G, unitary representations of G. A function f on G is said to be a class function if .f (s −1 gs) = f (g) for all .s, g ∈ G. Every finite dimensional representation .π naturally gives rise to a class function .χπ , called character, defined by .χπ (g) = tr (π(g)), g ∈ G, where .tr is the trace on .dπ × dπ matrices. Be aware that if G is an infinite group then in general a character is not in .ℓ2 (G). Observe that the character .χπ extends linearly to .C[G] as a linear functional. For a finite group, much is known about the structure of .C[G] and the set of class functions. In this

26

2 Finite Dimensional Group Representations

ˆ = {[π1 ], . . . , [πm ]} and set .dj = dπj . Then .{χπ1 , . . . , χπm } case, we may assume .G is an orthonormal basis for the subspace of class functions in .ℓ2 (G). The following facts are well-known, see for example Serre [201]. ˆ = {[π1 ], . . . , [πm ]}. Then Theorem 2.7 Let G be a finite group with .G (a) .C[G] ∼ = Md1 (C) ⊕ · · · ⊕ Mdm (C). (b) .χλG = d1 χπ1 + · · · + dm χπm on .C[G]. (c) Two representations .π and .ρ are equivalent if and only if .χπ = χρ . Exercise 2.8 1. Verify Proposition 2.1. 2. Prove Lemma 2.6. 3. Suppose a countable group G has∑ an element g of order .n > 1. Given a nth root i i of unity .ξ , we define .P (ξ ) = n1 ( n−1 i=0 ξ g ) ∈ C[G]. Show that the following hold: (a) .P (ξ ) is an idempotent, namely .P 2 (ξ ) = P (ξ ). (b) .P (ξ 0 ) + · · · + P (ξ n−1 ) = 1. (c) .P (ξ i )P (ξ j ) = 0 for .0 ≤ i < j ≤ n − 1. 4. Show that in Theorem 2.7 (a) implies (b). 5. Given a matrix .T ∈ Mk (C), we define the left multiplication operator Tˆ (M) = T M, M ∈ Mk (C).

.

Show that .det Tˆ = (det T )k and compute .tr Tˆ . 6. Prove that every irreducible representation of a finite group is finite dimensional.

2.2 Group Determinant Given a group G generated by a finite set .S = {g1 , . . . , gn } and a finite dimensional representation .π : G → Uk , we consider the linear pencil Aπ (z) := z0 I + z1 π(g1 ) + · · · + zn π(gn ), z ∈ Cn+1 .

.

The associated characteristic polynomial is defined as .Qπ (z) := det Aπ (z). When G is a finite group and .π is the regular representation .λG , we will simply write .QλG as .QG and call it the characteristic polynomial of G (with respect to the generating set S). Observe that .deg Qπ = dπ and, in particular, .deg QG = |G|. Three natural questions arise.

2.2 Group Determinant

27

Question 2.9 (a) Is .Qπ irreducible when .π is an irreducible representation? (b) Does .Qπ = Qρ imply .π ∼ = ρ? (c) Does .QG determine G up to isomorphism? We have seen in Sect. 1.1.2 that when .G = D∞ the answer is yes for questions (a) and (b) above. In general, the answer depends on the group G, the representation .π , as well as the generating set S. In light of Theorems 1.17 and 1.19, we have the following fact. Corollary 2.10 Let G be a finitely generated group and .π be a finite dimensional unitary representation. Then (a) .π contains a one-dimensional representation if and only if .Qπ has a linear factor. (b) .π(G) is abelian if and only if .Qπ is a product of linear factors. Proof The necessity of part (a) is trivial. To prove the sufficiency, we assume G is generated by .{g1 , . . . , gn }. Theorem 1.17 implies that .Qπ has a linear factor if and only if .π(gi ), 1 ≤ i ≤ n, have a common eigenvector v. It follows that .π(g)v = cg v for every .g ∈ G and some unimodular scalar .cg ∈ C. Therefore .π contains the one-dimensional representation defined by the map .g → cg . Part (b) is a direct consequence of Theorem 1.19 because unitary matrices are normal. ⨆ ⨅ In particular, if the representation .π has dimension .k = 2 or 3, then .π is irreducible if and only if the characteristic polynomial .Qπ (z) is irreducible. However, as we shall see in later sections, the situation regarding nonlinear factors of .Qπ is rather complicated.

2.2.1 An Old Theorem of Frobenius In the case .G = {1, g1 , g2 , . . . , gs }, its characteristic polynomial .QG (z) is called the group determinant (or Frobenius determinant) of G, and it is usually denoted by .DG (z) in the literature. Group determinant was originally studied by Dedekind and Frobenius [58, 90] in the late nineteenth century. In fact, their work is the original motivation for group representation theory. To read about this piece of history, we refer the reader to [40, 48, 64, 89] and the references therein. The following theorem is due to Frobenius [90], but it is stated in terms of characteristic polynomial. Its proof here is based on Theorem 2.7. Theorem 2.11 Let .G = {1, g1 , g2 , . . . , gs } be a finite group. Then QG (z) =

Π

.

ˆ [π ]∈G

(Qπ (z))dπ .

28

2 Finite Dimensional Group Representations

Moreover, .Qπ (z) is an irreducible polynomial of degree .dπ , and it determines .π . ˆ = {[π1 ], . . . , [πm ]} and shall use the same notations as that Proof We assume .G in Theorem 2.7. Pick any open set .Ω ⊂ Cs+1 on which .QG and .Qπj , 1 ≤ j ≤ m, do not vanish. Then the linear pencils .Aπ (z) and .Aπj (z) are invertible for each .z ∈ Ω. Consider the Maurer–Cartan form .ωA (z) = A−1 λ (z)dAλ (z) on .Ω. By Jacobi’s formula (Lemma 1.9), we have ⎛ ⎞ tr A−1 λ (z)dAλ (z) = d log det Aλ (z) = d log QG (z).

.

Parallel equalities hold for the irreducible representations .πj , j = 1, . . . , m. We denote .dπj by .dj for convenience. Theorem 2.7 (b) implies m ⎛ ⎞ ∑ ⎛ ⎞ tr A−1 (z)dA (z) = dj tr A−1 λ πj (z)dAπj (z) , λ

.

j =1

and hence d log QG (z) =

m ∑

.

dj d log Qπj (z).

j =1

Π dj on .Ω for some nonzero constant It follows that .QG (z) = α m j =1 (Qπj (z)) s+1 . .α. Since both sides of the equality are polynomials, the equality holds on .C s+1 Comparing the coefficient of .z0 on both sides of the equation, we get .α = 1. As a 2. consequence, we have .s + 1 = |G| = deg QG = d12 + · · · + dm To show that .Qπj (z) is irreducible for each j , it suffices to consider the case .j = 1. In light of Theorem 2.7, (.π1 is a )homomorphism from .C[G] onto .Md1 (C). It means that if we write .Aπ1 (z) = aij (z) ∈ Md1 (C), then the entries .aij , 1 ≤ i,( j ≤ ) d1 are linearly independent linear functions in .z0 , . . . , zs . Since .Qπ1 (z) = det aij , which is irreducible as a polynomial in .aij , it is irreducible in the variables .z0 , . . . , zs . To show that .Qπ determines .π , we expand .Qπ as in (1.8) with .k = dπ . Proposition 1.8 gives .q1 (z' ) = tr π(g1 )z1 +· · ·+tr π(gs )zs . If .ρ is any representation such that .Qρ = Qπ , then .tr ρ(gj ) = tr π(gj ) for each j , e.g., .χρ = χπ . Theorem 2.7 (c) then implies .ρ ∼ ⨆ ⨅ = π.

2.2.2 Group Determinant on a Generating Set For a proper generating subset .S = {g1 , . . . , gn } ⊂ G, the number n is usually much smaller than .|G|, hence the characteristic polynomial .Qπ (z) = det(z0 I + z1 λ(g1 ) + · · ·+zn λ(gn )) has a much smaller number of variables. A most telling example is the Fischer–Griess Monster group which is simple and has roughly .8 × 1053 elements, more than 5000 times the number of atoms in the Earth, but it is generated by only

2.2 Group Determinant

29

two of them. Thus a natural question is how much of Frobenius’ theorem remains valid. By setting .zn+1 = · · · = zs = 0 in Theorem 2.11, we have the factorization Qλ (z0 , . . . , zn ) =

Π

.

Qdππ (z0 , . . . , zn ).

(2.3)

ˆ π ∈G

However, the factors .Qπ in this case may not be irreducible nor must they uniquely determine the representations .π . The following two examples are due to Klep– Volˇciˇc [149]. Note that for finite groups, every representation is similar to a unitary representation.

2.2.2.1

The Alternating Group

The alternating group Alt.m is the group of even permutations on a set of m elements. Hence it is the kernel of the signature homomorphism .sgn : Sm → {±1}. It is known that Alt.m is simple when .m = 3 or .m ≥ 5. Alt.6 admits the presentation 2 4 2 5 5 .. If we set 1 2 2 ⎛

1 ⎜0 ⎜ ⎜0 ⎜ ⎜0 ⎜ ⎜ .A1 = ⎜ 0 ⎜ ⎜0 ⎜ ⎜0 ⎜ ⎝0 −1

0 0 1 0 0 0 0 0 −1

0 1 0 0 0 0 0 0 −1

0 0 0 0 1 0 0 0 −1

0 0 0 1 0 0 0 0 −1

0 0 0 0 0 0 0 1 −1

0 0 0 0 0 0 1 0 −1

0 0 0 0 0 1 0 0 −1

⎞ 0 0⎟ ⎟ 0⎟ ⎟ 0⎟ ⎟ ⎟ 0 ⎟, ⎟ 0⎟ ⎟ 0⎟ ⎟ 0⎠ −1

and ⎛ 0 ⎜0 ⎜ ⎜0 ⎜ ⎜0 ⎜ ⎜ .A2 = ⎜0 ⎜ ⎜1 ⎜ ⎜0 ⎜ ⎝0 0

1 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 1

0 1 0 0 0 0 0 0 0

0 0 1 0 0 0 0 0 0

0 0 0 1 0 0 0 0 0

0 0 0 0 1 0 0 0 0

0 0 0 0 0 0 0 1 0

⎞ 0 0⎟ ⎟ 0⎟ ⎟ 0⎟ ⎟ ⎟ 0⎟ , ⎟ 0⎟ ⎟ 1⎟ ⎟ 0⎠ 0

then the map .ρ : g1 → A1 , g2 → A2 extends to be a group representation .ρ : Alt6 → GL9 . Since Alt.6 is simple, the kernel of .ρ is .{1} which means .ρ is faithful.

30

2 Finite Dimensional Group Representations

Lemma 2.6 tells that .ρ is irreducible because it can be verified that .C[Alt6 ] ∼ = M9 (C). Setting .Aρ (z) = z0 I + z1 A1 + z2 A2 , one has .det Aρ (z) = p1 (z)p2 (z), where p1 (z)=z04 +2z03 z1 −2z0 z13 − z14 + z02 z1 z2 + 2z0 z12 z2 + z13 z2 − z0 z1 z22 + z1 z23 − z24 ,

.

p2 (z)=z05 −z04 z1 −2z03 z12 + 2z02 z13 + z0 z14 − z15 + z04 z2 − z03 z1 z2 − z02 z12 z2 + z0 z13 z2 − z0 z24 − z25 . 2.2.2.2

The Group GL3 (Z/3Z)

We now consider the finite linear group .G = GL3 (Z/3Z) which admits the presentation G = .

.

If we set ⎛ A1 =

.

− √1

2

− 12 +

− 12 − √1 2

i 2

i 2

⎛1

⎞ , A2 =

2

+

i 2

− √1

1 2 2

√1 2

−

⎛1

⎞ i 2

, A3 =

2

− √i

2

√i

i 2 1 2

⎞ 2

+

i 2

,

then the maps ρ± (g1 ) = ±A1 , ρ± (g2 ) = A2 , ρ± (g3 ) = A3

.

extend to two unitary representations .ρ+ , ρ− : G → U2 . Since ρ+ (C[G]) = M2 (C) = ρ− (C[G]),

.

both representations are irreducible according to Lemma 2.6. A simple computation gives Qρ+ (z) = Qρ− (z) = z02 + z0 (z2 + z3 ) − z12 + z22 + z32 .

.

If there were a unitary .U ∈ U2 such that .Uρ+ (g)U ∗ = ρ− (g), g ∈ G, then the linear transformation .T : M2 (C) → M2 (C) defined by .T (M) = U MU ∗ would have eigenvalue 1 with corresponding eigenvectors I , .A2 , and .A3 ; it would have eigenvalue .−1 with corresponding eigenvectors .A1 , .A1 A2 , and .A1 A3 , which is impossible because .dim M2 (C) = 4. This verifies the inequivalence of .ρ+ and .ρ− . Exercise 2.12 1. Compute the group determinants for the Klein 4-group .Z2 × Z2 and the cyclic group .Z4 , and compare your results.

2.3 Abelian Groups

31

2*. Verify that the matrices .A1 and .A2 in Sect. 2.2.2.2 regarding Alt.6 have no nontrivial joint reducing subspace in .C9 . 3. Let .λ be the left regular representation of a finite group G. Show that .Qλ (z) has integer coefficients, i.e., .Qλ ∈ Z[z]. (Hint: Observe that for every .g ∈ G, .λ(g) can be expressed as a permutation matrix.) 4. Compute the characteristic polynomial of the alternating group Alt.3 . Project 2.13 1. The quaternion group .Q8 = {1, i, j, k, −1, −i, −j, −k} satisfies the algebraic relations (−1)2 = 1, i 2 = j 2 = k 2 = i · j · k = −1.

.

Compute the characteristic polynomial of .Q8 with respect to the generating set {i, j, k} and find its factors. What if the generating set is .{i, j }? 2. Consider group .G = and representation .π : G → Uk . If .Qπ has an irreducible quadratic factor, does .π necessarily contain an irreducible two dimensional representation? 3. Suppose N is a normal subgroup of a finite group G with index .|G/N| = k. Study the relation between the characteristic polynomial of N and that of G. Is k visible in the two polynomials? .

2.3 Abelian Groups Although Question 2.9 has negative answers in general, it does have positive answers for certain types of groups. A good example is the class of finite abelian groups. A cyclic group .Zk is said to be primary if k is a power of a prime number. The fundamental theorem of finite abelian groups states that every finite abelian group G is isomorphic to a direct sum of primary groups. In other words, up to an isomorphism we can write G = Zp1 × · · · × Zpk ,

.

where .p1 , . . . , pk are powers of prime numbers (not necessarily distinct). Some preparation is needed for further discussion.

2.3.1 Word Length Metric Given a finitely generated group G, a generating set .S = {g1 , . . . , gn } is said to be symmetric if .g ∈ S implies .g −1 ∈ S. In this section, we assume S is symmetric. Then for every non-unit .g ∈ G, we can write .g = gi1 · · · gik , where .k ≥ 1 and

32

2 Finite Dimensional Group Representations

{gi1 , . . . , gik } ⊂ S. This expression of g is not unique, for example .g1 = g1 g2 g2−1 . But an economical (not necessarily unique!) expression can be found such that k is minimal. In this case, which we shall always assume, the word .gi1 · · · gik is said to be in a reduced form. The number k is called the word length of g and is denoted by .l(g) (or .lS (g) when more than one generating sets are considered). Consistent with this definition, we set .l(1) = 0. Moreover, we let .αi (g) be the number of times .gi occurs in the reduced form of g. The signature of g is the tuple sig.(g) := (α1 , . . . , αn ). Apparently, the word length .l(g) = |α1 |+· · ·+|αn | := |α|. The following properties readily follow.

.

Proposition 2.14 For any elements g and .g ' in a finitely generated group G, we have (a) .l(g) = l(g −1 ). (b) .l(gg ' ) ≤ l(g) + l(g ' ). The word length metric d on G is defined by .d(g, g ' ) = l(g −1 g ' ). Lemma 2.15 Given a finitely generated group G, if .π and .ρ are two finite dimensional representations of G such that .Qπ = Qρ , then (a) .χπ (g) = χρ (g) for every element .g ∈ G with .l(g) ≤ 2. (b) For every .α ∈ Zn+ with .|α| = 3, we have ∑ .

χπ (g) =

sig(g)=α

∑

χρ (g).

sig(g)=α

Proof Assume .S = {g1 , . . . , gn } is a generating set for G. For (a), note first that l(g) = 1 if and only if .g ∈ S, and .l(g) ≤ 2 if and only if .g = gi gj for some .1 ≤ i, j ≤ n. The assumption .Qπ = Qρ and Proposition 1.8 (a) then imply that .χπ (g) = χρ (g) for all .g ∈ S. Consequently, by Proposition 1.8 (b) we have .

n ∑ .

i,j =1

zi zj tr π(gi gj ) =

n ∑

zi zj tr ρ(gi gj ).

i,j =1

There are two cases concerning the coefficients of .zi zj : (1) When .i = j , we have .tr π(gi2 ) = tr ρ(gi2 ). (2) When .i < j , we have .tr (π(gi gj ) + π(gj gi )) = tr (ρ(gi gj ) + ρ(gj gi )). The trace property .tr (xy) = tr (yx) then implies .χπ (gi gj ) = χρ (gi gj ), i, j = 1, . . . , n. For (b), in light of Proposition 1.5, we have tr (z1 π(g1 ) + · · · + zn π(gn ))3 = tr (z1 ρ(g1 ) + · · · + zn ρ(gn ))3 .

.

(2.4)

2.3 Abelian Groups

33

The coefficient of the term .z1α1 · · · znαn in the left-hand side of (2.4) is of the form tr π(W ) + tr

⎛ ∑

.

⎞ π(g) ,

(2.5)

sig(g)=α

∑ where W can either be 0 or a finite sum . i ci wi ∈ C[G] of some elements .wi ∈ G such that .l(wi ) < 3 for each i. To understand why such W may appear in (2.5), we suppose .g12 = 1. Then the coefficient of .z12 z2 in .(z1 π(g1 ) + z2 π(g2 ))3 is π(g12 g2 + g1 g2 g1 + g2 g12 ) = 2π(g2 ) + π(g1 g2 g1 ),

.

in which case .W = 2g2 . To proceed with the proof, we observe that part (a) indicates that .tr π(W ) = tr ρ(W ). Hence, the corollary follows from equation (2.4) and the decomposition (2.5) for .π and .ρ. ⨆ ⨅

2.3.2 The Characteristic Polynomial Determines an Abelian Group The commutativity of a finite group is completely determined by its characteristic polynomial in the following sense. Theorem 2.16 Let G be a finite group. Then G is abelian if and only if its characteristic polynomial .QG is a product of linear factors. Proof Let .{g1 , . . . , gn } be any generating set of G. If G is abelian, then it follows from Proposition 1.18 that its characteristic polynomial is a product of linear factors. On the other hand, since .λG (gj ), j = 1, . . . , n are unitary matrices, they are of course normal. If the characteristic polynomial .QG is a product of linear factors, then the matrices are commuting by Theorem 1.19. Since the regular representation .λG is faithful, the elements .g1 , . . . , gn are commuting, i.e., G is abelian. ⨆ ⨅ Furthermore, the characteristic polynomial of a finite abelian group also determines the group up to an isomorphism. Theorem 2.17 Let G and .G' be two finite groups with .QG = QG' . If G is abelian, then G and .G' are isomorphic. Proof Assume .{g1 , . . . , gn } and .{g1' , . . . , gn' ' } are generating sets for G and .G' , respectively. First of all, the condition .QG = QG' evidently shows that .n = n' . Theorem 2.16 implies that .G' is abelian. We let .λ and .λ' be the regular representations of G and resp. .G' . Since .λ(g1 ), . . . , λ(gn ) are commuting normal matrices, up to a change of orthonormal bases, they can be written in the diagonal forms

34

2 Finite Dimensional Group Representations

⎛

0 ··· λi2 · · · .. . ··· 0 0 ···

0 0 .. .

λi1 ⎜0 ⎜ .λ(gi ) = ⎜ . ⎝ ..

⎞ ⎟ ⎟ ⎟ , i = 1, . . . , n, ⎠

(2.6)

λik

where .k = |G|. Then QG (z) =

k Π

.

j =1

⎛ z0 +

n ∑

⎞ λij zi .

i=1

Since .QG' = QG , Proposition 1.5 indicates that .σ (λ(gi )) = σ (λ(gi' )), 1 ≤ i ≤ n. Moreover, the factorization of .QG shows that, up to a simultaneous permutation of rows and columns depending on the order of the factors in .QG' , each .λ(gi' ) is in the same diagonal form as .λ(gi ). This concludes that there is a unitary matrix U such that .λ(gi ) = U λ(gi' )U ∗ , 1 ≤ i ≤ n. Hence the linear groups .λ(G) and .λ(G' ) are isomorphic. The injectivity of the regular representation then implies that G and .G' are isomorphic with the isomorphism induced by the map .gi → gi' , 1 ≤ i ≤ n. ⨅ ⨆ Since irreducible representations of an abelian group are all one-dimensional, Question 2.9 (a) is trivial in this case. Theorem 2.18 Let .π and .ρ be two finite dimensional representations of a finite abelian group G. Then .Qπ = Qρ if and only if .π ∼ = ρ. Proof Let .{g1 , . . . , gn } be any generating set of G. The sufficiency is obvious because characteristic polynomial is an invariant for group representations. Since k G is finite, for each .1 ≤ j ≤ n there exists an integer .kj ≥ 1 such that .gj−1 = gj j . Therefore, there is no need to assume that the generating set S is symmetric. For the necessity, in view of Theorem 2.7 (3), we show by induction that χπ (g) = χρ (g), g ∈ G.

.

(2.7)

The method is similar to that in the proof of Proposition 2.14 (b). First, Lemma 2.15 indicates that (2.7) holds for .l(g) ≤ 2. Assume it holds for .l(g) ≤ m − 1, where α1 αn .m ≥ 3. Pick any .g = g 1 · · · gn ∈ G such that .|α| = m. By Proposition 1.5, we have tr (z1 π(g1 ) + · · · + zn π(gn ))m = tr (z1 ρ(g1 ) + · · · + zn ρ(gn ))m

.

(2.8)

for all complex numbers .z1 , . . . , zn . Since G is abelian, every elements .g ' ∈ G with .sig(g ' ) = α is equal to g. Hence the coefficient of the term .z1α1 · · · znαn in the left-hand side of (2.4) is of the form

2.3 Abelian Groups

35

⎛ tr π(W ) + tr

∑

.

'

⎞

π(g ) = χπ (W ) + cχπ (g),

(2.9)

sig(g ' )=α

∑ where W can either be 0 or a finite sum . i ci wi ∈ C[G] of some elements .wi ∈ G with .l(wi ) < m for each i, and c is a positive integer independent of representation .π . Likewise, we have the same expression with respect to the representation .ρ. The induction assumption and equation (2.8) then imply .χπ (W ) = χρ (W ) and consequently .χπ (g) = χρ (g). The proof is thus completed by induction. ⨆ ⨅ Looking beyond abelian groups, we may consider the semidirect product of two abelian groups. Given a group G, we let .Aut(G) denote the group of automorphisms on G. We say that a group K acts on G if there is a group homomorphism .τ : K → Aut(G). If G is nonabelian, then a nontrivial action of G on itself is the inner automorphism defined by .τ (x)(g) = xgx −1 , x, g ∈ G. Definition 2.19 Given two groups H and K and an action .τ : K → Aut(H ), the corresponding semidirect product .H xτ K is the set .H × K equipped with the multiplication .· defined by (h, k) · (h' , k ' ) = (hτ (k)(h' ), kk ' ), h ∈ H, k ∈ K.

.

Check that .{(h, 1) | h ∈ H } is a normal subgroup of .H xτ K. On the other hand, if N is a normal subgroup of G, then G is isomorphic to .N xτ G/N, where .τ is the action of .G/N on N by inner automorphism. The semidirect product of two abelian groups can be nonabelian. Example 2.20 The infinite dihedral group .D∞ contains the normal subgroup .H = . The subgroup .K = {1, t}, which is isomorphic to .Z2 , acts on H by the map m → t (at)m t = (ta)m = (at)−m . Then one verifies that .D = H x K. .τ (t) : (at) ∞ τ It is a tempting project to investigate whether the results above extend to the semidirect product of finite abelian groups. Exercise 2.21 1. Prove Proposition 2.14. 2. Assume .G = Z2 × Z3 . Can you detect the numbers 2 and 3 in .QG ? Also study the question for .G = Zp × Zq , where p and q are arbitrary prime numbers. 3. If N is a normal subgroup of G, show that G is isomorphic to .N xτ G/N, where .τ is the action of .G/N on N by inner automorphism. Project 2.22 1. Do Theorems 2.17 and 2.18 hold if G is a semidirect product of two finite abelian groups?

36

2 Finite Dimensional Group Representations

2.4 The Coxeter Groups A Coxeter group W is generated by a subset .S = {g1 , . . . , gn } that satisfies the following relations: (gi gj )mij = 1, mii = 1, 2 ≤ mij ≤ ∞ (i /= j ),

.

i, j = 1, . . . , n.

(2.10)

The pair .(W, S) is called the Coxeter system, and the .n × n matrix .M = (mij ) is called the Coxeter matrix. Since .(gi gj )−1 = gj gi , the matrix M is symmetric. A Coxeter group is also called a reflection group because each element .gj can be realized as the reflection through a hyperplane in .Rn . Thus, the composition .gi gj is a rotation of angle . m2πij . Finite Coxeter groups are classified by the so-called Coxeter diagram into several one-parameter families: .An , Bn , Dn , and .I (n) plus six exceptional types .E6 , E7 , E8 , F4 , H3 , and .H4 . They are also connected with the classification of finite dimensional simple Lie algebras, which we will deliberate in the next chapter. For more details, we refer the reader to Coxeter’s original paper [42], Humphrey [125, 126], and Fulton–Harris [81].

2.4.1 A Complete Invariant The characteristic polynomial and its zero set (called projective spectrum) for ˇ ckovi´c–Stessin–Tchernev [53] in which Coxeter groups are studied in detail in Cuˇ Question 2.9 is answered almost completely. The following example serves to elucidate the main idea used to prove Theorem 2.25. Example 2.23 The finite dihedral group .Dm = , where .1 ≤ m < ∞, is a special type of the Coxeter groups. For each angle .θ = 2πmk with .0 < 2k m < 1, a faithful and irreducible representation .ρθ of .Dm is defined by sending ⎤ ⎡ ⎤ 0 eiθ 01 , t → A = . 2 e−iθ 0 10

⎡ a → A1 =

.

One observes that .A1 A2 + A2 A1 = 2 cos(θ )I which lies in the center of ρθ (Dm ), and the order m is determined by its eigenvalue .2 cos(θ ). Therefore, the characteristic polynomial

.

Qρθ (z) = z02 − z12 − z22 − 2z1 z2 cos θ

.

determines the representation .ρθ .

2.4 The Coxeter Groups

37

To proceed with the discussion, we recall that the Chebyshev’s polynomials of the first kind are defined as .T0 (x) = 1, T1 (x) = x and Tm (x) = 2xTm−1 (x) − Tm−2 (x), m ≥ 2, x ∈ C.

.

(2.11)

It is well-known that .Tm (cos θ ) = cos(mθ ). Therefore, the function .1−Tm has roots cos(2π k/m), k = 0, . . . , m − 1. For more details on the Tchebyshev’s polynomials, we refer the reader to Rivlin [192], and Mason–Handscomb [159].

.

Lemma 2.24 If .A1 and .A2 are two unitary matrices satisfying .A21 = A22 = I , then the following are equivalent: (a) .(A1 A2 )m = I . (b) .σ (A1 A2 + A2 A1 ) ⊂ {2 cos(2π k/m) | k = 0, . . . , m − 1}. ⎛ ⎞ 1 m m and verify readily Proof For each .m ≥ 0, we set .Rm = 2 (A1 A2 ) + (A2 A1 ) that R0 = I,

.

R1 =

1 (A1 A2 + A2 A1 ), 2

Rm = 2R1 Rm−1 − Rm−2 , m ≥ 2. Observe that the recursion relation for .Rm is the same as that of the Tchebyshev’s polynomials .Tm . This implies .Rm = Tm (R1 ), m ≥ 0. One may also check this by induction. If .(A1 A2 )m = I , then since .A2 A1 = (A1 A2 )−1 , we also have .(A2 A1 )m = I and consequently .Tm (R1 ) = Rm = I . Therefore, if .α is an eigenvalue for .R1 then by the spectral mapping theorem .Tm (α) = 1 which implies that .α = cos(2π k/m) for some .0 ≤ k ≤ m − 1. On the other hand, if .σ (R1 ) ⊂ {cos(2π k/m) | k = 0, . . . , m−1}, then spectral mapping theorem implies .σ (Rm ) = σ (Tm (R1 )) = {1}. Since .A1 and .A2 are unitary matrices, we see that .Rm is self-adjoint. Thus we must have ⎞ ⎛ 1 m m = I. (A1 A2 ) + (A2 A1 ) .Rm = 2 The fact .A1 A2 = (A2 A1 )−1 then implies ⎛ ⎞2 m .0 = Rm − I = (A1 A2 ) − I . It follows that .(A1 A2 )m − I = 0 since it is normal.

⨆ ⨅

38

2 Finite Dimensional Group Representations

The following theorem shows that the characteristic polynomial of a finite Coxeter group W is a complete invariant. Theorem 2.25 Let W be a finite Coxeter group defined by (2.10). If .W ' is a finite group such that .QW ' = QW , then it is isomorphic to W . Proof Assume .W ' is generated by the set .{g1' , . . . , gn' } with respect to which .QW ' = QW . We show that .g1' , . . . gn' must satisfy (2.10). We first make the argument for .g1 and .g2 . For convenience, we set .Ai = λW (gi ) and .A'i = λW ' (gi' ), i = 1, 2, where .λ stands for the regular representation. The condition .QλW = QλW ' implies .

det(z0 + z1 A1 + z2 A2 ) = det(z0 + z1 A'1 + z2 A'2 ), z ∈ C3 .

Proposition 1.5 indicates that .(z1 A1 + z2 A2 )2 and .(z1 A'1 + z2 A'2 )2 have the same set of eigenvalues for each .(z1 , z2 ) ∈ C2 , in particular, we have .σ (A'i ) = σ (Ai ) = {±1}, i = 1, 2. The normality of .A'1 and .A'2 then gives .(A'i )2 = I, i = 1, 2. Furthermore, since (z1 A1 + z2 A2 )2 = z12 + z22 + z1 z2 (A1 A2 + A2 A1 ), z1 , z2 ∈ C,

.

we must have .σ (A1 A2 + A2 A1 ) = σ (A'1 A'2 + A'2 A'1 ). Lemma 2.24 then concludes that .(A'1 A'2 )m12 = I , and consequently .(g1' g2' )m12 = 1 because the regular representation is faithful. Applying the same argument for each pair .(gi' , gj' ), 1 ≤ i, j ≤ n, we see that .g1' , . . . , gn' satisfy (2.10), and hence the map .π : gi → gi' induces a surjective homomorphism from W onto .W ' . Moreover, since .|W | = deg QW = deg QW ' = |W ' |, the map .π must also be injective. The theorem follows. ⨆ ⨅ A careful review of the proof above indicates that the injectivity of the regular representation .λ and the fact .dλ = |W | have played an indispensable role. Regarding other representations of the Coxeter group, the following holds. Theorem 2.26 Let W be a finite Coxeter group of type .An , Bn , Dn , or .I (n). If .π and .ρ are two representations of .(W, S) such that .Qπ = Qρ , then .π is equivalent to .ρ. The proof in [53], which aims to establish the equality .χπ = χρ on W , is highly technical. It is desirable if a simpler proof can be found. Further study along this line can be found in Stessin [212].

2.4.2 The Tits Representation Other than the regular representation, a well-studied representation for a Coxeter group is the Tits representation (also called geometric representation or reflection representation). Let .(W, S) be a Coxeter system as in (2.10), and let .{e1 , . . . , en } be a basis for .Rn . The Tits representation of W is defined by

2.4 The Coxeter Groups

39

ρ(gi )(ej ) = ej + 2 cos

.

π ei , 1 ≤ i, j ≤ n. mij

(2.12)

An associated bilinear form .(·, ·) on .Rn is defined by (ei , ej ) = − cos

.

π . mij

One verifies that ( ) ρ(gk )(ei ), ρ(gk )(ej ) = (ei , ej ), 1 ≤ i, j, k ≤ n.

.

(2.13)

) ( It is known that .ρ is irreducible if and only if the .n × n matrix . (ei , ej ) is nondegenerate, in which case (2.13) implies that .ρ is a unitary representation on .Rn with respect to the bilinear form .(·, ·). For more details, we refer the reader to Bourbaki [25]. The characteristic polynomial .Qρ also determines the group W . The case .n = 3 provides a simple illustration of this fact. Example 2.27 Let W be a Coxeter group with Coxeter generators .g1 , g2 , g3 that satisfy (2.10), and let .{e1 , e2 , e3 } be the standard basis for .R3 . Then the Tits representation .ρ of W on .R3 is defined by ρ(gi )(ej ) = ej + 2 cos

.

π ei , 1 ≤ i, j ≤ 3. mij

Setting .α = 2 cos mπ12 , β = 2 cos mπ13 , and .γ = 2 cos mπ23 , we can write down the matrix representations ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ −1 α β 1 0 0 1 0 0 .ρ(g1 ) = ⎣ 0 1 0 ⎦ , ρ(g2 ) = ⎣α −1 γ ⎦ , ρ(g3 ) = ⎣ 0 1 0 ⎦ . 0 0 1 0 0 1 β γ −1 Then the characteristic polynomial Qρ (z) = det(z0 I + z1 ρ(g1 ) + z2 ρ(g2 ) + z3 ρ(g3 ))

.

=(z0 − z1 + z2 + z3 )(z0 + z1 − z2 + z3 )(z0 + z1 + z2 − z3 ) − α 2 z1 z2 (z0 + z1 + z2 − z3 ) − β 2 z1 z3 (z0 + z1 − z2 + z3 ) − γ 2 z2 z3 (z0 − z1 + z2 + z3 ) + 2αβγ z1 z2 z3 . If we write .Qρ = z03 + q1 (z' )z02 + q2 (z' )z0 + q3 (z' ), where .z' = (z1 , z2 , z3 ), then q1 = z1 + z2 + z3 ,

.

q2 = −(z12 + z22 + z32 ) + (2 − α 2 )z1 z2 + (2 − β 2 )z1 z3 + (2 − γ 2 )z2 z3 .

40

2 Finite Dimensional Group Representations

Observe that the powers .mij are uniquely determined by the nonnegative values α, β, and .γ (or equivalently by .q2 ). This unveils the following general fact.

.

Theorem 2.28 A Coxeter system .(W, S) is uniquely determined, up to isomorphism, by its characteristic polynomial with respect to the Tits representation. Proof Let .(W, S) be a Coxeter system as in (2.10) with its Tits representation .ρ defined by (2.12). For simplicity, we denote .cos mπij by .αij , 1 ≤ i, j ≤ n. Since the Coxeter matrix M is symmetric, so is the matrix .α = (αij ). Moreover, .αii = −1 for each i. The definition (2.12) shows that the matrix representation of .ρ(gi ) is simply the identity matrix with its ith row being replaced by the ith row of .α, namely ⎛

1 ⎜ .. ⎜ . ⎜ .ρ(gi ) = ⎜αi1 ⎜ ⎜ . ⎝ .. 0

0 ··· 0 .. .. . ··· . αi2 · · · αi(n−1) .. .. . ··· . 0 ··· 0

⎞ 0 .. ⎟ . ⎟ ⎟ αin ⎟ ⎟ , i = 1, . . . , n. .. ⎟ . ⎠ 1

Then by Proposition 1.8 (b), for .i < j the coefficient of .zi zj in .q2 (z' ) is .

) 1( tr ρ(gi )tr ρ(gj ) − tr (ρ(gi )ρ(gj )) 2 ⎞ 1⎛ (n − 2)2 − (n − 4 + αij αj i ) = 2 ⎞ 1⎛ 2 n − 5n + 8 − αij2 . = 2

' }) is another Coxeter system with Tits representation .ρ ' such If .(W ' , {g1' , . . . , gm that .Qρ ' = Qρ , then we must have: (1) .m = deg Qρ = n; and (2) .q2' = q2 , which implies .αij = αij' and consequently .mij = m'ij , 1 ≤ i, j ≤ n. It follows that the ⨆ ⨅ map .gi → gi' induces an isomorphism between W and .W ' .

Exercise 2.29 1. Consider the dihedral group .Dm = , where .m ≥ 2, and let .π and .ρ be two unitary representations of .Dm (not necessarily irreducible). Show that if .Qπ = Qρ , then .π and .ρ are equivalent. 2. Regarding Example 2.27, determine whether .Qρ is irreducible. Project 2.30 1. To generalize exercise 2 above, consider a general finite Coxeter group W and let .ρ be the Tits representation. (a) Is .Qρ necessarily irreducible? (b) If the answer is no, then how is the factorization of .Qρ related to the Coxeter matrix?

Chapter 3

Finite Dimensional Lie Algebras

Another fertile field to explore the link between algebraic relations of the matrices and the properties of their characteristic polynomial is where the matrices form a basis for a finite dimensional complex Lie algebra .L. The study in this case is particularly useful because of the following two reasons. 1. Finite dimensional complex simple Lie algebras have been fully classified. This shall enable us to see how the characteristic polynomials are linked with the algebraic invariants. 2. On the other hand, a full classification of solvable Lie algebras remains a challenge, even in low dimensions such as 8 or 9. This offers an opportunity to explore the potential of characteristic polynomial. This chapter collects some recent ideas and results in this exploration. It contains a discussion about the automorphism group .Aut(L), irreducible representations of .sl2 , semidirect sum, a new perspective on the theorems of Killing and Cartan, as well as new spectral invariants of solvable Lie algebras.

3.1 Characteristic Polynomial for Lie Algebras Given a Lie algebra .L, its derived algebra .[L, L] is the linear span of all such elements .[s, t], s, t ∈ L. Clearly, the derived algebra is an ideal of .L and it is equal to .{0} if and only if .L is abelian. A nonabelian .L is said to be simple if it contains no proper ideal, and it is said to be semisimple if it is a direct sum of simple Lie algebras. In particular, we must have .L = [L, L] for simple or semisimple Lie algebras. When .[L, L] is a proper ideal of .L, we set .L0 = L = L0 and define Lk = [Lk−1 , Lk−1 ],

.

and Lk = [L, Lk−1 ], k = 1, 2, . . .

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 R. Yang, A Spectral Theory Of Noncommuting Operators, https://doi.org/10.1007/978-3-031-51605-4_3

41

42

3 Finite Dimensional Lie Algebras

The algebra .L is said to be solvable if .Lk = {0} for some .k ≥ 1, and it is said to be nilpotent if .Lk = {0} for some .k ≥ 1. The smallest number k such that .Lk = 0 is called the nilindex of .L. Since .Lk ⊂ Lk for each k, a nilpotent Lie algebra is necessarily solvable. The largest solvable (resp. nilpotent) ideal of .L is called the radical (resp. nilradical) of .L and is denoted by .rad(L) (resp. .nil(L)). Clearly, the quotient .L/rad(L) is semisimple. The space .gl(V ) of all linear endomorphisms of a vector space V has a natural Lie algebra structure with the bracket .[a, b] := ab − ba, a, b ∈ gl(V ). A representation of Lie algebra .L on a vector space V is a Lie algebra homomorphism .ρ : L → gl(V ), meaning that .ρ is a linear map from .L into .gl(V ) such that ρ([x, y]) = [ρ(x), ρ(y)] = ρ(x)ρ(y) − ρ(y)ρ(x), x, y ∈ L.

.

Two representations .ρ : L → gl(V ) and .ρ , : L → gl(V , ) are said to be equivalent if there exists an invertible linear map .U : V → V , such that Uρ(x)U −1 = ρ , (x), x ∈ L.

.

A representation .ρ is said to be irreducible if the linear transformations .ρ(x), x ∈ L, do not have a nontrivial common invariant subspace in V . Irreducible representations of a complex semisimple Lie algebra are classified in the theorem of the highest weight due to Élie Cartan. The most natural representation of a Lie algebra is the adjoint representation .ad : L → gl(L) defined by .ad x(y) = [x, y], y ∈ L. It is worth noting here that the adjoint representation of Lie algebras and the regular representation of groups are intrinsic to the algebraic structure of Lie algebras and resp. groups. Thus they hold more information than other representations. In this chapter, we shall frequently use the Einstein summation convention. Suppose .L has a basis .{x1 , . . . , xn }. Then for .1 ≤ i, j ≤ n, we can write .[xi , xj ] = Tijk xk . The numbers .Tijk , 1 ≤ i, j, k ≤ n, are called the structure constants of .L, as they completely determine the structure of .L. In particular, the matrix representation of .ad xi is given by the matrix .Ti := (Tijk )1≤k,j ≤n , whose rows and columns are indexed by k and j , respectively. For instance, if .L = span{x1 , x2 , x3 }, then 1 2 3 [x1 , x1 ] = T11 x1 + T11 x2 + T11 x3

.

1 2 3 [x1 , x2 ] = T12 x1 + T12 x2 + T12 x3 1 2 3 [x1 , x3 ] = T13 x1 + T13 x2 + T13 x3 ,

and ⎛

1 T1 T1⎞ T11 ⎛ ⎞ 12 13 2 T2 T2⎠ = Tk . .T1 = ⎝T 1j 11 12 13 3 T3 T3 T11 12 13

3.1 Characteristic Polynomial for Lie Algebras

43

For simplicity, we shall not distinguish .ad x from its matrix representation whenever a basis of .L is fixed. Engel’s theorem asserts that a finite dimensional Lie algebra .L is nilpotent if and only if for every element .x ∈ L the matrix .ad x is nilpotent, i.e., the spectrum .σ (ad x) = {0}. It follows that the nilradical .nil(L) is the set of all elements .x ∈ L whose adjoint representation .ad x is nilpotent. Definition 3.1 Given a complex Lie algebra .L with basis .{x1 , . . . , xn }, if .π is a finite dimensional representation of .L, then the characteristic polynomial of .L with respect to .π is defined as .Qπ (z) = det (z0 I + z1 π(x1 ) + · · · + zn π(xn )) . In the case .π = ad, we will write .Qad as .QL and call it the characteristic polynomial of .L. The following fact is easy to check. Proposition 3.2 Given a Lie algebra with a fixed basis, if its representations .π and π , are equivalent, then .Qπ = Qπ , .

.

3.1.1 Invariance Under the Automorphism Group Recall that .z, stands for the row vector .(z1 , . . . , zn ). The general linear group .GLn acts on the polynomial ring .C[z, ] by the map p(z, ) −→ p(z, B), p ∈ C[z, ], B ∈ GLn .

.

Polynomials that are invariant under this action are constants. However, if G is a subgroup of .GLn , then there could be nontrivial invariant polynomials under the action by G. For example, if G is the group of permutation matrices, then the invariant polynomials are symmetric polynomials. It is an interesting and wellstudied problem whether a given subgroup .G ⊂ GLn has nontrivial invariant polynomials. This is the subject of invariant theory. For more information, we refer the reader to Dieudonné [60], Dolgachev [67], and Sturmfels [214]. The action of .GLn above extends canonically to polynomials in .Cn+1 such that B ◦ Q(z0 , z, ) := Q(z0 , z, B), Q ∈ C[z], B ∈ GLn .

.

If Q is expanded as in (1.8), then B ◦ Q(z) = z0n + q1 (z, B)z0n−1 + · · · + qn−1 (z, B)z0 + qn (z, B).

.

(3.1)

Therefore, a polynomial .Q ∈ C[z] is invariant for a subgroup .G ⊂ GLn if and only if every .qj is invariant. If .{x1 , . . . , xn } and .{xˆ1 , . . . , xˆn } are two bases of Lie algebra .L, and .xˆi = bij xj , 1 ≤ i ≤ n, then ˆ L := det(z0 + zi ad xˆi ) = det(z0 + zi bij ad xj ) = B ◦ QL . Q

.

(3.2)

44

3 Finite Dimensional Lie Algebras

In other words, the characteristic polynomials of .L with respect to two different bases differ by an action of some .B ∈ GLn via change of variables. Lemma 3.3 If Lie algebras .L1 and .L2 are isomorphic, then up to a change of variables we have .QL1 (z) = QL2 (z). Proof Suppose .L1 = span{x1 , . . . , xn } and assume .φ : L1 → L2 is an isomorphism. Then .{xˆi = φ(xi ) | 1 ≤ i ≤ n} is a basis of .L2 , and we have Tˆijk xˆk = [xˆi , xˆj ] = φ([xi , xj ]) = Tijk xˆk ,

.

which implies .Tj = Tˆj , 1 ≤ j ≤ n, and therefore ) ( ) ( QL1 (z) = det z0 I + zj Tˆj = det z0 I + zj Tj = QL2 (z).

.

The lemma then follows from the preceding observation.

u n

Let .Aut(L) denote the group of automorphisms of a Lie algebra .L. Then with respect to a fixed basis .{x1 , . . . , xn }, every .φ ∈ Aut(L) has a matrix representation .Bφ = (bij ) ∈ GLn defined by .φ(xi ) = bij xj , 1 ≤ i ≤ n. Thus we can identify .Aut(L) as a subgroup of .GLn . Lemma 3.3 then reveals the following fact [7]. Theorem 3.4 The characteristic polynomial .QL is invariant for .Aut(L). If we expand .QL as .z0n + q1 (z, )z0n−1 + · · · + qn−1 (z, )z0 + qn (z, ). Then the following fact is immediate. Corollary 3.5 The polynomials .qj (z, ), j = 1, . . . , n are invariant for .Aut(L). The characteristic polynomial .QL thus gives us more insight about the group Aut(L). We use an example to show what this means. Recall that the orthogonal group of .Cn is .On = {B ∈ GLn | BB T = I }. The special orthogonal group .SOn is the subgroup of elements B in .On such that .det B = 1.

.

Example 3.6 The characteristic polynomial )of .su2 is calculated in Example 1.1.1, ( namely, .Qsu2 (z) = z0 z02 − 4(z12 + z22 + z32 ) . Since it is invariant for .Aut(su2 ) due to Theorem 3.4, so is the quadratic form .z12 +z22 +z32 . It implies that .Aut(su2 ) ⊂ O3 . We leave it as an exercise to prove that .Aut(su2 ) is in fact isomorphic to .SO3 . The following related fact is interesting. Proposition 3.7 If a n-dimensional Lie algebra .L is such that .tr ad x /= 0 for some x ∈ L, then the subgroup .Aut(L) ⊂ GLn has an invariant vector.

.

Proof Assume .L = span{x1 , . . . , xn }. The assumption .tr ad x /= 0 for some .x ∈ L implies that .tr ad xk /= 0 for some .1 ≤ k ≤ n. Suppose .B = (bij ) ∈ GLn , and .xˆi = bij xj . As before, we denote .ad xj by .Tj and .ad xˆj by .Tˆj . The structure constants with respect to the two bases are given by the equations

3.1 Characteristic Polynomial for Lie Algebras

45

[xi , xj ] = Tijk xk , [xˆi , xˆj ] = Tˆijk xˆk ,

.

(3.3)

respectively. Then one verifies that k [xˆi , xˆj ] = [biα xα , bjβ xβ ] = biα bjβ [xα , xβ ] = biα bjβ Tαβ xk .

.

Hence by (3.3), k biα bjβ Tαβ xk = Tˆijk xˆk = Tˆijk bkl xl .

.

l = Tˆ k b , which can be written Equating the coefficients of .xl , we have .biα bjβ Tαβ ij kl in matrix form as

biα Tα B T = B T Tˆi , 1 ≤ i ≤ n.

.

(3.4)

Rewriting it as .biα Tα = B T Tˆi (B T )−1 and applying the trace to both sides, we obtain biα tr Tα = tr (biα Tα ) = tr (B T Tˆi (B T )−1 ) = tr Tˆi , 1 ≤ i ≤ n,

.

or expressed in matrix form, ⎞ ⎛ ˆ ⎞ tr T 1 tr T1 ⎜tr T2 ⎟ ⎜tr Tˆ 2 ⎟ ⎟ ⎜ ⎟ ⎜ .B ⎜ . ⎟ = ⎜ . ⎟ . ⎝ .. ⎠ ⎝ .. ⎠ ⎛

tr Tn

tr Tˆ n

In the case .φ ∈ Aut(L) and .B = Bφ , we have .Ti = Tˆi for each i. Thus T .(tr T1 , . . . , tr Tn ) is a common eigenvector of .Bφ corresponding to the eigenvalue 1. u n However, since .ad[x, y] = [ad x, ad y], we have .tr ad[x, y] = 0, x, y ∈ L. If .L is simple, then .L = [L, L], and hence .ad x is a sum of commutators which implies .tr ad x = 0. Furthermore, if .L is nilpotent, then .ad x is a nilpotent matrix, in which case we also have .tr ad x = 0. Thus, the condition in Proposition 3.7 is met only for non-nilpotent solvable Lie algebras. The Killing form .K(·, ·) on a finite dimensional Lie algebra is a symmetric bilinear form defined by K(x, y) = tr (ad x ad y), x, y ∈ L.

.

It is said to be degenerate if there exists a nonzero .x ∈ L such that .K(x, y) = 0 for all .y ∈ L. The collection S of such .x ∈ L is called the radical of the Killing form. Be aware that S is in general different from .rad(L). The following fact is known as Cartan’s criterion.

46

3 Finite Dimensional Lie Algebras

Theorem 3.8 A finite dimensional Lie algebra .L is semisimple if and only if its Killing form is nondegenerate. Associated( with a fixed )n basis .{x1 , . . . , xn } for .L, there is a symmetric Killing matrix .K = K(xi , xj ) i,j =1 . Clearly, the Killing form is degenerate if and only if the matrix K is degenerate. The Killing matrix is closely connected with .q1 and .q2 in the characteristic polynomial .QL . Indeed, replacing .Aj by .ad xj in Proposition 1.8, we obtain the following equation q2 (z, ) =

.

⎞ 1⎛ T (q1 (z, ))2 − z, Kz, . 2

(3.5)

The following fact is a direct consequence of Cartan’s criterion. Its proof is left as an exercise. Corollary 3.9 A n-dimensional Lie algebra .L is semisimple if and only if .q1 = 0 and .q2 (z, ) is an irreducible quadric in n variables.

3.1.2 The Irreducible Representations of sl2 The Lie algebra .sl2 has a standard basis .{h, x, y} with brackets [h, x] = 2x, [h, y] = −2y, [x, y] = h.

.

It plays a particularly important role in Lie theory because its structure sheds light on that of a general simple Lie algebra, leading to a classification of them by Killing and Cartan [81]. In addition, every nonabelian simple Lie algebra contains a subalgebra that is isomorphic to .sl2 and hence is a .sl2 -module (see Theorem 3.2.1). The irreducible representations of .sl2 are well-understood. Theorem 3.10 Let .π be an irreducible representation of .sl2 on a complex vector space V of dimension .m + 1. Then there is an orthonormal basis .{v0 , . . . , vm } for V such that (a) .π(h)vi = (m − 2i)vi . (b) .π(y)vi = (i + 1)vi+1 . (c) .π(x)vi = (m − i + 1)vi−1 , where .0 ≤ i ≤ m and .v−1 , vm+1 are assumed to be .{0}. Therefore, up to equivalence there is a unique irreducible representation of .sl2 for every .m ≥ 1. The eigenvalues and the corresponding eigenspaces of .π(h) are called the weights and weight spaces of .π , respectively. The operators .π(x) and .π(y) are truncated backward and resp. forward weighted shifts. The characteristic polynomial with respect to .π is

3.1 Characteristic Polynomial for Lie Algebras

47

Qπ (z) = det(z0 + z1 ad h + z1 ad x + z3 ad y).

.

It is important to understand how the properties of .π are reflected by .Qπ (z). This knowledge will shed light on the connection for other Lie algebras. First of all, the matrix representations for .ad h, ad x, and .ad y are ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 00 0 0 01 0 −1 0 .Th = ⎝0 2 0 ⎠ , Tx = ⎝−2 0 0⎠ , Ty = ⎝0 0 0⎠ , 0 0 −2 0 00 2 0 0 respectively. Therefore, the characteristic polynomial of .sl2 is ⎛ ⎞ Qsl2 (z) = z0 z02 − 4(z12 + z2 z3 ) .

(3.6)

.

It follows from Theorem 3.4 and Corollary 3.5 that the quadric .z12 + z2 z3 is invariant for .Aut(sl2 ). If .π : sl2 → gl(m + 1) is an irreducible representation then so is .π ◦ φ for every .φ ∈ Aut(sl2 ). Since .m+1 dimensional irreducible representation is unique up to equivalence, .π and .π ◦ φ must be equivalent. Proposition 3.2 then implies Qπ (z) = Qπ ◦φ (z) = Bφ ◦ Qπ (z).

.

In other words, .Qπ is invariant for .Aut(sl2 ). Since every representation is direct sum of irreducible ones, this observation yields the following fact. Corollary 3.11 For every representation .π of .sl2 , the characteristic polynomial Qπ (z) is invariant for .Aut(sl2 ).

.

Indeed, we have the following fact due to Hu-Zhang [135] and Chen-Chen-Ding [30]. The proof here is a simplified version. Theorem 3.12 Let .π be an irreducible representation of .sl2 on a complex vector space V of dimension .m + 1. Then

Qπ (z) =

.

⎧(m−1)/2 ⎛ ⎞ ⎪ ⎪ || 2 2 2 ⎪ z − (m − 2j ) (z + z z ) , ⎪ 2 3 0 1 ⎪ ⎪ ⎨ j =0

m odd;

⎪ (m/2)−1 ⎞ ⎪ || ⎛ ⎪ ⎪ ⎪ z z02 − (m − 2j )2 (z12 + z2 z3 ) , m even. ⎪ 0 ⎩ j =0

Proof A standard realization of .sl2 is given by ( ⎞ ( ⎞ ( ⎞ 1 0 01 00 .h = , x= , y= . 0 −1 00 10

(3.7)

48

3 Finite Dimensional Lie Algebras

But it is not unique. In fact, if .h, ∈ GL2 is any matrix such that .det h, = −1 and , , .tr h = 0, then its eigenvalues are .±1. Hence there exists .S ∈ GL2 such that .h = −1 , −1 , −1 , , , ShS . If we let .x = SxS and .y = SyS , then .{h , x , y } is another realization of .sl2 , and the map .φ : (h, , x , , y , ) → (h, x, y) extends to be an automorphism of .sl2 . Since .π ◦ φ is also a representation of dimension .m + 1, the uniqueness property indicates that .π ◦ φ is equivalent to .π . In particular, .π(h, ) has eigenvalues .m − 2i, i = 0, . . . , m. Pick any open domain .o ⊂ C3 on which .z12 + z2 z3 /= 0 and set h, (z, ) =

.

z 1 h + z2 x + z3 y / , z, = (z1 , z2 , z3 ) ∈ o. 2 z 1 + z2 z 3

Then direct calculation verifies that .det h, = −1 and .tr h, = 0 on /.o. Using the prop-

erty of the classical characteristic polynomial for the matrix . z12 + z2 z3 π(h, (z, )) we have Qπ (z) = det(z0 I + z1 π(h) + z2 π(x) + z3 π(y)) ( ⎞ / , , 2 = det z0 I + z1 + z2 z3 π(h (z ))

.

=

⎞ m ( / || z0 + (m − 2j ) z12 + z2 z3 , j =0

which establishes the theorem for .z, ∈ o after multiplying the pairs / / z0 + (m − 2i) z12 + z2 z3 , and z0 + (m − 2j ) z12 + z2 z3

.

with .m − 2i = −(m − 2j ) /= 0, .0 ≤ i, j ≤ m. The analyticity of .Qπ implies that the theorem holds on .C4 . u n

3.1.3 Semidirect Sum This subsection considers the characteristic polynomial of a semidirect sum of two Lie algebras. Quite a few definitions are needed to proceed. Definition 3.13 A derivation D of a Lie algebra .L is a linear map .D : L → L satisfying the Leibnitz law with respect to the Lie bracket: D([x, y]) = [D(x), y] + [x, D(y)], x, y ∈ L.

.

3.1 Characteristic Polynomial for Lie Algebras

49

The set of all derivations of .L, denoted by .Der(L), is a vector space which can be given a Lie algebra structure. A derivation D is said to be inner if there exists an element .x ∈ L such that .D = ad x.The set .I nn(L) of inner derivations is an ideal of .Der(L). Definition 3.14 Let .L1 and .L2 be Lie algebras over the same field .F and .τ : L1 → Der(L2 ) be a fixed homomorphism. The semidirect sum .L1 +τ L2 is the direct sum of their underlying vector spaces .L1 ⊕ L2 with the Lie bracket defined as [(x1 , y1 ), (x2 , y2 )] = ([x1 , x2 ], τ (x1 )(y2 ) − τ (x2 )(y1 ) + [y1 , y2 ]) .

.

Evidently, the semidirect sum .L1 +τ L2 varies with the choice of .τ . In general .L1 is a subalgebra and .L2 is an ideal in .L1 +τ L2 . In the special case when .τ = 0, the semidirect sum turns into the direct sum .L1 ⊕ L2 . The following decomposition theorem is due to Levi [200]. Theorem 3.15 Every Lie algebra .L can be decomposed as .s +τ rad(L), where .s is semisimple. Suppose .L = L, +τ L,, is a semidirect sum of two Lie algebras. Assume , ,, .dim L = k > 0, .dim L = n − k ≥ 0, and let .{x1 , . . . , xk } and .{xk+1 , . . . , xn } be , ,, bases for .L and .L , respectively. To avoid confusion, we let .Ti denote the adjoint representation of .xi on .L, +τ L,, , 1 ≤ i ≤ n. Then we have the block matrices ⎞ ⎞ ( ( ad xi 0 0 0 , k + 1 ≤ j ≤ n, , 1 ≤ i ≤ k; Tj = .Ti = 0 τ (xi ) Xj ad xj where .τ (xi ) ∈ Der(L,, ), and .Xj is the adjoint representation of .xj on .L, , k + 1 ≤ j ≤ n. Hence (

) E z0 Ik + kj =1 zj ad xj 0 E E En .z0 I +zj Tj = , z0 In−k + nj=k+1 zj ad xj + ki=1 zi τ (xi ) j =k+1 zj Xj which implies ( ⎞ n k E E QL (z) = QL, (z) det z0 In−k + zj ad xj + zi τ (xi ) .

.

j =k+1

i=1

Remark 3.16 If .L,, is abelian, then .ad xj = 0 for .k + 1 ≤ j ≤ n, and hence ( ⎞ k E QL (z) = QL, (z) det z0 I + zi τ (xi ) ,

.

i=1

(3.8)

50

3 Finite Dimensional Lie Algebras

where the second factor is the characteristic polynomial of the matrices τ (x1 ), . . . , τ (xk ). On the other hand, if .τ = 0 then .L = L, ⊕ L,, and equation (3.8) becomes .QL (z) = QL, (z)QL,, (z).

.

Exercise 3.17 1. In a quantum system over .R, the position operator .xˆ and the momentum operator d , where .h¯ is the reduced Plank constant, satisfy the commutation .p ˆ = −i h¯ dx relation [x, ˆ p] ˆ = i hI, [x, ˆ I ] = 0 = [p, ˆ I ]. ¯

.

(a) Verify that the Lie algebra .h = span{x, ˆ p, ˆ I }, called the Heisenberg algebra, is nilpotent and find its nilindex. (b) Determine its automorphism group .Aut(h). 2. In a quantum system over .R3 , the angular momentum operators .Lx , Ly , and .Lz satisfy the commutation relation: [Lx , Ly ] = i hL ¯ z , [Ly , Lz ] = i hL ¯ x , [Lz , Lx ] = i hL ¯ y.

.

(a) Show that the Lie algebra .a = span{Lx , Ly , Lz } is simple. (b) Compute the characteristic polynomial .Qa . 3. Use an example to show that the converse of Proposition 3.2 is not true. 4. Show that the characteristic polynomial of a finite dimensional Lie algebra has the factor .z0 . (Hint: Show that .z1 ad(x1 )+· · ·+zn ad(xn ) has a nontrivial kernel.) ˆ are two Lie algebras such that .tr ad x = 0 for all .x ∈ L but 5. Suppose .L and .L ˆ are not isomorphic. ˆ Prove that .L and .L .tr ad x ˆ /= 0 for some .xˆ ∈ L. 6. For any Lie algebra .L, suppose .φ ∈ Aut L and .π is a representation of .L. Prove .Qπ ◦φ (z) = Bφ ◦ Qπ (z). 7. Complete Example 3.6 by showing that .Aut(su2 ) is isomorphic to .SO3 . 8. Prove (3.5) and Corollary 3.9. (Hint: Consider Exercise 1.34 1.) 9. The invariance of .z12 + z2 z3 under .Aut(sl2 ) also reveals more details about the latter. Prove that the group of unitary matrices .Bφ , where .φ ∈ Aut (sl2 ), is ⎧⎛ ⎨ 10 ⎝0 α . ⎩ 00

⎫ ⎞ ⎞ ⎛ 0 −1 0 0 || ⎬ 0 ⎠ , ⎝ 0 0 β ⎠ || α, β ∈ T . ⎭ α 0 β 0

Project 3.18 1. Compute the characteristic polynomials for finite dimensional simple Lie algebras and investigate their factorizations.

3.2 Simple Lie Algebras

51

3.2 Simple Lie Algebras Given a simple Lie algebra .L of dimension n, an element .x ∈ L is said to be semisimple if .ad x is diagonalizable. The Cartan subalgebra .h ⊂ L is a maximal abelian subalgebra consisting of semisimple elements. Thus, the matrices .ad x, x ∈ h, can be diagonalized simultaneously, meaning that they possess a set of common eigenvectors which form a basis for .L. Each common eigenvector y gives rise to a linear functional .y ∗ on .h defined by .ad x(y) = y ∗ (x)y, x ∈ h. Observe that the map .y → y ∗ is not injective. In fact, .y ∗ = 0 for every .y ∈ h due to the fact that .h is abelian. Further, .(cy)∗ = y ∗ for every nonzero scalar .c ∈ C. The set R of such linear functionals .y ∗ , excluding 0, is called the root system of .L, and the elements of R are called the roots of .L. Given a root .α ∈ R, we set Lα = {y ∈ L | ad x(y) = α(x)y, ∀x ∈ h},

.

and define .hα = [Lα , L−α ]. Interestingly, there exists a real-valued nondegenerate symmetric bilinear form .(·, ·) on .h∗ such that the values . := 2(β, α)/(α, α), α, β ∈ R, are all integers. Moreover, we have the following facts. Theorem 3.19 Let .L be a simple Lie algebra with root system R and .α ∈ R. (a) .Lα and .hα are both one-dimensional. (b) .hα ⊂ h, and there exists a unique element .Hα ∈ hα such that .α(Hα ) = 2. (c) For each .Xα ∈ Lα , there exists a unique element .Yα ∈ L−α such that [Xα , Yα ] = Hα ,

[Hα , Xα ] = 2Xα ,

.

[Hα , Yα ] = −2Yα ,

i.e., the subalgebra .hα ⊕ Lα ⊕ L−α is isomorphic to .sl2 . (d) If .β ∈ R and .α + β /= 0, then .α + β ∈ R and .[Lα , Lβ ] = Lα+β . (e) For .β ∈ R, we have .[Hα , Xβ ] = Xβ , and .[Hα , Yβ ] = −Yβ . Among other things, this theorem leads to the Cartan decomposition ( L=h⊕

.

⎞ ⊕α∈R Lα .

(3.9)

For proofs of these facts, we refer the reader to Humphrey [125] and Serre [200]. The general linear Lie algebra .glk contains the following classical subalgebras: slk = {x ∈ glk | tr x = 0},

.

sok = {x ∈ glk | x + x T = 0}, spk = {x ∈ glk | Jm x + x T Jm = 0}, k = 2m,

52

3 Finite Dimensional Lie Algebras

(

⎞ 0 I where .Jm stands for the block matrix . . Among them, −I 0 Ak := slk+1 , Bk := so2k+1 , Ck = sp2k , Dk := so2k , k ≥ 1

.

are simple, and they are called the classical simple Lie algebras. Do there exist other types of finite dimensional simple Lie algebras? We will address this question momentarily.

3.2.1 Root System The classification of simple Lie algebras is achieved through the classification of their root systems. Let V be a k-dimensional Euclidean vector space with inner product denoted by .(·, ·). Then every nonzero vector .α ∈ V gives rise to a reflection .σα defined by σα (β) = β − 2

.

(β, α) α, β ∈ V . (α, α)

Clearly, the reflecting hyperplane is .Pα := {β ∈ V | (β, α) = 0}. Definition 3.20 A subset R of a real Euclidean vector space V is called a root system if it satisfies the following conditions. (1) (2) (3) (4)

R is finite, spans V , and does not contain 0. If .α ∈ R, then the only multiples of .α in R are .±α. For every pair of roots .α, β ∈ R, we have .σα (β) ∈ R. If .α, β ∈ R, then . = 2(β, α)/(α, α) is an integer.

The dimension of V is called the rank of R. One observes that condition (4) implies that = 4 cos2 θ

.

(3.10)

||β|| is an integer, where .θ is the angle between .α and .β. Moreover, we have . = ||α||2 . Since . and . have the same sign, assuming .α /= ±β and .||β|| ≥ ||α||, we have the following table of possible values. A subset .A ⊂ R is called a base if it is a basis of V , and every root .β ∈ R can be written as a linear combination E .β = cα α 2

α∈A

3.2 Simple Lie Algebras Table 3.1 Values of and 2 2 .||β|| /||α|| ., , θ,

53

.

.

0 1 .−1 1 .−1 1 .−1

0 1 .−1 2 .−2 3 .−3

||β||2 ||α||2

.θ

.

.π/2

undetermined 1 1 2 2 3 3

.π/3 .2π/3 .π/4 .3π/4 .π/6 .5π/6

with either all nonnegative or all nonpositive integer coefficients .cα . It can be shown that every root system R has a base .A. Roots in .A are called simple roots. We fix an order of the simple roots .A = {α1 , . . . , αk }. The number k is the rank of R. If a root .β = ci αi is such that .ci ≥ 0 (.ci ≤ 0) for every .1 ≤ i ≤ k, then we say .β is positive (resp. negative) and write .β > 0 (resp. .β ≺ 0). This gives rise to a partial order on R such that .α ≺ β whenever .β − α > 0. We let .R + (resp. .R − ) denote the set of positive (resp. negative) roots in R. The Coxeter group W generated by the reflections .σα , α ∈ A, is called the Weyl group of R. The third column of Table 3.1 indicates that the angles between two roots .αi and .αj are rational multiples of .π . Thus the product .σαi σαj is a rotation of finite order, indicating that the Weyl group W is a finite Coxeter group. Apparently, it is a subgroup of the orthogonal group .Ok , and its Tits representation is just the identity map. Therefore, in light of Theorem 2.28, the characteristic polynomial .Qσ of the reflections .σαi , i = 1, . . . , k, determines the Weyl group up to isomorphism. However, inequivalent root systems may have isomorphic Weyl groups. It is thus a tempting question whether .Qσ also determines a root system (Project 1). A root system R is said to be irreducible if it cannot be written as a disjoint union .R = R1 ∪ R2 such that .R1 ⊥ R2 . The following are two well-known properties of a root system. Lemma 3.21 Let R be an irreducible root system with a fixed base .A = {α1 , . . . , αk }. (a) With respect to the partial order .≺ there exists a unique maximal root .β. Further, if .β = ci αi (in Einstein summation convention), then .ci > 0 for each .1 ≤ i ≤ k. (b) At most two root lengths are possible in R, and all roots of the same length are conjugate under group W . Note that (b) follows easily from the fourth column of Table 3.1, since the presence of three or more different root lengths would imply a ratio .3/2 in this column. If R has two distinct root lengths, then (b) divides R into two subsets, the set of long roots and that of the short roots. If R has only one root length, then every root in R is said to be long. It can be shown that the maximal root in (a) is long.

54

3 Finite Dimensional Lie Algebras

3.2.2 Dynkin Diagram Given a ) base .A = {α1 , . . . , αk } of a root system R, the .k × k matrix ( is called the Cartan matrix of R. By equation (3.10), the product .γij := is equal to one of the values .0, 1, 2, and 3. The Dynkin diagram of R is a graph with k vertices such that the ith and the j th vertices (corresponding to .αi and .αj , respectively) are joined by .γij edges. Moreover, if .αi and .αj are of different length, then an arrow is drawn pointing to the shorter root. For example, the diagram indicates: .k = 3, .γ12 = 1, .γ13 = 0, .γ23 = 2, and .||α1 || = ||α2 || > ||α3 ||. Two root systems .R ⊂ V and .R , ⊂ V , are said to be isomorphic if there is a vector space isomorphism .φ : V → V , such that .φ(R) = R , and .

= , α, β ∈ R.

.

Theorem 3.22 Up to an isomorphism, every irreducible root system of rank k has one of the Dynkin diagrams in Table 3.2 below. The root systems with Dynkin diagrams .G2 , F4 , E6 , E7 , or .E8 are said to be exceptional. Remarkably, we have the following theorem. Theorem 3.23 Let .L be any finite dimensional complex simple Lie algebra. (a) The root system R of .L is an irreducible root system whose Dynkin diagram is described in Theorem 3.22. (b) Isomorphic Lie algebras have isomorphic root systems. (c) Every Dynkin diagram in Table 3.2 corresponds to the root system R of a unique simple Lie algebra up to isomorphism. In other words, finite dimensional simple Lie algebras are completely characterized by the Dynkin diagram of their root systems. The simple Lie algebras with exceptional Dynkin diagrams are called the exceptional Lie algebras, and they are of dimensions .14, 52, 78, 133, and 248, respectively [200]. The classification Table 3.2 The Dynkin diagrams of root systems

(k ≥ 1): (k ≥ 2): .Ck (k ≥ 3): .Dk (k ≥ 4): .Ak

.Bk

: : .E6 :

.G2 .F4

.E7

:

.E8

:

3.2 Simple Lie Algebras

55

of complex finite dimensional simple Lie algebras is one of the most outstanding achievements in the 19th century mathematics. Example 3.24 The Lie algebra .G2 has Dynkin diagram√ which can be realized 3 3 by the root system with simple roots .A = {α1 = (− 2 , 2 ), α2 = (1, 0)} ⊂ R2 and positive roots R + = {α1 , α2 , α1 + α2 , α1 + 2α2 , α1 + 3α2 , 2α1 + 3α2 }.

.

The angle between .α1 and .α2 is . 5π 6 , and the composition .σα1 σα2 is a rotation of angle .− π3 . Thus, the Weyl group of .G2 is isomorphic to the dihedral group .D6 . Moreover, the algebra is 14-dimensional with basis .{h1 , h2 , xi , yi | i = 1, . . . , 6} satisfying: [xi , yi ] = hi

.

[hi , xi ] = 2xi

[hi , yi ] = −2yi ,

i = 1, 2,

and .

[x2 , x1 ] = x3

[x2 , x3 ] = x4

[x2 , x4 ] = x5

[x1 , x5 ] = x6

[y2 , y1 ] = y3

[y2 , y3 ] = y4

[y2 , y4 ] = y5

[y1 , y5 ] = y6 .

The algebra .G2 was first discovered by Killing in 1887 and shortly after wellstudied by Cartan, Engel and his student Reichel. In particular, the fact that it has a seven-dimensional complex irreducible representation was mentioned in Cartan’s thesis. Regarding this piece of history, as well as a wealth of algebraic and geometric properties of .G2 , we refer the reader to Agricola [1].

3.2.3 A Classification by Characteristic Polynomials Given a root .α ∈ R of a finite dimensional complex simple Lie algebra .L, we let Hα , Xα , and .Yα be the elements in .L that are uniquely selected by Theorem 3.19 (c). Then for each .β ∈ R, the reflection .σβ induces a map .φβ : L → L defined by

.

φβ (Hα ) = Hσβ (α) , φβ (Xα ) = Xσβ (α) , φβ (Yα ) = Yσβ (α) , α ∈ R.

.

Lemma 3.25 For a simple Lie algebra .L and a root .β ∈ R, the map .φβ is an automorphism of .L. The proof is a direct computation based on Theorem 3.19. For instance, among others, one can check that if .α, α , ∈ R are such that .α + α , ∈ R, then by Theorem 3.19 we have φβ ([Xα , Xα , ]) = φβ (Xα+α , )

.

56

3 Finite Dimensional Lie Algebras

= Xσβ (α+α , ) = Xσβ (α)+σβ (α , ) = [Xσβ (α) , Xσβ (α , ) ] = [φβ (Xα ), φβ (Xα )]. We leave the details of the proof as an exercise. The Weyl group of .L can thus be identified as a subgroup of .Aut(L). Pick any root .α ∈ R, and let .ρ : sl2 → L be an embedding defined by ρ(h) = Hα , ρ(x) = Xα , ρ(y) = Yα .

.

If .ad is the adjoint representation of .L, then .ad ◦ρ is a representation of .sl2 , which turns .L into a .sl2 -module. The characteristic polynomial of .sl2 with respect to .ad ◦ρ can be determined by Theorem 3.12. To be precise, assume .n ≥ 4 and .{Hα , Xα , Yα , x4 , . . . , xn } is a basis for .L. Then the characteristic polynomial .QL restricted to the plane .V := {z4 = · · · = zn = 0} is the characteristic polynomial of .sl2 with respect to the representation .ad ◦ρ. For convenience, we set fα (z) = det(z0 I + z1 ad Hα + z2 ad Xα + z3 ad Yα ) = QL (z) |V .

.

Although .fα depends on the choice of .α, there are at most two of them: either short root or long root. It is so because if two roots .α and .α , are of the same length, then there exists an element g in the Weyl group W of .R such that .g(α) = α , . By Theorem 3.4, the automorphism of .L induced by g due to Lemma 3.25 preserves the characteristic polynomial .QL (z). Thus we have .fα = fα , . Corollary 3.26 For a simple Lie algebra .L and a long root .α ∈ R, we have fα (z) = z0k0

.

3 ⎛ ⎞kj || z02 − j 2 (z12 + z2 z3 ) , j =1

where for each .0 ≤ j ≤ 3, the power .kj is the multiplicity of the eigenvalue j of ad Hα .

.

Proof First, since the representation .ad ◦ρ is a direct sum of irreducible representations of .sl2 , its associated characteristic polynomial .fα has possible irreducible factors .z0 and .z02 − λ2 (z12 + z2 z3 ) according to Theorem 3.12, where .λ is an eigenvalue (weight) of .ad Hα . Theorem 3.19 indicates that .λ is either 0 or . for some roots .β ∈ R. Since the latter has possible values .0, ±1, ±2, ±3, the likely irreducible factors of .fα are .z0 and .z02 − j 2 (z12 + z2 z3 ), j = 1, 2, 3. Moreover, with respect to any irreducible representation .π of .sl2 , the eigenvalues of .π(h) are distinct and each has multiplicity 1 (Theorem 3.10). Thus, the power .kj , 0 ≤ j ≤ 3, is equal to the multiplicity of the eigenvalue j of .ad Hα . u n Example 3.27 Now we take another look at Example 3.24. We pick the long root α = α1 and let .Xβ , Yβ , Hβ , β ∈ R + , be selected by Theorem 3.19. Then,

.

3.2 Simple Lie Algebras

57

Table 3.3 The values of . .β

.α1

.α2

.α1

.

2

.−1

1

+ α2

.α1

+ 2α2

0

.α1

+ 3α2

.−1

.2α1

+ 3α2

1

[Hα , Xβ ] = Xβ , [Hα , Yβ ] = −Yβ , β ∈ R + .

.

We have Table 3.3. If we let .Ej , 0 ≤ j ≤ 3 denote the eigenspaces of .ad Hα corresponding to the eigenvalues .0, 1, 2, 3, respectively, then E0 = span{Hα1 , Hα2 , Xα1 +2α2 , Yα1 +2α2 },

.

E1 = span{Xα1 +α2 , X2α1 +3α2 , Yα2 , Yα1 +3α2 }, E2 = span{Xα1 },

E3 = {0}.

Thus, .k0 = 4, k1 = 4, k2 = 1, and .k3 = 0. Note that it is not necessary to consider the negative roots because . = −. On the other hand, if we choose the short root .α = α2 , then similar computation verifies that .k0 = 4, k1 = 2, k2 = 1, and .k3 = 2. Since the characteristic polynomial .QL is invariant under .Aut L (Theorem 3.4), the tuple .(k0 , k1 , k2 , k3 ) in Corollary 3.26 is invariant under the isomorphism of Lie algebras. It is referred to as the power index of .L. It completely determines .L in the following sense. Theorem 3.28 Two simple Lie algebras are isomorphic if and only if their corresponding power indices are equal. Proof Theorem 3.22 indicates that finite dimensional simple Lie algebras are classified by their root systems and the associated Dynkin diagrams. Their power indices can be computed directly (in a way similar to that in Example 3.27), and they are shown in Table 3.4. Here .α refers to a long root and .γ refers to a short root. It is apparent that the five exceptional Lie algebras have distinct power indices. If the power index of .G2 were identical to that of .Am for some .m ≥ 2, then equating the values of .k1 , we see that .m = 4. But the values of .k0 are different for .G2 and .A4 . This shows that the power index of .G2 is not identical to that of any .Am for .m ≥ 2. The same method can verify, albeit tediously, that no power index of the exceptional Lie algebras coincides with that of any classical simple Lie algebras. Likewise, it can be verified that no two non-isomorphic classical simple Lie algebras have the same power index. We leave the verification as an exercise. u n In fact, the proof shows that the pair .(k0 , k1 ) alone is enough to distinguish all simple Lie algebras. The power index includes .k2 and .k3 for the sake of completeness. In view of Theorem 3.23, any finite dimensional complex simple Lie algebra .L is isomorphic to a unique one in Table 3.4. We shall use the standard basis for .L as described in [125] and let .QL be the corresponding characteristic polynomial.

58

3 Finite Dimensional Lie Algebras

Table 3.4 The power indices of simple Lie algebras

Type .An .Bn .Cn

.Dn .G2

.F4

.E6 .E7 .E8

Root .α .α .γ .α .γ .α .α .γ .α .γ .α .α .α

.k0

.k1

− 2n + 2 − 7n + 10 2 .2n − 3n + 2 2 .2n − 3n + 2 2 .2n − 7n + 10 2 .2n − 9n + 14 4 4 22 22 36 67 134 2 .n

2

.2n

.2n .4n

−2 −6

0 −2 −8 .4n − 8 4 2 14 8 20 32 56 .2n

.4n

.k2

.k3

1 1 .2n − 1 1 3 1 1 1 1 7 1 1 1

0 0 0 0 0 0 0 2 0 0 0 0 0

Since the polynomial .fα in Corollary 3.26 is the restriction of .QL , the following fact is a direct consequence of the preceding theorem. Corollary 3.29 Let .L1 and .L2 be two finite dimensional complex simple Lie algebras. Then they are isomorphic if and only if .QL1 = QL2 . For more details about the content of this subsection, we refer the reader to GengLiu-Wang [102]. Exercise 3.30 1. Prove that if .L is simple then .tr ad x = 0 for every .x ∈ L. 2. Compute .QL for .L = su3 and find its irreducible factors. 3. Compute the characteristic polynomial of the Weyl group of .G2 with respect to the base .A in Example 3.24. 4. In Example 3.27, choose the short root .α = α2 and compute the eigenspaces .Ej , 0 ≤ j ≤ 3 of .ad Hα . 5. Let R be the root space of a simple Lie algebra .L with Cartan subalgebra .h. Assume that .Hα , Xα , Yα , α ∈ R are selected as in Theorem 3.19. Prove that the map φ : Hα → −Hα , Xα → −Yα , Yα → −Xα , α ∈ R

.

induces an automorphism on .L such that .φ 2 = I . 6. Prove Lemma 3.25. 7. Let .{e1 , . . . , e8 } be an orthonormal basis of .R8 . The root system E R of the exceptional Lie algebra .E8 consists of vectors .±(ei ±ej ), i /= j , and . 12 8i=1 (−1)mi ei , E where .mi = 0 or 1 and . 8i=1 mi is even. Fix the base { } 1 A = e1 + e2 , e2 − e1 , . . . , e7 − e6 , (e1 + e8 − (e2 + · · · + e7 )) . 2

.

3.3 Solvable Lie Algebras

59

(a) Determine the set .R + of positive roots. (b) Verify that the power index of .E8 is .(134, 56, 1, 0). 8. Determine the Weyl groups for the simple Lie algebras .B3 and .C3 and show that they are isomorphic. 9. This exercise completes the proof to Theorem 3.28. (a) It is known that .A3 ∼ = D3 and .B2 ∼ = C2 . Verify that the two pairs of simple Lie algebras have the same power indices. (b) Check that no other two classical simple Lie algebras have the same power index. Project 3.31 1. Let R be a root system and .A = {α1 , . . . , αk } be the set of simple roots. Does the characteristic polynomial of the reflections .σα1 , . . . , σαk determine R up to isomorphism? 2*. Determine the irreducible factors of .QL (z) for simple Lie algebras .L. (a) What do the factors tell about the structure of .L? (b) Do we have anything similar to the Frobenius Theorem 2.11?

3.3 Solvable Lie Algebras By Levi decomposition theorem (Theorem 3.15), every Lie algebra is a semidirect sum of a semisimple Lie subalgebra and a solvable ideal. Since simple Lie algebras are classified by the Dynkin diagram and Theorem 3.28, the success of classifying all Lie algebras hinges on a classification of solvable ones. For low dimensions, achievements have been made in [111, 171, 199, 221]. However, solvable Lie algebras in high dimensions are very difficult to classify. A seemingly insurmountable challenge is the sheer number of inequivalent classes of them. This section describes how the characteristic polynomial is linked with the structure of a solvable Lie algebra. Some more flexible invariants for solvable Lie algebras will be introduced along the way. The main reference is [7]. The following characterization of solvable Lie algebras is due to Hu-Zhang [135]. Theorem 3.32 A finite dimensional Lie algebra .L is solvable if and only if .QL is a product of linear factors. Proof The case .dim L ≤ 2 is left as an exercise. We assume .dim L = n ≥ 3 and QL is a product of linear factors. Recall that Levi’s decomposition theorem asserts that every Lie algebra .L can be written as a semidirect sum .L, +τ L,, , where .L, is a semisimple subalgebra and .L,, = rad(L). If .L is not solvable, then .L, is nontrivial. Then, as noted in Sect. 3.1.2, there are elements .h, x, y ∈ L, such that

.

60

3 Finite Dimensional Lie Algebras

the subalgebra .S spanned by .{h, x, y} is isomorphic to .sl2 . If .{h, x, y, x4 , . . . , xn } is a basis for .L, then .ad : S → I nn(L) is a n-dimensional representation of .sl2 , and hence by Theorem 3.12 the polynomial .QL (z0 , z1 , z2 , z3 , 0, . . . , 0) contains an irreducible quadratic factor of the form .z02 − k(z12 + z2 z3 ), where .k > 0. This contradicts the assumption that .QL is a product of linear factors. On the other hand, if .L is solvable, then by Lie’s theorem there exists a basis .{x1 , . . . , xn } for .L such that .ad x1 , . . . , ad xn are all upper triangular [125]. Hence, with respect to this basis the characteristic polynomial .QL is a product of linear factors. u n The following example and theorem provide a good connection with the earlier study of simple Lie algebras. Example 3.33 If .L is a simple Lie algebra (with root system R and Cartan ) subalgebra .h, then the direct sum .B = h ⊕ ⊕α∈R + Lα is called the Borel subalgebra of .L. Since .h is abelian and .R + is a finite set, Theorem 3.19 implies that .B is solvable. Suppose the Cartan subalgebra .h = span{h1 , . . . , hk }. Then .fh := det(z0 I + z1 ad h1 + · · · + zk ad hk ) is a product of linear factors. Although the function .fh depends on the choice of basis for .h, it captures a fundamental property of .L. In light of Lemma 1.14, every factor of .fh is of the form .z0 + α1 z1 + · · · + αk zk , where .αj is an eigenvalue of .ad hj . The map .hj → αj , 1 ≤ j ≤ k, defines a unique linear functional, or a root, in .h∗ . On the other hand, for every root .α of .L, there is a common eigenvector .y ∈ L such that .ad hj (y) = α(hj )y for each j . Thus .z0 + α(h1 ) ad h1 + · · · + α(hk ) ad hk is a factor of .fh . Therefore, the function .fh determines the root space of .L, with the number of distinct factors of .fh , excluding .z0 , equal to the rank of .L. Since a simple Lie algebra is determined by its root system, the following is immediate. Theorem 3.34 Let .L and .L, be simple Lie algebras with Cartan subalgebras .h and resp. .h, . Then .L and .L, are isomorphic if and only if .fh = fh, up to a linear change of variables. This gives another proof to Corollary 3.29.

3.3.1 Spectral Matrix In view of Theorem 3.32, given a solvable Lie algebra .L with a fixed basis .S = {x1 , . . . , xn }, we may write QL (z) =

n || (

.

j =1

) z0 + λij zi .

(3.11)

3.3 Solvable Lie Algebras

61

| Observe that Lemma 1.14 implies .σ (Ti ) = {λij | 1 ≤ j ≤ n} for each i, where .Ti = ad xi . Definition 3.35 The spectral matrix for .L with respect to the basis S is defined as λL = (λij )n×n , 1 ≤ i, j ≤ n.

.

We shall write .λL simply as .λ whenever there is no risk of confusion. Since the ith row of .λ is the set of eigenvalues of .Ti , counting multiplicity, the order of .λ’s rows is in line with the order .T1 , . . . , Tn . Moreover, the order of factors in factorization (3.11) determines the order .λi1 , . . . , λin for every .1 ≤ i ≤ n. In other words, it determines the order of columns in the spectral matrix. Since .z0 is always a factor of .QL , we take it to be the first factor. Thus, the first column in .λ is the zero vector, e.g., .λi1 = 0, .1 ≤ i ≤ n. Lemma 3.36 Let .{x1 , . . . , xn } and .{xˆ1 , . . . , xˆn } be two bases for a solvable Lie algebra .L and assume .λ and .λˆ are the corresponding spectral matrices. Then .λ = B λˆ , where .B = (bij ) is such that .xi = bij xˆj for .1 ≤ i, j ≤ n. ˆ Proof For simplicity, we let .Q(z) and .Q(z) denote the characteristic polynomials ˆ ˆ with respect to the two bases. Then by (3.2), we have .Q(z) = Q(w) = B ◦ Q(z), where .w0 = z0 and .wk = zi bik , 1 ≤ k ≤ n. In other words, we have n || .

j =1

(z0 + λij zi ) =

n ||

(z0 + λˆ kj wk ) =

j =1

n ||

(z0 + λˆ kj zi bik ) =

j =1

ˆ which shows .λij = bik λˆ kj , i.e., .λ = B λ.

n || ( ) z0 + (bik λˆ kj )zi , j =1

u n

ˆ are isomorphic solvable Lie algebras of dimension n In the case .L and .L with corresponding spectral matrices .λ and .λˆ , respectively, Lemma 3.3 shows the ˆ In particular, this suggests that the rank of existence of .B ∈ GLn such that .λ = B λ. .λ is an invariant with respect to the isomorphism of solvable Lie algebras. It is thus interesting to determine this rank. Definition 3.37 Let .L be a n-dimensional Lie algebra and .1 ≤ k ≤ n. The elements f1 , . . . , fk in .L areEsaid to be nil-independent if there exist no scalars .c1 , . . . , ck , not all 0, such that . ki=1 ci fi ∈ nil(L).

.

In other words .f1 , . . . , fn are nil-independent if and only if .[fi ], 1 ≤ i ≤ k, are linearly independent in the quotient algebra .L/nil(L). Proposition 3.38 Given a solvable Lie algebra .L, we have .rank λ = dim (L/ nil(L)).

.

Proof The proposition holds trivially when .L is nilpotent, in which case both sides of the equation are equal to 0. Thus we consider a non-nilpotent .L with dimension n. Assume its nilradical .nil(L) has a basis .{x1 , . . . , xk }, where .k < n, and .xk+1 , . . . , xn are elements in .L such that .{[xk+1 ], . . . , [xn ]} is a basis for the quotient .L/nil(L). Then .{x1 , . . . , xn } is a basis for .L. The matrices .Ti = ad xi , 1 ≤

62

3 Finite Dimensional Lie Algebras

i ≤ k, are nilpotent by Engel’s theorem. This means that the first k rows in the spectral matrix .λ are 0s, and hence .rank λ ≤ n − k. Suppose .t = rank λ < n − k. Then there exists an invertible .n × n matrix .B = (bij ) such that .Bλ has precisely t nonzero rows. If we set .x{ˆi = bij xj , 1| ≤ i ≤ n, }then .λˆ = Bλ by Lemma 3.36, | and it follows that the set . Tˆi = ad xˆi | 1 ≤ i ≤ n contains only t non-nilpotent matrices, or equivalently, the basis .{xˆi : 1 ≤ i ≤ n} has t non-nilpotent elements. This contradicts the assumption that .xk+1 , . . . , xn are nil-independent. u n Example 3.33 and Proposition 3.38 lead us to the following fact. Corollary 3.39 For a simple Lie algebra .L with Borel subalgebra .B, it holds that rank λB = rank L.

.

For a general solvable Lie algebra .L, it is shown in Šnobl [208] that .

dim(L/nil(L)) ≤ dim nil(L) − dim[nil(L), nil(L)].

Adding .dim(L/nil(L)) to the both sides above, we obtain the following fact. Corollary 3.40 Given a solvable Lie algebra .L, we have .rank λ ≤

dim L 2 .

3.3.2 Spectral Invariants Although classifying high dimensional solvable Lie algebras remains a challenge, one fruitful approach is to classify solvable Lie algebras with the same nilradical, or in other words, to classify different extensions of a given nilpotent Lie algebra. The reader can find more details along this line in [154, 198, 209, 237] and the references therein. It is an appealing question whether characteristic polynomial and spectral matrix may be useful for this effort. This subsection collects some preliminary work in this direction. If .L is nilpotent, then its characteristic polynomial is a power of .z0 (exercise), and the spectral matrix is 0. This fact provides convenience for the study of solvable extensions of nilpotent Lie algebras. Indeed, for a general solvable Lie algebra .L, a close observation of Lemma 3.3 and factorization (3.11) suggests that the number of distinct irreducible factors, which we denote by .k(L), in .QL is invariant with respect to a change of bases or an automorphism of .L. Recall that .|S| denotes the cardinality of a finite set S. Proposition 3.41 Let .L be a finite dimensional solvable Lie algebra. Then k(L) ≥ max{|σ (ad x)| | x ∈ L}.

.

Proof Assume .{x1 , . . . , xn } is a basis for .L. Then obviously .1 ≤ |σ (ad x)| ≤ n for every .x ∈ L. Without loss of generality, we assume .|σ (ad x1 )| = max{|σ (ad x)| | x ∈ L} =: t. Given the factorization

3.3 Solvable Lie Algebras

63

QL (z) =

.

⎞ n ( || z0 + λij zi , j =1

| | we have .λ = (λij )n×n , and .σ (ad xi ) = {λij | j = 1, 2, . . . , n}. Suppose .λ1j1 , . . . , λ1jt are the distinct eigenvalues in .σ (ad x1 ). Then the factors z0 + λ1j z1 + · · · + λnj zn , j ∈ {j1 , j2 , . . . , jt }

.

in the factorization above are distinct because the coefficients of the variable .z1 are distinct. This implies that .QL has at least t distinct factors, i.e., we have .k(L) ≥ t. u n It is not hard to see that the equality in Proposition 3.41 holds when .L is a onedimensional extension of a nilpotent Lie algebra. But it is not clear whether this is true for all solvable Lie algebras. The invariant .k(L) seems hard to detect from an algebraic point of view. It is an interesting problem for investigation. Corollary 3.42 Let .N be a .(n−1)-dimensional nilpotent Lie algebra, where .n ≥ 2, and let .L = Cxn + N and .L, = Cxn, + N be two 1-dimensional solvable extensions of .N. If .L and .L, are isomorphic, then .σ (Tn ) = βσ (Tn, ) for some nonzero .β ∈ C. Proof Let .S = {x1 , . . . , xn−1 } be a basis for .N. Then .S ∪ {xn } and .S ∪ {xn, } are bases for .L and .L, , respectively. Their corresponding spectral matrices .λ and , .λ have nonzero entries only at the bottom row. Suppose the bottom row of .λ is .(0, μ1 , . . . , μn−1 ) and that of .λ, is .(0, μ,1 , . . . , μ,n−1 ). If .φ : L → L, is an isomorphism, then Lemma 3.36 implies .λ = Bλ, for some .B = (bij ) ∈ GLn . Therefore, we have .μj = bnn μ,j , 1 ≤ j ≤ n − 1. Moreover, since .ad xn is not nilpotent, we have .bnn /= 0, and this completes the proof with .β = bnn . u n Corollary 3.42 suggests that, up to a nonzero scalar multiple, the spectrum .σ (Tn ) is an invariant for the isomorphism classes of 1-dimensional extensions of a .(n − 1)dimensional nilpotent Lie algebra. We use an example to show the usefulness of this fact. Example 3.43 Consider a two-parameter solvable Lie algebra .Aa,b = span {x1 , . . . , x4 } with nonzero brackets .[x1 , x4 ] = ax1 , .[x2 , x4 ] = bx2 − x3 , .[x3 , x4 ] = x2 + bx3 , .a > 0. This algebra is of type .L4,6 according to [172], and it is a onedimensional extension of the three-dimensional abelian subalgebra generated by .{x1 , x2 , x3 }. One verifies easily that .

⎞ ⎛ 0 0 az1 z0 − az4 ⎜ 0 z0 − bz4 −z4 bz2 + z3 ⎟ ⎟, .z0 I + zj Tj = ⎜ ⎝ 0 z4 z0 − bz4 −z2 + bz3 ⎠ 0 0 0 z0

64

3 Finite Dimensional Lie Algebras

and hence the characteristic polynomial is Q(z) = z0 (z0 − az4 )(z0 − (b + i)z4 )(z0 − (b − i)z4 ),

.

which indicates that .σ (T4 ) = {0, −a, −b − i, −b + i}. If .Aa , ,b, is isomorphic to Aa,b , then by Corollary 3.42 (and its proof) there exists a nonzero scalar .β such that

.

βa , = a, β(b, + i) = b + i, β(−b, + i) = −b + i.

.

Adding the last two equations one obtains .β = 1, and hence .a = a , , b = b, . This shows that .Aa , ,b, is isomorphic to .Aa,b only if .a = a , , b = b, . Exercise 3.44 1. Prove that the Borel subalgebra of a simple Lie algebra is solvable. 2. Consider the Lie algebra .G2 in Example 3.24 and let .h be its Cartan subalgebra. (a) Compute .fh . (b) How are the linear factors of .fh related to the roots given in the example? 2. Let .La,b = span{x1 , x2 , x3 } be a complex Lie algebra with nonzero brackets: [x3 , x1 ] = x2 , [x3 , x2 ] = ax1 + bx2 , b /= 0.

.

(a) Verify that .La,b is solvable but non-nilpotent. (b) Determine whether .La,b ∼ = La , ,b, if and only if .a = a , and .b = b, . (c) Use Theorem 3.4 to determine the automorphism group .Aut(La,b ). 3. Prove that a n-dimensional Lie algebra .L is nilpotent if and only if .QL (z) = z0n . 4. Suppose .L is solvable with .dim L = n ≥ 2 and .dim nil(L) = n − 1. Show that .Aut(L) ∩ U (n) = Aut(nil(L)) ∩ U (n − 1) ⊕ 1. 5. Four-dimensional solvable Lie algebras .L are classified in [111]. Compute the invariant .k(L) for all equivalent classes of .L. 6. For any Lie algebra .L, its characteristic polynomial .QL (z) has the factor .z0 (Exercise 3.17 4). Show that if .QL (z)/z0 is irreducible then .L is simple. (Hint: Use factorization (3.8) and Theorem 3.32.) Project 3.45 1. Determine whether the equality in Proposition 3.41 holds in general. 2. A classification of six-dimensional solvable Lie algebras .L with a fivedimensional nilradical is given in Mubarakzyanov [171]. Compute the invariant .k(L) for all equivalent classes of .L. 3*. Given any .n × n matrix M with the first column equal to 0 and .rank M ≤ n/2, does there exist a solvable Lie algebra whose spectral matrix is equal to M?

Chapter 4

Projective Spectrum in Banach Algebras

Consider a Banach algebra .B with unit I . The classical spectrum of an element A ∈ B is the set .σ (A) = {λ ∈ C | A − λI is not invertible in B}, and it is one of the main subjects in operator theory. For several elements .A1 , . . . , An in .B, is there a proper notion of their joint spectrum? This problem is natural and important, but it seems difficult even for .2 × 2 matrices, particularly so when the matrices are not commuting. An ideal definition is expected to provide an efficient mechanism for our study on multi-operator systems. This chapter first briefly reviews various notions of joint spectrum in the literature for commuting operators, with an emphasis on their connections. Then it defines the projective spectrum for elements in a Banach algebra. The simplicity and symmetry of this definition makes it easy to compute in an abundance of examples, including the Cuntz algebra, the irrational rotation algebra, and the free group von Neumann algebra. .

4.1 Joint Spectra Let .B be an abelian Banach algebra. A nontrivial bounded linear functional .φ on B is said to be multiplicative if .φ(AB) = φ(A)φ(B), A, B ∈ B, i.e., .φ is an algebra homomorphism from .B to .C. Hence, .ker φ is an ideal in .B, and it is maximal because .B/ ker φ is 1-dimensional. For this reason, the set of multiplicative linear functionals, denoted by .MB , is called the maximal ideal space. A very useful fact is that, due to the Banach–Alaoglu theorem, .MB is compact with respect to the weak.∗ topology on the dual space .B∗ . Given .B ∈ B, there is a continuous function .Bˆ on .MB defined by .

ˆ B(φ) = φ(B), φ ∈ MB .

.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 R. Yang, A Spectral Theory Of Noncommuting Operators, https://doi.org/10.1007/978-3-031-51605-4_4

65

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4 Projective Spectrum in Banach Algebras

The map .B → Bˆ is called the Gelfand map, and we have the following spectral theorem. ˆ Theorem 4.1 Let B be an element in .B. Then .σ (B) = ran B. Hence, an element B is non-invertible in .B if and only if there exists .φ ∈ MB such that .φ(B) = 0. For more discussion on abelian Banach alegebras, we refer the reader to Douglas [66]. The following notion of joint spectrum is a natural extension of the classical spectrum, and it has been well studied in Gamelin [94] and Hörmander [123]. Definition 4.2 For a tuple .A = (A1 , . . . , An ) of elements in .B, the algebraic joint spectrum .Sp(A) is the collection of .λ = (λ1 , . . . , λn ) ∈ Cn such that the ideal generated by .A1 − λ1 I, . . . , An − λn I is proper in .B. In other words, .λ is not in .Sp(A) if and only if there are elements .B1 , . . . , Bn in .B such that .(A1 − λ1 I )B1 + · · · + (An − λn I )Bn = I . Theorem 4.3 For a tuple .A = (A1 , . . . , An ) of elements in .B, we have .

( ) Sp(A) = { Aˆ1 (φ), . . . , Aˆn (φ) ∈ Cn | φ ∈ MB }.

The spectrum .Sp(A) is a nonempty compact subset of .Cn . In particular, it satisfies the projection property that .pi (Sp(A)) = σ (Ai ), where .pi (z) = zi is the projection to the ith coordinate, .1 ≤ i ≤ n. For a compact subset .K ⊂ Cn , we let .Hol(K) denote the algebra of functions that are holomorphic in a neighborhood of K. Functional calculus for the tuple A holds in the following sense. Theorem 4.4 There is a homomorphism .ρ : Hol(K) → B such that ρ(1) = I, and ρ(zi ) = Ai , i = 1, . . . , n.

.

4.1.1 The Koszul Complex Taylor spectrum was introduced in [216, 217] using the Koszul complex. Although less intuitive at the first sight, it has later become one of the central topics in multivariable operator theory. Let .{e1 , . . . , en } be a basis of a complex vector space V . Then the wedge product .∧ on V is a bilinear operation such that u ∧ v + v ∧ u = 0, u, v ∈ V .

.

(4.1)

Note that .u ∧ u = 0. For .1 ≤ p ≤ n, the space of p-forms, which we denote by .Ap (V ), is spanned by .ei1 ∧ · · · ∧ eip , 1 ≤ i1 < · · · < ip ≤ n. It follows that .Ap (V ) = {0} for .p > n. For consistency, the field .C is denoted by .A0 (V ). The direct sum .⊕0≤p≤n Ap (V ) is called the exterior algebra on V . Given a Hilbert

4.1 Joint Spectra

67

space .H, the tensor product .H ⊗ Ap (V ) is endowed with a natural inner product defined by { =

.

0

(i1 , . . . , ip ) = (j1 , . . . , jp ) (i1 , . . . , ip ) /= (j1 , . . . , jp ). (4.2)

A tuple .A = (A1 , A2 , . . . , An ) of commuting bounded linear operators on .H gives rise to a cochain complex .E(H, A): d−1

d0

d1

dn−1

dn

0 −→ H ⊗ A0 −→ H ⊗ A1 −→ · · · −→ H ⊗ An −→ 0,

.

(4.3)

where .d−1 = dn = 0, and .dp : H ⊗ Ap (V ) → H ⊗ Ap+1 (V ), 0 ≤ p ≤ n − 1, is a bounded linear map defined by dp (x ⊗ ω) =

n E

.

Ai x ⊗ (ei ∧ ω), x ∈ H, ω ∈ Ap (V ).

i=1

The reader shall gain some insight by looking at the special cases .n = 1 and 2. Proposition 4.5 For every commuting tuple A, we have .dp+1 dp = 0, 0 ≤ p ≤ n − 1. Its proof is left as an exercise. Proposition 4.5 implies the inclusion .ran dp ⊂ ker dp+1 . The complex .E(H, A) is said to be exact if .ran dp = ker dp+1 for all .−1 ≤ p ≤ n − 1. Observe in particular that the exactness of .E(H, A) implies that ran .dp is closed for each p because the kernel of a bounded linear map is always closed. It can be checked that the adjoint of .dp∗ : H ⊗ Ap+1 → H ⊗ Ap with respect to the inner product (4.2) is given by dp∗ (x ⊗ ei1 ∧ · · · ∧ eip+1 ) =

p+1 E

.

(−1)m−1 A∗im x ⊗ ei1 ∧ · · · ∧ eim ∧ · · · ∧ eip+1 ,

m=1

where .aˆ stands for the omission of a. The self-adjoint operators ∗ Dp = dp∗ dp + dp−1 dp−1 : H ⊗ Ap → H ⊗ Ap , 0 ≤ p ≤ n,

.

are called the Laplacians of the complex .E(H, A). In particular, D0 = d0∗ d0 = A∗1 A1 + · · · + A∗n An ,

.

∗ Dn = dn−1 dn−1 = A1 A∗1 + · · · + An A∗n .

68

4 Projective Spectrum in Banach Algebras

To gain a better understanding of the exactness of .E(H, A), we consider the ˜ = ⊕n H ⊗ Ap and accordingly set orthogonal direct sum .H p=0 ⎛

0 ⎜d0 ⎜ ⎜ .δ = ⎜ 0 ⎜. ⎝ ..

0 0 d1 .. .

··· ··· ···

0 0 0 .. .

⎞ 0 0⎟ ⎟ 0⎟ ⎟. .. ⎟ .⎠

··· 0 0 · · · dn−1 0

Then Proposition 4.5 implies that .δ 2 = 0. Proposition 4.6 The following are equivalent for a commuting tuple A: (a) .E(H, A) is exact. ˜ (b) .δ + δ ∗ is invertible on .H. (c) .Dp is invertible for each .0 ≤ p ≤ n. Proof To show the equivalence of (a) and (b), we observe that the fact .δ 2 = 0 implies .δ ∗2 = 0 and consequently .ran δ ⊂ ker δ and .ran δ ∗ ⊂ ker δ ∗ . If .E(H, A) is exact, then .ran dp−1 = ker dp , p = 0, . . . , n, and it follows that .ran δ = ker δ. Parallelly, we have .ran δ ∗ = ker δ ∗ . It gives rise to the decomposition ˜ = ran δ ∗ ⊕ ker δ = ker δ ∗ ⊕ ran δ. H

.

Observe that the two summands are closed, and we have the following two bounded bijections: δ : ran δ ∗ → ran δ = ker δ,

.

δ ∗ : ran δ → ran δ ∗ = ker δ ∗ .

˜ This implies that .δ + δ ∗ is invertible on .H. ∗ ˜ Conversely, if .δ + δ is invertible, then for every .η ∈ ker δ there exists .ξ ∈ H ∗ 2 ∗ such that .η = (δ + δ )ξ . Since .δ = 0, we have .η − δ ξ ∈ ker δ, which is orthogonal to .ran δ ∗ . Therefore, we must have .δ ∗ ξ = 0, and hence, .η = δξ ∈ ran δ. This shows .ker δ = ran δ, or equivalently, .ran dp−1 = ker dp , p = 1, . . . , n. To show the equivalence of (b) and (c), one only needs to observe that since .δ+δ ∗ is self-adjoint, it is invertible if and only if .(δ + δ ∗ )2 = δδ ∗ + δ ∗ δ is invertible. A direct computation yields .δδ ∗ + δ ∗ δ = ⊕np=0 Dp , which concludes that (b) is equivalent to (c). u n Proposition 4.6 (c) readily implies the following fact. Corollary 4.7 If one of the elements in the tuple A is invertible, then .E(H, A) is exact.

4.1 Joint Spectra

69

4.1.2 Taylor Spectrum For a vector .λ = (λ1 , . . . , λn ) ∈ Cn , we let .A − λ stand for .(A1 − λ1 I, . . . , An − λn I ). Definition 4.8 The Taylor spectrum of a commuting tuple A is defined as σT (A) = {λ ∈ Cn | E(H, A − λ) is not exact}.

.

Taylor spectrum indeed extends the classical spectrum because it is clear that in the case .A = A1 we have .σT (A) = σ (A). It is worth mentioning that the exactness of the complex .E(H, A) is not dependent on the choice of the basis .{e1 , . . . , en } for V or the ordering of .(A1 , . . . , An ), which means it is intrinsic to the tuple A. The following property is important. Theorem 4.9 For a commuting tuple of bounded linear operators .A = (A1 , . . . , An ) on .H, its Taylor spectrum .σT (A) is a nontrivial compact subset of .Cn . In the sequel, we assume .B is any abelian Banach subalgebra in .B(H) that contains all .Aj and is closed under inversion in the sense that if .a ∈ B is invertible in .B(H), then .a −1 ∈ B. Such subalgebra .B is not unique, but since it is inversion closed, the spectra .σT (A) and .Sp(A) are independent of the choice of such .B. A connection between the two spectra is as follows. Proposition 4.10 Let A be a tuple of elements in .B. Then .σT (A) ⊂ Sp(A). Proof We show that .Spc (A) ⊂ σTc (A), and without loss of generality, we prove for / Sp(A), then .0 ∈ / σT (A). Assume .B1 , . . . , Bn are the case .λ = 0, namely, if .0 ∈ elements in .B such that .A1 B1 + · · · + An Bn = I . We set .t0 = 0 and for .1 ≤ p ≤ n define .tp : H ⊗ Ap → H ⊗ Ap−1 by tp (x ⊗ ei1 ∧ · · · ∧ eip ) =

p E

.

(−1)m−1 Bim x ⊗ ei1 ∧ · · · ∧ eim ∧ · · · ∧ eip .

m=1

Then a direct computation will verify that .t1 d0 (x) = x, and for .1 ≤ p ≤ n − 1, (tp+1 dp +dp−1 tp )(x⊗ei1 ∧· · ·∧eip ) =

n E

.

Aj Bj x⊗ei1 ∧· · ·∧eip = x⊗ei1 ∧· · ·∧eip .

j =1

Since all involved maps are linear, we have (tp+1 dp + dp−1 tp )h = h, h ∈ H ⊗ Ap .

.

(4.4)

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4 Projective Spectrum in Banach Algebras

Consequently, for each .0 ≤ p ≤ n, if .dp (h) = 0, then .h ∈ ran dp−1 , which shows that .E(H, A) is exact. u n Functional calculus for a commuting tuple A of elements in .B is a natural extension of the functional calculus for a single operator. First, Corollary 4.7 implies that .σT (A) ⊂ σ (A1 ) × · · · × σ (An ). Let .o be a domain in .Cn that contains the Cartesian product above. Recall that .pj (z) = zj . For each .1 ≤ j ≤ n, we fix a piecewise-smooth simple closed curve .γj in .pj (o) that encloses .σ (Aj ). Then for every holomorphic function on .o, we can define 1 (2π i)n

f (A) =

.

f

f ··· γ1

n ||

(zj − Aj )−1 f (z)dz1 · · · dzn .

γn j =1

It can be shown that the map .τ : f → f (A) is a homomorphism from .Hol(o) into B. Moreover, we have .τ (1) = I and .τ (pj ) = Aj for each j . An important property of this functional calculus is the following spectral mapping theorem.

.

Theorem 4.11 For a holomorphic map .f = (f1 , . . . , fm ) : o → Cm , we have σT (f (A)) = f (σT (A)).

.

In particular, the theorem implies .σ (Aj ) = pj (σT (A)), 1 ≤ j ≤ n. So the functional calculus can in fact be defined on any domain .o that contains .σT (A).

4.1.3 Fredholm Tuples Taylor spectrum motivated the study of multivariable index theory. This section gives a brief review of the development. For more details, we refer the reader to Curto [45]. Here, we assume the Hilbert space .H is infinite dimensional, and let .H1 denote the closed unit ball of .H. An operator .K ∈ B(H) is said to be compact if .K(H1 ) is compact in .H. Compact operators form a closed twosided ideal .K(H) ⊂ B(H) that is a non-unital .C ∗ -algebra. The quotient algebra .C(H) = B(H)/K(H) is called the Calkin algebra, and it gives rise to the short exact sequence i

ρ

0 −→ K(H) −→ B(H) −→ C(H) −→ 0,

.

(4.5)

where i is the inclusion, and .ρ is the quotient map. Definition 4.12 A bounded linear operator T on .H is said to be Fredholm if .ρ(T ) is invertible in the Calkin algebra .C(H). We let .F (H) denote the set of Fredholm operators on .H. Since the set of invertible elements in .C(H) is open, the set .F (H) is open in .B(H), and it is a group with

4.1 Joint Spectra

71

respect to the product of linear operators. The following Atkinson theorem is often used as an alternative definition of Fredholm operators. Theorem 4.13 A bounded linear operator T on .H is Fredholm if and only if it has closed range with finite dimensional .ker T and .ker T ∗ . In terms of Koszul complex, we can express the Fredholmness of T by the short sequence d−1

d0

d1

0 −→ H ⊗ A0 −→ H ⊗ A1 −→ 0,

.

(4.6)

where .d0 x = T x ⊗ e1 , x ∈ H, and .d0 has a finite dimensional kernel and a closed range with finite codimension. In this case, the index of T is defined as .

ind T = dim ker T − dim ker T ∗ = dim(ker d0 o ran d−1 ) − dim(ker d1 o ran d0 ).

The index possesses three very useful properties: (1) Invariance under compact perturbation, namely, for every compact operator K, the sum .T + K is also Fredholm and .ind(T + K) = ind T . (2) Strong invariance under homotopy, namely, .ind T1 = ind T2 if and only if .T1 and .T2 are in the same path-connected component of .F (H). (3) The index map .ind : F (H) → Z is a surjective group homomorphism. For a tuple A of elements in .B(H) that is not necessarily commuting, the cochain (4.3) can still be defined, though .dp dp−1 may not be equal to 0. To make a meaningful definition of Fredholm tuple in this case, we shall first filter out the perturbation by compact operators. A tuple A is said to be almost commuting (a.c. for short) if the commutators .[Ai , Aj ] are compact for all .1 ≤ i, j ≤ n. It is easy to see that if A is a.c. then any compact perturbation .(A1 + K1 , . . . , An + Kn ) is a.c. In this case, one can verify that .dp dp−1 is compact for each .0 ≤ p ≤ n. Definition 4.14 An almost commuting tuple A is said to be Fredholm if .ran dp is closed and .ker dp ∩ (ran dp−1 )⊥ is finite dimensional for each .0 ≤ p ≤ n. ∗ h = 0, which Observe that .h ∈ ker dp ∩(ran dp−1 )⊥ if and only if .dp h = 0 and .dp−1 is true if and only if .Dp h = 0. Thus we see that .

ker Dp = ker dp ∩ (ran dp−1 )⊥ , 0 ≤ p ≤ n.

Indeed, it holds that the tuple A is Fredholm if and only if each .Dp is Fredholm. In this case, the index of A is defined as

.

ind A =

n E p=0

(−1)p dim ker Dp .

(4.7)

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4 Projective Spectrum in Banach Algebras

If we let .Fn (H) be the set of a.c. Fredholm n-tuples of operators on .H, then it is relatively open in the set of all a.c. n-tuples with respect to the norm topology on n .(B(H)) . Theorem 4.15 The function .ind : Fn (H) → Z is continuous, surjective, and invariant under compact perturbation. In particular, .ind is constant on pathconnected components of .Fn (H). However, there are two unsettled questions. Question 4.16 1. Can .Fn (H) be endowed with a group structure so that .ind is a homomorphism? 2. (Douglas [66]) Given two tuples .A, B ∈ Fn (H) with the same index, do they necessarily lie in the same path-connected component of .Fn (H)? Regarding the second question above, partial results have been obtained in the case of essentially normal tuples. A tuple .A = (A1 , . . . , An ) is said to be essentially normal if the commutators .[Ai , Aj ] and .[Ai , A∗j ] are compact for all .1 ≤ i, j ≤ n. Essential normality is an important subject of study for many natural tuples of operators on holomorphic function spaces. The following two theorems are thus of interest. Theorem 4.17 If .A ∈ Fn (H) is essentially normal with all commutators in the trace class, then .ind A = 0. In fact, it is reasonable to suspect that the theorem still holds if only one of the elements in the tuple A is essentially normal with trace class commutators. Theorem 4.18 If .A, B ∈ Fn (H) are essentially normal with the same index, then they are in the same path-connected component.

4.1.4 Essential Taylor Spectrum The concept of Fredholm tuple makes the following definition natural. Definition 4.19 The essential Taylor spectrum of an a.c. tuple .A = (A1 , . . . , An ) is defined as .σT ,e (A) = {λ ∈ Cn | A − λ is not Fredholm}. Taylor spectrum and essential spectrum can be calculated for many operator tuples on Hilbert spaces of holomorphic functions. The classical Hardy space .H 2 (T) is the subspace of .L2 (T) consisting of functions that have analytic extension to the disc .D. A standard orthonormal basis of .H 2 (T) is .{w k | k ≥ 0, |w| = 1}. The Hardy space over the n-torus .Tn is equal to the tensor product of n copies of 2 2 2n−1 ) over the unit .H (T). Another important example is the Hardy space .H (S n 2 2n−1 sphere in .C . Likewise, it is the subspace of .L (S ) consisting of functions that have holomorphic extension to the open unit ball .Bn .

4.1 Joint Spectra

73

Example 4.20 For each variable .wj , 1 ≤ j ≤ n, where .n ≥ 2, we let .Sj and .Wj stand for the multiplication operators .f (w) → wj f (w) on .H 2 (Tn ) and that on 2 2n−1 ), respectively. Then for the two tuples S and W , it can be shown that: .H (S (a) .σT (S) = Dn , .σT ,e (S) = ∂Dn , where .∂ stands for the boundary. (b) .σT (W ) = Bn , .σT ,e (W ) = S2n−1 . (c) .ind S = ind W = −1. It is an excellent exercise to verify the three claims in the example. The tuple S is not essentially normal; for instance, .S1∗ S1 − S1 S1∗ is the orthogonal projection f (z1 , z2 , . . . , zn ) → f (0, z2 , . . . , zn ), f ∈ H 2 (Tn ),

.

which is clearly not compact. On the other hand, a direct computation can verify that although the tuple W is essentially normal, the commutators .[Wj∗ , Wj ], j = 1, . . . , n are not trace class. Indeed, they are in the Schatten-p class for .p > n. Hence, the fact .ind W = −1 indicates that the trace class condition in Theorem 4.17 is indispensible. Observe also that the indices of S and W do not reflect the topological difference between .Tn and .S2n−1 .

4.1.5 Harte Spectrum For a tuple A of bounded linear operators on .H, not necessarily commuting, the notion of Harte spectrum .σH (A) was defined in [118]. It is the set of vectors .λ ∈ Cn such that at least one of the following two conditions holds: 1. There exists a sequence of unit vectors .{xk } ⊂ H such that .

lim ||(Aj − λj )xk || = 0, 1 ≤ j ≤ n.

k→∞

(4.8)

2. The vector space sum .(A1 − λ1 )H + · · · + (An − λn )H is not equal to .H. A connection between Harte spectrum and Taylor spectrum is described as follows. Proposition 4.21 For any commuting tuple A on .H, we have .σH (A) ⊂ σT (A). Proof Without loss of generality, we show that if the 0 vector is in .σH (A), then it is in .σT (A). According to the definition, there are three cases when .0 ∈ σH (A): (i) There exists a nontrivial vector .x ∈ H such that .Ai x = 0 for all .1 ≤ i ≤ n. Clearly, this implies .d0 (x) = 0, and hence .x ∈ σT (A). (ii) It holds that .∩ni=1 ker Ai = {0}, but there exists a sequence of unit vectors .{xk } ⊂ H such that (4.8) holds for .λ = 0. Then we must have .ran d0 /= ker d1 in this case, for otherwise .ran d0 would be closed, and it would follow that the map .d0 : H ⊗ A0 → ran d0 is invertible, implying that

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4 Projective Spectrum in Banach Algebras

||d0 x||2 =||A1 x ⊗ e1 + · · · + An x ⊗ en ||2

.

=||A1 x||2 + · · · + ||An x||2 is bounded below by a positive constant for all unit vectors x. This would contradict (4.8) for .λ = 0. (iii) The space .A1 H + · · · + An H is not equal to .H, in which case .dn−1 is not surjective. In all three cases, the Koszul complex .E(H, A) is not exact, and therefore, .0 ∈ σT (A). u n If A is commuting and only condition (4.8) is met, then the vector .λ is said to be in the joint approximate point spectrum of A that is denoted by .σπ (A), Dash [56]. For a tuple of normal operators, the following theorem is due to Cho–Takaguchi [44]. Theorem 4.22 If A is a tuple of commuting normal operators, then .σπ (A) = Sp(A). Proof First of all, the Fuglede theorem [92] states that if T and N are commuting operators on .H, and N is normal, then T also commutes with .N ∗ . Since the .Aj s are commuting and are all normal, we have .Ai A∗j = A∗j Ai for all .1 ≤ i, j ≤ n. Thus the .C ∗ -algebra .B generated by .I, A1 , . . . , An is abelian. Moreover, it is known that every unital .C ∗ -algebra is inversion closed [66]. It is clear that .σπ (A) ⊂ Sp(A). To show the inclusion in the other direction, without loss of generality, we assume .0 ∈ / σπ (A). In this case, there is a constant En 2 ≥ α||x||2 for all nonzero x, which implies that .α > 0 such that . ||A x|| j j =1 En ∗ . j =1 Aj Aj is bounded below and hence invertible. If we set Bk = A∗k

n (E

.

Aj A∗j

)−1

, k = 1, . . . , n,

j =1

then .Bk ∈ B and .A1 B1 + · · · + An Bn = I . This shows .0 ∈ / Sp(A).

u n

Summarizing the above observations, we arrive at the following appeasing fact. Corollary 4.23 If A is a tuple of commuting normal operators, then .σπ (A) = σH (A) = σT (A) = Sp(A). For commuting tuples, functional calculus and spectral mapping theorem can be established as well for Harte spectrum. A subset .K ⊂ D2 is said to be dominating if for every bounded holomorphic function f on .D2 , we have .||f ||∞ = sup{|f (z)| | z ∈ K ∩ D2 }. A linear operator .T ∈ B(H) is called a contraction if .||T || ≤ 1. The following deep result is due to Eschmeier [76]. Theorem 4.24 If .A = (A1 , A2 ) is a pair of commuting contractions on .H such that σH (A) is dominating for .D2 , then .A1 and .A2 have a nontrivial common invariant subspace.

.

4.2 Projective Spectrum

75

Literature Note For a comprehensive treatise on Banach algebras and Fredholm operator, we refer the reader to Douglas [66]. For more details about the joint spectrum in Definition 4.2, we refer the reader to Šilov [202]. Curto [45] is a treatise on the subject of Fredholm operator tuples. Two other extensive treatments of Taylor spectrum are Curto [47] and Eschmeier–Putinar [75]. Proposition 4.6 is due to Vasilescu [224] and Källström–Sleeman [147]. An effort to extend Taylor spectrum to noncommuting operators was made in Taylor [218]. For Harte spectrum, more details can be found in Harte’s original papers [118, 119]. Further study on its functional calculus was made in Choi and Davis [33], Turovskii [222], and Vasilescu [225]. However, this is only an incomplete list of literatures on this extensive subject. Exercise 4.25 1. Verify Proposition 4.5. 2. In the case .dim H < ∞ and .A = (A1 , A2 ) is a commuting pair, show that the following conditions are equivalent: (a) (b) (c) (d)

E(H, A) is exact. ker A1 ∩ ker A2 = {0}. .ran d0 = ker d1 . ∗ ∗ .ker A ∩ ker A = {0}. 1 2 . .

3. Prove Corollary 4.7. 4. Verify the claims in Example 4.20 for the case .n = 2. 5. For an almost commuting tuple A, show that .dp dp−1 is compact for each .0 ≤ p ≤ n.

4.2 Projective Spectrum While the spectral theory of commuting or almost commuting operators has taken a strong foothold, such a theory for noncommuting operators is conspicuously lacking. In an effort to address this issue, the notion of projective spectrum was introduced in 2009 [243]. It was motivated by a change of view on the classical spectrum .σ (A): Instead of regarding it as a property of A, we can see it as a reflection of the interaction between A and the unit I . This change, albeit small, makes it natural to consider the invertibility of the linear pencil .A1 − λA2 and, in more generality, the invertibility of the multiparameter pencil .A(z) = z1 A1 + · · · + zn An for .n ≥ 2. Therefore, we are inevitably led to the following definition. Definition 4.26 For elements .A1 , . . . , An in a Banach algebra .B, their projective spectrum is defined as .P (A) = {z ∈ Cn | A(z) is not invertible}. The projective resolvent set refers to the complement .P c (A) = Cn \ P (A). Here, the letter “A” in “.P (A)” stands for the tuple .(A1 , . . . , An ) or the pencil .A(z). Since .A(z) is homogeneous, projective spectrum and resolvent set can be defined in

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4 Projective Spectrum in Banach Algebras

the projective space .Pn−1 , namely, .p(A) := φ(P (A)) and .pc (A) := φ(P c (A)), where .φ stands for the canonical projection .Cn → Pn−1 . More details about projective space are given in Example 7.11. Earlier attempts on the spectral theory for noncommuting operators can be found in Sleman [206], Volkmer [229], Jeffries [136], as well as the references therein. The work directly led to the definition of projective spectrum is [242]. Some notable features of .P (A) are worth highlighting: 1. .P (A) is well-defined for general elements in .B, commuting or not. 2. It is consistent with the definition of characteristic polynomial for matrices. 3. Rather than considering the invertibility of the tuple .(A1 − λ1 I, . . . , An − λn I ) in earlier notions of joint spectra, projective spectrum places no emphasis on the unit I . Instead, it treats the elements in an equal footing by assigning a variable .zj to each .Aj . 4. Last but not least, it can be computed in an abundance of examples. Indeed, these features underpin many developments in the theory of projective spectrum. To avoid trivial complications, we shall always assume that the elements .A1 , . . . , An are linearly independent in .B. Hence, they span a n-dimensional subspace of .B. It is a classical fact that for every .a ∈ B, its spectrum .σ (a) is a nonempty compact subset of .C. A standard proof of this fact uses Liouville’s theorem in complex analysis. The same fact holds for .p(A). Its proof requires the following version of Hartogs extension theorem [146, Theorem 1.2.6]. Theorem 4.27 Suppose K is a compact subset of a domain .o ⊂ Cn such that .o\K is connected. Then every holomorphic function on .o \ K extends holomorphically to .o. An immediate consequence is that no holomorphic function of more than one variable has isolated singularities. The extension from .o \ K to .o is achieved one variable at a time by means of the Cauchy integral formula. More details will be given later in Example 4.31. Theorem 4.27 therefore remains valid for vector-valued holomorphic functions. Proposition 4.28 For any elements .A1 , . . . , An in a Banach algebra .B, their projective spectrum .p(A) is a nontrivial compact subset of .Pn−1 . Proof If .p(A) = Pn−1 , then there is nothing more to prove. Thus we assume .pc (A) is nonempty. To show .p(A) is closed, we assume .w ∈ pc (A). Then since .A(z) is continuous in z, there exists a neighborhood V of w such that .||A−1 (w)(A(z) − A(w))|| < 1 for all .z ∈ V . It follows that ( ) A(z) = A(w) I + A−1 (w)(A(z) − A(w))

.

is invertible for all .z ∈ V . This shows that .pc (A) is open in .Pn−1 . Since .Pn−1 is compact, the spectrum .p(A) is a compact subset. It remains to show that .p(A) is nontrivial, or equivalently, the set .P (A) contains nonzero vectors. On .P c (A), we have

4.2 Projective Spectrum

77

A−1 (z1 , z2 , . . . , zn ) − A−1 (z1' , z2 , . . . , zn ) ( ) = A−1 (z) A(z1' , z2 , . . . , zn ) − A(z) A−1 (z1' , z2 , . . . , zn )

.

= −(z1 − z1' )A−1 (z)A1 A−1 (z1' , z2 , . . . , zn ). This shows that .A−1 (z) is analytic in .z1 , and likewise, it is analytic in all other variables. Moreover, the calculation above implies that .

∂ −1 A (z) = −A−1 (z)Aj A−1 (z), 1 ≤ j ≤ n. ∂zj

(4.9)

Since the vector 0 cannot be an isolated singularity of .A−1 (z) due to Theorem 4.27, there must exist .z /= 0 for which .A(z) is not invertible. It follows that .p(A) is nontrivial. u n Proposition 4.28 is an interesting fact because the elements .A1 , . . . , An may have nothing to do with each other. Note that in the special case .A(z) = z0 I +z1 a, a ∈ B, it provides an alternative proof to the nonemptiness of the classical spectrum .σ (a).

4.2.1 Connection with Taylor Spectrum The case when the elements are commuting is of particular interest. Since Taylor spectrum is well-defined in this case, it is a natural question whether or how the projective spectrum .P (A) is related to the Taylor spectrum .σT (A). Proposition 4.29 Let .A = (A1 , . . . , An ) be a commuting tuple of operators on a Hilbert space .H. Then (a) .P (A) = ∪w∈σT (A) Hw , where .Hw = {z ∈ Cn | z1 w1 + · · · + zn wn = 0}. (b) If .o is an open neighborhood of .σT (A) and .f : o → Cm is holomorphic, then P (f (A)) = ∪w∈σT (A) Hf (w) .

.

Proof The proof is a simple application of the spectral mapping theorem (Theorem 4.11). Indeed, if we consider the linear function .fz (w) = z1 w1 + · · · + zn wn , w ∈ Cn , then .A(z) = fz (A), and hence .σ (A(z)) = fz (σT (A)) for each n .z ∈ C . This shows that .A(z) is not invertible if and only if there exists a .w ∈ σT (A) such that .fz (w) = 0. Part (a) follows. If .f : o → Cm is holomorphic, then by functional calculus .f (A) is a m-tuple of commuting operators. The spectral mapping theorem gives σT (f (A)) = {f (w) ∈ Cm | w ∈ σT (A)},

.

and hence, part (b) follows from part (a).

u n

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4 Projective Spectrum in Banach Algebras

In particular, projective spectrum of a commuting tuple is a union of hyperplanes. We shall have more to say about this in the next chapter. A simple example helps to illustrate Proposition 4.29. Example 4.30 The disk algebra .A(D) is the subalgebra of continuous functions on the closed unit .D that are analytic on .D. Its maximal ideal space M can be identified with .D by the map .ξ → φξ , where φξ (f ) := f (ξ ), f ∈ A(D), ξ ∈ D.

.

To avoid confusion, we let a stand for the identity function .a(ξ ) = ξ . Then it is clear that .σ (a) = D. Consider the tuple .A = (1, a, . . . , a m ) and the vector-valued function .h(ξ ) = (1, ξ, . . . , ξ m ). It follows from the spectral mapping theorem that σT (A) = σT (h(a)) = h(σ (a)) = {(1, ξ, . . . , ξ m ) | |ξ | ≤ 1}.

.

Hence, in light of Proposition 4.29, we have P (A) =

U

.

{z ∈ Cm+1 | z0 + z1 ξ + · · · + zm ξ m = 0}.

|ξ |≤1

A spectral mapping theorem for noncommuting operators is far from reach at this point. It is not even clear how to establish a proper functional calculus in this case.

4.2.2 Stein Domain The proof of Hartogs extension theorem (Theorem 4.27) is an application of the Cauchy integral formula, as illustrated in the following example concerning the polydisc. Example 4.31 For .0 < r, we set .Dr = {w ∈ C | |w| < r} and consider the Hartogs figure (Fig. 4.1) ) ( D2 := (D1/2 × D) ∪ D × (D \ D1/2 ) .

.

Every function f holomorphic on .D2 can be extended to the whole bidisc .D2 by using the Cauchy integral formula 1 fˆ(z) = 2π i

f

.

|ξ |=r

f (z1 , ξ )dξ , max{1/2, |z2 |} < r < 1. ξ − z2

It is not hard to verify that .fˆ agrees with f on .D2 (Fig. 4.1).

4.2 Projective Spectrum

79

Fig. 4.1 The Hartogs figure

This example directly motivated two definitions about complex domains in .Cn . We let ) ( n−1 n−1 .Dn := (D × (D \ D1/2 ) 1/2 × D) ∪ D be the generalized Hartogs figure in .Cn . A complex domain .o ⊂ Cn is said to be pseudoconvex if for every injective holomorphic map .h : Dn → Cn , the condition n .h(Dn ) ⊂ o implies .h(D ) ⊂ o, Oka [178], and Takeuchi [215]. In other words, .o is pseudoconvex if whenever it contains a holomorphic copy of the Hartogs figure, it contains a holomorphic copy of the polydisc that envelopes it. A domain .o is called a domain of holomorphy if there exists a function f holomorphic on .o that does not have holomorphic extension to any domain larger than .o. In a seemingly different approach, a domain .o ⊂ Cn is said to be Stein if it is holomorphically convex, namely, for every compact subset .K ∈ o, its holomorphic convex hull Kˆ := {λ ∈ o | |f (λ)| ≤ sup |f (z)|, ∀f ∈ Hol(o)}

.

z∈K

is also a compact subset of .o. Remarkably, the following is true. Theorem 4.32 The following statements are equivalent for a complex domain: (a) It is pseudoconvex. (b) It is a domain of holomorphy. (c) It is Stein. For details on this development, we refer the reader to Siu [204] and Krantz [146]. Other equivalent definitions have been shown later, one of which is by the following local extension property [146, Theorem 3.4.5 (3)]. Theorem 4.33 A complex domain in .o is not Stein if and only if there exists a λ ∈ ∂o, a connected neighborhood V of .λ, and an open subset .V1 ⊂ V ∩ o with

.

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4 Projective Spectrum in Banach Algebras

λ ∈ ∂V1 such that for every function .f ∈ Hol(o) there exists a .g ∈ Hol(V ) such that .f = g on .V1 .

.

In other words, every .f ∈ Hol(o) above extends holomorphically to V through V1 . A good example to illustrate the need of .V1 for the extension is the set .o = C \ [0, ∞) with .λ = 1. In this case, there is no holomorphic function on any neighborhood of 1 that agrees with .log z on both sides of the x-axis. Since every domain in the complex plane .C is Stein (why?), any Cartesian product of domains in .C is Stein. The following are some well-known properties of Stein domains. .

Proposition 4.34 The following hold for a complex domain: (a) The union of an increasing sequence of Stein domains is Stein. (b) If .o = ∩∞ j =1 oj , where.{oj } is a sequence of Stein domains, then .o is Stein. (c) If .f : o → Cm is a holomorphic map and .f (o) is Stein, then .o is Stein. Another important property of Stein domain is that its de Rham cohomology can be computed by holomorphic differential forms. We will return to this point in Chap. 6. Regarding projective spectrum, it is possible that .P (A) = Cn . For example, if A is a tuple of compact operators on an infinite dimensional Hilbert space, then the pencil .A(z) is compact and hence not invertible for any .z ∈ Cn . In what follows, we shall assume .P c (A) is nonempty. A subset S in a Banach algebra .B is said to be weakly bounded if .φ(S) is bounded in .C for every .φ ∈ B∗ . The Uniform Boundedness Principle asserts that every weakly bounded subset S is bounded in norm, Rudin [196, Theorem 3.18]. We are now in position to state the following property of projective spectrum. Theorem 4.35 For a tuple A of elements in .B, every path-connected component of P c (A) is Stein.

.

Proof We will prove by contradiction. Suppose .o is a path-connected component of .P c (A) and it is not Stein. Let .λ ∈ ∂o, neighborhood V , and open set .V1 be as in Theorem 4.33. Note first that .λ ∈ P (A). Pick a small .r > 0 such that the closed ball B(λ, r) := {z ∈ Cn | |z − λ| ≤ r} ⊂ V .

.

Now for every .φ ∈ B∗ , since .φ(A−1 (z)) is holomorphic on .o, it is holomorphic on .V1 . So by Theorem 4.33 there exists a holomorphic function .Fφ (z) on V such that −1 (z)) = F (z) on .V , i.e., the function .F (z) is a holomorphic extension of .φ(A φ 1 φ −1 (z)) to V through .V . Since .F (z) is bounded on the compact set .B(λ, r), .φ(A 1 φ so is the function .φ(A−1 (z)) on .B(λ, r) ∩ V1 . This shows that .A−1 (z) is weakly bounded and hence bounded in norm on .B(λ, r) ∩ V1 . Therefore, there exists a constant .M > 0 such that .||A−1 (z)|| ≤ M for any .z ∈ B(λ, r) ∩ V1 . Pick any sequence .{λn } in .B(λ, r) ∩ V1 convergent to .λ. Then for any positive integers n and m, we have

4.2 Projective Spectrum

81

||A−1 (λn ) − A−1 (λm )|| = ||A−1 (λn )(A(λm ) − A(λn ))A−1 (λm )||

.

≤ ||A−1 (λn )||||A(λm ) − A(λn )||||A−1 (λm )|| ≤ M 2 ||A(λm − λn )||. This means .{A−1 (λn )} is a Cauchy sequence in .B and therefore converges to an element B in .B. Now since ||A−1 (λn )A(λ) − I || = ||A−1 (λn )(A(λ) − A(λn ))|| ≤ M||A(λ − λn )||,

.

we have .BA(λ) − I = limn→∞ A−1 (λn )A(λ) − I = 0. Similar argument shows that .A(λ)B = I , and it follows that .A(λ) is invertible in .B. This contradicts the assumption .λ ∈ ∂o ⊂ P (A). u n This theorem is instrumental for our study in Chap. 6 on the topology of .P c (A). The proof above does not rely too much on the pencil .A(z). With a small modification, it works for general .B-valued holomorphic functions. Moreover, it is worth noting that the condition .B being a Banach algebra is indispensible for the theorem. One important class of counterexamples is the group algebra. A countable group G is said to be torsion free if it contains no element of finite order greater than 1. Kaplansky’s unit conjecture claims that if G is torsion free, then every invertible element in .C[G] is of the form .αg for some .g ∈ G and .α ∈ C× . These issues are discussed further in exercise 9. Projective spectrum can be computed in many important examples, as we have already seen in the case with characteristic polynomial in the finite dimensional cases. In the following sections, we will look at some infinite dimensional cases associated with the Cuntz algebra, the irrational rotation algebra, and the free group von Neumann algebra. Exercise 4.36

E 1. Given a sequence .{an } ⊂ D satisfying . ∞ j =1 1 − |aj | < ∞, the Blaschke ||∞ w−aj product .B(z) := j =1 1−aj w is a nontrivial holomorphic function on .D. Show that if .{an } converges to .λ ∈ T, then .B(z) has no holomorphic extension to a neighborhood of .λ. 2. Use the above exercise to construct a holomorphic function f on .D that has no holomorphic extension to any larger domain. 3. Prove Proposition 4.34 using the definition of Stein domain. 4. Let h be a plurisubharmonic function on .Cn and let .o be the set of points n .z ∈ C for which h is pluriharmonic in a neighborhood of z. Show that .o is Stein (if it is nonempty). (Hint: Show that .o is pseudoconvex.) 5. Suppose .A∗ (z) = I + z1 A1 + · · · + zn An , where .Aj ∈ K(H) for each j : (a) Show that .A∗ (z) has nontrivial kernel and cokernel at every .z ∈ P (A∗ ). (b) Show that .P c (A∗ ) is path-connected.

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4 Projective Spectrum in Banach Algebras

6. Find an operator tuple .A = (A1 , A2 , A3 ) such that .P c (A) has at least two path-connected components. (Hint: Use Fredholm operators and consider their indices.) 7*. Let .An = {z ∈ Cn | Im zj > 0, j = 1, . . . , n} and suppose .A = (A1 , . . . , An ) is a tuple of positive operators in .B(H). Prove that .P (A) is either .Cn or .P (A) ∩ An = ∅. 8. To generalize Theorem 4.35, let .f : Cn → B be a holomorphic function and define .P (f ) = {z ∈ Cn | f (z) is not invertible}. Show that every pathconnected component of .P (f ) is Stein. (Note: A study of this topic can be found in [15, 247].) 9. Theorem 4.35 does not hold for a general complex algebra. Consider the infinite cyclic group .G = and the pencil .A(z) = z0 + z1 t ∈ C[G]. Show that c 2 .P (A) = {(z0 , 0), (0, z1 ) ∈ C | z0 z1 /= 0}. Project 4.37 1. For a n-tuple A of linear operators, is the complex dimension of .P (A) necessarily greater than or equal to .n − 1? 2. Two operators .S, T ∈ B(H) are said to commute essentially if their commutator ∗ in which T is said to be .[S, T ] is compact. A special case is when .S = T essentially normal. Is there a condition on the projective spectrum of .(I, S, T ) that is not satisfied when S and T essentially commute? 3. How to rephrase Theorem 4.24 in terms of the projective spectrum of .(I, A1 , A2 )?

4.3 Two Important C ∗ -Algebras Given any abelian monoid (a unital semigroup) .G+ , the quotient G = (G+ × G+ )/{(x, x) | x ∈ G+ }

.

is called the Grothendieck group of .G+ in which every element .[(x, y)] has an inverse .[(y, x)]. In a .C ∗ -algebra .B (not necessarily unital), an element p is called a projection if .p = p∗ and .p2 = p, and an element .s ∈ B is said to be a partial isometry if .s ∗ s is a projection. Note that this implies that .ss ∗ is also a projection. We let .P(B) denote the set of projections in .B. Two projections p and q are said to be equivalent (denoted by .p ∼ q) if there is a partial isometry s such that ∗ ∗ = q. However, the sum of two projections is not necessarily .s s = p and .ss a projection. To make a valid definition of addition, we consider the direct limit .M∞ (B) of matrix algebras .Mk (B) under the embedding .a → diag(a, 0). Then the set of equivalence classes of projections in .P(M∞ (B)) becomes an abelian monoid with addition defined by .[p] + [q] = [p ⊕ q]. When .B is unital, the K-group .K0 (B) is the Grothendieck group of .P(M∞ (B)), and it plays an important role in the study of .C ∗ -algebras. We refer the reader to Karoubi [142], Blackadar [19], and Davidson [57] for an introduction to this subject.

4.3 Two Important C ∗ -Algebras

83

4.3.1 The Cuntz Algebra For .n ≥ 2, the Cuntz algebra .On [49] is the .C ∗ -algebra generated by n isometries .S = (S1 , . . . , Sn ) satisfying .

Si∗ Sj = δij I n E

for 1 ≤ i, j ≤ n, .

Si Si∗ = I.

(4.10) (4.11)

i=1

Note that such isometric tuple S does not exist in the finite dimensional case because every isometric matrix is in fact a unitary and hence (4.11) is not satisfied. Therefore, if .On is represented as a subalgebra of .B(H), then .ker(Sj∗ ) = ∞ for each j . Thus, the structure of .On is independent of the choice of the isometries. In fact, the Cuntz algebra is the first concrete example of a separable infinite simple .C ∗ algebra. Setting .pj = Sj Sj∗ , we observe that .pj ∼ I for each j . Moreover, since .pi pj = 0 for .i /= j due to (4.10), we have n[I ] = [p1 ] + · · · + [pn ] = [p1 + · · · + pn ] = [I ],

.

∼ Zn−1 . which means .(n−1)[I ] = 0. Indeed, it is shown in Cuntz [50] that .K0 (On ) = In this section, we compute the projective spectrum of the Cuntz tuple S. To this end, we first fix a faithful irreducible representation .π of .On on a Hilbert space .H. Clearly, an element .a ∈ On is invertible if and only if .π(a) is invertible as an operator on .H. In other words, the discussion of the projective spectrum is not affected by the choice of such representations. Lemma 4.38 The projective spectrum .P (S) = Cn . Proof We set .S(z) = z1 S1 + · · · + zn Sn . It is obvious that .S(0) = 0, so in what follows we assume that .z /= 0. There are two cases to consider. Case 1: n is even. For a nonzero .x ∈ H, we let y = (¯z2 S1 − z¯ 1 S2 + · · · + z¯ n Sn−1 − z¯ n−1 Sn )x.

.

Conditions (4.10) and (4.11) imply that the .Sj s are isometries on .H with mutually orthogonal ranges. Therefore, ||y||2 = ||(¯z2 S1 − z¯ 1 S2 + · · · + z¯ n Sn−1 − z¯ n−1 Sn )x||2

.

= |z2 |2 ||S1 x||2 + |z1 |2 ||S2 x||2 + · · · + |zn |2 ||Sn−1 x||2 + |zn−1 |2 ||Sn x||2 = (|z1 |2 + · · · + |zn |2 )||x||2 /= 0. Moreover, we have

84

4 Projective Spectrum in Banach Algebras

S ∗ (z)y = (¯z1 S1∗ + · · · + z¯ n Sn∗ )(¯z2 S1 − z¯ 1 S2 + · · · + z¯ n Sn−1 − z¯ n−1 Sn )x

.

∗ = z¯ 1 z¯ 2 (S1∗ S1 − S2∗ S2 )x + · · · + z¯ n−1 z¯ n (Sn−1 Sn−1 − Sn∗ Sn )x = 0,

and hence, .S(z) is not invertible. Thus the lemma holds if n is even. Case 2: n is odd. For .x /= 0, let y = (¯z2 S1 − z¯ 1 S2 + · · · + z¯ n−1 Sn−2 − z¯ n−2 Sn−1 )x.

.

Then .||y||2 = (|z1 |2 + · · · + |zn−1 |2 )||x||2 . If one of .z1 , . . . , zn−1 is nonzero, then .y /= 0, and a direct calculation verifies S ∗ (z)y = (¯z1 S1∗ + · · · + z¯ n Sn∗ )(¯z2 S1 − z¯ 1 S2 + · · · + z¯ n−1 Sn−2 − z¯ n−2 Sn−1 )x

.

∗ ∗ = z¯ 1 z¯ 2 (S1∗ S1 − S2∗ S2 )x + · · · + z¯ n−2 z¯ n−1 (Sn−2 Sn−2 − Sn−1 Sn−1 )x

= 0, which indicates that .S(z) is not invertible. If .z1 = · · · = zn−1 = 0, then .zn /= 0. But in this case, we have .S(z) = zn Sn , which is not onto. Thus the lemma also holds in this case. u n Since .P (S) = Cn , the resolvent set .P c (S) is empty. To make the study more interesting, we consider the projective spectrum of the pencil .S∗ (z) = I + z1 S1 + · · · + zn Sn . Given a Hilbert space .H with orthonormal basis .{ek }k≥0 , the unilateral shift W is the linear operator that sends .ek → ek+1 for each k. If .K is another Hilbert space with the identity map denoted by .IK , then .W ⊗ IK on .H ⊗ K is called a unilateral shift of multiplicity .dim K. The following lemma, known as the von Neumann–Wold decomposition, is needed to establish the next theorem. We leave its proof as an exercise. Also recall that .Bn stands for the open unit ball of .Cn . Lemma 4.39 Every isometry V on a Hilbert space .H is of the form .V = U ⊕ W , where U is unitary and W is a unilateral shift of multiplicity .dim(H o V H). Theorem 4.40 The projective resolvent set .P c (S∗ ) = Bn . Proof .S∗ (0) is obviously invertible. In what follows, we assume .z /= 0. First, (4.10) and (4.11) imply S ∗ (z)S(z) = (¯z1 S1∗ + · · · + z¯ n Sn∗ )(z1 S1 + · · · + zn Sn )

.

= |z1 |2 S1∗ S1 + · · · + |zn |2 Sn∗ Sn = (|z1 |2 + · · · + |zn |2 )I = ||z||2 I. Hence, if we let .V (z) = S(z) ||z|| , then .V (z) is an isometry for every .z /= 0. By Lemma 4.39, we can write .V (z) = Uz ⊕ Wz , where .Uz is unitary and .Wz is a unilateral shift. By Lemma 4.38, the isometry .V (z) is not invertible and hence not unitary. It follows that .Wz /= 0. Consequently, we have .D = σ (Wz ) ⊆ σ (V (z))

4.3 Two Important C ∗ -Algebras

85

(exercise 2). But .V (z) is a contraction; hence, we must also have .σ (V (z)) ⊆ D. This concludes that .σ (V (z)) = D. Moreover, since (

) 1 .S∗ (z) = I + ||z||V (z) = ||z|| V (z) + I , ||z|| it is invertible if and only if .||z|| < 1, or equivalently .z ∈ Bn .

u n

4.3.2 The Irrational Rotation Algebra Now we consider .L2 (T, m), where m is the normalized Lebesgue measure on the unit circle .T. Fix a .θ ∈ (0, 1) and consider the two unitary operators defined by Uf (w) = f (e2π iθ w), Vf (w) = wf (w), f ∈ L2 (T, m).

.

(4.12)

Then the commutation relation .U V = e2π iθ V U holds. The operator V is called the bilateral shift, and it will play an important role in Sect. 5.1. The irrational rotation algebra .Aθ is the .C ∗ -algebra generated by U and V when .θ is irrational. It is universal in the sense that every pair of unitaries u and v satisfying this commutation relation generates a .C ∗ -algebra that is isomorphic to .Aθ through the map .u → U, v → V . Elements of the form E .a = am,n U m V n , where .{am,n } is a finite set of complex numbers, make a dense subset of .Aθ . A canonical trace .tr on .Aθ can be defined by .tr a = a0,0 , and it gives rise to a beautiful identity. Theorem 4.41 .tr K0 (Aθ ) = Z + Zθ. Moreover, .tr P(Aθ ) = (Z + Zθ ) ∩ [0, 1]. It follows readily that two irrational rotation algebras .Aθ1 and .Aθ2 are isomorphic if and only if .θ1 = θ2 or .θ1 + θ2 = 1. Two .C ∗ -algebras .A and .B are said to be stably isomorphic if .A⊗K(H) and .B⊗ K(H) are isomorphic, where .K(H) is the .C ∗ -algebra of compact operators on .H. For unital .C ∗ -algebras, stable isomorphism is equivalent to the Morita equivalence [17]. In order to describe when two irrational rotation algebras .Aθ1 and .Aθ2 are ˆ = C ∪ {∞} by fractional stably isomorphic, we consider the action of .GL2 (Z) on .C linear maps, namely, ( .

) aw + b ab ˆ · w := , w ∈ C. cd cw + d

86

4 Projective Spectrum in Banach Algebras

Observe that since .a, b, c, d are integers, . aw+b cw+d is irrational if and only if w is irrational. Define the subgroup .G = {M ∈ GL2 (Z) | det M = ±1}. Theorem 4.42 Two irrational rotation algebras .Aθ1 and .Aθ2 are stably isomorphic if and only if .θ1 = θ2 modulo the action of G on the set of irrational numbers. For details about the aforementioned results, we refer the reader to Rieffel [193]. The projective spectrum .P (A) for the pencil .A(z) = z0 I + z1 U + z2 V in .Aθ can be readily computed based on the following theorem due to Fang–Jiang–Lin–Xu [79]. Theorem 4.43 For any irrational .θ , we have ⎧ T ⎪ ⎪ ⎨ .σ (U + αV ) = D ⎪ ⎪ ⎩ αT

0 < α < 1, α = 1, α > 1.

Observe that the spectrum above is independent of the choice of .θ . Corollary 4.44 For .θ irrational, we have P (A) = {|z0 | = |z1 | ≥ |z2 |} ∪ {|z1 | = |z2 | ≥ |z0 |} ∪ {|z0 | = |z2 | ≥ |z1 |}.

.

Proof We first show that .P (A) is invariant under the action of .T on each coordinate. Let .λ = e2π iθ . Since .θ is irrational, the set .A = {λn | n = 1, 2, . . .} is dense in the unit circle. Assume .z0 I + z1 U + z2 V is not invertible. Then the identities z0 U + z1 U 2 + z2 U V = z0 U + z1 U 2 + z2 λV U = (z0 I + z1 U + z2 λV )U

.

imply that .z0 I +z1 U +z2 λV is not invertible. Repeating this argument continuously, we see that .z0 I + z1 U + z2 ηV is not invertible for each .η ∈ A. Since .A is dense in the unit circle and .P (A) is closed, .z0 I + z1 U + z2 ηV is not invertible for every .η ∈ T. Using similar arguments, the same conclusion can be reached for .z0 I + z1 ηU + z2 V , and consequently, .z0 ηI + z1 U + z2 V . In light of this fact, we may assume in the sequel that .z0 , z1 , z2 are nonnegative real numbers. Assume .z1 > 0. Then .z0 I + z1 U + z2 V is not invertible if and only if ( ) z2 . − z0 ∈ σ (z1 U + z2 V ) = z1 σ U + V . z1 Theorem 4.43 and the fact .σ (U ) = σ (V ) = T (exercise) lead to the following three cases: (1) When .0 ≤ z2 < z1 , we have .z0 = z1 . (2) When .z1 = z2 , we have .z0 ≤ z1 . (3) When . zz12 > 1, we have .z0 = z2 .

4.4 Free Group von Neumann Algebras

87

Observe that the case .z1 = 0 falls in the third case. Taking into account of the invariance under the action of .T, the three cases above give the three components in .P (A). u n It follows that the resolvent set .P c (A) is the following disjoint union of pathconnected components: {|z0 | > max{|z1 |, |z2 |}} ∪ {|z1 | > max{|z0 |, |z2 |}} ∪ {|z2 | > max{|z0 |, |z2 |}}. (4.13)

.

Exercise 4.45

n k 1. Prove Lemma 4.39. (Hint: Set .H1 = ∞ k=0 V H and analyze V on .H1 and on .H o H1 .) 2. Prove that for the unilateral shift W , we have .σ (W ) = D. (Hint: Show that .W − λI is not onto for every .λ ∈ D.) 3. Let .M = (S1 + · · · + Sn )/n, where .S1 , . . . , Sn are the generators of the Cuntz algebra. Compute the classical spectrum .σ (M) using Theorem 4.40. 4. The operator U in (4.12) is an example of composition operator. Show that .σ (U ) = σ (V ) = T when .θ is irrational. What if .θ is rational? ( )−1 as a series in U and V . 5. Let U and V be as in (4.12). Express . I + 41 (U + V ) Can you recover .θ from this expression? Project 4.46 1. Compute the spectrum .σ (U + αV ) in Theorem 4.43 in the case .θ is rational.

4.4 Free Group von Neumann Algebras Nonabelian free groups play an important role in group theory as well as in operator algebras. One example is its use in the Banach–Tarski paradox, which led von Neumann to define the notion of amenable groups. A property relevant to the study in this section is the Nielsen–Schreier theorem which asserts that every subgroup of a free group is itself isomorphic to a free group. In particular, no element of finite order (other than the unit) exists in a free group. This section determines the projective spectrum of free groups. Let .G = Fn be the free group on n generators .g1 , . . . , gn , where .n ≥ 2, and let the Hilbert space .HG , the regular representation .λ, the reduced .C ∗ -algebra .Cλ∗ (G), the von Neumann algebra .L(G), as well as the canonical trace .tr on .L(G) be as defined in Sect. 2.1. For convenience, we denote .L(G) by .Bn . Both .Bn and .Cλ∗ (G) are well-studied examples of operator algebras. The following are some well-known facts due to Powers [186], Pimsner–Voiculescu [185], and Ge [96, 97]: (1) .Cλ∗ (G) is simple and has a unique trace, namely the restriction of .tr to .Cλ∗ (G). (2) .Cλ∗ (G) contains no nontrivial idempotents. (3) .Bn is not isomorphic to a tensor product of two factors of type II.1 .

88

4 Projective Spectrum in Banach Algebras

An intriguing unsolved problem posed by Kadison in the 1960s is whether .B2 is isomorphic to .B3 . At present, it is known that either .Bn is isomorphic to .Bm for all .m, n ≥ 2 or no two of them are isomorphic, Radulescu [188] and Voiculescu– Dykema–Nica [232].

4.4.1 Noncommutative Probability Space The free probability theory was developed initially by Voiculescu [230, 231] in an effort to solve Kadison’s problem. Its main setting is the noncommutative probability space defined as follows. Definition 4.47 A noncommutative probability space .(A, τ ) consists of: (1) An associative complex algebra .A with unit I , and (2) A fixed linear function .τ : A → C such that .τ (I ) = 1 The pair .(Bn , tr ) is a good example of noncommutative probability space. Although Kadison’s problem remains open, free probability theory has found applications in solving some other problems. For more details, we refer the reader to [170, 232]. For convenience, we set .Uj = λ(gj ), j = 1, . . . , n, where .λ is the regular representation of G. Although the elements .U1 , . . . , Un are apparently “free” to one another in .Bn , it is rather technical to define the freeness for general elements in a noncommutative probability space .(A, τ ). To serve the purpose here, we assume in the sequel that .A is a von Neumann algebra with a faithful normalized trace .τ . An element .a ∈ A is said to be centered if .τ (a) = 0. The von Neuman subalgebra generated by .I, a and .a ∗ is denoted by .A(a). Definition 4.48 A family .{aj | j ∈ J } of centered elements in .A are said to be ∗-free if for all .k ≥ 1 we have

.

τ (xj1 · · · xjk ) = 0, ∀ xjs ∈ A(ajs ), 1 ≤ s ≤ k,

.

whenever .j1 , . . . , jk ∈ J are such that .js /= js+1 , s = 1, . . . , k − 1. Further, a unitary element .u ∈ A is called a Haar unitary if .τ (uk ) = 0 for all integers .k /= 0. Evidently, the .∗-freeness depends on the choice of .τ . Intuitively speaking, if .τ cannot “see” any interaction among the elements in the family, then these elements are regarded as being .∗-free to each other. Example 4.49 One verifies that the elements .U1 , . . . , Un are .∗-free in .(Bn , tr ), which conforms with our understanding. Moreover, since every .g /= 1 in G has infinite order, we have tr (λk (g)) = tr (λ(g k )) = = 0, k /= 0,

.

i.e., .λ(g) is a Haar unitary in .(Bn , tr ).

4.4 Free Group von Neumann Algebras

89

The notions of .∗-freeness and Haar unitary give rise to the definition of an important class of elements in .(A, τ ). Definition 4.50 An element .T ∈ (A, τ ) is said to be R-diagonal if T √ has polar decomposition .U |T | such that U is a Haar unitary and is free from .|T | = T ∗ T . It follows in particular that every Haar unitary itself is R-diagonal. It is also clear that if T is R-diagonal then so is .αT for any nonzero .α ∈ C. Proposition 4.51 Let a and b be .∗-free R-diagonal elements in .A. Then both .a + b and ab are R-diagonal. The following theorem determines the spectrum of R-diagonal elements. For √ simplicity, we set .||T ||2 = τ (T ∗ T ). Theorem 4.52 Let T be an R-diagonal element in .(A, τ ). Then: (a) .σ (T ) = {w ∈ C | |w| ≤ ||T ||2 } if T is not invertible. (b) .σ (T ) = {w ∈ C | ||T −1 ||−1 2 ≤ |w| ≤ ||T ||2 } if T is invertible. It follows readily that in this case the spectral radius .r(T ) = ||T ||2 . R-diagonal element was initially defined through the R-transform of joint distributions for a pair of elements in .(A, τ ). For details regarding this subject as well as Proposition 4.51 and Theorem 4.52, we refer the reader to Haagerup–Larsen [121] and Nica– Speicher [177].

4.4.2 Spectral Properties Regarding the projective spectrum of the pencil .U (z) = z1 U1 + · · · + zn Un in .Bn , we have the following fact. Theorem 4.53 The projective resolvent set .P c (U ) = ∪nj=1 oj , where oj = {z ∈ Cn | 2|zj |2 > ||z||2 }, j = 1, . . . , n.

.

Proof First, using Proposition 4.51 repeatedly we see that .U (z) = z1 U1 + · · · + zn Un is R-diagonal for all nonzero .z ∈ Cn . Since .g1 , . . . , gn are free and G is an orthonormal basis for .HG , for every .k ≥ 1, we have ||U k (z)||22 =

.

= = ||z||2k . In particular, Theorem 4.52 implies that the spectral radius .r(U (z)) = ||U (z)||2 = ||z||. It follows that .αI + U (z) is invertible in .Bn whenever .|α| > ||z||, and the series

90

4 Projective Spectrum in Banach Algebras

(αI + U (z))−1 = α −1

(E ∞

.

(−1)k

k=0

U k (z) αk

)

( ) ∗ m k is convergent in norm. Moreover, since .tr (U (z)) U (z) = 0 when .m /= k, we have ||(αI

.

+ U (z))−1 ||22

= |α

−1 2

|

(E ∞ k=0

= |α −1 |2

∞ ( E k=0

||U k (z)||22 |α|2k ||z|| |α|

)2k

)

= (|α|2 − ||z||2 )−1 .

(4.14)

We can now prove the corollary by induction. For .n = 2, we have .U (z) = U1 (z1 + z2 U1−1 U2 ), which is invertible if and only if .|z1 | > |z2 | or .|z2 | > |z1 |. Hence, the corollary holds in this case. Assume it holds for .n = m and consider the pencil .U (z) = z1 U1 + · · · + zm+1 Um+1 . Without loss of generality, we assume |z1 | ≥ max{|z2 |, . . . , |zm+1 |}.

.

(4.15)

Write U (z) = Um+1 (z1 U1' + · · · + zm Um' + zm+1 I ) = Um+1 (U ' (z' ) + zm+1 I ),

.

−1 where .z' = (z1 , . . . , zm ) ∈ Cm . Then .Uj' = Um+1 Uj , j = 1, . . . , m are also .∗-free Haar unitaries, and hence, .U ' (z' ) is R-diagonal for all nonzero vector .z' . In view of Theorem 4.52, there are two cases:

(1) If .U ' (z' ) is not invertible, then .U (z) is invertible when .|zm+1 |2 > ||U ' (z' )||22 = |z1 |2 + · · · + |zm |2 . But this would contradict the assumption (4.15). Hence, .U (z) is never invertible in this case. (2) If .U ' (z' ) is invertible, then the induction assumption and (4.15) indicate that ' .z ∈ o1 for .n = m. It follows from Theorem 4.52 (b) that .U (z) is invertible if ' ' and only if .||(U ' (z' ))−1 ||−1 2 > |zm+1 |. Factoring .U1 out of .U (z ) from the left and using (4.14) for .α = z1 , we have 2 2 2 1/2 ||(U ' (z' ))−1 ||−1 > |zm+1 |. 2 = (|z1 | − |z2 | − · · · − |zm | )

.

In summary, with the assumption (4.15), we see that .U (z) is invertible if and only if .z ∈ o1 . The proof is thus completed by induction and the symmetry of the elements .U1 , . . . , Un . u n For the free group .Fn+1 = , n ≥ 1, we consider the pencil Bλ (z) = z0 λ(g1 ) + · · · + zn λ(gn+1 ), where .λ is the regular representation of .Fn+1 .

.

4.4 Free Group von Neumann Algebras

91

Clearly, it is invertible if and only if the pencil .Aλ (z) = z0 I + z1 λ(g1−1 g2 ) + · · · + zn λ(g1−1 gn+1 ) is invertible. Since .{g1−1 g2 , . . . , g1−1 gn+1 } generates .Fn , the projective spectrum .P (Aλ ) for .Fn is determined by Theorem 4.53 for .Fn+1 . Corollary 4.54 Consider the free group .Fn with generators .g1 ,n . . . , gn and the pencil .Aλ (z) = z0 I + z1 λ(g1 ) + · · · + zn λ(gn ). Then .P (Aλ ) = nj=0 ocj , where n+1 | 2|z |2 > ||z||2 }, j = 0, 1, . . . , n. .oj = {z ∈ C j Given a group G generated by a finite set .S = {g1 , . . . , gn } and a representation (π, H) of G, the associated Markov operator is defined as

.

Mπ =

.

1 (π(g1 ) + · · · + π(gn )). n

Apparently, .Mπ depends on the generating set S, but what is important is that it possesses properties that are intrinsic to G and .π. We will have more to say about Markov operator in Chap. 9. The following is a simple consequence of Corollary 4.54. Corollary 4.55 Let .Mλ be the Markov operator of .Fn with respect to the regular representation .λ, where .n ≥ 2. Then .σ (Mλ ) = {w ∈ C | |w| ≤ √1n }. It is worth noting that the spectrum of the “real part” of .Mλ , namely .Re Mλ := + Mλ∗ ), has been determined in Kesten [143]. [ √ ] √ Theorem 4.56 .σ (Re Mλ ) = − n1 2n − 1, n1 2n − 1 . 1 2 (Mλ

In general, for a non-normal operator T , it is not clear how .σ (T ) and .σ (Re T ) are related. Exercise 4.57 1. Prove that if T is a bounded normal operator, then .σ (Re T ) = {Re z | z ∈ σ (T )}. 2. Let .ξ be a nontrivial nth root of unity, i.e., .ξ n = 1 and .ξ /= 1. Compute the spectrum of the operator .U1 + ξ U2 + · · · + ξ n−1 Un . 3. Let .H0 be the hyperplane in .Cn+1 determined by the equation .z0 + · · · + zn = 0. Describe the set .H0 ∩ P (Aλ ), where .P (Aλ ) is the projective spectrum in Corollary 4.54. 4. Prove Corollary 4.55. Project 4.58 1. Does Corollary 4.55 imply Theorem 4.56? 2. Let V be the Volterra operator on .L2 [0, 1] defined by f

x

Vf (x) =

.

f (t)dt, f ∈ L2 [0, 1].

0

It is well-known that V is quasinilpotent, for instance, see [66]. Determine σ (Re V ).

.

Chapter 5

The C ∗ -Algebra of the Infinite Dihedral Group D∞ .

.

The infinite dihedral group .D∞ = is isomorphic to the free product .Z2 ∗Z2 . As the simplest nonabelian infinite group, it plays an important role in group theory as well as in several other areas of mathematics. Geometrically, the generators a and t can be realized as reflections through two hyperplanes (lines) in 2 ∗ ∗ .R in generic position. Since .D∞ is amenable, its reduced .C -algebra .Cr (D∞ ) ∗ ∗ and its full .C -algebra .C (D∞ ) are isomorphic. Hence, we shall denote both of them simply by .C ∗ (D∞ ). Topics in this chapter include projective spectrum, projections in generic position, Fuglede–Kadison determinant, and an application to the Grigorchuk group of intermediate growth. A good reference of this chapter is [108].

5.1 Projective Spectrum of the Generators D∞ contains the normal subgroup N = (which is isomorphic to Z) with D∞ /N ∼ = Z2 . So elements in D∞ are of the form (at)k or t (at)k , where k ∈ Z. For example, a = t (at)−1 . Hence, the group algebra C[D∞ ] consists of elements in the form E E .h = hk (at)k + h,j t (at)j . k

j

The canonical trace tr on C[D∞ ], defined by tr (h) = h0 , is a faithful tracial state on C[D∞ ], and we have the Hilbert space decomposition HD∞ = HN ⊕ tHN , where { HN = f =

∞ E

.

−∞

| ∞ } | E 2 | fj (at) | |fj | < ∞ j

−∞

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 R. Yang, A Spectral Theory Of Noncommuting Operators, https://doi.org/10.1007/978-3-031-51605-4_5

93

5 The .C ∗ -Algebra of the Infinite Dihedral Group .D∞

94

with inner product = tr (f ∗ h) =

∞ E

.

fj hj .

−∞ dθ Clearly, the map U : (at)k → eikθ , k ∈ Z, identifies HN with L2 (T, 2π ), and the multiplication by at on HN , denoted by T , is thus unitarily equivalent to the bilateral shift operator f → eiθ f on L2 (T). For convenience, we shall denote the bilateral shift also by T .

5.1.1 Regular Representation via the Bilateral Shift Since the left regular representation of .λ : D∞ → U (HD∞ ) is the multiplication (Sect. 2.1), the decomposition .HD∞ = HN ⊕ tHN gives a(at)j = t (at)j −1 ∈ tHN , and a[t (at j )] = (at)j +1 ∈ HN .

.

It follows that ) 0 Tt , .λ(a) = tT ∗ 0 (

) 0t . λ(t) = t 0 (

Now consider the map .V : HD∞ → HN ⊕ HN defined by ( V =

.

) I0 0 , 0 t

where .I0 is the identity operator on .HN . It is clear that V is unitary and .V 2 = I . Further, one verifies that ) 0 T V, T∗ 0

(

( λ(a) = V

.

λ(t) = V

) 0 I0 V. I0 0

Thus, the representation .λ can be equivalently described on .L2 (T) ⊕ L2 (T) as ) 0 T , T∗ 0

(

( λ(a) =

.

λ(t) =

) 0 I0 . I0 0

(5.1)

This enables us to compute the projective spectrum of the pencil .R(z) = z0 + z1 λ(a) + z2 λ(t) in .C ∗ (D∞ ) by functional calculus. Consider a general normal operator N on .H with its spectral decomposition

5.1 Projective Spectrum of the Generators

95

f N=

ξ dE(ξ ).

.

σ (N )

We denote the .C ∗ -algebra generated by I and N by .C ∗ (N ). Then by functional calculus, the map .κ : C(σ (N)) → C ∗ (N ) defined by f κ(f ) = f (N) :=

f (ξ )dE(ξ )

.

σ (N )

is a .∗-isomorphism. Given any .k ∈ N, .κ extends naturally to a map .κˆ : C(σ (N )) ⊗ Mk (C) → C ∗ (N ) ⊗ Mk (C) such that f ( ( ) ( ) ) fij (ξ ) dE(ξ ). .κ ˆ (fij ) = fij (N ) := σ (N )

( ) In particular, the block matrix (Lemma)5.1 The map .κˆ is a .∗-isomorphism. fij (N ) is invertible if and only if .det fij (ξ ) does not vanish on .σ (N).

.

The proof is left as an exercise. The projective spectrum of the pencil .R(z) can be computed with the assistance of Lemma 5.1. U Theorem 5.2 .P (R) = −1≤x≤1 {z ∈ C3 | z02 − z12 − z22 − 2z1 z2 x = 0}. Proof It is well-known that the bilateral shift T is a unitary with spectrum .σ (T ) = T, and its spectral decomposition is given by f T =

.

T

(5.2)

ξ dE(ξ ),

where E is the associated projection-valued spectral measure such that .E(S) is the multiplication by the characteristic function .χS on .L2 (T) for any Lebesgue measurable subset .S ⊂ T. Using (5.1), we can now write (

z0 z1 T + z2 z1 T ∗ + z2 z0 f ( ) z0 z1 T ξ dE(ξ ) + z2 f = z1 T ξ¯ dE(ξ ) + z2 z0 ) f ( z0 z1 ξ + z2 = dE(ξ ), ¯ z z0 ξ + z 1 2 T

)

R(z) = z0 + z1 λ(a) + z2 λ(t) =

.

where the last integral can be viewed as a spectral decomposition of .R(z). It follows from Lemma 5.1 that .R(z) is invertible if and only if (

z0 z1 ξ + z2 . det ¯ z1 ξ + z2 z0

)

= z02 − (z1 ξ + z2 )(z1 ξ¯ + z2 )

5 The .C ∗ -Algebra of the Infinite Dihedral Group .D∞

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= z02 − z12 − z22 − z1 z2 (ξ + ξ¯ ) /= 0, ∀ξ ∈ T. The proof is thus completed by setting .x = Re ξ .

u n

5.1.2 Connectedness of the Resolvent Set The following fact will be needed later. Corollary 5.3 The resolvent set .P c (R) is path-connected. Proof We first look at the case .z0 /= 0. Since .R(z) is homogeneous, without loss of generality, we assume .z0 = 1 and set .R∗ (z) = 1 + z1 λ(a) + z2 λ(t). Then Theorem 5.2 implies that P (R∗ ) =

U

.

{z ∈ C2 | 1 − z12 − z22 − 2z1 z2 x = 0}.

−1≤x≤1

We show that .P c (R∗ ) is path-connected. To this end, we check that every point c .λ = (λ1 , λ2 ) ∈ P (R∗ ) is path-connected to .(0, 0). By possibly choosing a point in a small ball centered at .λ and using the symmetry of .P (R∗ ) in .z1 and .z2 , we may assume without loss of generality that .|λ1 | > |λ2 | > 0. Consider the complex line .Cλ = Cλ = {(wλ1 , wλ2 ) | w ∈ C}. The intersection .Cλ ∩ P c (R∗ ) contains c .(0, 0) and .(λ1 , λ2 ). It is thus sufficient to check that .Cλ ∩ P (R∗ ) is path-connected. Observe that U .Cλ ∩ P (R∗ ) = {(wλ1 , wλ2 ) | 1 − w2 (λ21 + λ22 ) − 2w2 λ1 λ2 x = 0}. −1≤x≤1

Solving for .w 2 , we have .w 2 = (λ21 + λ22 + 2λ1 λ2 x)−1 , and hence, we have two solution curves in .C: L± = {w± (x) := ±(λ21 + λ22 + 2λ1 λ2 x)−1/2 | −1 ≤ x ≤ 1}.

.

Now we verify the following two claims: 1. .L+ ∩ L− = ∅. This is because if there are .x1 , x2 ∈ [−1, 1] such that .w+ (x1 ) = 2 (x ) = w 2 (x ), which implies w− (x2 ), then .w+ 1 − 2 λ21 + λ22 + 2λ1 λ2 x1 = λ21 + λ22 + 2λ1 λ2 x2 ,

.

and hence, .x1 = x2 . It follows that .w+ (x1 ) = w− (x1 ), which happens only when both are 0, which is impossible.

5.2 Two Projections in Generic Position

97

2. Neither .L+ nor .L− self-intersects. If there are .x1 , x2 ∈ [−1, 1] such that .w+ (x1 ) = w+ (x2 ) (or .w− (x1 ) = w− (x2 )), then by similar arguments we have .x1 = x2 . In summary, the set .Cλ ∩ P (R∗ ) is a disjoint union of two simple curves in .C. Hence, its complement in .Cλ is path-connected. This concludes that .P c (R∗ ), and hence, .P c (R) ∩ {z0 /= 0} is path-connected. For the case .z0 = 0, we see by Theorem 5.2 that a fixed .(0, z1 , z2 ) ∈ P c (R) if and only if .z12 + z22 + 2z1 z2 x /= 0 for all .x ∈ [−1, 1]. Since .[−1, 1] is compact, there is a constant .δ such that .|z12 + z22 + 2z1 z2 x| ≥ δ > 0, x ∈ [−1, 1]. Pick .ξ such that 2 c .0 < |ξ | < δ; then the path .(sξ, z1 , z2 ), 0 ≤ s ≤ 1 lies in .P (R) by Theorem 5.2, and it connects .(0, z1 , z2 ) to .(ξ, z1 , z2 ), which in turn is connected to .(1, 0, 0) by the preceding argument. This completes the proof. u n Exercise 5.4 1. Prove Lemma 5.1. 2. Show that the projective spectrum .P (R) in Theorem 5.2 contains the hyperplane .{z0 + z1 + z2 = 0}. 3. Use Theorem 5.2 to compute .σ (λ(a) + αλ(t)), where .α is a complex constant. 4. Consider the finite dihedral group .Dn = . Compute the projective spectrum of the pencil .z0 I + z1 λ(a) + z2 λ(t), where .λ is the left regular representation of .Dn . Project 5.5 1. Given a finite abelian group K and an action .τ : K → Aut(Z), using Lemma 5.1 to investigate the projective spectrum of .Z xτ K.

5.2 Two Projections in Generic Position Given two projections P and Q in a Hilbert space .H, we set .M = ran P and .N = ran Q. The size of the intersections M ∩ N, M ⊥ ∩ N, M ∩ N ⊥ , M ⊥ ∩ N ⊥

.

(5.3)

reflects the relative positions of M and N . We may decompose .H in the following way: L = (L ∩ N ) ⊕ (L ∩ N ⊥ ) ⊕ M0 and L⊥ = (L⊥ ∩ N) ⊕ (L⊥ ∩ N ⊥ ) ⊕ M1 ,

.

where .M0 and .M1 are closed subspaces of L and .L⊥ , respectively. Therefore, H = (L ∩ N ) ⊕ (L ∩ N ⊥ ) ⊕ (L⊥ ∩ N) ⊕ (L⊥ ∩ N ⊥ ) ⊕ M0 ⊕ M1 .

.

(5.4)

5 The .C ∗ -Algebra of the Infinite Dihedral Group .D∞

98

On the part (L ∩ N) ⊕ (L ∩ N ⊥ ) ⊕ (L⊥ ∩ N) ⊕ (L⊥ ∩ N ⊥ ),

.

(5.5)

we have .P = (1, 1, 0, 0), .Q = (1, 0, 1, 0), where .(α00 , α01 , α10 , α11 ) stands for α00 I(L∩N ) ⊕ α01 I(L∩N ⊥ ) ⊕ α10 I(L⊥ ∩N ) ⊕ α11 I(L⊥ ∩N ⊥ ) .

.

With respect to the decomposition (5.4), Halmos [115] shows the following fact. Theorem 5.6 We have .dim M0 = dim M1 , and if it is nonzero, then up to unitary equivalence, ( ) ( ) √ H I 0 H (1 − H ) , Q = (1, 0, 1, 0) ⊕ √ , .P = (1, 1, 0, 0) ⊕ 00 H (1 − H ) H where the operator .H := P QP |M0 is a positive, self-adjoint contraction such that 0 ≤ H ≤ I and .ker H = ker(I − H ) = {0}.

.

According to Halmos, the part (5.5) is “thoroughly uninteresting.” The spaces M and N are said to be in generic position if this part is trivial. Two projections P and Q on .H are said to be in generic position if .ran P and .ran Q are in generic position. Such pairs have been well studied in the literature. For more details, we refer the reader to Böettcher-Spitkovsky [27].

5.2.1 Universal Projections in C ∗ (D∞ ) Returning to .C ∗ (D∞ ), we set .p = (I − λ(a))/2 and .q = (I − λ(t))/2. Then p and q are two projections. We leave it as an exercise to verify that p and q are in generic position. They are universal in the following sense [195]: If .(P , Q) is an arbitrary pair of projections in a Hilbert space .H, then there exists a unital .∗ homomorphism .π : C ∗ (D∞ ) −→ B(H) such that .π(p) = P and .π(q) = Q. Therefore, if .A(z) = z0 + z1 p + z2 q is invertible in .C ∗ (D∞ ), then π(A(z)) = z0 π(I ) + z1 π(p) + z2 π(q) = z0 I + z1 P + z2 Q =: AP ,Q (z)

.

is invertible in .B(H). To determine the projective spectrum .P (A), we write ( ) z1 + z2 z1 z2 A(z) = z0 + − λ(a) − λ(t). 2 2 2

.

It follows from Theorem 5.2 that .A(z) is not invertible if and only if

5.2 Two Projections in Generic Position

99

( ) ( )2 ( )2 z1 z2 z1 z2 z 1 + z2 2 z0 + cos θ − − − 2 2 2 2 z1 z2 (1 − cos θ ) = z02 + z0 (z1 + z2 ) + 2 θ = z02 + z0 (z1 + z2 ) + z1 z2 sin2 = 0, 2

.

for some .0 ≤ θ < 2π . Summarizing these facts and setting .x = sin2 θ2 , we arrive at the following discovery. Corollary 5.7 For the pencil .A(z) = z0 + z1 p + z2 q, we have U

P (A) =

.

{z ∈ C3 | z02 + z0 (z1 + z2 ) + z1 z2 x = 0}.

0≤x≤1

Moreover, for an arbitrary pair of projections .(P , Q), we have .P (AP ,Q ) ⊂ P (A). Considering the case .x = 0 in Corollary 5.7, we see that .P (A) contains the slice {z ∈ C3 | z0 (z0 + z1 + z2 ) = 0},

.

i.e., the hyperplanes .{z0 + z1 + z2 = 0} and .{z0 = 0} are in .P (A). In particular, this shows that .z1 p + z2 q is not invertible for any .z1 and .z2 . Given an arbitrary pair of projections P and Q in generic position, the operator .H = P QP is a positive contraction, and it reflects the angle between .ran P and .ran Q in the sense that .

sup{|| | u ∈ ran P , v ∈ ran Q, ||u|| = ||v|| = 1} = sup{|

| | u, v ∈ H, ||u|| = ||v|| = 1} = ||QP || = ||H ||1/2 .

Thus, if .||H || < 1, then .ran P and .ran Q have a positive angle. With respect to the decomposition .H = ran P ⊕ ran(I − P ), we can write (up to a unitary equivalence) P =

.

( ) I 0 , 00

) ( √ H (I − H ) H . Q= √ H (I − H ) I −H

(5.6)

This makes it convenient to determine the invertibility of the pencil of .z1 P + z2 Q. For simplicity, a pair of projections in generic position is also called a normal pair. Lemma 5.8 Let P and Q be a normal pair of projections: (a) If .||H || = 1, then .z1 P + z2 Q is not invertible for all .z1 , z2 ∈ C. (b) If .||H || < 1, then .z1 P + z2 Q is not invertible if and only if .z1 z2 = 0.

5 The .C ∗ -Algebra of the Infinite Dihedral Group .D∞

100

Proof If .z1 z2 = 0, then .z1 P + z2 Q is clearly not invertible. Thus, without loss of generality, we assume .z2 = 1 and .z1 /= 0. For (a), since H is self-adjoint, there exists a sequence of unit vectors .{xn } ⊂ ran(I − P ) such that .(I − H )xn → 0. It follows that )( ) ( ) ( √ 0 0 z1 I + H H (I − H ) .(z1 P + Q) = √ xn xn H (I − H ) I −H (√ ) H (I − H )xn = → 0, (I − H )xn which indicates that .z1 P + Q is not invertible for any .z1 . In case (b), .I − H is invertible. If we set 1 .D = z1

(

) / I − H (I − H )−1 / , − H (I − H )−1 (z1 I + H )(I − H )−1

(5.7)

then a direct computation verifies that .(z1 P + Q)D = D(z1 P + Q) = I . Thus z1 P + Q is invertible for all .z1 /= 0. u n

.

For the pencil .AP ,Q (z) = z0 I + z1 P + z2 Q, we have the following fact due to Wu–Jiang–Ruan–Qian [238]. Theorem 5.9 If P and Q are a normal pair of projections and .H = P QP , then U

P (AP ,Q ) =

.

{z ∈ C3 | z02 + z0 (z1 + z2 ) + z1 z2 x = 0}.

x∈σ (I −H )

Proof The theorem is easy to verify if either .z1 = 0 or .z2 = 0. Thus we assume z1 z2 /= 0 and set .ξi = zz2i , i = 0, 1 in the sequel. In view of (5.6), we need to determine the invertibility of the operator

.

) ( √ (ξ0 + ξ1 )I + H H (I − H ) √ . .M = ξ0 I + ξ1 P + Q = H (I − H ) (ξ0 + 1)I − H If we set ( R=

.

) √ (ξ0 + 1)I − H − H (I − H ) √ , − H (I − H ) (ξ0 + ξ1 )I + H

then ) (ξ0 + ξ1 )(ξ0 + 1)I − ξ1 H 0 . .RM = MR = 0 (ξ0 + ξ1 )(ξ0 + 1)I − ξ1 H (

5.2 Two Projections in Generic Position

101

Observe that RM commutes with R. Thus if .(ξ0 + ξ1 )(ξ0 + 1)I − ξ1 H is invertible, then so is M. On the other hand, since H is normal, if .(ξ0 + ξ1 )(ξ0 + 1)I − ξ1 H is not invertible, then there exists a sequence .{xn } ⊂ ran P of unit vectors such that (ξ0 + ξ1 )(ξ0 + 1)xn − ξ1 H xn → 0.

.

(5.8)

It follows that ) ( ) ( (ξ0 + ξ1 )(ξ0 + 1)xn − ξ1 H xn ((ξ0 + 1)I − H )xn √ = → 0. .M − H (I − H )xn 0 If M were invertible, then we would have .limn→∞ (ξ0 + 1)xn − H xn = 0. Another look at (5.8) gives .ξ0 (ξ0 + 1)xn → 0, which leads to two cases: .ξ0 = 0 or .ξ0 = −1. The first case implies .||H || = 1 and .M = ξ1 P + Q. In the second case, we have , , , .M = ξ1 P −(I −Q) and .H xn → 0. If we set .Q = I −Q and .H = P Q P = P −H , then P and .Q, are a normal pair and .(I −H , )xn = H xn → 0, and hence, .||H , || = 1. Thus, in either case, Lemma 5.8 indicates that M is not invertible, leading to a contradiction. In summary, in the case .z1 z2 /= 0, pencil .AP ,Q (z) is not invertible if and only if (z02 + z0 (z1 + z2 ))I + z1 z2 (I − H )

.

is not invertible, or equivalently, .−(z02 + z0 (z1 + z2 ))/(z1 z2 ) ∈ σ (I − H ), and the theorem follows. u n Since H is a positive contraction, .σ (I − H ) is a subset of .[0, 1]. Therefore, the spectrum .P (AP ,Q ) is a subset of the spectrum .P (A) in Corollary 5.7.

5.2.2 Two Projection Matrices In the case .dim H < ∞, and P and Q are two general projections, not necessarily in generic position, Theorem 5.6 enables one to compute the characteristic polynomial .Φ(z) := det(z0 + z1 P + z2 Q). Indeed, the theorem gives the decomposition ) ( ) ( Q1 0 P1 0 , and Q = , .P = 0 P2 0 Q2 where .P1 = (1, 1, 0, 0), .Q1 = (1, 0, 1, 0), .

) ( ) ( √ H (1 − H ) I 0 H , P2 = , and Q2 = √ H (1 − H ) I −H 00

where .H = P2 Q2 P2 . Then we have

5 The .C ∗ -Algebra of the Infinite Dihedral Group .D∞

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det(z0 I + z1 P + z2 Q) ( ) z0 I + z1 P1 + z2 Q1 0 = det 0 z0 I + z1 P2 + z2 Q2

.

= det(z0 I + z1 P1 + z2 Q1 ) det(z0 I + z1 P2 + z2 Q2 ). A simple calculation gives .

det(z0 I + z1 P1 + z2 Q1 ) = z0k1 (z0 + z1 )k2 (z + z2 )k3 (z0 + z1 + z2 )k4 ,

(5.9)

where .k1 , k2 , k3 , and .k4 are the dimensions of .L⊥ ∩ N ⊥ , L ∩ N ⊥ , L⊥ ∩ N , and .L ∩ N , respectively. Furthermore, we have ( det(z0 I + z1 P2 + z2 Q2 ) = det

.

) √ z0 I + z1 I + z2 H z2 H (I − H ) √ . z2 H (I − H ) z0 I + z2 (I − H ).

Since all the block entries are commuting and H is diagonalizable, the above determinant is equal to ( ) det (z0 I + z1 I + z2 H )(z0 I + z2 (I − H ) − z22 H (I − H )) ( ) = det (z02 + z0 z1 + z0 z2 )I + z1 z2 (I − H ) ) || ( z02 + z0 (z1 + z2 ) + z1 z2 x , =

.

(5.10)

x∈σˆ (I −H )

where .σˆ (I − H ) stands for the set of eigenvalues of .I − H counting multiplicity. Summarizing the computations above, we can state the following theorem [133]. Theorem 5.10 Let P and Q be projection matrices. Then their characteristic polynomial Φ(z) = z0k1 (z0 +z1 )k2 (z+z2 )k3 (z0 +z1 +z2 )k4

||

.

(

) z02 +z0 (z1 +z2 )+z1 z2 x ,

x∈σˆ (I −H )

where .(k1 , k2 , k3 , k4 ) and H are as defined before. Since .ker H = ker(I − H ) = {0}, we have .σˆ (I − H ) ⊂ (0, 1). It is a pleasure to check that the factor .z02 + z0 (z1 + z2 ) + z1 z2 x in Theorem 5.10 is irreducible for every .x ∈ (0, 1). Since the degree of the characteristic polynomial is equal to the size of the involved matrices, the theorem exposes an interesting phenomenon essentially due to the fact .dim M0 = dim M1 in Theorem 5.6. Corollary 5.11 No two projections in .Mk (C) are in generic position if k is odd.

5.2 Two Projections in Generic Position

103

Literature Note . The study of pencils of projections and idempotents is extensive. To close this section, we mention some interesting results. A contraction on H is said to be generic if it does not have eigenvalues ±1. The following fact due to Shi–Ji–Du [205] characterizes the difference of two projections P and Q. Theorem 5.12 Up to unitary equivalence, a self-adjoint contraction A can be written in the form P −Q if and only if A = 0⊕I ⊕−I ⊕A+ ⊕−A+ , where I stands for the identity operator on a subspace, and A+ is a generic positive contraction. Thus, P − Q is compact if and only if A+ is compact and the components I and −I are of finite rank. Further, if P − Q is in the Schattern-(2n + 1) class, where n ∈ N, then we have the following formula due to Avron–Seiler–Simon [5]: tr (P − Q)2m+1 = dim(ker Q ∩ ran P ) − dim(ker P ∩ ran Q), m ≥ n.

.

Moreover, it follows that if P and Q are in generic position, then tr (P −Q)2m+1 = 0 for all m ≥ n. In contrast to the difference of two projections, pencils of three or more projections can be a rather general. Stampfli [211] made a surprising discovery that every operator in B(H) is a pencil of 257 projections. The number was quickly reduced to 16 in Pearcy–Topping [184]. If we only allow sums of projections, then Fillmore [78] proved the following fact. Theorem 5.13 Any self-adjoint operator on H with spectrum in [3, 5] is a sum of eight projections. A different spectral condition was used in Choi–Wu [51]. Recall that given an operator T ∈ B(H), its essential norm ||T ||e = inf{||T + K|| | K ∈ K(H)}.

.

It is not hard to see that if T is a sum of a finite number of projections and dim H = ∞, then ||T ||e ≥ 1. Theorem 5.14 Any positive operator T with ||T ||e > 1 is the sum of finitely many projections. It is not known whether the number of projections required in this case may be estimated in terms of ||T ||e . In comparison, if we consider sums of idempotents, [211] offered another interesting result. Theorem 5.15 Every operator in B(H) is the sum of eight idempotents. It is also not clear whether the number eight above is minimal. Exercise 5.16 1. Use Theorem 5.9 to determine σ (P + αQ), where α ∈ C, and study the special case P = p, Q = q. 2. Show that the projections p and q in Corollary 5.7 are in generic position.

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3. Consider two projection matrices P , Q ∈ Mk (C) and Theorem 5.10. (a) Show that they are in generic position if and only if Φ(z) has no linear factor. (b) Prove Corollary 5.11. (c) Compute det(z0 I + z1 P + z2 Q + z3 P Q). 4. Show that Theorem 5.10 implies Theorem 1.24. Project 5.17 1. Consider three projection matrices p = (p1 , p2 , p3 ) of equal size: (a) Can their characteristic polynomial Qp be computed? (b) What are the possible degrees of Qp ’s irreducible factors? (c) Is Qp a complete unitary invariant of the tuple p? 2*. Assume G = , and λ is its regular representation. (a) Can the projective spectrum of z0 I + z1 λ(a1 ) + z2 λ(a2 ) + z3 λ(a3 ) be determined? (b) If pi = (I − λ(ai ))/2, i = 1, 2, 3, then is the pencil z1 p1 + z2 p2 + z3 p3 non-invertible for any z ∈ C3 ?

5.3 Fuglede–Kadison Determinant A von Neumann algebra .B ⊂ B(H) is said to be a factor if it has trivial center. It is said to be finite if every isometry in .B is a unitary. It is known that .B is finite if and only if it possesses a faithful tracial state that is weakly continuous on the unit ball of .B. In this case, given any two equivalent projections p and q with .p ≤ q, we must have .p = q (why?). An infinite dimensional finite factor is called a type II.1 factor. The group von Neuman algebras .L(G), equipped with the canonical trace .tr defined in Sect. 2.1, provide abundant examples of such factors. For details, we refer the reader to Murray–Neumann [165, 166], Dixmier [63], and Kadison–Ringrose [145].

5.3.1 Basic Properties and Harpe–Skandalis Extension We fix a type II.1 factor .B with a tracial state .φ and let .GL(B) denote the group of non-singular elements in .B. The Fuglede–Kadison determinant (FK determinant for short) for √ elements .x ∈ GL(B) is defined as .detF K x := exp(φ(log |x|)) [82]. Here .|x| = x ∗ x, and .log |x| is defined by functional calculus, i.e., f .

log |x| =

log λdE(λ), σ (|x|)

(5.11)

5.3 Fuglede–Kadison Determinant

105

where E is the spectral measure of .|x|. The non-singularity condition on x ensures that .σ (|x|) is bounded away from 0, and thus the integral is convergent. It follows that f .φ(log |x|) = log λdφ(E(λ)). (5.12) σ (|x|)

The following properties are listed in [82]: (1) .detF K (xy) = detF K xdetF K y. (2) .detF K : (GL(B), || · ||) → (0, ∞) is continuous. (3) .detF K x ≤ r(x), where .r(x) is the spectral radius of x. The FK determinant has an interesting consequence about the trace .φ. Theorem 5.18 For any .a ∈ B, the value of .φ(a) lies inside the convex hull of .σ (a). As a particular example, if a is quasinilpotent, i.e., .σ (a) = {0}, then .φ(a) = 0. FK determinant can be extended as an upper semi-continuous function to singular elements in .B, in which case the convergence of the integral in (5.11) becomes a subtle issue because now 0 may be in .σ (|x|). If the integral is convergent, then .detF K x /= 0 (even though x may be singular); if the integral is equal to .−∞, then .det x = 0. The following definition aims to distinguish these two kinds of singular elements in .B. Definition 5.19 A singular element .x ∈ B is said to be .φ-singular if .detF K x = 0. It is known [82, Lemma 5] that if .{xn } ⊂ B converges to x in norm, then .

lim sup detF K xn ≤ detF K x. n

In particular, if .detF K x = 0, then .limn detF K xn = 0. Corollary 5.20 FK determinant is continuous at .φ-singular elements in .B. The existence of spectral measure E on .σ (|x|) relies on the fact that the von Neumann algebras .B have a sufficient amount of projections. This requirement prevents a direct extension of FK determinant to general .C ∗ -algebras. Indeed, many ∗ .C -algebras do not have any nontrivial projections, for instance .C(T). A way to get around this difficulty is described in Harpe–Skandalis [122]. Let .B be a unital .C ∗ -algebra and .GL0 (B) denote the path-connected component of .GL(B) that contains I . Given an element .x ∈ GL0 (B) and a piecewise-smooth path in 0 .GL (B) parametrized by .γ (s), 0 ≤ s ≤ 1, such that .γ (0) = I and .γ (1) = x, the integral Aγ (x) :=

.

1 2π i

f

1 0

( ) φ γ −1 (s)dγ (s)

(5.13)

5 The .C ∗ -Algebra of the Infinite Dihedral Group .D∞

106

depends only on the homotopy class of the path .γ . Whenever the value of detH S x := exp(2π iAγ (x))

.

(5.14)

is independent of the choice of .γ , it defines the Harpe–Skandalis (HS) determinant of x. In particular, in the case .B is a type II.1 factor and .φ is the canonical tracial state, the modulus .| detH S x| coincides with .detF K x [117, Corollary 14]. So from this point of view, HS determinant is an extension of FK determinant. Observe that (5.14) implies ( ) d log detH S γ (s) = φ γ −1 (s)dγ (s) .

.

(5.15)

The following example draws a connection between FK determinant and the ordinary determinant for a matrix. Its proof is left as an exercise. Example 5.21 Assume .B = Mk (C) and .φ = k1 tr . Then .φ is a tracial state on .B, and we have .detF K x = | det x|1/k , x ∈ B. Hence x is .φ-singular if and only it is singular. For more details on this topic, we refer the reader to Harpe [117].

5.3.2 A Two-Variable Jacobi’s Formula The differential form .γ −1 (s)dγ (s) in (5.13) is a case of the Maurer–Cartan form. In the matrix case formula, (5.15) is Jacobi’s formula. This subsection computes the formula explicitly for the pencil .R(z) = z0 + z1 λ(a) + z2 λ(t), where .λ is the representation of .D∞ given in (5.1). The Maurer–Cartan form for pencils of general operators will be the focus of the next chapter. Thus, this section can be viewed as a precursor. For simplicity, we consider the pencil .R∗ (z) = I + z1 λ(a) + z2 λ(t), and we shall write it as .I + z1 a + z2 t in the sequel. The Maurer–Cartan form of the pencil .R∗ is defined as ωR∗ (z) = R∗−1 (z)dR∗ (z) = R∗−1 (z)(adz1 + tdz2 ), (z1 , z2 ) ∈ P c (R∗ ).

.

In order to compute .tr ωR∗ , equation (5.1) suggests that we first need to transplant the canonical trace .tr on .L(D∞ ) to the von Neumann algebra (denoted by .L{T }) generated by the bilateral shift T . Recall that .N = {(at)k | k ∈ Z} is a normal subgroup of .D∞ , and the unitary dθ map .U : HN → L2 (T, 2π ) defined by .U (at)k = ekiθ , .k ∈ Z, furnishes a unitary equivalence between the bilateral shift operator T on .L2 (T) and the multiplication by at on .HN . Therefore, we have .U ∗ L{T }U = L(N ), and we can define a tracial state .φ on .L{T } by .φ(A) := tr (U ∗ AU ).

5.3 Fuglede–Kadison Determinant

107

Lemma 5.22 Let E be the spectral measure of the bilateral shift T in (5.2). Then dθ φ(dE(eiθ )) = 2π .

.

Proof For any f in the group algebra .C[N ] ⊂ HN , Uf is a polynomial in .e±iθ , and we have f = =

.

2π

(Uf )(eiθ )

0

dθ . 2π

(5.16)

On the other hand, .(Uf )(T ) is a polynomial in T and .T ∗ , and hence by functional calculus = tr (f ) = φ ((Uf )(T )) ) f ( f 2π iθ iθ (Uf )(e )dE(e ) = =φ

.

0

2π

(Uf )(eiθ )φ(dE(eiθ )).

0

Comparing this with (5.16), we obtain f

2π

.

(Uf )(eiθ )

0

f

dθ = 2π

This implies .φ(dE(eiθ )) = weak topology.

dθ 2π

2π

(Uf )(eiθ )φ(dE(eiθ )), f ∈ C[N ].

0

because .C[N ] is dense in .L(N ) with respect to the u n

The tracial state .φ on .L{T } can be lifted to .L{T } ⊗ M2 (C) so that ) ( 1 a11 a12 = φ(a11 + a22 ), aij ∈ L{T }. .φ a21 a22 2

(5.17)

We are now fully equipped to establish the following formula for the pencil .R∗ . Lemma 5.23 On the projective resolvent set .P c (R∗ ), we have ( φ(ωR∗ (z)) = ∂

.

where .∂ =

∂ ∂z1 dz1

+

1 4π

f

2π 0

) log(1 − z12 − z22 − 2z1 z2 cos θ )dθ ,

∂ ∂z2 dz2 .

Proof By (5.1), we have ] z 1 T + z2 I0 . .R∗ (z) = z 1 T ∗ + z2 I0 [

Since all the entries are commuting, setting .K(z) = I0 − (z1 T + z2 )(z1 T ∗ + z2 ), we have

5 The .C ∗ -Algebra of the Infinite Dihedral Group .D∞

108

−1 .R∗ (z)

=K

−1

[

] I0 −(z1 T + z2 ) (z) . (z1 T ∗ + z2 ) I0

Expressing .R∗−1 (z) as a block matrix .(R j k (z))2×2 , we can write ] [ 0 T 0 dz1 + .ωR∗ (z) = T∗ 0 I0 [ 12 [ 12 ∗ 11 ] R R T R T dz + = 1 22 ∗ 21 R T R T R 22 R∗−1 (z)

([

] ) I0 dz2 0 ] R 11 dz2 . R 21

Using the spectral decomposition (5.2) and Lemma 5.22, we obtain 1 1 φ(R 12 T ∗ + R 21 T )dz1 + φ(R 12 + R 21 )dz2 2 2 f (z1 ξ + z2 )ξ¯ + (z1 ξ¯ + z2 )ξ −1 = φ(dE(ξ ))dz1 2 T 1 − z12 − z22 − z1 z2 (ξ + ξ¯ ) f (z1 ξ + z2 ) + (z1 ξ¯ + z2 ) −1 + φ(dE(ξ ))dz2 2 T 1 − z12 − z22 − z1 z2 (ξ + ξ¯ ) f (2z1 + z2 (ξ + ξ¯ ))dz1 + (z1 (ξ¯ + ξ ) + 2z2 )dz2 −1 = φ(dE(ξ )) 2 T 1 − z12 − z22 − z1 z2 (ξ + ξ¯ ) f 2π 1 ∂ log(1 − z12 − z22 − 2z1 z2 cos θ )dθ = 4π 0 ( ) f 2π 1 =∂ log(1 − z12 − z22 − 2z1 z2 cos θ )dθ . 4π 0

φ(ωR∗ (z)) =

.

u n

5.3.3 On the Fundamental Group of the Resolvent Set In order to derive a formula for .detF K R∗ (z), we first need to check whether definition (5.14) is independent of the choice of path in this case. Therefore, some topological information about the resolvent set .P c (R∗ ) is required. First, recall that c .P (R∗ ) is path-connected due to Corollary 5.3. Let .γ (s), 0 ≤ s ≤ 1, be a piecewisesmooth loop in .P c (R∗ ) such that .γ (0) = γ (1) = (0, 0). Theorem 5.2 implies that for every fixed .x ∈ [−1, 1] the function .Lx (z) := 1 − z12 − z22 − 2z1 z2 x does not vanish on .P c (R∗ ). In particular, we have Lx (γ (s)) = 1 − z12 (s) − z22 (s) − 2z1 (s)z2 (s)x /= 0, ∀s ∈ [0, 1].

.

5.3 Fuglede–Kadison Determinant

109

We define the winding number .W (γ ) of .γ around .P (R∗ ) as the winding number of Lx (γ ) around 0, namely,

.

1 2π i

W (γ ) :=

.

1 2π i

=

f f

Lx (γ ) 1 0

1 dw w

1 1 dLx (z(s)) = Lx (z(s)) 2π i

f 0

1

d log Lx (z(s))ds. ds

It is connected with the integral .Aγ in (5.13). Lemma 5.24 For any piecewise-smooth loop .γ in .P c (R∗ ), we have .Aγ (I ) = W (γ ) 2 . Proof Observe that since .Lx (z) is linear in x, the above integrals show that .W (γ ) is continuous with respect to x. But since .W (γ ) is integer-valued, it is a constant with respect to the change of x, i.e., the value of .W (γ ) is independent of the choice of .x ∈ [−1, 1]. Lemma 5.23 then implies Aγ (I ) =

.

= = = =

1 2π i

f γ

φ(ωR∗ (z))

) f 2π f ( 1 1 2 2 log(1 − z1 − z2 − 2z1 z2 cos θ )dθ ∂ 4π 0 2π i γ ) f 1 f 2π ( 1 d 1 2 2 log(1 − z1 (s) − z2 (s) − 2z1 (s)z2 (s) cos θ )ds dθ 2π i 0 ds 4π 0 ) f 1 f 2π ( 1 1 d log Lcos θ (z(s))ds dθ 2π i 0 ds 4π 0 f 2π 1 W (γ ) . W (γ )dθ = 2 4π 0 u n W (γ ) 2

in Lemma 5.24 depends only on the Since the map .A : γ → Aγ (I ) = homotopy class of .γ , it induces a map (also denoted by .A) from the fundamental group .π1 (P c (R∗ )) to the group . 12 Z. Corollary 5.25 The map .A : π1 (P c (R∗ )) → 12 Z is an epimorphism. Proof By Lemma 5.24, .A([γ ]) = W 2(γ ) ∈ 12 Z for every .[γ ] ∈ π1 (P c (R∗ )). The fact that .A is a group homomorphism is a standard property of the winding number. Thus, it only remains to check that .A is surjective. To this end, we consider the path γ = {γ (s) = (1 + e2π is /2, 0) | 0 ≤ s ≤ 1},

.

5 The .C ∗ -Algebra of the Infinite Dihedral Group .D∞

110

in which case .R∗ (γ (s)) = I + (1 + e2π is /2)λ(a). Since the classical spectrum .σ (λ(a)) = {±1}, we have .(±1, 0) ∈ P (R∗ ). Observe that .γ is a loop that only encloses .(1, 0). Apparently, for every .x ∈ [−1, 1], Lx (γ (s)) = 1 − (1 + e2π is /2)2 = −e2π is − e4π is /4 /= 0, s ∈ [0, 1].

.

Hence by Theorem 5.2, the path .γ is inside .P c (R∗ ). Its winding number W (γ ) =

.

1 2π i

1 = 2π i

f

1

d log Lx (γ (s))ds ds

1

−2(1 + e2π is /2)π ie2π is ds = −ei2π s − e4π is /4

0

f

0

f 0

1

(1 + e2π is /2)ds . 1 + e2π is /4

Using the geometric series for .(1 + e2π is /4)−1 , we verify easily that .W (γ ) = 1. Then Lemma 5.24 gives .A([γ ]) = W 2(γ ) = 12 , and consequently, .A(n[γ ]) = nW (γ ) = n2 , n ∈ Z. This completes the proof. u n 2 It is worth mentioning that although . 12 Z is isomorphic to .Z, the factor . 12 tells more about the map .A and the property of .D∞ , in particular, the fact .a 2 = 1.

5.3.4 The FK Determinant of C ∗ (D∞ ) We are now in position to prove the following fact. Theorem 5.26 For the pencil .R∗ (z) = I + z1 λ(a) + z2 λ(t), we have (

1 .detF K R∗ (z) = exp 4π

f

2π 0

) log |1 − z12

− z22

− 2z1 z2 cos θ|dθ ,

and it is continuous on .C2 . Proof Since .P c (R∗ ) is path-connected, for every .(z1 , z2 ) ∈ P c (R∗ ), there is a piecewise-smooth path .γ0 = {z(s) = (z1 (s), z2 (s)) | 0 ≤ s ≤ 1} in .P c (R∗ ) connecting .z(0) = (0, 0) to .z(1) = (z1 , z2 ). Then Lemma 5.23 implies ) ( f 2π ( ) d 1 φ ωR∗ (z(s)) = log(1 − z12 (s) − z22 (s) − 2z1 (s)z2 (s) cos θ )dθ ds. ds 4π 0 (5.18) Changing the order of integration gives .

5.3 Fuglede–Kadison Determinant

111

2π iAγ0 (R∗ (z)) f 1 ( ) φ ωR∗ (z(s)) =

.

0

1 = 4π =

1 4π

f

2π

0

f

(f

1 0

2π

0

) d 2 2 log(1−z1 (s) −z2 (s) −2z1 (s)z2 (s) cos θ )ds dθ ds

log(1 − z12 − z22 − 2z1 z2 cos θ )dθ,

(5.19)

which is dependent only on the homotopy class of .γ0 . If .γ0 and .γ1 are two nonhomotopic paths connecting .(0, 0) to .(z1 , z2 ), then .γ := γ0 − γ1 is a loop in c .P (R∗ ) that goes from .(0, 0) to .(z1 , z2 ) along .γ0 and returns to .(0, 0) along .γ1 , and it represents a nontrivial element in the fundamental group .π1 (P c (R∗ )). For simplicity, we also denote the parametrization of .γ by .z(s) = (z1 (s), z2 (s)), 0 ≤ s ≤ 1. It follows from Lemma 5.24 that Aγ0 (R∗ (z)) − Aγ1 (R∗ (z)) = Aγ (I ) =

.

W (γ ) . 2

(5.20)

In view of (5.19), we have (

( ) ∗ .detF K R∗ (z) = |detH S R∗ (z)| = exp Re 2π iAγ0 (R (z)) (

1 = exp 4π

f

2π

0

log |1 − z12

− z22

)

) − 2z1 z2 cos θ |dθ .

(5.21)

Apparently, it is independent of the path .γ0 and thus is well-defined for every .z ∈ P c (R∗ ). For .z ∈ P (R∗ ) such that .z1 z2 /= 0, after factoring out .2z1 z2 , the integral in (5.21) is of the form f

2π

.

log |β − cos θ |dθ,

0

which is convergent and continuous with respect to .β. Thus .detF K R∗ (z) extends continuously to z. For .z ∈ P (R∗ ) such that .z1 z2 = 0, Theorem 5.2 implies that (z1 , z2 ) ∈ {(±1, 0), (0, ±1)},

.

in which case the integral is .−∞, and hence, .detF K R∗ (z) = 0. Therefore, detF K R∗ (z) extends continuously to z due to Corollary 5.20. In conclusion, 2 .detF K R∗ (z) is well-defined and continuous on the entire .C . u n .

5 The .C ∗ -Algebra of the Infinite Dihedral Group .D∞

112

Remark 5.27 Observe that equation (5.20) indicates that the HS determinants ( ) detH S R∗ (z) = exp 2π iAγj (R∗ (z)) , j = 0, 1,

.

have two possible values that differ by a negative sign depending on whether .W (γ0 − γ1 ) is even or odd, and therefore, it is not well-defined on .P c (R∗ ). Corollary 5.28 The FK determinant of .R(z) = z0 +z1 λ(a)+z2 λ(t) is well-defined and continuous on .C3 . Moreover, (

1 .detF K R(z) = exp 4π

f

2π

0

) log |z02

− z12

− z22

− 2z1 z2 cos θ|dθ .

Theorem 5.26 has two interesting special cases. Example 5.29 Consider the quadratic surface .S = {(z1 , z2 ) ∈ C2 | 1 − z12 − z22 = 0}. It is clear that .S ⊂ P (R∗ ) by Theorem 5.2. Using the fact (exercise) f

π/2

.

log cos θ dθ = −

0

π log 2, 2

(5.22)

we obtain ( detF K R∗ (z) = exp

.

1 4π

f

2π

0

) log |2z1 z2 | + log | cos θ|dθ

) f 2π 1 log | cos θ|dθ 4π 0 ( f π/2 ) / / 1 = 2|z1 z2 | exp log cos θ dθ = |z1 z2 |, z ∈ S. π 0 /

=

(

2|z1 z2 | exp

This provides an illuminating example of FK determinant being nonzero at singular elements. The other case concerns with the Mahler measure for polynomials. Example 5.30 For a complex polynomial P (w) = α0 (w − α1 )(w − α2 ) · · · (w − αn ),

.

|| its Mahler measure is defined as .M(P ) = |α0 | |αj |≥1 |αj |. Lehmer [153] conjectured that there exists a universal constant .μ > 1 such that for all polynomials P with integer coefficients, either .M(P ) = 1 or .M(P ) ≥ μ. The problem is still open. According to Smyth [207], .μ is possibly equal to .μ = 1.176280818 . . ., which is achieved by the polynomial

5.4 An Application to Group of Intermediate Growth

113

P (w) = w 10 + w 9 − w 7 − w 6 − w 5 − w 4 − w 3 + w + 1.

.

It follows from Jensen’s formula [196] that (

1 .M(P ) = exp 2π

f

2π

) log |P (e )|dθ . iθ

0

The connection with the FK determinant is thus apparent. Here, if we let .Pz (w) = / w(1 − z12 − z22 ) − z1 z2 (w 2 + 1), then .Pz is a nontrivial polynomial when .z ∈ {(±1, 0), (0, ±1)}. On the unit circle |Pz (eiθ )| = |(1 − z12 − z22 ) − z1 z2 (eiθ + e−iθ )| = |1 − z12 − z22 − 2z1 z2 cos θ |,

.

/ and Theorem 5.26 yields .detF K R∗ (z) = M(Pz ), z ∈ C2 . Thus, if Lehmer’s conjecture is true, then .detF K R∗ (z) is either equal to 1 or greater than .μ for all .z ∈ Z × Z \ {(±1, 0), (0, ±1)}. Exercise 5.31 1. 2. 3. 4. 5.

Suppose .B = Mk (C) and let .φ = k1 tr . Prove that .detF K x = | det x|1/k . Theorem 5.18 is [82, Theorem 2]. Read its proof. Use Theorem 5.26 to determine the set of .φ-singular points in .P (R∗ ). Prove Corollary 5.28. (Hint: It remains to check the case .z0 = 0.) Verify the integral formula (5.22).

Project 5.32 1. Determine .π1 (P c (R∗ )). Moreover, the Hurewicz theorem indicates that the homology group .H1 (P c (R∗ )) is isomorphic to the abelianization of .π1 (P c (R∗ )). Is .H1 (P c (R∗ )) isomorphic to . 21 Z? 2. As a follow-up to Example 5.30 and the discussion of Lehmer’s conjecture, determine the minimum number .μ > 1 such that .detF K R∗ (z) ≥ μ for all .z ∈ Z × Z \ {(±1, 0), (0, ±1)}. Furthermore, are there integer points z for which .detF K R∗ (z) = 1?

5.4 An Application to Group of Intermediate Growth Consider a finitely generated group G with a symmetric generating set .S = {g1 , . . . , gn }. For any .k ∈ N, we let .B S (k) = {g ∈ G | l(g) ≤ k} be the ball in G centered at 1 with radius k, where l is the word length of g defined in Sect. 2.3. The growth function of G with respect to S is defined as .ΓGS (k) = |B S (k)|. Clearly, S .Γ G is bounded if and only if G is a finite group. For a finitely generated infinite group, the growth function contains structural information about the group.

5 The .C ∗ -Algebra of the Infinite Dihedral Group .D∞

114

For two functions .Γ, Γ , : N → N, we write .Γ < Γ , if there exist constants , , .C, a > 0 such that .Γ (k) ≤ CΓ (ak) for all k. We say that .Γ and .Γ are equivalent , , , and write .Γ ∼ Γ if .Γ < Γ and .Γ < Γ . The following fact is not hard to check. Lemma 5.33 If S and .S , are two finite symmetric generating sets of the group G, , then .ΓGS ∼ ΓGS . Thus, if our concern is only the growth rate of .ΓGS , we may drop the superscript S and write the function as .ΓG . Definition 5.34 A finitely generated group G is said to be of polynomial growth if ΓG (k) < k α for some constant .α > 0; and it is said to be of exponential growth if k .α < ΓG (k) for some constant .α > 1. .

It is not hard to check that abelian groups have polynomial growth, and nonabelian free groups have exponential growth. Given a subgroup .K ⊂ G, its index is .|G/K|. We say that G is a finite extension of K if .|G/K| < ∞. If .|G/K| = d < ∞, then it can be checked that .ΓK (k) ≤ ΓG (k) ≤ dΓK (k), k ≥ 1, which implies .ΓK ∼ ΓG .

5.4.1 The Growth of Solvable and Nilpotent Groups A group G is said to be solvable (or soluble) if there exists an abelian series, namely, a sequence of normal subgroups {1} = G0 < G1 < · · · < Gr = G

.

(5.23)

such that each quotient group .Gi+1 /Gi , 0 ≤ i ≤ r − 1, is abelian. It is said to be nilpotent if in (5.23) each quotient group .Gi+1 /Gi is in the center of .G/Gi . In this case, (5.23) is often called a central series. Apparently, every abelian group is nilpotent, and every nilpotent group is solvable. The smallest example of a non-nilpotent solvable group is the permutation group .S3 that has trivial center. The infinite dihedral group .D∞ is also non-nilpotent but solvable because it has the abelian series .{1} < < D∞ and a trivial center. The class of solvable groups is closed with respect to the formation of finite extensions, subgroups, and homomorphic images of its members; and the class of nilpotent groups is closed under the formation of subgroups, homomorphic images, and finite direct products. Solvable groups have been studied extensively. We refer the reader to [156] for more details. The following interesting result is discovered in Feit–Thompson [91]. Theorem 5.35 Every finite group of an odd order is solvable. Regarding the growth function, the following result is due to Wolf [239]. Proposition 5.36 Every finitely generated nilpotent group is of polynomial growth.

5.4 An Application to Group of Intermediate Growth

115

Hence, a finite extension of a finitely generated nilpotent group is also of polynomial growth. Remarkably, the converse is also true, Gromov [112]. Theorem 5.37 If a finitely generated group is of polynomial growth, then it contains a nilpotent subgroup of finite index. However, there exist finitely generated solvable groups of exponential growth. In fact, the following is shown in Milnor [162]. Theorem 5.38 A finitely generated solvable group has exponential growth unless it contains a nilpotent subgroup of finite index. Thus, every finitely generated solvable group is either of polynomial growth or of exponential growth. Regarding groups of matrices (linear groups), the following fact is proved in Tits [220] in response to a question by Bass and Serre. Theorem 5.39 Over a field of characteristic 0, a linear group either contains a nonabelian free subgroup or possesses a solvable subgroup of finite index. Therefore, a linear group is also either of polynomial growth or of exponential growth. Indeed, until 1968 it appeared that all known finitely generated groups obey this dichotomy. This prompted Milnor [163] to formally pose the natural question whether there exists a group whose growth is faster than that of a polynomial but slower than that of an exponential function. In other words, are there groups of intermediate growth? This problem is solved affirmatively in Grigorchuk [103, 104].

5.4.2 The Grigorchuk Group A better way to introduce this group .G is through its measure-preserving action on the interval .[0, 1). But we shall wait until Chap. 10 to do that. Here, we only mention that it is generated by four involutions .a, b, c, d satisfying the following infinite set of algebraic equations: a 2 = b2 = c2 = d 2 = bcd = 1,

.

σ k ((ad)4 ) = σ k ((adacac)4 ) = 1, k = 0, 1, 2, · · · , where .σ is the substitution:. a → aca, b → d, c → b, d → c. Despite √ its complexity, it was shown that the growth function .ΓG (k) is faster than .e k log 31 but slower than .ek 32 . Recently, the precise growth rate of .ΓG is determined in Erschler–Zheng [77]. Theorem 5.40 The growth function .ΓG satisfies

5 The .C ∗ -Algebra of the Infinite Dihedral Group .D∞

116

lim

.

k→∞

log log ΓG (k) log 2 ≈ 0.7674, = log k log λ0

where .λ0 is the positive root of the polynomial .X3 − X2 − 2X − 4. The measure-preserving action of .G on .[0, 1) gives rise to a unitary representation .π (called Koopman representation) of .G on .L2 [0, 1). The spectral property of the pencil .Rπ (z) = z0 I + z1 π(a) + z2 π(b) + z3 π(c) + z4 π(d) was first considered in Bartholdi–Grigorchuk [16]. Although the full picture of the projective spectrum .P (Rπ ) seems far from reach at this time, some interesting partial results have been obtained. To make a connection with the infinite dihedral group .D∞ , observe that the relations .b2 = c2 = d 2 = bcd = 1 show that .b, c, and d pairwise commute. Define the element .u = 12 (b + c + d − 1) ∈ C[G], one checks easily that .u2 = 1. The group generated by a and u is isomorphic to .D∞ . For the pencil ˆ π (z) = z0 I + z1 π(a) + z2 π(u), it is shown in [113] that the projective spectrum .R ˆ π ) coincides with that in Theorem 5.2, namely, .P (R P (Rˆ π ) =

U

.

{z ∈ C3 | z02 − z12 − z22 − 2z1 z2 x = 0}.

(5.24)

−1≤x≤1

Let .Mπ = 14 π(a + b + c + d) be the Markov operator of .G with respect to .π. Corollary 5.41 The spectrum .σ (Mπ ) = [− 12 , 0] ∪ [ 12 , 1]. Proof If we write .4Mπ − ζ I = (1 − ζ )I + π(a) + 2π(u), then the spectrum .σ (Mπ ) can be determined by (5.24). Indeed, substituting .(z0 , z1 , z2 ) by .(1 − ζ, 1, 2), one has (1 − ζ )2 − 5 − 4x = 0, x ∈ [−1, 1],

.

and the range of .ζ is easily determined to be .[−2, 0] ∪ [2, 4].

u n

To look into the projective spectrum .P (Rπ ), we consider the subspace .Mˆ = {z ∈ C5 | z2 = z3 = z4 } and define the map .X : C3 → Mˆ by ( ) w2 w2 w 2 w 2 X(w0 , w1 , w2 ) = w0 − , w1 , , , . 2 2 2 2

.

(5.25)

Then the pencils .Rˆ π and .Rπ are related by the equation .Rˆ π (w) = Rπ (X(w)), w ∈ C3 . This offers us a glimpse of .P (Rπ ). Corollary 5.42 .P (Rπ ) ∩ Mˆ = X(P (Rˆ π )). ˆ But this has been difficult. It would be interesting to find some points in .P (Rπ ) \ M. When the coefficients are all real, the following fact is discovered in Grigorchuk– Lenz–Nagnibeda [109].

5.4 An Application to Group of Intermediate Growth

117

Theorem 5.43 For distinct real coefficients .xi , 1 ≤ i ≤ 4, the classical spectrum σ (x1 π(a) + x2 π(b) + x3 π(c) + x4 π(d))

.

is a Cantor set with measure 0. Exercise 5.44 1. 2. 3. 4. 5.

Prove Lemma 5.33. Compute the growth function of .Zn . Compute the growth function .ΓG (k) of the free group .G = Fn , n ≥ 2. Verify that .Rˆ π (w) = Rπ (X(w)), w ∈ C3 , and prove Corollary 5.42. The subgroup .K = {1, b, c, d} ⊂ G is called the Klein four group. Verify that K is abelian and determine the projective spectrum of the pencil .λ(z0 I + z1 b + z2 c + z3 d), where .λ is the regular representation of K.

Project 5.45 1. Determine whether the subgroup . ⊂ G is isomorphic to .D∞ . 2. Compute the projective spectrum of the pencil .z0 +z1 π(a)+z2 π(b). This could ˆ help us to find points in .P (Rπ ) \ M. 3*. The ultimate goal is to determine the projective spectrum .P (Rπ ).

Chapter 6

The Maurer–Cartan Form of Operator Pencils

Given a Lie group G, its Maurer–Cartan form .ω(g) = g −1 dg, g ∈ G, contains important information about the property of G. Since .0 = d(g −1 g) = (dg −1 )g + g −1 dg, we have .dg −1 = −g −1 dgg −1 and consequently the following Maurer– Cartan equation: dω + ω ∧ ω = 0.

.

(6.1)

The 2-form .dω + ω ∧ ω is the curvature form of .ω. Hence, the Maurer–Cartan form .ω provides a flat connection on G. When .A1 , . . . , An are several elements in a Banach algebra .B and .A(z) = z1 A1 + · · · + zn An , Theorem 4.35 indicates that every path-connected component of the projective resolvent set .P c (A) is a Stein domain. The .B-valued 1-form .ωA (z) = A−1 (z)dA(z), z ∈ P c (A), is called the Maurer–Cartan form of the pencil .A(z). When .B is a matrix algebra, the resolvent set .pc (A) = {z ∈ Pn−1 | det A(z) /= 0} is a hypersurface complement, and its topology has been well-studied in algebraic geometry. However, when .B is infinite dimensional, new tools are required. In geometry, the Chern–Weil theorem relates connection and curvature on a smooth manifold M to the de Rham cohomology of M, and it is viewed as a grand generalization of the classic Gauss–Bonnet theorem for surfaces. It is thus an appealing question whether there is a similar phenomenon in the operator setting here, i.e., whether topological information about .P c (A) could be read from .ωA . This chapter aims to address this question.

6.1 Curvature and Chern–Weil Theorem Given a smooth manifold .o of real dimension n, the de Rham complex is the cochain complex

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 R. Yang, A Spectral Theory Of Noncommuting Operators, https://doi.org/10.1007/978-3-031-51605-4_6

119

120

6 The Maurer–Cartan Form of Operator Pencils d

d

d

d

{0} → A0 (o) → A1 (o) → · · · → An (o) → {0},

.

where .Ap (o), 0 ≤ p ≤ n, are the vector spaces of smooth p-forms, and p p+1 (o) is the exterior derivative. In a local chart .U ⊂ o with .d : A (o) → A coordinates .x1 , . . . , xn , we have d

( E

.

) fJ dxJ

|J |=p

=

n E E ∂fJ |J |=p i=1

dxi

dxi ∧ dxJ ,

where .J = (j1 , . . . , jp ) is a multi-index and .dxJ stands for .dxj1 ∧ · · · ∧ dxjp . It is a good exercise to check that .d 2 = 0. A p-form f is said to be closed if .df = 0, and it is exact if .f = dh for some .h ∈ Ap−1 (o). Thus every exact form is closed. The pth de Rham cohomology space of .o is defined as p

Hd (o, C) := {closed p-forms}/{exact p-forms}, p ≥ 0.

.

p

The Betti numbers of .o are defined as .bp (o) := E dim Hd (o, C) for each p. And the Euler characteristic of .o is defined as .χ (o) = np=0 (−1)p bp . The reader should pause here for a moment and take another look at the definition of Taylor spectrum and the index of Fredholm tuples in Sect. 4.1. If .o is a complex domain in .Cn , then we have two natural differential operators ∂f =

.

E ∂f E ∂f dzj , ∂f = dzj , ∂zj ∂zj j =1

j =1

and therefore .d = ∂ + ∂. A smooth function f is said to be holomorphic if .∂f = 0 p on E .o. A differential form .ω ∈ A (o) is said to be holomorphic if it is of the form . |J |=p fJ (z)dzJ , where each .fJ is holomorphic. It is known that if .o is a Stein domain, then its de Rham cohomology can be computed by holomorphic differential forms, Range [187]. Theorem 6.1 If .o is a Stein domain in .Cn , then p

Hd (o, C) = {holomorphic closed p-forms}/{holomorphic exact p-forms}, p ≥ 0.

.

p

In particular, it follows that .Hd (o, C) = {0} for .p > n even though .o is of real dimension 2n.

6.1.1 Connection and Curvature The Chern–Weil theorem extracts topological information about a manifold from the connections on its vector bundles. We shall briefly review the construction here

6.1 Curvature and Chern–Weil Theorem

121

for complex manifolds, as it will illuminate our later discussion on the Maurer– Cartan form of operator pencils. Let .o be a complex manifold of dimension n and .π : E → o be a complex vector bundle of rank k. The space of smooth sections of p-forms is denoted by p .A (o, E). The notation .End(E) stands for the vector bundle of endomorphisms of E, i.e., for every .z ∈ o, the fiber .End(E)z is the set of linear transformations on the fiber .Ez . A connection on E is a linear map .D : A0 (o, E) → A1 (o, E) that satisfies the Leibniz rule D(f · s) = df · s + f · Ds, f ∈ C ∞ (o), s ∈ A0 (o, E).

.

Ds is called the exterior covariant derivative of s. The connection extends canonically to .Ap (o, E) for each .p ≥ 1 such that for every .η ∈ Ap (o) and .s ∈ A0 (o, E) we have D(η · s) = (dη) · s + (−1)p η ∧ (Ds).

.

(6.2)

Definition 6.2 The curvature of a connection D is the differential operator D 2 : A0 (o, E) → A2 (o, E).

.

Connection and curvature can be expressed more explicitly on a local chart .U ⊂ o, where we have the identifications E |U ∼ = U × Ck , and End(E) |U ∼ = U × Mk (C).

.

Then for each .p ≥ 0, Ap (o, E) |U ∼ = Ap (o) |U ⊗Ck , and Ap (o, End(E)) |U ∼ = Ap (o) |U ⊗Mk (C). (6.3)

.

Every section .ω ∈ A1 (o, End(E)) gives rise to a connection .Dω defined locally by Dω s = ds + ω · s, s ∈ A0 (o, E).

.

(6.4)

Indeed, for every smooth function f on .o, one verifies that Dω (f · s) = d(f · s) + ω · (f · s)

.

= df · s + f · ds + f · (ω · s) = df · s + f · (Dω s), which is the Leibniz rule. It is worthwhile to make the following clarifications.

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6 The Maurer–Cartan Form of Operator Pencils

1. The differential ds is properly defined only when a local chart U and the identification (6.3) are specified so that s becomes a vector-valued function. If such an identification is not used, then the partial difference quotient . s(z)−s(w) zi −wi (and hence ds) is undefined for any i because .s(z) and .s(w) belong to different fibers for distinct .z, w ∈ o. 2. Due to the chain rule, the differential ds is independent of the choice of local chart. On the other hand, if a frame is fixed for E, then it can be shown that every connection D on E is locally of the form .Dω , where .ω is obtained by the action of D on the sections in the frame. Observe that if .ω' is another element in 1 ' .A (o, End(E)), then it holds globally that .(Dω − Dω' )s = (ω − ω ) · s. Therefore, expression (6.4) has the following notable consequence. Proposition 6.3 Let .D0 be a fixed connection on the bundle E. Then the set of all connections on E is the affine space .{D0 + ω | ω ∈ A1 (o, End(E))}. To compute the curvature of .Dω in a local chart, for any .s ∈ A0 (o, E) we have Dω2 s = Dω (ds + ω · s)

.

= d(ds + ω · s) + ω ∧ (ds + ω · s) = dω · s − ω ∧ ds + ω ∧ ds + ω ∧ (ω · s) = (dω + ω ∧ ω) · s.

(6.5)

The 2-from .Oω := dω + ω ∧ ω is called the curvature form of the connection .Dω . Note that (6.5) indicates that the curvature is a multiplication operator with symbol .Oω . The following Bianchi identity is not hard to check: dOω = Oω ∧ ω − ω ∧ Oω .

.

(6.6)

A connection .Dω := d + ω (expressed locally) on E naturally induces a connection (also denoted by .Dω ) on the bundle .End(E). Indeed, if .F ∈ Aq (o, End(E)) and .η ∈ Ap (o, E), then Dω (F ∧ η) = d(F ∧ η) + ω ∧ (F ∧ η)

.

= dF ∧ η + (−1)q F ∧ dη + (ω ∧ F ) ∧ η. On the other hand, since the induced connection .Dω on .Aq (o, End(E)) needs to satisfy the Leibniz rule, we must have Dω (F ∧ η) = (Dω F ) ∧ η + (−1)q F ∧ (Dω η)

.

= (Dω F ) ∧ η + (−1)q F ∧ (dη + ω ∧ η).

6.1 Curvature and Chern–Weil Theorem

123

Equating the right-hand sides of the above identities, we obtain the following definition. Definition 6.4 Let .Dω = d + ω be a connection on a vector bundle E. Then the induced connection on .End(E) is defined locally by Dω F = dF + ω ∧ F − (−1)q F ∧ ω, F ∈ Aq (o, End(E)), 0 ≤ q ≤ n.

.

Hence, the Bianchi identity can be written as .Dω Oω = 0. We leave it as an exercise to check that this .Dω indeed satisfies the Leibniz rule.

6.1.2 Invariant Linear Functional Linear functionals serve as a useful tool to extract topological invariants from a connection .Dω and its curvature .Oω . Definition 6.5 A nontrivial continuous p-linear functional F on a Banach algebra B is said to be invariant if

.

F (a1 , . . . , ap ) = F (ga1 g −1 , . . . , gap g −1 )

.

(6.7)

for all .g, a1 , . . . , ap in .B with g being invertible. It is not hard to see that an invariant 1-linear functional F is just a trace, i.e., it satisfies .F (ab) = F (ba), a, b ∈ B. An algebraic structure exists for the space of invariant linear functionals. If .F1 and .F2 are invariant p-linear and, respectively, q-linear functionals, then an associative product .F1 × F2 can be defined by (F1 × F2 )(a1 , . . . , ap+q ) := F1 (a1 , . . . , ap )F2 (ap+1 , . . . , ap+q ).

.

Clearly, the product .F1 ×F2 is an invariant .(p+q)-linear functional. We set .F 0 = C and let .F p be the vector space of invariant p-linear functionals on .B for .p ≥ 1. The direct sum F ∗ (B) :=

∞ O

.

F p (B)

p=0

is a graded algebra over .C with respect to addition and the product .×. Lemma 6.6 For any .η ∈ A1 (o, B) and .F ∈ F p (B), we have Ep (a) . i=1 (−1)i F (a1 , a2 , . . . , ai ∧ η + η ∧ ai , . . . , ap ) = 0, a1 , . . . , ap ∈ 1 A (o, B). E p (b) . i=1 F (a1 , a2 , . . . , ai ∧ η − η ∧ ai , . . . , ap ) = 0, a1 , . . . , ap ∈ A2 (o, B).

124

6 The Maurer–Cartan Form of Operator Pencils

Proof For any .g ' ∈ B with .||g ' || < 1, the element .g = I − g ' is invertible with g −1 = I + g ' + g ' + · · · . 2

.

To illustrate how the proof works, we first consider the case F is 1-linear. For .a ∈ B, we write gag −1 = (I − g ' )a(I + g ' + g ' + · · · ) 2

.

= a + (ag ' − g ' a) + (ag ' − g ' ag ' ) + (ag ' − g ' ag ' ) + · · · . 2

3

2

Since F is invariant, we have .F (gag −1 ) = F (a) and .F (g ' ag ' j −1 ) = F (g ' j a) for each .j ≥ 1. It follows that F (a) = F (a) +

∞ E

.

F ([a, g ' ]). j

(6.8)

j =1

Replacing .g ' by .wg ' , where w is a complex variable with .|w| < 1, (6.8) gives ∞ E .

F ([a, g ' ])w j = 0, j

j =1

which implies that .F ([a, g ' j ]) = 0 for each .j ≥ 1. If F is p-linear, then setting ' −1 , ga g −1 , . . . , ga g −1 ) .g = wg , considering the power expansion of .F (ga1 g 2 p with respect to w, and checking the coefficient of w, we obtain p E .

F (a1 , a2 , . . . , [ai , g ' ], . . . , ap ) = 0.

(6.9)

i=1

By linearity, (6.9) remains true when .a1 , . . . , ap are .B-valued differential forms. Furthermore, since the set of invertible elements in .B is dense, (6.9) also holds for an arbitrary element .g ' ∈ B. To prove part (a), using the linearity of F , we may simply assume .η = g ' dzs , where .1 ≤ s ≤ n. Then since each .aj is a 1-form, we have F (a1 , a2 , . . . , ai ∧ η + η ∧ ai , . . . , ap )

.

= F (a1 , a2 , . . . , ai ∧ g ' dzs + g ' dzs ∧ ai , . . . , ap ) = F (a1 , a2 , . . . , ai g ' − g ' ai , . . . , ap )(−1)p−i dzs . Part (a) then follows from (6.9). In the case each .aj is a 2-form, we have F (a1 , a2 , . . . , ai ∧ η − η ∧ ai , . . . , ap )

.

6.1 Curvature and Chern–Weil Theorem

125

= F (a1 , a2 , . . . , ai ∧ g ' dzs − g ' dzs ∧ ai , . . . , ap ) = F (a1 , a2 , . . . , ai g ' − g ' ai , . . . , ap )dzs , u n

and hence part (b) also follows from (6.9).

6.1.3 The Chern Class Let D be a connection on a rank-k vector bundle E over a smooth manifold o ⊂ Cn with curvature .O. For .B = Mk (C) and any .F ∈ F p (B), the evaluation .F (O, . . . , O) is a scalar-valued smooth 2p-form on .o. It is thus a natural question 2p whether it is closed. To answer the question, we define .τ : F ∗ (B) → ⊕∞ p=0 A (o) such that .τ (1) = 1 and for .p ≥ 1, .

τ (F ) = F (O, . . . , O), F ∈ F p (B).

(6.10)

.

It is clear that .τ is linear on .F p (B) for each p, and for .F1 ∈ F p (B) and .F2 ∈ F q (B) we have .τ (F1 × F2 ) = τ (F1 ) ∧ τ (F2 ). In other words, .τ is a homomorphism of graded algebras. 2p

Theorem 6.7 Then the map .τ is a homomorphism into .⊕np=0 Hd (o, C). Proof It only remains to show that for each .p ≥ 1 and .F ∈ F p (B), the 2p-form .τ (F ) is closed. Assume D is expressed locally as in (6.4). Then using the Bianchi identity (6.6) and Lemma 6.6 (b), we observe that dτ (F ) =

p E

.

F (O, . . . , dO, . . . , O)

i=1

=

p E

F (O, . . . , O ∧ ω − ω ∧ O, . . . , O) = 0.

i=1

u n The map .τ is referred to as the Chern–Weil homomorphism. As we will see here and later in the chapter, there are several ways to construct invariant multilinear functionals. The following one has a prominent application. Definition 6.8 A polynomial .Q : Mk (C) → C is said to be invariant if Q(ST S −1 ) = Q(T ), T ∈ Mk (C), S ∈ GLk .

.

Trace and determinant are familiar examples of invariant polynomials on .Mk (C). If Q is an invariant polynomial, then we may define an invariant p-linear functional ˆ on .Mk (C) by the polarization identity .Q

126

6 The Maurer–Cartan Form of Operator Pencils

ˆ 1 , . . . , ap ) = 1 .Q(a k!

E J ⊂{1,...,p}

(−1)

p−|J |

Q

(E

) aj , aj ∈ Mk (C).

(6.11)

j ∈J

In the particular case .p = k, the identity can be expressed alternatively as ∂k ˆ 1 , . . . , ak ) = 1 Q(t1 a1 + · · · + tk ak ). Q(a k! ∂t1 · · · ∂tk

.

ˆ is also invariant (or symmetric) with respect to a permutation of Observe that .Q coordinates. On the other hand, if F is an invariant p-linear functional on .Mk (C), then Q(a) := F (a, . . . , a), a ∈ Mk (C),

.

is an invariant homogeneous polynomial of degree p. We thus arrive at the following Chern–Weil theorem. Theorem 6.9 Let D be a connection on a vector bundle E of rank k over the manifold .o, and let .Q : Mk (C) → C be a homogeneous invariant polynomial of degree p. Then 2p

(a) The 2p-form .Q(O) is closed, i.e., .Q(O) ∈ Hd (o, C). (b) The cohomology class .[Q(O)] is independent of the choice of the connection D. Therefore, .[Q(O)] is a topological invariant of the bundle E. Part (a) is a direct consequence of Lemma 6.6 and the Bianchi identity (6.6). Part (b) follows from Proposition 6.3. For details, we refer the reader to Chern [35] and Milnor [162]. Another important example of invariant polynomial is the characteristic polynomial .

det(w + T ) = wk + Q1 (T )w k−1 + · · · + Qk (T ),

(6.12)

where .Qp is a degree-p homogeneous polynomial in the entries of T . One may compare this expansion with (1.8). Since .det(w + ST S −1 ) = det(w + T ), S ∈ GLk , each polynomial .Qp is invariant. Theorem 6.9 then gives rise to the following important topological invariants of the bundle E. ( i ) Definition 6.10 For .0 ≤ p ≤ k, the closed 2p-form .cp (E) := Qp 2π O is called 2p

the pth Chern form, and the equivalence class .[cp (E)] ∈ Hd (o, C) is called the pth Chern class of the bundle E. i In particular, .c0 (E) = 1, .c1 (E) = 2π tr O, and .cp (E) = 0 when .p > n. The total Chern form of E is ( ) i .c(E) := det 1 + O = c0 (E) + · · · + ck (E) ∈ Hd∗ (o, C). 2π

6.1 Curvature and Chern–Weil Theorem

127

In the case E is the tangent bundle .T (o), it is customary to write .cp (T (o)) simply as .cp (o) and call .[cp (o)] the pth Chern class of the manifold .o. For more information on the Chern classes, we refer the reader to Chern [34] and Milnor– Stasheff [164]. Example 6.11 One simple case helps to illuminate the Chern classes. Suppose E is a trivial bundle .o × V , where V is any vector space, and then the exterior derivative d itself is a connection with curvature .d 2 = 0. Hence .cp (E) = 0 for each .p ≥ 1.

6.1.4 Chern–Simons Forms Although the Chern–Weil theorem only produces even-degree forms on the manifold .o, there is a somewhat indirect way to produce odd-degree ones from the curvature. Assume the connection .Dω = d + ω is expressed locally as in (6.4). For an integer .p ≥ 1, the map .T → tr (T p ) is obviously an invariant degree-p p homogeneous polynomial on .Mk (C). Hence, the 2p-form .tr (Oω ) is closed. It turns out that it is actually exact. The Chern–Simons form .η2p−1 ∈ A2p−1 (o, End(E)) is a solution to the equation dη2p−1 = tr (Opω ).

.

(6.13)

There is an interesting way to compute such .η2p−1 . If we set .ωt := tω, then its curvature .Ft := tdω + t 2 (ω ∧ ω). It can be shown that f

1

η2p−1 = p

.

0

) ( p−1 dt. tr ω ∧ Ft

It is thus easy to verify that η1 = tr (ω), ) ( 1 3 η3 = tr Oω ∧ ω − ω , 3 ) ( 1 1 η5 = tr O2ω ∧ ω − Oω ∧ ω3 + ω5 . 10 2

.

In the case .Dω is a flat connection, we have .Oω = dω + ω ∧ ω = 0, and therefore .η2p−1 is closed for each .p ≥ 1 by (6.13). Moreover, since in this case 2 .Ft = (t − t)ω ∧ ω, it follows that f η2p−1 = p

.

0

1

) ( p−1 dt tr ω ∧ Ft

128

6 The Maurer–Cartan Form of Operator Pencils

f =p

1

) ) ( ( (t 2 − t)p−1 dt · tr ω2p−1 = αp tr ω2p−1 ,

(6.14)

0

where .αp = (−1)p−1 pB(p, p) and f

1

B(s, t) =

.

x s−1 (1 − x)t−1 dx, s, t > 0,

0

is the Beta function. In mathematics, the Chern–Simons forms are used to study low-dimension topology. In physics, they have important applications to fields such as topological quantum field theory, condensed matter phenomenon, and high temperature super conductivity. We refer the reader to [37, 73, 235] for more information.

6.1.5 Chern–Simons Forms of Operator Pencils Consider the pencil .A(z) = z1 A1 + · · · + zn An of elements in a Banach algebra B, and let .oA be a path-connected component of the projective resolvent set c .P (A). Then .oA is an open complex manifold of dimension n, and it is Stein by Theorem 4.35. The Maurer–Cartan form of the pencil .A(z) is defined as .

ωA (z) = A−1 (z)dA(z) = A−1 (z)

n E

.

Aj dzj ∈ A1 (oA , B).

j =1

If we let .E := oA × B be the trivial bundle over .oA , then .ωA gives rise to a connection .DωA on E defined by (6.4). However, the Maurer–Cartan equation (6.1) indicates that the curvature OωA = dωA + ωA ∧ ωA = 0.

.

(6.15)

Hence in this case the Chern–Weil theorem does not produce any interesting topological information. An alternative idea is to retrieve the information directly from .ωA using invariant multilinear functionals. First, given a functional .φ ∈ B∗ , φ(ωA (z)) =

n E

.

φ(A−1 (z)Aj )dzj ,

j =1

which is a holomorphic 1-form on .oA . For multilinear functionals, similar to (6.10) we define a map .κ : F ∗ (B) → ⊕np=0 Ap (o) by .κ(1) = 1, and for each .p ≥ 1,

6.1 Curvature and Chern–Weil Theorem

129

κ(F ) = F (ωA (z), ωA (z), . . . , ωA (z)), F ∈ F p (B).

.

(6.16)

The following fact is parallel to Theorem 6.7. Theorem 6.12 .κ is a homomorphism from .F ∗ (B) into .Hd∗ (oA , C). Proof It is easy to see that .κ is linear and .κ(F1 × F2 ) = κ(F1 ) ∧ κ(F2 ), meaning that .κ is a homomorphism of graded algebras. To prove that .κ maps invariant linear functionals to closed forms, we set .ai = η = ωA , i = 1, . . . , p, in Proposition 6.6 (a) and use equation (6.15) to obtain dF (ωA (z), ωA (z), . . . , ω(z))

.

p E (−1)i−1 F (ωA (z), ωA (z), . . . , dωA (z), . . . , ωA (z))

=

i=1 p E = (−1)i F (ωA (z), ωA (z), . . . , ωA (z) ∧ ωA (z), . . . , ωA (z)) = 0. i=1

u n It is worth pointing out a clear difference between the map .κ above and the map .τ in (6.10): the range of .τ consists of even forms that are not necessarily holomorphic, while the range of .κ consists of holomorphic forms of possibly any order. Example 6.13 If .B is a Banach algebra with a trace .tr , then for any .p ≥ 1, the p-linear functional .F (a1 , . . . , ap ) := tr (a1 · · · ap ), ai ∈ B, is invariant, and hence p p for any fixed elements .A1 , . . . , An in .B, we have .κ(F ) = tr ωA (z) ∈ Hd (oA , C). In particular, if p is even, say .p = 2s, then equation (6.15) implies that p

tr ωA (z) = (−1)s tr (dωA (z))s = (−1)s+1 dtr (ωA (z))2s−1 = 0.

.

p

However, we shall see later that .tr ωA (z) can be nontrivial when p is odd. We conclude this section with the following definition motivated by (6.14) and Park [181]. Definition 6.14 Given elements .A1 , . . . , An in .B, the Chern–Simons forms of the 2s+1 pencil .A(z) are defined on .P c (A) to be .tr ωA (z), s = 0, 1, . . .. The total Chern character of the pencil is defined to be Ch(A) =

∞ E

.

s=0

(−1)s

s! 2s+1 . tr ωA (2s + 1)!

Observe that the series of .Ch(A) is a finite sum. Since .oA is a Stein domain of 2s+1 complex dimension n, Theorems 6.1 and 6.12 indicate that .tr ωA = 0 when .2s + 1 > n.

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6 The Maurer–Cartan Form of Operator Pencils

Exercise 6.15 1. Verify equation (6.9). 2. Prove the Bianchi identity (6.6). 3. Show that every connection D on .A1 (o, End(E)) can be expressed on a local chart as .Dω for some .ω ∈ A1 (o, End(E)). 4. Consider any .ω ∈ A1 (o, End(E)). Show that the connection .Dω in Definition 6.4 satisfies the Leibniz rule and compute its curvature. 5. Show that an invariant 1-linear functional F satisfies .F (ab) = F (ba), a, b ∈ B. 6. Let .σ (z) = z0 I + z1 σ1 + z2 σ2 + z3 σ3 be the pencil of Pauli matrices considered in Sect. 1.1.1, and let .ωσ (z) be its Maurer–Cartan form. (a) Compute .Ch(σ ). (b) Observe that .σ : P c (σ ) → GL2 is a homeomorphism. Based on part (a), what can you say about the de Rham cohomology of .GL2 ? (c) Show that .κ : F ∗ (M2 (C)) → Hd∗ (P c (σ ), C) is surjective. 7. Let .Aj , 1 ≤ j ≤ 4, be any matrices in .M2 (C). Use exercise 6 to show that ∗ ∗ c .κ : F (M2 (C)) → H (P (A), C) is surjective. d 8. Let .Qj be the invariant polynomials defined in (6.12). Evaluate .Qj (ωσ (z)) and compare them with the terms in .Ch(σ ). Project 6.16 1. Study generalizations of exercise 7 to higher order matrix algebras. 2. To generalize exercise 8 above, for .A1 , . . . , An ∈ Mk (C) one is tempted to define the differential form .ηA := det(I + 2π1 i ωA ). Investigate this definition by considering more examples.

6.2 Trace Formula and Hyperplane Arrangement A crucial question regarding Theorem 6.12 is whether the map .κ defined in (6.16) is nontrivial. To address this question, we shall recall Jacobi’s formula in Lemma 1.9 which links the trace of Maurer–Cartan form to the determinant, namely, if .o is a complex domain in .Cn and .f : o → GLk is differentiable, then ( ) −1 .tr f (z)df (z) = d log det f (z), z ∈ o. In particular, if .A1 , . . . , An are .k × k matrices, then tr ωA = d log det A(z), z ∈ P c (A).

.

6.2 Trace Formula and Hyperplane Arrangement

131

One should observe that, since .log det A(z) is not globally defined on .P c (A), .tr ωA is closed but not exact. For a general Banach algebra .B, we have the following generalization. Lemma 6.17 Given elements .A1 , . . . , An ∈ B, if .φ ∈ B∗ is such that .φ(I ) /= 0, then there is no holomorphic function .f (z) on .oA such that .df (z) = φ(ωA (z)). Proof Since .ωA (z) is homogeneous of degree 0, we have .ωA (tz) = ωA (z) for all t ∈ C× . If there is a holomorphic function .f (z) on .oA such that .df (z) = φ(ωA (z)), then .d(f (tz)) = df (z). It follows that .f (tz) − f (z) is a constant, say .c(t). To determine its value, we observe that

.

.

∂f (z) = φ(A−1 (z)Aj ), 1 ≤ j ≤ n. ∂zj

It follows that c' (t) =

n E

.

j =1

=t

−1

E ∂f (tz) = t −1 zj φ(A−1 (z)Aj ) ∂zj n

zj

j =1

( φ A

−1

(z)

(E n

)) zj Aj

= t −1 φ(I ).

j =1

Since .c(1) = 0, .c(t) = φ(I ) log t, and therefore .f (tz) − f (z) = φ(I ) log t. But since f is holomorphic on .oA and .tz ∈ oA for all .t ∈ C× , the function .φ(I ) log t would be holomorphic on .C× , which is a contradiction. u n Corollary 6.18 Let .B be a Banach algebra with a trace .φ and .oA be a connected component of .P c (A). Then .φ(ωA (z))|oA is a nontrivial element in .Hd1 (oA , C). In particular, .oA is not simply connected. Proof By Theorem 4.35, domain .oA is a Stein. Hence .Hd1 (oA , C) can be calculated by holomorphic forms. The fact that .φ is a trace implies .φ(I ) /= 0 and .φ ∈ F 1 . Theorem 6.12 and Lemma 6.17 conclude that .φ(ωA (z))|oA is a nontrivial element in .Hd1 (oA , C). u n

6.2.1 An Example with the Free Groups It is insightful to consider an example where the Banach algebra .B is infinite dimensional. The free group von Neumann algebra .Bn studied in Sect. 4.4 has a canonical trace .tr. Using the same notations, we let .U (z) = z1 U1 + · · · + zn Un and consider .ωU (z) = U −1 (z)dU (z), z ∈ P c (U ). Theorem 4.53 states that the projective resolvent set is the disjoint union .P c (U ) = ∪nj=1 oj , where

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6 The Maurer–Cartan Form of Operator Pencils

oj = {z ∈ Cn | 2|zj |2 > ||z||2 , j = 1, . . . , n}.

.

To compute .tr ωU on .o1 , we use Einstein’s summation convention to write U

.

−1

Setting .ξi = fact

(z)dU (z) = (zi Ui ) zi+1 z1 , 1

−1

( (Ui dzi ) =

zi ∗ U Ui z1 1

)−1 ( ) dzi ∗ U 1 Ui . z1

≤ i ≤ n − 1, we have .ξ ∈ o1 if and only if .||ξ || < 1. Using the

dξi =

.

dz1 dz1 dzi+1 dzi+1 − zi+1 2 = − ξi , z1 z1 z1 z1

we can write ( ) )−1 ( dz1 ∗ ∗ ∗ U1 Ui+1 dξi + (I + ξi U1 Ui+1 ) .ωU = I + ξi U1 Ui+1 z1 )−1 ( ∗ ) dz1 ( U1 Ui+1 dξi + I. = I + ξi U1∗ Ui+1 z1 For simplicity, we denote the sum .ξi U1∗ Ui+1 by .W (ξ ). Then ( ) dz1 −1 .tr ωU = + tr (I + W (ξ )) dW (ξ ) . z1 For convenience, we denote the second summand by .h(ξ ). When .||ξ || is small enough so that .||W (ξ )|| < 1, we can write .(I + W (ξ ))−1 = (−1)j W j (ξ ), and it follows that .h(ξ ) = 0 because .U1 , U2 , . . . , Un are .∗-free Haar unitaries. Since h is holomorphic, it must vanish for all .||ξ || < 1. In conclusion, on .o1 we have dz1 .tr ωU = z1 . In fact, a careful look at .o1 shows that o1 = C× × {ξ ∈ Cn−1 | ||ξ || < 1}.

.

1 Since the unit ball is contractable, we have .Hd1 (o1 , C) = C dz z1 . By symmetry, we have

Hd1 (oj , C) = C

.

dzj = Ctr ωU |oj , j = 1, 2, . . . , n. zj

It is interesting to observe that .tr ωU appears differently on different components of P c (U ).

.

6.2 Trace Formula and Hyperplane Arrangement

133

6.2.2 Theorems of Arnold and Brieskorn Given several elements .A1 , . . . , An in .B, Theorem 6.12 establishes a homomorphism .κ from .F ∗ to .Hd∗ (P c (A), C), and Corollary 6.18 indicates that .κ is nontrivial in order 1. Is it nontrivial in higher orders? We shall address this question in this subsection. First, we consider the case when .B is abelian. Then Proposition 4.29 indicates that .P (A) is a union of hyperplanes in .Cn . When the union is finite, it is also called a hyperplane arrangement. In this case, the de Rham cohomology of c .P (A) has interesting properties, and it has been extensively studied. Let .{Hj | j = 1, . . . , m} be a finite set of hyperplanes in .Cn defined by Um n linear equations .aj (z) = 0. We set .M = C \ ( j =1 Hj ). Then the function ||m −1 is holomorphic on M and cannot be extended holomorphically to .( j =1 aj (z)) any larger domain, meaning that M is a domain of holomorphy. Thus the de Rham p cohomology .Hd (M, C) = {0} for .p > n. The Poincaré polynomial .Poin(M, t) is p the generating function of the Betti numbers .bp (M) = dim Hd (M, C), 0 ≤ p ≤ n, namely,

.

Poin(M, t) =

n E

bp (M)t p .

p=0

A well-studied example is the braid arrangement consisting of the hyperplanes Hj,k = {z ∈ Cn | zj − zk = 0}, 1 ≤ j /= k ≤ n.

.

Associated with each hyperplane, there is a canonical holomorphic 1-form ωj,k (z) =

.

1 dzj − dzk 1 d log(zj − zk ), z ∈ M. = 2π i 2π i zj − zk

A direct computation verifies that these 1-forms satisfy the algebraic equations ωi,j ∧ ωj,k + ωj,k ∧ ωk,i + ωk,i ∧ ωi,j = 0, 1 ≤ i, j, k ≤ n.

.

The following elegant result was shown by Arnold in [6]. Theorem 6.19 Let M be the complement of the braid arrangement in .Cn . (a) The de Rham cohomology .Hd∗ (M, Z) is generated by 1 and the 1-forms .ωj,k . (b) .Poin(M, t) = (1 + t)(1 + 2t) · · · (1 + (n − 1)t). Furthermore, Arnold conjectured that (a) holds for any complex hyperplane arrangement. This was proved by Brieskorn [22] in a 1971 Bourbaki seminar.

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6 The Maurer–Cartan Form of Operator Pencils

Theorem 6.20 Let .Hj be hyperplanes in .Cn with defining linear equations .aj (z) = da 0, 1 ≤ j ≤ m. Then .Hd∗ (M, Z) is generated by 1 and the 1-forms . 2π1 i ajj . The braid arrangement has a close connection with braid groups (hence the name) and knot theory. A good reference on this subject is [180] by Orlik and Terao.

6.2.3 Abelian Banach Algebras When .B is abelian, .gag −1 = a for all .a, g ∈ B with g being invertible. Hence every multilinear functional on .B is invariant, meaning that .F ∗ (B) is just the graded algebra of multilinear functionals on .B. To reveal a connection with Arnold and Brieskorn theorems, we shall focus on multiplicative linear functionals on .B. Following the notation in Sect. 4.1, we let .MB denote the collection of such linear functionals (the maximal ideal space). An element .a ∈ B is not invertible if and only if there exists a .φ ∈ MB such that .φ(a) = 0, for example, see Douglas [66]. Therefore, given elements .A1 , . . . , An ∈ B, the pencil .A(z) is not invertible if and only if there exists a .φ ∈ MB such that φ(A(z)) = φ(A1 )z1 + · · · + φ(An )zn = 0.

.

(6.17)

We denote the hyperplane defined by the equation above by .Hφ . Then Proposition 4.29(a) can be rewritten as a union: .P (A) = ∪φ∈MB Hφ . When .B is infinite dimensional, the union can well be an uncountable union, and hence .P c (A) may not be path-connected. Let .oA be a path-connected component of .P c (A). Since every multiplicative linear functional .φ ∈ MB is a trace, Corollary 6.18 indicates that .φ(ωA (z)) is a nontrivial element in .Hd1 (oA , C). Furthermore, φ(ωA (z)) = φ(A−1 (z))φ(dA(z)) =

.

dφ(A(z)) . φ(A(z))

(6.18)

If .φ1 , . . . , φp are multiplicative linear|| functionals on .B, we can define an invariant p k-linear functional .F (a1 , . . . , ap ) := j =1 φj (aj ). Hence by Theorem 6.12, κ(F ) = φ1 (ωA (z)) ∧ · · · ∧ φp (ωA (z)), z ∈ P c (A),

.

is a closed p-form. When .B is an abelian subalgebra in .Mk (C), the set .MB is finite. Indeed, in this case matrices in .B can be simultaneously upper triangulized. Then for .1 ≤ m ≤ k, a multiplicative linear functional .φm can be defined by ( ) φm (aij )k×k = amm , (aij ) ∈ B.

.

6.3 Jacobi’s Formula in Higher Orders

135

Hence .MB = {φ1 , . . . , φk }. In this case, .P (A) = ∪km=1 Hφm is a hyperplane arrangement with complement .M = P c (A). Then, in view of (6.18) and Theorem 6.20, we see that .H ∗ (P c (A), C) is generated by the 1-forms .φm (ωA ), 1 ≤ m ≤ k. This observation can be summarized as follows. Corollary 6.21 Let .B be an abelian Banach algebra of matrices. Then for any elements .A1 , . . . , An in .B , the map .κ : F ∗ (B) → H ∗ (P c (A), C) is surjective. Evidently, the map .κ is highly nontrivial in this case. It is not clear whether Corollary 6.21 holds for a nonabelian or an infinite dimensional .B. Exercise 6.22 1. Consider the braid arrangement .H = {z1 = z2 } ∪ {z1 = z3 } ∪ {z2 = z3 } in .C3 . (a) Find diagonal matrices .A1 , A2 , A3 ∈ M3 (C) such that .P (A) = H . (b) Let .B be the algebra generated by .I, A1 , A2 , and .A3 . Describe .F ∗ (B). (c) Determine the kernel of the homomorphism .κ : F ∗ (B) → Hd∗ (M, C). 2. Pick any .T ∈ M3 (C) and let .Ai = T i , i = 0, 1, 2. (a) Study the problems (b) and (c) above for this case. (b) Consider the special case T is normal or nilpotent. Project 6.23 1. Study exercise 2 above for a general matrix .T ∈ Mk (C). 2. Can Corollary 6.21 be generalized to an infinite dimensional abelian Banach algebra .B? A good case to consider is Example 4.30. (a) Compute .Hd∗ (P c (A), C) and .F ∗ (B). (b) Determine whether the homomorphism .κ : F ∗ (B) → Hd∗ (P c (A), C) is surjective. 3. Let .B be abelian and .A1 , . . . , An ∈ B. Study the kernel of the homomorphism ∗ → H ∗ (P c (A), C). .κ : F d 4. Let .Aθ be the irrational rotation algebra in Sect. 4.3. Determine .F ∗ (Aθ ).

6.3 Jacobi’s Formula in Higher Orders In the cases .B is a matrix algebra, Lemma 1.9 gives a description for the first Chern– Simons form .tr ωA of a linear pencil .A(z). It makes one wonder if there are similar p expressions for .tr ωA , p ≥ 2. An example shall provide some insight.

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6 The Maurer–Cartan Form of Operator Pencils

6.3.1 su2 and the Chern Character We take another look at Example 1.1.1 regarding the Lie algebra .su2 . Let .σi , i = 1, 2, 3, be the Pauli matrices and set .σ (z) = z0 I + z1 σ1 + z2 σ2 + z3 σ3 . Then the characteristic polynomial is Qσ (z) = det σ (z) = z02 − z12 − z22 − z32 ,

.

and the projective spectrum .P (σ ) is the zero set of .Qσ . Since .σ : C4 → M2 (C) is a vector space isomorphism, it identifies .P c (σ ) with the general linear group .GL2 . For convenience, we set .dz := dz0 ∧ dz1 ∧ dz2 ∧ dz3 and let .d zˆj be the 3-form with .dzj being omitted from dz, .j = 0, 1, 2, 3. Moreover, setting S(z) := z0 d zˆ0 − z1 d zˆ1 + z2 d zˆ2 − z3 d zˆ3 ,

.

we have .dS(z) = 4dz. If we let .ωσ = σ −1 dσ , then the Chern–Simons forms are m c .tr ωσ , 1 ≤ m ≤ 4. Because of the identification of .P (σ ) with .GL2 , they can be regarded as the Chern–Simons forms for .GL2 . Jacobi’s formula gives .tr ωσ = dQσ /Qσ , Example 6.13 indicates that .tr ωσ2 = tr ωσ4 = 0, and a direct computation shows that tr ωσ3 (z) =

.

12i S(z), z ∈ P c (σ ). Q2σ (z)

(6.19)

To see that .tr ωσ3 (z) is not exact, we observe that .GL2 = C× × SU2 . If we write .zj = xj + iyj , where .j = 0, 1, 2, 3 and .xj and .yj are real, then the pre-image of the Lie group .SU (2) under the map .σ is S := {(x0 , ix1 , ix2 , ix3 ) ∈ C4 | x02 + x12 + x22 + x32 = 1},

.

(6.20)

which is the unit 3-sphere. We leave the verification of this claim as an exercise. It follows that .tr (ωσ3 (z))|S = −12S(x0 , x1 , x2 , x3 ), which up to a scalar multiple is the standard 3-form on the unit 3-sphere [24]. In summary, the Chern–Simons forms generate the de Rham cohomology of .GL2 . In this case the total Chern character Ch(σ ) =

.

2i dQσ (z) − 2 S(z), z ∈ P c (σ ), Qσ (z) Qσ (z)

which contains complete topological information about .GL2 .

6.3 Jacobi’s Formula in Higher Orders

137

6.3.2 An Extension to General Matrices Formula (6.19) can be viewed as a case of Jacobi’s formula in order 3. It is not coincidental to the Pauli matrices. Theorem 6.24 If .A1 , . . . , A4 are elements in a Banach algebra .B with trace .tr , then 3 tr ωA (z) = g(z)S(z), z ∈ P c (A),

(6.21)

.

where .g(z) is a holomorphic scalar function on .P c (A). Proof To simplify the computation, we set .Bj (z) := A−1 (z)Aj , j = 1, . . . , 4. Then it holds that z1 B1 + · · · + z4 B4 = I.

(6.22)

.

A straightforward calculation using the properties of trace yields the formula 3 tr ωA =

E

Iij k dzi ∧ dzj ∧ dzk ,

.

(6.23)

1≤i |z1 | + |z2 |} ⊂ P c (A). In view of Corollary 6.39, we have κ(φj ) = φj (ωA , ωA ) = gj (z)S(z), j = 1, 2,

.

where .g1 and .g2 are holomorphic on D. Can .g1 and .g2 be determined?

Chapter 7

Hermitian Metrics on the Resolvent Set

For elements .A1 , . . . , An in a unital Banach algebra .B, every nonempty pathconnected component .oA of the resolvent set .P c (A) is a Stein domain. Moreover, the previous chapter has shown that some topological information about .oA can be retrieved from the Maurer–Cartan form .ωA by pairing it with linear functionals. In this chapter, we assume .B is a .C ∗ -algebra and consider some Hermitian metrics ∗ ∧ ω . Topics include the on .oA naturally induced through the .(1, 1)-form .−ωA A fundamental form of linear operators, Kähler metric, extremal arc length, and singularities.

7.1 Hermitian Vector Bundle Let .o be a complex manifold of dimension n. If .z = (z1 , . . . , zn ) is the coordinate in a local chart U , then .∂i stands for . ∂z∂ i , and .∂¯i stands for . ∂∂z¯ i . As in Sect. 6.1, we let E E .∂ = ∂i dzi , ∂¯ = ∂¯i d z¯ i , i

i

¯ A smooth function f on .o is holomorphic if .∂f ¯ (z) = and therefore .d = ∂ + ∂. 2 2 ¯ ¯ ¯ 0, z ∈ o. Since .∂ f is a .(2, 0)-form, .∂ f is a .(0, 2)-form, and .(∂ ∂ + ∂∂)f is a 2 2 ¯ 2 = ∂ ∂¯ + ∂∂ ¯ = 0. .(1, 1)-form, the fact .d = 0 implies .∂ = ∂ Definition 7.1 A smooth function f on .o is said to be pluriharmonic if .∂∂f = 0 or, equivalently, .∂j ∂¯k f = 0 for all .1 ≤ j, k ≤ n. Note that .∂∂f = .∂∂f for smooth functions. Proposition 7.2 A smooth real-valued function f on .o is pluriharmonic if and only if it is locally the real part of a holomorphic function. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 R. Yang, A Spectral Theory Of Noncommuting Operators, https://doi.org/10.1007/978-3-031-51605-4_7

145

146

7 Hermitian Metrics on the Resolvent Set

Proof The sufficiency is obvious. For the necessity, we assume D is a small polydisc in .o and set .g = i(∂f − ∂f ) which is real. The fact f is pluriharmonic implies dg = i(∂ + ∂)(∂f − ∂f )

.

= i(∂∂f − ∂∂f ) = 0. Since D is simply connected, the Poincaré lemma ensures the existence of a real smooth function .h on D such that .g = dh = ∂h + ∂h. Equating the two .(0, 1)forms in g, we must have .∂h = i∂f , and it follows that .i(h − if ) = f + ih is holomorphic and its real part is f . u n

7.1.1 The Dolbeault Operator For integers .s, t ≥ 0, a smooth .(s, t)-form .η over .o can be written locally as E

η=

.

fI,J dzI ∧ d z¯ J , fI,J ∈ C ∞ (o),

|I |=s,|J |=t

where .I = (i1 , . . . , is ) stands for a multi-index of s strictly increasing integers between 1 and n, and .dzI stands for .dzi1 ∧ · · · ∧ dzis . We denote the space of all smooth .(s, t)-forms by .As,t (o). Likewise, given a rank-k complex vector bundle s,t (o, E) denote the space of smooth type-.(s, t) sections .π : E → o, we shall let .A k ∼ of E. If .ψ : E |U = U × C is a trivialization of E over the local chart U , then As,t (o, E)|U ∼ = As,t (o)|U ⊗ Ck .

.

Thus, globally we have the decompositions Ap (o) =

O

.

s+t=p

As,t (o), Ap (o, E) =

O

As,t (o, E), 0 ≤ p ≤ 2n.

s+t=p

(7.1) Given a connection D on E, it can be expressed locally as .D = d + ω (see (6.4)), where .ω ∈ A1 (o, End(E)). In view of (7.1), we can decompose .ω = ω1,0 + ω0,1 , where .ω1,0 ∈ A1,0 (o, End(E)) and .ω0,1 ∈ A0,1 (o, End(E)). Consequently, we can write D = (∂ + ω1,0 ) + (∂¯ + ω0,1 ) := D 1,0 + D 0,1 .

.

(7.2)

It is easy to check that both .D 1,0 and .D 0,1 satisfy the Leibniz rule and hence are connections themselves.

7.1 Hermitian Vector Bundle

147

Definition 7.3 A complex vector bundle .π : E → o is said to be holomorphic if for every local chart U the trivialization map .ψ : E |U ∼ = U × Ck is biholomorphic. It follows that if .φ : E |V ∼ = V × Ck is a trivialization of E over another local chart V , the transition function ψ ◦ φ −1 : (U ∩ V ) ⊗ Ck → (U ∩ V ) ⊗ Ck

.

is a matrix-valued holomorphic function over .U ∩ V such that for every section s of .E |U ∩V one has .ψ(s(z)) = (ψ ◦ φ −1 )φ(s(z)), z ∈ U ∩ V . A unique connection ¯E can be defined on a holomorphic vector bundle as follows. Given any section .∂ 0 .s ∈ A (o, E) and a local trivialization .ψ : E |U ∼ = U × Ck , we define ( ) −1 ¯ ¯ .∂E s = ψ ∂ψ(s) . If .φ is another trivialization of .E|V , then since .ψ ◦ φ −1 is holomorphic on .U ∩ V , we have ( ( ) ) −1 ¯ −1 ¯ −1 .ψ ∂ψ(s) = ψ ∂[(ψ ◦ φ )φ(s)] ( ) ¯ ¯ ◦ φ −1 )φ(s) + (ψ ◦ φ −1 )∂φ(s) = ψ −1 ∂(ψ ( ( ) ) ¯ ¯ = φ −1 ∂φ(s) = ψ −1 ψ ◦ φ −1 ∂φ(s) . In other words, the section .∂¯E s is independent of the choice of local trivialization, and hence ∂¯E : A0 (o, E) → A0,1 (o, E)

.

is a well-defined connection on E. This connection is often called the Dolbeault operator. It is not hard to verify that .∂¯E2 = 0. This gives rise to the Dolbeault cohomology of .o with coefficients in E. The connection .∂¯E leads to two natural definitions. Definition 7.4 A section .s ∈ A0 (o, E) is said to be holomorphic if .∂¯E s = 0. It is worth noting that not every complex vector bundle possesses nonconstant holomorphic sections. ˆ × V be a rank-k trivial ˆ be the Riemann sphere and .E = C Example 7.5 Let .C ˆ ¯ bundle over .C. Then the Dolbeault operator .∂E on E coincides with the traditional .∂¯ operator on .C. Thus every holomorphic section of E is a vector-valued holomorphic ˆ which is constant by Liouville’s theorem. function on .C

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7 Hermitian Metrics on the Resolvent Set

However, nonconstant holomorphic sections always exist locally, and it thus makes a perfect sense to speak of holomorphic local frames over a chart U , a fact that we will refer to a little while later. The Dolbeault operator gives rise to the following type of connections. Definition 7.6 A connection D on a holomorphic vector bundle E is said to be compatible with the complex structure if .D 0,1 = ∂¯E .

7.1.2 Hermitian Metric A Hermitian structure h on .E → o is an inner product .hz on each fiber .Ez which depends differentiably on .z ∈ o. The pair .(E, h) is called a Hermitian vector bundle. If .ψ : E |U ∼ = U × Ck is a trivialization of E over a local chart .U ⊂ o and .{s1 , . . . , sk } is a local frame of .E |U , we may evaluate .hi j¯ (z) := h(si , sj )|z . ( )k Then the metric matrix function .H (z) = hi j¯ (z) i,j =1 is positive definite and differentiable at every .z ∈ U . Hermitian structure h on E always exists. A canonical one is constructed by the map .ψ −1 : U × Ck → E |U , which transports the Euclidean inner product in .Ck to each fiber .Ez , z ∈ U . Then by means of partition of unity associated with the local charts on .o, a global Hermitian structure on E can be constructed. Observe that for any two sections s and .s ' of E, the function ' 0 .h(s, s ) ∈ A (o). Definition 7.7 Let .(E, h) be a Hermitian vector bundle over .o. A connection D on E is said to be Hermitian (or compatible with h) if for arbitrary sections s and .s ' of E we have dh(s, s ' ) = h(Ds, s ' ) + h(s, Ds ' ).

.

If D is a Hermitian connection, then in a local chart we have ¯ (∂ + ∂)h(s, s ' ) = h(Ds, s ' ) + h(s, Ds ' )

.

= h(D 1,0 s + D 0,1 s, s ' ) + h(s, D 1,0 s ' + D 0,1 s ' ) ) ( ) ( = h(D 1,0 s, s ' ) + h(s, D 0,1 s ' ) + h(D 0,1 s, s ' ) + h(s, D 1,0 s ' ) . Comparing the type of differential forms on both sides of the equation, we obtain ∂h(s, s ' ) = h(D 1,0 s, s ' ) + h(s, D 0,1 s ' ).

.

(7.3)

Here, since .h(s, s ' ) is conjugate linear in the second variable, the summand 0,1 s ' ) above is indeed a .(1, 0)-form. .h(s, D

7.1 Hermitian Vector Bundle

149

Theorem 7.8 Given a Hermitian holomorphic vector bundle, there exists a unique Hermitian connection that is compatible with the complex structure. Proof Let .(E, h) be a rank-k holomorphic Hermitian vector bundle over .o, and assume D is a Hermitian connection on E that is compatible with the complex structure. Then .D 0,1 = ∂¯E , and hence for a local holomorphic frame .{s1 , . . . , sk } over U , we have D(si ) = D 1,0 si + ∂¯E si = D 1,0 si .

.

If we write .D 1,0 si = ωki sk , then .ω = (ωki ) is a matrix-valued .(1, 0)-form, and dh(si , sj ) = h(ωki sk , sj ) + h(si , ωkj sk ),

.

or in matrix form .dH = ωT H + H ω, where H is the metric matrix function. ¯ = H ω, or equivalently Comparing the type of the forms on both sides, we see .∂H ω = H¯ −1 ∂ H¯ .

.

Hence the connection .D = d + ω is uniquely determined by the metric h.

(7.4) u n

The unique connection given by Theorem 7.8 is called the Chern connection. To compute its curvature .O, we observe from (7.4) that .ω is a Maurer–Cartan form with respect to .∂. Thus the corresponding Maurer–Cartan equation implies O = dω + ω ∧ ω

.

¯ + (∂ω + ω ∧ ω) = ∂ω. ¯ = ∂ω

(7.5)

Example 7.9 Let E be a holomorphic line bundle over .o with a local frame s. A Hermitian metric h on E is then given locally by the positive function .H (z) = h(s, s)|z . Thus, the Chern connection matrix is the scalar .(1, 1)-form .ω = H −1 ∂H = ∂ log H . It follows that the Chern connection on E is .D = d + ∂ log H and its curvature ¯ = ∂∂ ¯ log H. O = ∂ω

.

7.1.3 Kähler Metric and the Ricci Form A scalar-valued .(1, 1)-form .ω(z) = 2i hj k¯ (z)dzj ∧ d z¯k on .o is said to be real if .ω(z) = ω(z), and this occurs when the matrix function .H (z) = (hj k¯ (z)) is Hermitian at every .z ∈ o. Such a form induces a Hermitian bilinear form h on the tangent bundle .T (o) over the chart such that

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7 Hermitian Metrics on the Resolvent Set

h(∂j , ∂k ) = hj k¯ (z) = h(∂¯k , ∂¯j ), h(∂¯j , ∂k ) = 0 = h(∂j , ∂¯k ), 1 ≤ j, k ≤ n.

.

We say that .ω induces a Hermitian metric on .o if h defines a Hermitian metric on T (o), or equivalently, .H (z) is smooth and positive definite at each .z ∈ o. In this case, .ω(z) is called the fundamental form of the metric. There are several elemental definitions about such a metric.

.

Definition 7.10 For a complex manifold .o with the fundamental form .ω, the Hermitian metric on .o induced by .ω is said to be Kähler if .dω = 0. In this case, .o is called a Kähler manifold, and the cohomology class .[ω] ∈ Hd2 (o, C) is called the Kähler class. By the Poincaré lemma, locally there exists a smooth 1-form .η = η1,0 + η0,1 such that .

− iω = dη ¯ 1,0 + η0,1 ) = (∂ + ∂)(η ¯ 1,0 + ∂η ¯ 0,1 . = ∂η1,0 + ∂η0,1 + ∂η

¯ 0,1 . Another application But since .ω is a .(1, 1)-form, we must have .∂η1,0 = 0 = ∂η of the Poincaré lemma gives ¯ '' η1,0 = ∂ρ ' , and η0,1 = ∂ρ

.

for some local smooth functions .ρ ' and .ρ '' . Setting .ρ = ρ ' + ρ '' , we arrive at ¯ 1,0 ) = i∂ ∂ρ. ¯ ω = i(∂η0,1 + ∂η

.

(7.6)

This .ρ is the Kähler potential of .ω. However, it is not unique, and it may not exist globally. But it is important to observe that a Kähler metric can be expressed locally by a scalar function rather than the metric matrix. This is exactly why it is more tractable than a general Hermitian metric. Example 7.11 Consider the complex projective space .Pn with coordinate .z = [z0 : · · · : zn ]. For .0 ≤ j ≤ n, we let .Uj = {z ∈ Pn | zj /= 0} and define .φj : Uj → Cn by ( φj (z) =

.

zj zn z0 ,..., ,..., zj zj zj

) =: (w1 , . . . , wn ),

where .x stands for the omission of x. Then .{(U0 , φ0 ), . . . , (Un , φn )} is a local chart of .Pn . On each .Uj , we define ¯ log(1 + |w1 |2 + · · · + |wn |2 ), ωj = i ∂∂

.

7.1 Hermitian Vector Bundle

151

where .∂ is the differential with respect to the variables .wj (not .zj ). We leave it as an exercise to check that .ωj = ωk on .Uj ∩ Uk , and thus they patch up to be a global Kähler form .ω on .Pn . Observe that .ω’s Kähler potential is not defined globally. The metric on .Pn defined by .ω is called the Fubini–Study metric. An important closed .(1, 1)-form naturally associated with the curvature is the Ricci form. Definition 7.12 For a Hermitian vector bundle .(E, h) with the Chern connection D and its curvature .O, the Ricci form is defined as .R = itr O. The connection D is said to be Ricci-flat if .R = 0. i tr O. Hence .R = 2π c1 (E) which is closed Recall the first Chern form .c1 (E) = 2π (Theorem 6.9). Moreover, using Jacobi’s formula (Lemma 1.9), formula (7.4), and the fact .H (z) is everywhere positive definite, one checks that

( ) −1 ¯ ¯ ¯ .R(z) =itr O = itr ∂ H ∂H ( ) ¯ ¯ log det H (z) =i ∂tr H¯ −1 ∂ H¯ = i ∂∂ = − i ∂¯k ∂j log det H (z)dzj ∧ d z¯ k =: i Ricj k¯ dzj ∧ d z¯ k , where the negative sign in the front of the line above is due to the change from d z¯ k ∧ dzj to .dzj ∧ d z¯ k . Thus, the scalar function .log det H (z) is a potential function for .R. The tensor .Ric(z) = Ricj k¯ dzj ⊗ d z¯ k is called the Ricci curvature tensor. We have just seen that for a Kähler manifold .(o, ω), the Ricci form .R represents the first Chern class .[c1 (o)]. On the other hand, if .R is any real .(1, 1)-form with this property, is there necessarily a Kähler form .ω on .o for which .R is the associated Ricci form? It was conjectured yes for compact manifolds in 1954 by Calabi. A proof was given 20 years later in Yau [245].

.

Theorem 7.13 Let .(o, ω) be a compact Kähler manifold and .R' be a real .(1, 1)form that represents .[c1 (o)]. Then there exists a Kähler form .ω' on .o whose associated Ricci form is equal to .R' , and moreover .[ω' ] = [ω]. In view of the observations preceding Example 7.11, the fact .[ω' ] = [ω] in Theorem 7.13 implies the existence of a global potential function .ρ such that ' ¯ In other words, for a compact Kähler manifold .o, its Kähler form, .ω = ω + i∂ ∂ρ. up to a global potential function, is uniquely determined by the first Chern class of .o. In particular, any compact Kähler manifold with trivial first Chern class has a unique Ricci flat Kähler metric in every Kähler class. Such manifolds are called Calabi–Yau manifolds. Primary examples of such manifolds are smooth projective varieties .(X, ω|X ), where .X ⊂ Pn and .ω is the Kähler form in Example 7.11. Since for a Kähler manifold .(o, ω), both .ω and .R = 2π c1 (o) are .(1, 1)-forms, it naturally makes one wonder whether the two may coincide (up to a scalar .λ). Indeed, the equation .R = λω is called the Kähler–Einstein equation, and it was

152

7 Hermitian Metrics on the Resolvent Set

motivated by Einstein’s field equations in general relativity theory. Note that since ω and .R are real forms, it is only necessary to consider the equation for a real scalar .λ. Clearly, the case .λ = 0 reduces to the Calabi–Yau manifolds. Such manifolds play an important role in superstring theory. We refer the reader to Yau [246] and Hübsch [127] for more information on this exciting application to modern physics. For an introduction to complex geometry and Kähler manifolds, we refer the reader to Ballmann [8] and Huybrechts [128]. .

Exercise 7.14 1. Consider the Dolbeault operator .∂¯E : A0 (o, E) → A0,1 (o, E). (a) Verify that it is a connection. (b) Show that .∂¯E2 = 0. 2. Prove by direct computation that the form .ω = 2i hj k¯ (z)dzj ∧ d z¯k is closed if and only if .∂k hi j¯ (z) = ∂i hk j¯ (z), 1 ≤ i, j, k ≤ n, z ∈ o. 3. In Example 7.11, verify that .ωj = ωk on .Uj ∩ Uk . Also, check that .ω defines a Hermitian metric on .Pn , i.e., the metric matrix .H (z) = (hj k¯ (z)) is positive definite at every .z ∈ Pn . 4. Continuing with exercise 3, determine the Ricci curvature .R of the metric .ω. 5. Consider a Kähler manifold .(o, ω) with metric matrix H . Show that the Ricci | |2 curvature .R = 0 if and only if it holds locally that .det H (z) = |ef (z) | for some holomorphic function .f (z). 6. Let .f (z) be a homogeneous polynomial in .Cn+1 and set .X = {z ∈ Pn | f (z) = 0}. If the differential .df (z) /= 0 for each .z ∈ X, then we say that X is a smooth variety. (a) Prove that in this case X is a compact submanifold of .Pn . (b) Show that the restriction .ω|X , where .ω is the Kähler form in Example 7.11, is a Kähler form on X. (c) Verify that X is Calabi–Yau.

7.2 The Fundamental Form of Operator Pencils For elements .A1 , . . . , An in a .C ∗ -algebra .B, we consider the pencil A(z) = z0 I + z1 A1 + · · · + zn An

.

(7.7)

as we did for the characteristic polynomial of matrices. For convenience, we denote I by .A0 and use Einstein’s summation convention in many of the computations. In this section, we shall always assume that .oA is the path-connected component of c n+1 that contains .(1, 0, 0, . . . , 0). The adjoint of .ω (z) = A−1 (z)dA(z) .P (A) ⊂ C A is the .B-valued .(0, 1)-form

7.2 The Fundamental Form of Operator Pencils

153

∗ ωA (z) = dA∗ (z)(A−1 (z))∗ = (A−1 (z)Ak )∗ dzk .

.

This gives rise to the following definition. Definition 7.15 The fundamental form of the pencil .A(z) is defined as IA =

.

−i ∗ i ωA ∧ ωA = (A−1 (z)Ak )∗ A−1 (z)Aj dzj ∧ dzk . 2 2

(7.8)

The factor . −i 2 is used to make .IA real in the sense that IA∗ =

.

i ∗ (ω ∧ ωA )∗ = IA . 2 A

(7.9)

Moreover, in the case .n = 0 and .A(z) = zI , we have .P c (A) = C \ {0} and dz .ωA (z) = z . Writing .z = x + yi, we have IA =

.

dx ∧ dy i(dz ∧ dz) = 2 . 2|z|2 x + y2

A linear functional .φ ∈ B∗ is called a state if .φ(I ) = 1 and .φ(a ∗ a) ≥ 0 for all .a ∈ B. One verifies that if .φ is a faithful state, then φ(IA ) =

.

) i ( −1 φ (A (z)Ak )∗ A−1 (z)Aj dzj ∧ dzk 2

defines a Hermitian form on the tangent bundle of .oA , thus giving a Hermitian metric on .oA . The connection between this metric and the algebraic properties of .A1 , . . . , An shall be the primary concern of this section. For more details, we refer the reader to [69].

7.2.1 Operator-Valued Differential Forms The space of .B-valued smooth .(p, q)-forms on a complex domain .o ⊂ Cn is denoted by .Ap,q (o, B). Thus Am (o, B) =

O

.

Ap,q (o, B), 0 ≤ m ≤ 2n.

p+q=m

The notions of Hermitian form and Kähler form can be generalized to the .B-valued setting.

154

7 Hermitian Metrics on the Resolvent Set

Definition 7.16 A smooth .B-valued .(1, 1)-form .O(z) = 2i Oj k (z)dzj ∧ dzk on .o is said to be Hermitian if for every .z ∈ o and any set of n complex numbers .v1 , v2 , . . . , vn the double sum .vj Oj k (z)v ¯k is a positive element in .B. Two nuances are worth mentioning. First, there is a difference between a self-adjoint B-valued .(1, 1)-form defined in (7.9) and a Hermitian .B-valued .(1, 1)-form defined above: the latter requires positivity of .B-valued matrix .(Oj k (z)). Second, in the scalar-valued case, for a Hermitian metric defined by .ω = 2i hj k¯ (z)dzj ∧ d z¯ k , the complex metric matrix .(hj k¯ (z)) needs to be positive definite. In the .B-valued case, it is only required that .vj Oj k (z)vk ≥ 0 in .B for every vector .v ∈ Cn . Given any elements .A1 , . . . , An in .B and complex numbers .v0 , v1 , . . . , vn , one checks the double sum

.

( )∗ ( ) A−1 (z)A(v) ≥ 0, v¯k A∗k (A−1 (z))∗ A−1 (z)vj Aj = A−1 (z)A(v)

.

(7.10)

showing that the fundamental form .IA is Hermitian. Definition 7.17 A .B-valued Hermitian .(1, 1)-form .O(z) is said to be Kähler if dO(z) = 0.

.

One observes that for every .φ ∈ B∗ and .ωp,q ∈ Ap,q (o, B), the pairing p,q (z)) produces a scalar-valued smooth .(p, q)-form. The fact .φ(ω dφ(ωp,q (z)) = φ(dωp,q (z))

.

is nicely expressed in the following commuting diagram: Λ p (Ω, B)

d

φ

φ .

Λ p (Ω, C)

Λ p +1 (Ω, B)

d

Λ p +1 (Ω, C).

(7.11)

Since an element .a ∈ B is 0 if and only if .φ(a) = 0 for every .φ ∈ B∗ , the following lemma is clear. Lemma 7.18 A .B-valued form .ωp,q (z) = 0 if and only if .φ(ωp,q (z)) = 0 for every ∗ .φ ∈ B . Therefore, .ωp,q (z) is closed if and only if .φ(ωp,q (z)) is closed for every .φ ∈ B∗ . Moreover, if .φ is positive, (7.9) gives .φ(IA ) = φ(IA∗ ) = φ(IA ), indicating that .φ(IA ) is a real .(1, 1)-form. The following fact is a bit surprising. Proposition 7.19 For elements .A1 , . . . , An in .B, the following statements are equivalent: (a) The elements pairwise commute.

7.2 The Fundamental Form of Operator Pencils

155

(b) The Maurer–Cartan form .ωA (z) is closed. (c) The fundamental form .IA (z) is Kähler. Proof Since .ωA (z) is holomorphic on .oA , the Maurer–Cartan equation gives ∂ωA = −ωA ∧ ωA

.

= −A−1 (z)Aj A−1 (z)Ak dzj ∧ dzk E =− [A−1 (z)Aj , A−1 (z)Ak ]dzj ∧ dzk , z ∈ oA .

(7.12)

0≤j β > 0 such that .βI ≤ (A−1 (z))∗ A−1 (z) ≤ αI . Note that .α and .β depend on z, but it is not of concern here. For every nonzero vector n+1 , we check that .v = (v0 , v1 , . . . , vn ) ∈ C vj hj k¯ (z)v¯k = vj φ(A∗k (A−1 (z))∗ A−1 (z)Aj )v¯k

.

( ) =φ(v¯k A∗k (A−1 (z))∗ A−1 (z)vj Aj )=φ A∗ (v)(A−1 (z))∗ A−1 (z)A(v) .

Since βA∗ (v)A(v) ≤ A∗ (v)(A−1 (z))∗ A−1 (z)A(v) ≤ αA∗ (v)A(v)

.

and .φ is positive, it follows that

7.2 The Fundamental Form of Operator Pencils

157

βφ(A∗ (v)A(v)) ≤ vj hj k¯ (z)v¯k ≤ αφ(A∗ (v)A(v)).

.

Hence .H (z) = (hj k¯ (z)) is positive definite for every .z ∈ oA if and only if .φ is faithful on .HA . u n Example 7.22 The following two well-known types of states on .B produce useful Hermitian metrics on .oA . (a) If .φ is a faithful tracial state, then it is clearly a faithful state on .HA . (b) Assume .B is a .C ∗ -algebra of operators acting on a Hilbert space .H. For a unit vector .x ∈ H, let .φx be a vector state defined on .B by .φx (a) = , a ∈ B. If .{x, A1 x, . . . , An x} is linearly independent, then for any nonzero vector n+1 , we have .φ ((c A )∗ (c A )) = ||c A x||2 > 0, .(c0 , c1 , . . . , cn ) ∈ C x i i i i i i indicating that .φ is faithful on .HA . If .A1 , · · · , An are commuting elements of .B and state .φ is faithful on .HA , then Proposition 7.19 implies .dφ(IA ) = φ(dIA ) = 0, i.e., .φ(IA ) defines a Kähler metric on .oA . Interestingly, the converse is also true if .φ is faithful on the entire .B. Theorem 7.23 Let .φ be a faithful state on .B. Then .A1 , . . . , An pairwise commute if and only if .φ(IA ) is Kähler on .oA . Proof It only remains to check the sufficiency. If .φ(IA ) is Kähler, then we have 0 = dφ(IA ) = φ(dIA ). Then (7.13) gives

.

∗ 0 = φ(ωA ∧ ∂IA ) ) EE ( φ (A−1 (z)Ak )∗ [A−1 (z)Ai , A−1 (z)Aj ] d z¯ k ∧ dzi ∧ dzj , =

.

k

i |z1 |2 + |z2 |2 }. 4*. Do a parallel study for the irrational rotation algebra .(Aθ , tr ) in Sect. 4.3 with respect to the pencil .A(z) = z0 I + z1 U + z2 V . Does the Hermitian form .tr (IA ) depend on .θ ? Project 7.25 1. Find conditions on commuting elements .A1 , . . . , An ∈ B and state .φ faithful on .HA such that .φ(IA ) defines a Ricci-flat metric on .oA . 2. In the case .A1 , . . . , An are commuting, is there a .B-valued potential function for the fundamental form .IA ? 3. Are there noncommuting matrices .A1 , A2 ∈ M2 (C) and state .φ on .M2 (C) for which .φ(IA ) is a Kähler form on .P c (A)? One may start the investigation by considering the Pauli matrices .Ai = σi , i = 1, 2, and the pencil .A(z) = z0 I + z1 A1 + z2 A2 .

7.3 The Issue of Completeness Suppose a globally defined but locally expressed fundamental form . 2i gj k¯ (z)dzj ∧ d z¯k induces a Hermitian metric on a complex manifold .o with metric matrix .g(z) = (gi j¯ ). If .γ : [a, b] → o is a differentiable curve, then its tangent vector γ ' (t) =

.

(

dzn dz1 ,..., dt dt

)

7.3 The Issue of Completeness

159

can be identified with the section . length is

dzj dt ∂j

in the tangent bundle .T (o). Thus its square

||γ ' (t)||2 = gj k¯ (z)

.

dzj d z¯ k . dt dt

(7.17)

Hence, the infinitesimal arc length ds is given by .ds 2 = ||γ ' (t)||2 dt 2 = gj k¯ (z)dzj d z¯k , and therefore the arc length of .γ is f L(γ ) =

.

a

b

'

f

||γ (t)||dt = a

b

( )1/2 dzj d z¯ k gj k¯ (z) dt. dt dt

If p and q are two points in .o, then their distance is .dist(p, q) = infγ L(γ ), where γ is any piecewise-smooth path in .o parametrized by .γ (t), 0 ≤ t ≤ 1 such that .γ (0) = p and .γ (1) = q. If .o is embedded in a larger space, then the completion of .o with respect to the metric g shall be denoted by .[o]. In other words, the set .[o] consists of the limits of all Cauchy sequences in .o with respect to the metric .dist induced from the fundamental form. .

7.3.1 Distance in the Resolvent Set Assume .B is a .C ∗ -algebra with a tracial state .φ. In this subsection, we shall study the metric on .oA defined by the fundamental form .φ(IA ), where “A” stands for the pencil (7.7). A notable feature of this metric is that it has singularities at the projective spectrum .P (A). So the completeness of the metric is a delicate issue. The reader shall find it helpful to review Sect. 5.3 at this time. We assume that .|detH S x| is well-defined for every .x ∈ GL(B). Then the FK determinant is also well-defined with .detF K (x) = |detH S x|. Moreover, recall that a point .p ∈ P (A) is said to be .φ-singular if the linear pencil .A(p) is .φ-singular. Since .ωA (z) = A−1 (z)dA(z) resembles the derivative of logarithmic function, it is probably not a surprise that .log plays an important role here. Lemma 7.26 For any tuple A of elements in .B, if .p, q ∈ oA , then .

dist(p, q) ≥| φ(log |A(p)|) − φ(log |A(q)|) | .

) ( Proof First of all, by (7.14) we have .gi j¯ (z) = φ (A−1 (z)Aj )∗ A−1 (z)Ai . Let p and q be two points in .oA , and let .γ = {z(t)| 0 ≤ t ≤ 1} be a piecewise-smooth path such that .z(0) = p and .z(1) = q. Then on .γ , the derivative .dA(z(t))/dt = zj' (t)Aj , and the length of .γ can be computed as

160

7 Hermitian Metrics on the Resolvent Set

f

1/

L(γ ) =

.

f

0

= f

0 1

= f

1/

0

=

zi' (t)gi j¯ (z(t))zj' (t)dt ) ( zi' (t)φ (A−1 (z)Aj )∗ A−1 (z)Ai zj' (t)dt

/ ( ) φ (A−1 (z(t))zj' (t)Aj )∗ A−1 (z(t))zi' (t)Ai dt

1/

φ

(( )∗ ) A−1 (z(t))A' (z(t)) A−1 (z(t))A' (z(t)) dt.

(7.18)

0

Since .φ is a state, for every .a, b ∈ B we have .|φ(ab)|2 ≤ φ(a ∗ a)φ(b∗ b). In particular, .|φ(a)|2 ≤ φ(a ∗ a). It follows that f

f ( )| | | |φ A−1 (z(t))A' (z(t)) | dt =

1|

L(γ ) ≥

.

0

f =

0

|f 1 | |φ (ωA (z(t)))| ≥ ||

0

1 0

1|

( )| | | |φ A−1 (z(t))dA(z(t)) |

| | φ (ωA (z(t)))|| .

Using (5.15), we have |f 1 | | | L(γ )≥ || d log detH S A(z(t))|| = |log detH S A(p) − log detH S A(q)|

.

0

|( ) ( )| = | log |detH S A(p)| − log |detH S A(q)| +i Arg detH S A(p)− Arg detH S A(q) | ≥ |log |detH S A(p)| − log |detH S A(q)|| = |log detF K A(p) − log detF K A(q)| . (7.19) Since .γ is arbitrary, we have .

dist(p, q) = inf L(γ ) ≥ |log detF K A(p) − log detF K A(q)| γ

= |φ(log |A(p)|) − φ(log |A(q)|)| .

(7.20) u n

In light of Corollary 5.20, the proof above can be modified a little to accommodate the case in which q is a boundary point of .oA . Theorem 7.27 For elements .A1 , . . . , An ∈ B, if .q ∈ ∂oA is .φ-singular, then .q ∈ / [oA ]. Proof First, we observe that in this case .q ∈ P (A). For any piecewise-smooth path γ = {z(t) | 0 ≤ t ≤ 1} such that .z(t) ∈ oA , 0 ≤ t < 1, and .q = z(1) ∈ P (A), the proof of Lemma 7.26 implies

.

7.3 The Issue of Completeness

f L(γ ) ≥ lim

.

s→1− 0

s

161

|f | |φ(ωA (z(t))| ≥ lim sup || − s→1

0

s

| | φ(ωA (z(t))||

≥ lim sup | log detF K A(p) − log detF K A(z(s)) | . s→1−

(7.21)

Since q is .φ-singular, we have .detF K A(z(1)) = detF K A(q) = 0. Corollary 5.20 then implies .lims→1 detF K A(z(s)) = 0. Therefore, .L(γ ) = ∞ holds for any such path .γ in .oA connecting p to q. Consequently, .dist(p, q) = infγ L(γ ) = ∞. Since q has infinite distance to every .p ∈ oA , it holds that .q ∈ / [oA ]. u n This theorem applies well to the matrix algebra. Example 7.28 For .B = Mk (C), we let .tr and .det stand for the trace and, respectively, determinant of .k × k matrices. Then .φ := k1 tr is a tracial state on 1/k (Sect. 5.3, exercise 1). Consider the .Mk (C), and it holds that .detF K x = | det x| characteristic polynomial .QA of matrices .A1 , . . . , An ∈ B. Then the projective spectrum P (A) = {z ∈ Cn+1 | QA (z) = 0},

.

which is the eigensurface of the matrices. Since .P c (A) is path-connected, we have c c .oA = P (A) and .∂P (A) = P (A) in this case. Moreover, every point in .P (A) is c .φ-singular. By Theorem 7.27, the resolvent set .P (A) is complete with respect to the metric defined by .φ(IA ). Furthermore, if .{A1 , . . . , An } is a basis for .Mk (C), where .n = k 2 , then every matrix is of the form .A(z) = z1 A1 + · · · + zn An for some z. Hence .z ∈ P c (A) if and only if .A(z) ∈ GLk , thereby identifying .P c (A) with .GLk . This means .φ(IA ) in fact defines a Hermitian metric on .GLk . Moreover, for every fixed .T ∈ GLk and c .z ∈ P (A), we can write T A(z) = z1 T A1 + · · · + zn T An = A(w),

.

for some unique .w ∈ P c (A). Denoting this w by .ρT (z), then .T A(z) = A(ρT (z)), and the map .ρ : T → ρT defines a natural left action of .GLk on .P c (A). Theorem 7.29 Let .{A1 , . . . , An } be a basis for .Mk (C). Then .φ(IA ) defines a complete, left .GLk -invariant, Ricci flat metric on .P c (A). The proof is given as an exercise. Note that the metric in Theorem 7.29 is not Kähler due to Theorem 7.23.

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7 Hermitian Metrics on the Resolvent Set

7.3.2 Another Example with D∞ The group .C ∗ -algebra .C ∗ (D∞ ) helps to illustrate the definition and computation of the Hermitian metric defined earlier. For simplicity, we consider the pencil .R∗ (z) = I + z1 λ(a) + z2 λ(t) as we did in Sect. 5.3, where .λ is the left regular representation of .D∞ on .L2 (T) ⊕ L2 (T) described in Sect. 5.1, namely ] [ ] 0 I0 0 T , λ(t) = . .λ(a) = I0 0 T∗ 0 [

Then by Theorem 5.2, we have U

P (R∗ ) =

{z ∈ C3 | 1 − z12 − z22 − 2z1 z2 x = 0}.

.

(7.22)

−1≤x≤1

For convenience, in the sequel we shall frequently write .λ(a) and .λ(t) as a and, respectively, t. If .tr is the canonical trace on .C ∗ (D∞ ), then 2itr (IR∗ )

.

∗ = tr [wR∗ ∧ wR∗ ]

= tr [(R∗−1 (z)a)∗ (R∗−1 (z)a)]dz1 ∧ d z¯ 1 + tr [(R∗−1 (z)t)∗ (R∗−1 (z)t)]dz2 ∧ d z¯ 2 + tr [(R∗−1 (z)a)∗ (R∗−1 (z)t)]dz2 ∧ d z¯ 1 + tr [(R∗−1 (z)t)∗ (R∗−1 (z)a)]dz1 ∧ d z¯ 2 . Writing .tr (IR∗ ) = 2i gj k¯ (z)dzj ∧ d z¯ k , the metric matrix is g(z) =

.

) ( tr [(R∗−1 (z)a)∗ (R∗−1 (z)a)] tr [(R∗−1 (z)t)∗ (R∗−1 (z)a)] . tr [(R∗−1 (z)a)∗ (R∗−1 (z)t)] tr [(R∗−1 (z)t)∗ (R∗−1 (z)t)]

(7.23)

Lemma 7.30 For .z ∈ P c (R∗ ), we have f g11 (z) = g22 (z) =

.

0

2π

1 + |z1 |2 + |z2 |2 + (¯z1 z2 + z1 z¯ 2 ) cos θ dθ . 2π |1 − z12 − z22 − 2z1 z2 cos θ |2

Proof Since .λ(a) is unitary, the trace property gives g11 = tr [(R∗−1 (z)a)∗ (R∗−1 (z)a)] = tr [a ∗ R∗−1 (z)∗ R∗−1 (z)a] = tr [R∗−1 (z)∗ R∗−1 (z)].

.

The block matrix ( R∗ (z) = I + z1 λ(a) + z2 λ(t) =

.

z1 T + z2 I0 I0 z1 T ∗ + z2 I0 I0

) (7.24)

7.3 The Issue of Completeness

163

has commuting entries, and thus R∗−1 (z) =

.

(

) I0 + (z1 T + z2 )K −1 (z)(z1 T ∗ + z2 ) −(z1 T + z2 )K −1 (z) , K −1 (z) −K −1 (z)(z1 T ∗ + z2 )

( ) where .K(z) = I0 − (z1 T + z2 )(z1 T ∗ + z2 ). Denoting .R∗−1 (z) by . R ij (z) 2×2 , we have ) 2 ( ) 1 (E (R ij (z))∗ R ij (z) . tr (R∗−1 (z))∗ R∗−1 (z) = tr 2

.

i,j =1

There is a factor of . 12 in the equation above because .tr is normalized such that .tr I = 1. To continue, we use the spectral decomposition f

2π

T =

.

eiθ dE(eiθ )

0

and recall from Lemma 5.22 that .tr dE(eiθ ) = have f K(z) =

.

2π

dθ 2π .

Then by functional calculus, we

( ) 1 − (z1 eiθ + z2 )(z1 e−iθ + z2 ) dE(eiθ ),

(7.25)

0

and hence, K −1 (z) =

f

2π

.

0

dE(eiθ ) . 1 − (z1 eiθ + z2 )(z1 e−iθ + z2 )

It follows that ∗

f

2π

(R (z)) R (z) =

.

11

11

(R 12 (z))∗ R 12 (z) = (R 21 (z))∗ R 21 (z) = (R 22 (z))∗ R 22 (z) =

f f f

0 2π

|z1 eiθ + z2 |2 dE(eiθ ) . |1 − (z1 eiθ + z2 )(z1 e−iθ + z2 )|2

2π

|z1 e−iθ + z2 |2 dE(eiθ ) . |1 − (z1 eiθ + z2 )(z1 e−iθ + z2 )|2

2π

dE(eiθ ) . |1 − (z1 eiθ + z2 )(z1 e−iθ + z2 )|2

0

0

0

| | iθ −iθ + z ) |2 | 2 | dE(eiθ ). |1 + (z1 e + z2 )(z1 e | 1 − (z1 eiθ + z2 )(z1 e−iθ + z2 ) |

Applying .tr to all four equations above and simplify, we arrive at

164

7 Hermitian Metrics on the Resolvent Set

−1 ∗ −1 .tr [(R∗ (z)) R∗ (z)]

1 = tr 2 f

(E 2

∗

ij

i,j =1

1 + |z1 |2 + |z2 |2 + (¯z1 z2 + z1 z¯ 2 ) cos θ dθ . 2π |1 − z12 − z22 − 2z1 z2 cos θ|2

2π

=

) (R (z)) R (z) ij

0

u n

The computation for .g22 is similar.

The component .g12 (which is equal to .g 21 ) can also be computed this way, and we leave it as an exercise. Observe that Lemma 7.30 clearly shows the singularity of the metric matrix at .z ∈ P (R∗ ). Recall from Theorem 5.26 that ( detF K R∗ (z) = exp

.

1 4π

f

2π

0

) log |1 − z12 − z22 − 2z1 z2 cos θ|dθ .

(7.26)

We are ready to describe the completion of .P c (R∗ ) with respect to the metric defined by the fundamental form .IR∗ . Theorem 7.31 .[P c (R∗ )] = C2 \ {(±1, 0), (0, ±1)}. It is worth noting that the four points .(±1, 0) and .(0, ±1) correspond to the classical spectra .σ (λ(a)) = σ (λ(t)) = {±1}. The proof of this theorem is too long to be included here. It consists of the following three steps [114]. 1. In view of (7.22), the four points .(0, ±1) and .(±1, 0) are inside .P (R∗ ). Then formula (7.26) verifies that they are .tr-singular. Thus Theorem 7.27 confirms that they are not in .[P c (R∗ )]. 2. If .q = (q1 , q2 ) ∈ P (R∗ ) is not one of the four points above, then we construct a particular path .γ such that .γ (t) ∈ P c (R∗ ) for .0 < t ≤ 1, and .γ (0) = q. There are two cases to consider: (a) in the case .q1 q2 is not real, we use the path √ ( √ ) γ (t) = q1 t + 1, q2 t + 1 , t ∈ [0, 1];

.

(b) in the case .q1 q2 is real, we shall use the path √ ) ( √ γ (t) = q1 it + 1, q2 it + 1 , t ∈ [0, 1].

.

3. Use Lemma 7.30 to verify the arc length f L(γ ) =

.

0

1(

dzj d z¯ k gj k¯ (z) dt dt

)1/2 dt < ∞.

To do so, we observe that gj k¯ (z)

.

dzj d z¯ k = ≤ ||g(γ (t))||||γ ' (t)||2 . dt dt

(7.27)

7.4 The Fundamental Form of a Single Operator

165

Since .||γ ' (t)|| is bounded over .[0, 1], what remains is to check f

1

.

||g(γ (t))||1/2 dt < ∞.

0

In the end, since .L(γ ) < ∞, the sequence .{γ ( n1 ) | n = 1, 2, . . .} is Cauchy and convergent to q, proving .q ∈ [P c (R∗ )]. Exercise 7.32 1. Let .φ be a state on a unital .C ∗ -algebra .B. Show that for every .a, b ∈ B one has |φ(ab)|2 ≤ φ(a ∗ a)φ(b∗ b).

.

2. Given any homogeneous polynomial .p(z), z ∈ Cn , show that the complement of the zero set .{p = 0} in .Pn−1 is path-connected. 3. This exercise breaks down the proof to Theorem 7.29 into three steps. (a) For any fixed .T ∈ GLk , the map .ρT : P c (A) → P c (A) is defined by .T A(z) = A(ρT (z)). Show that .ωA (ρT (z)) = ωA (z), and hence the metric form .φ(IA ) is invariant under the action by .ρT . i , 1 ≤ m ≤ k, is the (b) For .1 ≤ i ≤ k 2 , we write .Ai = (a1i , · · · , aki ), where .am mth column of .Ai . Show that k ) E ( j i gi j¯ := tr (A−1 (z)Aj )∗ A−1 (z)Ai = Ck .

.

m=1

( ) (c) For the metric matrix .g(z) = gi j¯ , show that .det g(z) = C| det A−1 (z)|2k for some constant .C > 0. This implies that the Ricci tensor is 0. 4. Compute .g12¯ in the metric matrix (7.23). 5. Consider the path .γ defined in (7.27). Show that .γ (t) ∈ P c (R∗ ) for .0 < t ≤ 1.

7.4 The Fundamental Form of a Single Operator The fundamental form .IA considered in the preceding sections is also valid for a single operator A. It gives rise to a Hermitian metric on the classical resolvent set .ρ(A). It is thus of particular interest to investigate what this new metric can reveal about A. In addition to showing some general facts, this section will take a closer look at the unilateral shift operator and quasinilpotent operators. For reference on the topics in this section, we refer the reader to [70, 157, 223]. Consider a unital .C ∗ -algebra .B and an element .A ∈ B. For the pencil .A(z) = z − A, its associated Maurer–Cartan form is

166

7 Hermitian Metrics on the Resolvent Set

ωA (z) = (z − A)−1 d(z − A) = (z − A)−1 dz, z ∈ ρ(A),

.

which is a key ingredient in the functional calculus of A. Indeed, given a function f analytic on an open domain .o containing .σ (A) and a simple loop .γ ⊂ o enclosing .σ (A), we have f f 1 1 f (z)(z − A)−1 dz = f (z)ωA (z). (7.28) .f (A) = 2π i γ 2π i γ ∗ (z) = (¯ The adjoint of .ωA (z) is the .B-valued .(0, 1)-form .ωA z − A∗ )−1 dz, and thus the fundamental form of the pencil .A(z) is

i ∗ i IA = − ωA ∧ ωA = (¯z − A∗ )−1 (z − A)−1 dz ∧ dz. 2 2

.

(7.29)

Suppose .B acts on a Hilbert space .H. Then given any unit vector .x ∈ H and its associated vector state .φx on .B, we have φx (IA ) =

.

i ||(z − A)−1 x||2 dz ∧ d z¯ . 2

(7.30)

In this case, the metric matrix is the scalar function .gx (z) = ||(z − A)−1 x||2 , and it defines a Hermitian metric on .ρ(A) with infinitesimal arc length ds given by ds 2 = gx (z)|dz|2 = gx (z)(du2 + dv 2 ), z ∈ ρ(A),

.

where .z = u + vi and .u, v ∈ R. Apparently, unlike the usual Euclidean metric on .ρ(A), the metric .gx depends on A as well as the choice of x, and it may have singularities at points in .σ (A).

7.4.1 Two Examples To get a better feeling for the metric .gx , let us look at two simple examples. Example 7.33 If .A = 0, then for every unit vector .x ∈ H we have .gx (z) = 1/|z|2 , which defines a flat metric on the punctured complex plane .C× = C \ 0. Moreover, .ds = |dz|/|z| is invariant with respect to the scalar multiplication .z → αz, where 1 .α /= 0, as well as the inversion .z → z . In other words, the metric is invariant with × respect to the group operations of .C . Consequently, the measure induced by the metric form .

i dz ∧ d z¯ du ∧ dv = 2 , z /= 0, 2 2 |z| u + v2

7.4 The Fundamental Form of a Single Operator

167

is a Haar measure on .C× . If A is densely defined (possibly unbounded) on .H, then .ρ(A) is the set of .z ∈ C for which .z − A has a bounded inverse. Thus the metric function .gx (z) = ||(z − A)−1 x||2 remains well-defined. Recall that a compact √ operator .T ∈ B(H) is said to be in the trace class if the eigenvalues of .|T | = T ∗ T are summable, and in this case its trace is defined by

.

Tr T =

∞ E , j =1

where .{e1 , e2 , . . .} is an orthonormal basis of .H. Operator T is said to be Hilbert– Schmidt if .T ∗ T is in the trace class. In this norm .||T ||2 = √ E∞case, its Hilbert–Schmidt 2 . Be aware that the trace .Tr Tr(T ∗ T ), or equivalently, .||T ||22 = ||T e || j j =1 here and the tracial state .tr in earlier sections are different things. An illuminating instance is with the identity operator I . As an element in .B(H), where .dim H = ∞, I is not in the trace class and .Tr I = ∞. However, for the identity operator ∗ .I ∈ C (D∞ ), we have .tr I = 1. dθ ) with orthonormal basis .{einθ | n ∈ Z} Example 7.34 Consider .L2 (T, 2π and the densely defined differential operator .Df (θ ) = −idf/dθ . Then D has eigenvalues n with corresponding eigenfunctions .einθ , n ∈ Z. Therefore, we have 1 inθ −1 einθ = , which implies that .||(z − D)−1 || < ∞ for every .(z − D) z−n e inθ , then the metric function .g (z) = 1/|z − n|2 . .z ∈ ρ(D) = C \ Z. So if .xn = e xn Moreover, it is not hard to check that

(¯z − D ∗ )−1 (z − D)−1 einθ =

.

1 einθ , z ∈ ρ(D), n ∈ Z. |z − n|2

Hence .(z − D)−1 is in fact Hilbert–Schmidt for every .z ∈ ρ(D), and .

i ||(z − D)−1 ||22 dz ∧ d z¯ 2 ) ( ∞ E 1 i i = dz ∧ d z¯ =: g(z)dz ∧ d z¯ . 2 2 n=−∞ |z − n| 2

Tr ID =

Observe that the metric function g has a pole at every point in .σ (D) = Z.

7.4.2 Ricci Curvature and Eigenvector For a Hermitian metric .g(z) defined on a complex domain .o ⊂ C, its Ricci ∂2 curvature is .R(z) = − ∂z∂ z¯ log g(z). Recall that the metric g is said to be flat at

168

7 Hermitian Metrics on the Resolvent Set

z ∈ o if .R(z) = 0, and it is said to be flat on D if it is flat at every point of .o, which means .log g(z) is harmonic on D. If .γ = {z(t) | 0 ≤ t ≤ 1} is a piecewise-smooth path in .o, then its arc length is

.

f L(γ ) =

1/

.

g(z(t))|z' (t)|dt.

(7.31)

0

Assume A is densely defined on .H and .x ∈ H with .||x|| = 1. The metric defined by .gx (z) = ||(z − A)−1 x||2 provides a direct connection between operator theory and geometry. Proposition 7.35 The metric .gx has non-positive Ricci curvature. Moreover, the following are equivalent: (a) .gx is flat at a point .z ∈ ρ(A). (b) .gx is flat on .ρ(A). (c) x is an eigenvector of A. ∂ (A − z)−1 = (A − z)−2 Proof Regarding the Ricci curvature of .gx , the fact that . ∂z yields

R(z) = −

.

∂2 log ||(A − z)−1 x||2 ∂z∂ z¯

=−

∂ ∂z ||(A − z)−1 x||2

=−

||(A − z)−2 x||2 ||(A − z)−1 x||2 − ||2 . ||(A − z)−1 x||4

The Cauchy–Schwarz inequality then implies .R(z) ≤ 0 for all .z ∈ ρ(A). To check (a) ⇔ (c), notice that the curvature .R(z) vanishes at some point .z0 if and only if −1 x = λ(A − z )−2 x, or equivalently .(A − z )x = λx, for some scalar .(A − z0 ) 0 0 .λ. So x is an eigenvector of A with corresponding eigenvalue .z0 + λ. Next, since .(b) contains .(a), we only need to check .(c) ⇒ (b). If .Ax = λx for some scalar .λ, then .(A − z)x = (λ − z)x and hence .(A − z)−1 x = (λ − z)(A − z)−2 x for every .z ∈ ρ(A). Thus .R(z) = 0 by the computation above. u n .

Proposition 7.35 seems to suggest that the metric .gx on .ρ(A) is canonical in a certain sense.

7.4.3 Non-Euclidean Circles The singularity of .gx on the spectrum .σ (A) leads to some outlandish phenomena. But interestingly, circles still play a prominent role [70].

7.4 The Fundamental Form of a Single Operator

169

Theorem 7.36 If .γ is a piecewise-smooth loop in .(ρ(A), gx ) that encloses .σ (A), then (a) .L(γ ) ≥ 2π . (b) .L(γ ) = 2π if and only if x is an eigenvector of A and .γ is a circle centered at the corresponding eigenvalue. Proof For part (a), let .r be a closed piecewise-smooth path in .ρ(A) that encloses σ (A) and is enclosed by .γ . In other words, the path .r sits between .γ and .σ (A), and .r ∩ γ is empty. Then by functional calculus .

I=

.

1 2π i

f

(λ − A)−1 dλ,

(7.32)

(λ − z)−1 (λ − A)−1 dλ.

(7.33)

r

and for every fixed .z ∈ γ , (z − A)−1 =

.

1 2π i

f r

Assuming .γ is parametrized by .z(t), 0 ≤ t ≤ 1, then by (7.31) its arc length with respect to the metric .gx satisfies

||f 1 || || || ' −1 || .L(γ ) = |z (t)|||(z(t) − A) x||dt ≥ || z (t)(z(t) − A) xdt || || 0 0 || || ) f (f 1 || || 1 ' −1 z (t)(λ − z(t)) dt (λ − A)−1 xdλ|| = || || || 2π i r 0 || || ) f (f 1 || || 1 −d log(λ − z(t)) (λ − A)−1 xdλ|| (7.34) = || || . || 2π i r 0 f

1

'

−1

170

7 Hermitian Metrics on the Resolvent Set

Clearly, the inside integral with respect to t is .2π iW (γ , λ), where .W (γ , λ) is the winding number of .γ around .λ. Since .r is connected and .γ properly encloses .r, the number .W (γ , λ) is independent of .λ and is indeed equal to the winding number of .γ around .σ (A) which is equal to 1 since .γ is simple. Therefore by (7.33) and (7.34), we have || || f || || 1 −1 || || .L(γ ) ≥ 2π || 2π i (λ − A) xdλ|| = 2π ||x|| = 2π. r For part (b), if .γ is a circle in .ρ(A) centered at .λ and .Ax = λx, then we can parametrize .γ by .z(t) = λ + re2π it for some small .r > 0. Hence for every nonzero −1 x = 1 x, and direct computation gives .z ∈ ρ(A), we have .(A − z) λ−z f

| | ' | | z (t) | dt = 2π. | λ − z(t) |

1|

L(γ ) =

.

0

For the other direction, if .L(γ ) = 2π , then by the proof of part (a) we must have f

1

.

'

|z (t)|||(z(t) − A)

−1

0

||f || x||dt = || ||

1

'

z (t)(z(t) − A)

0

−1

|| || xdt || || ,

(7.35)

and the triangle inequality implies that .z' (t)(z(t) − A)−1 x = η(t)y for some fixed element .y ∈ H and a scalar-valued piecewise-smooth function .η ∈ L1 [0, 1]. By functional calculus f

1

.

0

f η(t)ydt =

1

'

z (t)(z(t) − A)

−1

f xdt =

0

(z − A)−1 xdz = 2π ix,

γ

so by possibly rescaling .η we can assume .y = x and obtain f .

1

η(t)dt = 2π i.

(7.36)

0

Furthermore, putting this observation into equation (7.35), we have f

1

.

0

|f | | η(t) | dt = ||

0

1

| | η(t)dt || .

On the other hand, from the equation .z' (t)(z(t) − A)−1 x = η(t)x, we get ( .

z(t) −

) z' (t) x = Ax, η(t)

(7.37)

7.4 The Fundamental Form of a Single Operator

171

'

(t) must be a constant .λ which is an eigenvalue of A corresponding and hence .z(t)− zη(t) '

z (t) . By (7.37), .η(t) is of the to the eigenvector x. Solving for .η, we have .η(t) = z(t)−λ form .αη∗ (t), where .α is a unimodular constant and .η∗ (t) ≥ 0 almost everywhere on .[0, 1]. In view of (7.36), we have .α = i, i.e., the function .η(t) is purely imaginary. It follows that f t f t z(t) − λ z' (s) ds = log . η(s)ds = z(0) − λ 0 0 z(s) − λ

is purely imaginary. This concludes that .|z(t) − λ| = |z(0) − λ| for all .t ∈ [0, 1], showing that .γ is a Euclidean circle centered at .λ. n u Theorem 7.36 (a) shows that the length of any path enclosing .σ (A) has a universal lower bound, regardless of how small .σ (A) may be in the Euclidean metric. It is worth taking another look at Example 7.33. Here, the Euclidean circle 2π it | 0 ≤ t ≤ 1} in .(ρ(A), g ) has circumference .Cr = {re x f Lx (Cr ) =

.

1/

gx (z(t))|z' (t)|dt = 2π r

f

1

( )−1 || re2π it − A x|| dt.

(7.38)

0

0

Example 7.37 In the case .A = 0, we have .(ρ(A), gx ) = (C× , 1/|z|2 ), and therefore .Lx (Cr ) = 2π which is independent of r. Since the metric function .1/|z|2 is radial, the radius of .Cr is equal to the length of the interval .(0, r] in .(ρ(A), gx ), which is f

1

.

0

ds = ∞. s

This example reveals an interesting phenomenon in the manifold .(C× , 1/|z|2 ): there exist circles with infinite radius but finite circumference. This happens because the metric function .1/|z|2 has a pole at 0. Exercise 7.38 1. Suppose .σ (A) is not path-connected, and the loop .γ in .ρ(A) only encloses a component of .σ (A). Derive a lower bound for .L(γ ). 2. This problem concerns with a simple case of exercise 1. Consider the matrix A=

.

( ) 01 , 10

and let C be the circle .{|z − 1| = 1/2}. Study how the circumference .Lx (C) varies with respect to the choice of unit vector .x ∈ C2 . In particular, when does .Lx (C) obtain its maximum or minimum? 3. Verify the claims in Example 7.37.

172

7 Hermitian Metrics on the Resolvent Set

4. Suppose .B has a faithful state .φ. Show that the metric defined by ) ( g(z) := φ (¯z − A∗ )−1 (z − A)−1 , z ∈ ρ(A),

.

is Ricci flat if and only if A is a scalar multiple of the identity operator I . Project 7.39 1. Study the completeness problem of .(ρ(T ), gx ) for a normal operator T .

7.5 Extremal Equation and the Unilateral Shift Assume A is densely defined on .H and x is a unit vector in .H. Given a fixed curve C ∈ (ρ(A), gx ), its length .Lx (C) depends on the choice of x. Does .Lx (C) have a maximum or a minimum value? If yes, which x will give such a value? This section aims to study these questions. Assume .σ (A) ⊂ rD for some .r > 0. Then the circle .Cr = {reiθ | 0 ≤ θ ≤ 2π } lies inside .(ρ(A), gx ). For convenience, we denote the unit circle .C1 by C. Setting iθ − A)−1 , the circumference of C can be expressed as .A(θ ) = (e .

f Lx (C) =

.

2π

||A(θ )x|| dθ.

0

Since .gx (eiθ ) = ||A(θ )x||2 ≤ ||A(θ )||2 which is bounded, the set Sr (A) := {Lx (Cr ) | x ∈ H, ||x|| = 1}

.

(7.39)

is a bounded subset of .(0, ∞). Moreover, it is compact when .dim H < ∞. This is because the closed unit ball of .H is compact, and the metric function .gx is norm continuous in x. Thus, the values .sup Sr and .inf Sr are both obtainable in this case. This may not be true when .dim H = ∞.

7.5.1 Variational Calculus The method of variational calculus (also called the calculus of variations) is often used to find the extremal solution to an integral or differential equation. When the integral is the arc length, the solution gives a geodesic. When the integral is related to the total energy over a period of time, it leads to the “principle of least action.” Some fundamental equations in physics can be derived through this method, for example, the Euler–Lagrange equation, the Schrödinger equation, and the Yang–

7.5 Extremal Equation and the Unilateral Shift

173

Mills equations. We refer the reader to [23, 36, 150, 189] for more information about this important subject. Variational calculus also sheds light on the question regarding .sup S1 and .inf S1 . Theorem 7.40 Suppose a unit vector .x ∈ H is such that .Lx (C) is extremal in .S1 . Then x satisfies the extremal equation f

2π

.

0

A∗ (θ )A(θ )x dθ = Lx (C)x. ||A(θ )x||

(7.40)

Proof Let x be a vector such that .Lx (C) = sup S1 . The discussion for the case Lx (C) = inf S1 is parallel. For any fixed nonzero vector .y ∈ H, we set

.

x(t) =

.

x + ty , t ∈ C, ||x + ty||

and let f F (t) = Lx(t) (C) =

.

2π

||A(θ )x(t)|| dθ.

0

Since .F (0) is extremal, we must have | | ∂F (t) || ∂F (t) || =0= , . ∂t |t=0 ∂ t¯ |t=0 and the two equations are equivalent because F is | | real. We use several steps to ∂||x + ty|| || ∂F (t) || . First, we compute . compute . | . To this end, we write ∂t |t=0 ∂t t=0 .

∂/ ∂||x + ty|| =

∂t ∂t ( ) 1 ∂ = −1/2 + + + |t|2 2 ∂t + t¯ . = √ 2

Setting .t = 0 and using the fact that .||x|| = 1, we obtain | ∂||x + ty|| || 1 = . . | ∂t 2 t=0 Second, we set

(7.41)

174

7 Hermitian Metrics on the Resolvent Set

f L(x + ty) =

2π

.

||A(θ )(x + ty)|| dθ

0

| ∂L(x + ty) || and compute . | . A similar calculation yields ∂t t=0 f

∂ . L(x + ty) = ∂t

2π

0

f

=

2π

0

f =

2π

0

∂ ||A(θ )(x + ty)|| dθ ∂t / > ∂ < A(θ )(x + ty), A(θ )(x + ty) dθ ∂t < > A(θ )y, A(θ )x + t¯||A(θ )y||2 dθ. √ 2

Setting .t = 0, we have > | f 2π < A(θ )y, A(θ )x ∂L(x + ty) || = . dθ. | ∂t 2 ||A(θ )x|| 0 t=0 Since .

∂ L(x + ty) ∂F (t) = ∂t ∂t ||x + ty|| =

||x + ty||

∂ ∂ L(x + ty) − L(x + ty) ||x + ty|| ∂t ∂t , ||x + ty||2

setting .t = 0 and using the observations above, we arrive at the equation f .

0

2π

dθ − Lx (C) = 0, ∀y ∈ H, ||A(θ )x||

which implies f

2π

.

0

A∗ (θ )A(θ )x dθ = Lx (C)x. ||A(θ )x|| u n

Observe that if we set f TA :=

.

0

2π

A∗ (θ )A(θ ) dθ, ||A(θ )x||

7.5 Extremal Equation and the Unilateral Shift

175

then .TA is a positive linear operator, and Proposition 7.40 indicates that when .Lx (C) is extremal, the vector x is an eigenvector of .TA with corresponding eigenvalue .Lx (C). The following elementary example helps to illustrate the theorem. Example 7.41 Consider the following operator and vectors in .C2 : ( ) ( ) ( ) 01 1 0 .A = , x1 = , x2 = . 00 0 1 In this case we have A(θ ) = (e − A) iθ

.

−1

=e

−2iθ

(

) eiθ 1 , 0 eiθ

and f TA =

2π

.

0

A∗ (θ )A(θ ) dθ = 2π(I + A∗ A). ||A(θ )x1 ||

Since .Ax1 = 0, we have .Lx1 (C) = 2π which is the minimal arc length of C by Theorem 7.36. Thus, .TA x1 = 2π x1 . The discussion with respect to .x2 is left as an exercise.

7.5.2 Inner Functions and the Extremal Length of Circles The unilateral shift S on a Hilbert space .H with orthonormal basis .{ek }k≥0 is the linear operator which sends .ek → ek+1 . It is one of the most important examples in operator theory. This subsection takes a look at the extremal value problem for S. The classical Hardy space is defined as } { E 2 iθ ikθ . .H (T) = f ∈ L (T) | f (e ) = ck e 2

k≥0

If we let .w = eiθ , then .{w k | k ≥ 0} is an orthonormal basis for .H 2 (T). Since w has an obvious extension into .D, so does every function in .H 2 (T). Thus the Hardy space is also denoted by .H 2 (D). The reproducing kernel of .H 2 (T) is K(λ, w) =

∞ E

.

k=0

λ¯ k w k =

1 1 − λw

, |λ| < 1, |w| ≤ 1.

The multiplication operator .f → wf, f ∈ H 2 (T), which we also denote by S, is a representation of the unilateral shift. One should compare S with the bilateral shift

176

7 Hermitian Metrics on the Resolvent Set

on .L2 (T) considered in Sect. 5.1. It is not hard to verify that .σ (S) = D. A function 2 .f ∈ H (T) is said to be inner if .|f (w)| = 1 almost everywhere on .T. Beurling’s theorem [11] describes the invariant subspaces of S. Theorem 7.42 M is an invariant subspace of S if and only if .M = f H 2 (T) for some inner function f . An epiphanic observation is that this theorem cannot be properly stated for S in the general Hilbert space .H because of the lack of two necessary ingredients: (1) multiplication of vectors in .H and (2) an analogue of inner functions. In other words, the invariant subspaces of S cannot be inherently described in the abstract setting. Indeed, Beurling’s theorem, as simple as it appears, has a far-reaching impact on operator theory as a whole. In this sense, it is a powerful example of representation theory. We refer the reader to Brodskiˇi–Livsˇic [20], Garnett [93], Nagy–Foias–Bercovici–Kerchy [176], Martínez–Avendaño–Rosenthal [169], as well as the references therein for more details. The subsequent discussion can be found in [223]. Given .f ∈ H 2 (T) with .||f || = 1, the metric function .gf on .ρ(S) is || ||2 f |f (w)|2 || || dm(w), |z| > 1, g (z) = ||(S − z)−1 f || = 2 T |w − z|

. f

dθ for .w = eiθ . Let the circle .Cr be parametrized by .re2π it , 0 ≤ 2π t ≤ 1. If .r > 1, then .Cr ∈ (ρ(S), gf ), and its arc length is

where .dm(w) =

1(f

f Lf (Cr ) = 2π r

T

0

)1/2 dm(w) dt. | | |w − re2π it |2 |f (w)|2

.

(7.42)

Surprisingly, the extremal value problem for the set .Sr defined in (7.39) rediscovers the inner functions. Theorem 7.43 For the unilateral shift S and any .r > 1, we have .

/ sup Sr = 2π r/ r 2 − 1,

and it is obtained precisely at inner functions. Proof First, using the Cauchy–Schwarz inequality for the outside integral in (7.42), we have (f f Lf (Cr ) ≤ 2π r

.

T 0

1

|f (w)|2

| | dm(w)dt |w − re2π it |2

Fourier series and Parseval’s identity give

)1/2 .

(7.43)

7.5 Extremal Equation and the Unilateral Shift

f

1

.

0

1 1 = | |2 dt = 2 r −1 |w − re2π it |

177

f T

1 | | dm(w). |w − re2π it |2

(7.44)

Therefore, using the fact .||f || = 1, we have / Lf (Cr ) ≤ 2π r/ r 2 − 1,

.

(7.45)

and the equality holds when f is inner. Now suppose for some unit vector .f ∈ H 2 (T) the equality in (7.45) holds. Then by the Cauchy–Schwarz inequality the inside integral in (7.42) (which is .gf (re2π it )) is constant with respect to t. Writing f g (re

. f

2π it

)=

T

|f (w)|2 ) dm(w) ( )( w − re2π it w − re−2π it

E and expanding it as a Fourier series . k∈Z ak e2π kit , we verify that ak =

.

f

1 r |k| (r 2 − 1)

T

|f (w)|2 w −k dm(w), k ∈ Z.

Since .gf (re2π it ) is constant, we must have .ak = 0 for all .k /= 0, indicating that .|f (w)| must be constant 1 almost everywhere on .T. Hence f is inner. u n A natural question is whether .inf Sr is also obtainable and if so by what type of functions in .H 2 (T). Recall that a function .f ∈ H 2 (T) is said to be outer if span.{w k f | k ≥ 0} is dense in .H 2 (T), or in other words, it is a cyclic vector for S. A typical example is the normalized reproducing kernel function .kλ (w) = / 1 − |λ|2 /(1 − λw), |λ| < 1. Theorem 7.44 For the unilateral shift S and any .r > 1, we have f .

inf Sr = 2π 0

1|

|−1 | 2π it | − 1/r | dt, |e

and it is unattainable. Proof For any .f ∈ H 2 (T) with .||f || = 1, the measure .dmf (w) := |f (w)|2 dm(w) is a probability measure on the unit circle. Hence by (7.42) and the Cauchy–Schwarz inequality, one has 1(f

)1/2 f 1f dmf (w) dmf (w) dt dt > 2π r 2π it |2 |w − re2π it | |w − re T T 0 0 f f 1 = 2π |f (w)|2 |we ¯ 2π it − 1/r|−1 dtdm(w). f

Lf (Cr ) = 2π r

.

T

0

178

7 Hermitian Metrics on the Resolvent Set

For simplicity, we denote the right-hand side of the equation in the theorem by .I (r). If .w = e2π it0 , then a change of variable .t → t + t0 will show that the right-hand side of the last equality above is equal to .I (r)||f ||2 = I (r). Hence .Lf (Cr ) > I (r), showing that .I (r) is not attained at any f . Moreover, it follows that .inf Sr ≥ I (r). To prove .I (r) ≥ inf Sr , we observe that Theorem 7.43 suggests that a nonconstant outer function f will give .Lf (Cr ) < sup Sr . To gauge the value of .inf Sr , consider the outer function .kλ (w) for which the metric function is g (z) = ||(S − z)−1 kλ ||2 =

. kλ

f T

1 − |λ|2 dm(w), |z| > 1. |1 − λ¯ w|2 |w − z|2

(7.46)

Letting .β = 1/z and using the power series of .(1 − λ¯ w)−1 (w − z)−1 with respect to w, we compute that g (z) = ||(S − z)−1 kλ ||2 | |2 |2 | ( ) | | | | 2 2| 2 2 | | | = (1 − |λ| )|β| 1 + |λ + β | + |λ + λβ + β | + · · ·

. kλ

| ∞ ∞ | k 2 2 E E | λ − β k |2 k | | = (1 − |λ| )|β| |λ − β k |2 . = (1 − |λ| )|β| | λ−β | 2 |λ − β| k=1 k=1 2

2

The infinite sum can be computed as

.

∞ | |2 E | | k |λ − β k | = k=1

=

|λ|2 1 − |λ|2

−

λβ 1 − λβ

−

|β|2 λβ + 1 − λβ 1 − |β|2

| [ | ] |λ − β |2 1 − |λβ|2 (1 − |λ|2 )|1 − λβ|2 (1 − |β|2 )

.

Therefore, g (z) =

. kλ

[ ] |β|2 1 − |λβ|2 |1 − λβ|2 (1 − |β|2 )

=

|z|2 − |λ|2 , |z| > 1. (|z|2 − 1)|z − λ|2

(7.47)

Clearly, the metric .gkλ has singularities on the unit circle. By (7.42), we have Lkλ (Cr ) = √

.

2π r r2 − 1

f 0

1

√

2π r r 2 − λ2 | dt = √ | |re2π it − λ| r2 − 1

f 0

1

/

1 − |α|2 | dt, | |e2π it − α | (7.48)

where .α = λ/r ∈ D. Observe that the integrand in (7.48) is the square root of the Poisson kernel on the unit disc. Hence by the Cauchy–Schwarz inequality we have

7.5 Extremal Equation and the Unilateral Shift

f

2π r Lkλ (Cr ) < √ r2 − 1

1

.

0

179

1 − |α|2 2π r . | |2 dt = √ 2 |e2π it − α | r −1

This is consistent with Theorem 7.43. The integral in (7.48), which we denote by J (α), is related to the elliptic integral of the first kind. The following chart shows the values for .J (α) based on various choices of .r > 1 and .λ ∈ D.

.

f .J (α)

1

= 0

= 1.1 = 1/8 .λ = 1/4 .λ = 1/3 .λ = 1/2 .λ = 2/3 .λ = 3/4 .λ = 8/9 .λ = 9/10 .λ = 1

= 1.25 = 1/8 .λ = 1/4 .λ = 1/3 .λ = 1/2 .λ = 2/3 .λ = 3/4 .λ = 8/9 .λ = 9/10 .λ = 1

/ 1 − |α|2 | | dt, α = λ/r |e2π it − α | =5 = 1/8 .λ = 1/4 .λ = 1/3 .λ = 1/2 .λ = 2/3 .λ = 3/4 .λ = 8/9 .λ = 9/10 .λ = 1

= 13 = 1/8 .λ = 1/4 .λ = 1/3 .λ = 1/2 .λ = 2/3 .λ = 3/4 .λ = 8/9 .λ = 9/10 .λ = 1

.r

.J (α)

.r

.J (α)

.r

.J (α)

.r

.J (α)

.λ

0.996 0.986 .0.978 .0.942 .0.888 .0.850 .0.754 .0.743 .0.615

.λ

0.997 0.989 0.981 0.956 0.918 0.891 0.832 0.826 0.762

.λ

0.999 0.999 0.998 0.997 0.995 0.994 0.991 0.991 0.989

.λ

0.999 0.999 0.999 0.999 0.999 0.999 0.998 0.998 0.998

The chart indicates that when r is fixed and .|λ| approaches 1 the value of .J (α) is decreasing. Hence by (7.48) we have 2π r

.

inf Sr ≤ lim inf Lkλ (C) = √ |λ|→1 r2 − 1

f 0

1

/ 1 − 1/r 2 | | dt = I (r), 2π |e it − 1/r | u n

and this completes the proof. Exercise 7.45

x2 also satisfies the extremal equation (7.40) and 1. For Example 7.41, verify that .√ that the arc length .Lx2 (C) = 2 2π is maximal. 2. Similar to Example 7.41, study the extremal value problem of .Lx (C) with respect to the matrix ⎞ ⎛ 010 .A = ⎝0 0 1⎠ . 000 3. Show that the set .Sr (A) defined in (7.39) is a unitary invariant of A in the sense that .Sr (U AU ∗ ) = Sr (A) for any unitary operator U . 4. Let .A ∈ Mk (C) be a normal matrix with no eigenvalues in .T. Determine .sup S1 and .inf S1 . At which unit vectors .x ∈ Ck are the extremal values obtained?

180

7 Hermitian Metrics on the Resolvent Set

5. Show that a unit function .f ∈ H 2 (T) is inner if and only if .f ∈ M O wM for some invariant subspace M of the unilateral shift S. 6. Prove .σ (S) = D by showing that .ker(S ∗ − λ) is nontrivial for any .λ ∈ D. 7. For any .A ∈ B(H) and .r > ||A||, show that the set .Sr is an interval. (Hint: The metric function .gx is continuous in x.) Project 7.46 1. Given an inner function .θ ∈ H 2 (T), the associated model operator is defined as S(θ )f = P wf, f ∈ H 2 (T) O θ H 2 (T),

.

where P is the orthogonal projection onto .H 2 (T) O θ H 2 (T). Determine .Sr for .r > 1. Are .sup Sr and .inf Sr obtainable? 2. Given .f ∈ L∞ (T), the Toeplitz operator .Tf on .H 2 (T) is defined by Tf (h) = P (f h), h ∈ H 2 (T),

.

where .P : L2 (T) → H 2 (T) is the orthogonal projection. For .r > ||f ||∞ , is .sup Sr always obtainable for .Tf ? 3. The classical Volterra operator on .L2 [0, 1] is defined by f

t

Vo f (t) =

.

f (s)ds, f ∈ L2 [0, 1],

0

and it is well-known that .σ (Vo ) = {0} [66]. Determine .sup S1 and .inf S1 . 4. Do a similar investigation for the Volterra operator on .H 2 (D) defined by f Vh f (w) =

.

w

f (s)ds, f ∈ H 2 (D), w ∈ D.

0

7.6 The Power Set of Quasinilpotent Operators The most well-known unsolved problem in operator theory is the invariant subspace problem which asks whether every bounded linear operator A on an infinite dimensional separable complex Hilbert space .H has a nontrivial invariant subspace. The answer is yes for normal operators, and it is a direct consequence of the spectral decomposition theorem. For other operators, if the spectrum .σ (A) is a disjoint union of closed subsets .F1 ∪ F2 , then given any piecewise-smooth path .γ ⊂ ρ(A) that encloses .F1 but not .F2 , functional calculus asserts that f 1 . (λ − A)−1 dλ 2π i γ

7.6 The Power Set of Quasinilpotent Operators

181

is an idempotent whose range is a nontrivial invariant subspace of A. The first nontrivial result is due to von Neumann for quasinilpotent compact operators. Stronger results along this line are obtained in Aronszajn–Smith [4] and Lomonosov [155]. For subnormal operators, i.e., the restriction of a normal operator to an invariant subspace, the answer is also yes [26]. Although Brown’s proof is rather technical, the starting point of his method is the following simple lemma. Lemma 7.47 Given a contraction .A ∈ B(H), if there exist .x, y ∈ H and .λ ∈ D such that . = 1 and . = λn for .n ≥ 1, then A has a nontrivial invariant subspace. We leave its proof as an exercise. Brown’s method was later refined to prove stronger results, for instance, the following theorem regarding contractions with a “rich” spectrum. Theorem 7.48 If A is a contraction and .T ⊆ σ (A), then A has a nontrivial invariant subspace. The result was generalized to polynomially bounded operators in [3] by Ambrozie– Müller. On the other hand, quasinilpotent operators have the singleton spectrum .{0} which is extremely “meager.” They seem to be the “backbone” of the invariant subspace problem, where spectral analysis offers little help. So far, no effective general method has been found to investigate their invariant subspaces, although some partial success has been reported, for example, in Foias–Pearcy [85], Foias– Jung–Ko–Pearcy [80], and Tcaciuc [219]. It is worth noting that, on some Banach spaces, counterexamples to the invariant subspace problem have been constructed in Read [191] and Enflo [74]. For more details regarding the invariant subspace problem, we refer the reader to the survey [9] and the books Bercovici–Foias– Pearcy [10], Chalendar–Partington [43], and Rosenthal–Radjavi [194]. If V is quasinilpotent, the metric function .gx (z) = ||(V − z)−1 x||2 is defined on × .C , and it has only one possible singularity at .z = 0 in the sense that .lim sup gx (z) can be .∞ (or “blow up”) as z approaches 0. Does this singular behavior of .gx reveal any information about the invariant subspaces of V ? It appears to be so, but there is no definite answer at this time. This section is a brief survey on some recent work related to this question.

7.6.1 Gauging the Singularities One way to gauge the “blow-up rate” of .gx (z) near 0 is to compare it with that of the function .g(z) = ||(V − z)−1 ||2 which also blows up at 0 [197]. For a unit vector .x ∈ H, we define kx = lim sup

.

z→0

log gx (z) . log g(z)

182

7 Hermitian Metrics on the Resolvent Set

Roughly speaking, as .z → 0 the metric function .gx (z)’s blow-up rate is comparable to that of .g(z)kx . Since .gcx (z) = |c|2 gx (z) for a nonzero scalar c, we have .kx = kcx . This fact gives us the liberty to include non-unital x in the discussions of .kx . Definition 7.49 Given a quasinilpotent operator V on .H, its power set is defined as A(V ) = {kx | x ∈ H, x /= 0}.

.

It is not hard to check that .A(V ) ⊆ [0, 1]. The power set is an invariant for similar quasinilpotent operators. Proposition 7.50 Two similar quasinilpotent operators have the same power set. Proof Let .V1 and .V2 be quasinilpotent operators on .H, and assume .T ∈ B(H) is invertible and .T −1 V1 T = V2 . Then for .z /= 0 we have ||(V2 − z)−1 x|| = ||T −1 (V1 − z)−1 T x|| ≤ ||T −1 ||||(V1 − z)−1 T x||,

.

(7.49)

and ||(V2 − z)−1 || = ||T −1 (V1 − z)−1 T || ≥

.

||(V1 − z)−1 || . ||T ||||T −1 ||

(7.50)

If .||(V1 − z)−1 T x|| is bounded near .z = 0, then since both .||(V1 − z)−1 || and −1 || tend to .∞ as .z → 0, we easily obtain .k (V ) = k (V ) = 0. .||(V2 − z) x 2 Tx 1 In the case .||(V1 − z)−1 T x|| is unbounded near .z = 0, (7.49) and (7.50) give .

log ||T −1 || + log ||(V1 − z)−1 T x|| log ||(V2 − z)−1 x|| ≤ . log ||(V2 − z)−1 || log ||(V1 − z)−1 || − log(||T ||||T −1 ||)

Taking .lim sup as .z → 0 on both sides, we have that .kx (V2 ) ≤ kT x (V1 ). Moreover, it follows that .kx (V2 ) = kT −1 T x (V2 ) ≥ kT x (V1 ), and hence .kx (V2 ) = kT x (V1 ) for every .x /= 0. This proves .A(V1 ) = A(V2 ). u n To get a sense of how the power set .A(V ) is related to the structure of V , we consider an elementary example. Example 7.51 Assume V is nilpotent with .V n = 0 but .V n−1 /= 0 for some .n ≥ 1. Then (V − z)−1 = −

.

) )−1 ( ( 1 1 1 1 1 I− V = − I + V + · · · + n−1 V n−1 . z z z z z

(7.51)

For any nonzero .x ∈ H, there exists a number .1 ≤ k ≤ n such that .V k−1 x /= 0 but k .V x = 0. As .z → 0, (7.51) implies that the principle parts of .gx (z) and .g(z) are k−1 x||2 /|z|2k and .||V n−1 ||2 /|z|2n , respectively. Therefore, .||V

7.6 The Power Set of Quasinilpotent Operators

kx = lim sup

.

z→0

183

k log(||V k−1 x||/|z|k ) log gx (z) = lim sup = . n−1 n log g(z) n log(||V ||/|z| ) z→0

It follows that .A(V ) = { nk | 1 ≤ k ≤ n}. On the other hand, if .Hk := {x ∈ H | V k x = 0}, then {0} = H0 ⊂ H1 ⊂ · · · ⊂ Hn = H

.

is a complete chain of invariant subspaces for V . Thus, the power set .A(V ) precisely depicts the chain for every nilpotent operator V .

7.6.2 The Volterra Operators A distinguished example of quasinilpotent operator is the Volterra operator on L2 [0, 1] defined by

.

f

x

Vo f (x) =

.

f (t)dt, x ∈ [0, 1].

0

There is a vast amount of work in the literature on .Vo and its generalizations. A classical reference is [98]. For .0 ≤ α ≤ 1, we let .χ[α,1] be the characteristic function for the interval .[α, 1]. The invariant subspace of .Vo is described in Donoghue [72]. Theorem 7.52 M is an invariant subspace of .Vo if and only if .M = χ[α,1] L2 [0, 1] for some .0 ≤ α ≤ 1. In other words, the spaces .χ[α,1] L2 [0, 1], 0 ≤ α ≤ 1, form a complete decreasing chain of invariant subspaces of V . The Volterra integral operator on .H 2 (D) is defined by f

w

Vh f (w) =

.

f (t)dt, f ∈ H 2 (D), w ∈ D.

0 n+1

Obviously, .Vh w n = wn+1 for each .n ≥ 0. We leave it as an exercise to check that .Vh is compact and quasinilpotent. The following theorem due to [2] characterizes the invariant subspaces of .Vh . Theorem 7.53 A closed subspace M is invariant for .Vh if and only if .M = w n H 2 (T) for some integer .n ≥ 0. The power sets of .Vo and .Vh are computed in [70, 157] and [140]. Proposition 7.54 (a) .A(Vo ) = (0, 1];

(b) .A(Vh ) = {1}.

184

7 Hermitian Metrics on the Resolvent Set

Observe that .A(Vo ) depicts the invariant subspace lattice of .Vo , but it is not the case for .A(Vh ). Efforts have been made to generalize Proposition 7.54 (b). For an operator .T ∈ B(H), we let .Aw (T ) denote the commutative subalgebra generated by the identity I and T that is closed with respect to the weak operator topology. In general, it is not equal to the von Neumann algebra .A(T ) generated by .I, T , and ∗ .T . Definition 7.55 An operator T is said to be strictly cyclic if there exists a vector f ∈ H such that .Aw (T )f = H. In this case, the vector f is called a strictly cyclic vector for T .

.

An operator T on a Hilbert space .H with orthonormal basis .{e0 , e1 , . . .} is said to be a unilateral weighted shift if .T ej = αj ej +1 , j ≥ 0, for some sequence of numbers .{α0 , α1 , . . .}. The following theorem is proved in [134, 138]. Theorem 7.56 If T is a strictly cyclic quasinilpotent unilateral weighted shift, then A(T ) = {1}.

.

An operator .T ∈ B(H) is said to be strongly strictly cyclic if its restriction to every invariant subspace is also strictly cyclic. The operator .Vh is an example of such operators. Regarding quasinilpotent operators that may not be a weighted shift, the following still holds. Theorem 7.57 If T is strongly strictly cyclic and quasinilpotent, then .A(T ) = {1}. Thus, according to Proposition 7.54 (a), the Volterra operator .Vo is not strongly strictly cyclic. Indeed, it is conjectured in [134] that the power set of every strictly cyclic quasinilpotent operator is .{1}. For general quasinilpotent operators, the following surprising fact is discovered in [140]. Theorem 7.58 Suppose .V ∈ B(H) is quasinilpotent. (a) .A(V ) is right-closed in the sense that .sup A(V ) ∈ A(V ). (b) For any right-closed subset .E ⊆ [0, 1] containing 1, there is a quasinilpotent operator V for which .A(V ) = E. Regarding the Volterra operator .Vh on .H 2 (D), the extremal arc length of the unit circle .C ⊂ (C× , gx ) is studied in [157]. Theorem 7.59 The following hold for .Vh : (a) .inf S1 = 2π , and it is not attainable ( E∞ 1 )1/2 (b) .sup S1 = 2π , and it is obtained precisely when x is a k=0 (k!)2 unimodular constant.

7.6 The Power Set of Quasinilpotent Operators

185

7.6.3 A Touch on the Hyper-Invariant Subspace Example 7.51 and Proposition 7.54 (a) suggest a connection between the power set and the lattice of invariant subspaces. For a general quasinilpotent operator V on .H and a fixed .0 ≤ τ ≤ 1, we set Mτ = {x ∈ H | x /= 0, kx ≤ τ } ∪ {0}.

.

Clearly, .M1 = H, and .Mτ1 ⊂ Mτ2 whenever .τ1 < τ2 . The following holds [69]. Proposition 7.60 The set .Mτ is a subspace, and for every .T ∈ B(H) that commutes with V , we have .T (Mτ ) ⊂ Mτ . Proof We first check that .Mτ is a subspace. As remarked earlier, for any .c ∈ C× we have .kcx = kx . Thus, .x ∈ Mτ implies .cx ∈ Mτ . Furthermore, if x and y are nonzero vectors in .Mτ , then for any .e > 0 there exists .δ > 0 such that for all z with .0 < |z| < δ, we have .

log gy (z) log gx (z) < τ + e, and < τ + e, log g(z) log g(z)

or equivalently, .||(V − z)−1 x|| < ||(V − z)−1 ||τ +e , and ||(V − z)−1 y|| < ||(V − z)−1 ||τ +e . It follows that .

( ) log ||(V − z)−1 (x + y)|| ≤ log ||(V − z)−1 x|| + ||(V − z)−1 y|| ( ) ≤ log 2||(V − z)−1 ||τ +e = log 2 + (τ + e) log ||(V − z)−1 ||,

and hence .

log 2 log ||(V − z)−1 (x + y)|| . ≤τ +e+ −1 log ||(V − z)−1 || log ||(V − z) ||

Since .log ||(V −z)−1 || goes to .∞ as z goes to 0, the above inequality implies .kx+y ≤ τ + e. It follows that .kx+y ≤ τ , showing that .Mτ is a subspace. For any nonzero vector .x ∈ Mτ , since T commutes with V , we have kT x = lim sup

.

z→0

≤ lim sup z→0

log ||(V − z)−1 T x|| log ||T (V − z)−1 x|| = lim sup log ||(V − z)−1 || log ||(V − z)−1 || z→0 log ||T || + log ||(V − z)−1 x|| = kx ≤ τ. log ||(V − z)−1 ||

186

7 Hermitian Metrics on the Resolvent Set

Hence .T x ∈ Mτ .

u n

A closed subspace .M ⊂ H is said to be hyperinvariant for an operator .A ∈ B(H) if it is invariant for every operator .B ∈ B(H) that commutes with A. Therefore, if .Mτ is closed, then it is a hyperinvariant subspace of V . In particular, if .A(V ) contains two distinct points .τ1 < τ2 such that .Mτi , i = 1, 2, are both closed, then V has a nontrivial hyperinvariant subspace. However, there exists a quasinilpotent operator V for which .A(V ) = [0, 1], and .Mτ is dense in .H but not closed for every .τ ∈ [0, 1) [138]. Exercise 7.61 1. Prove Lemma 7.47. (Hint: Consider the closure of .span{(A − λ)An x | n ≥ 0}.) 2. Show that if A is a normal matrix, then every unit eigenvector x satisfies the extremal equation (7.40). 3. For any quasinilpotent operator V , show that its power set is .A(V ) ⊆ [0, 1]. 4. Show that the Volterra operator .Vo on .L2 [0, 1] is compact. The fact that .Vo is quasinilpotent is harder to prove, but it is worth trying. 5. Show that the Volterra operator .Vh on .H 2 (D) is quasinilpotent and strongly strictly cyclic. Project 7.62 1. Is .sup A(V ) = 1 for all quasinilpotent operator V ? 2. Algebraic properties of the power set .A(V ) are largely unknown. Here are two natural questions. (a) How is .A(V 2 ) related to .A(V )? (b) Are there connections between .A(V ∗ ) and .A(V )? 3. Characterize quasinilpotent operators for which .Mτ is closed for all .0 ≤ τ ≤ 1.

Chapter 8

Compact Operators and Kernel Bundles

As a bridge between matrices (or finite rank operators) and linear operators on an infinite dimensional Hilbert space .H, compact operators play an important role in operator theory and beyond. Indeed, the study of their invariant subspaces has led us to some of the deepest theorems, and the Calkin algebra .B(H)/K(H) is the ground for the definition of Fredholm operators which are fundamental to spectral theory, index theory, and noncommutative geometry. It is therefore of great interest to gain a somewhat in-depth understanding of the projective spectrum of compact operators.

8.1 The Fredholm Determinant and Logarithmic Integral Given .p ≥ 1, a compact operator T is said to be in the Schatten p-class if .|T |p is in the trace class. In this case, its Schatten p-norm is .||T ||p = (Tr|T |p )1/p . If .Sp denotes the set of Schatten p-class operators, then .(Sp , || ||p ) is a Banach space. In particular, .S1 is the space of trace class operators, and .S2 is the space of Hilbert– Schmidt operators. Moreover, ||T || ≤ ||T ||1

.

and

|TrT | ≤ ||T ||1 , T ∈ S1 .

If .I + T is invertible on .H, then .(I + T )−1 − I = −T (I + T )−1 ∈ S1 , indicating that .(I + T )−1 ∈ I + S1 . This shows that the set of invertible elements in .I + S1 is a group. Definition 8.1 Given .T ∈ S1 , its Fredholm determinant is defined as (E k ) k+1 Tr(T ) . . det(I + T ) = exp (−1) k k≥1

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 R. Yang, A Spectral Theory Of Noncommuting Operators, https://doi.org/10.1007/978-3-031-51605-4_8

187

188

8 Compact Operators and Kernel Bundles

We let .λ1 (T ), λ2 (T ), . . . stand for the eigenvalues of T , counting multiplicity. The following are some well-known properties. Details can be found in [203]. Proposition 8.2 For .T , T1 , T2 ∈ S1 , the following hold: || (a) .det(I + T ) = k≥1 (1 + λk (T )). ( ) (b) .det (I + T1 )(I + T2 ) = det(I + T1 ) det(I + T2 ). (c) .det(I + wT ) is an entire function on .C with .| det(I + wT )| ≤ exp(|w|||T ||1 ). (d) .I + wT is invertible if and only if .det(I + wT ) /= 0. Moreover, given a complex domain .o in .C and a differentiable function .g : o → S1 , Jacobi’s formula takes the form ( ) −1 .Tr (I + g(w)) dg(w) = d log det(I + g(w)), (8.1) wherever .I + g(w) is invertible [101]. This is an extension of Lemma 1.9.

8.1.1 The Argument Principle for Operator Functions Let .o be a bounded domain in .C with piecewise-smooth and positively oriented boundary .∂o, and let f be a meromorphic function on .o that extends analytically to .∂o and nonvanishing there. Then the argument principle states that the logarithmic integral 1 . 2π i

f ∂o

f , (w) dw = N0 − N∞ , f (w)

where .N0 and .N∞ are the number of zeros and, respectively, poles of f inside .o. In particular, if f is analytic on .o, then the logarithmic integral is equal to zero if and only if f does not vanish on .o. Much effort has been made to extend this theorem to operator-valued functions. For more details about the topics in this section, we refer the reader to Mittenthal [160], Gohberg–Kaashoek–Lay [99], Bart–Ehrhardt– Silbermann [13, 14], and the references therein. Suppose .B is a unital Banach algebra. A function .f : o → B is said to be normal if it is analytic on .o and takes invertible values except at a possibly finite number of points .w1 , . . . , wm (called normal points) inside .o. If we let .ωf (w) = f −1 (w)df (w) be the Maurer–Cartan form of f , then there are two natural questions regarding the (right) logarithmic integral R(f ; o) :=

.

1 2π i

f ωf . ∂o

(8.2)

8.1 The Fredholm Determinant and Logarithmic Integral

189

1. What are its possible values? 2. In the case its value is 0, is .f (w) necessarily invertible for every .w ∈ o? The answer depends on f as well as on the Banach algebra .B. But something general can be said if .f −1 (w) has a simple pole at normal points. We shall treat normal points one at a time. Let .D(w0 , r) stand for a small disc with center .w0 and radius r. Definition 8.3 Suppose f is normal on .D(w0 , r) with a unique normal point .w0 . Then the residue of f at .w0 is defined as R(f ; w0 ) =

.

1 2π i

f |w−w0 |=r

ωf .

Proposition 8.4 Suppose f is normal on .D(w0 , r) and .f −1 (w) has a simple pole at .w0 . Then .R(f ; w0 ) is a nonzero idempotent. Proof Since f is analytic on .D(w0 , r), we can write f (w) = a0 + a1 (w − w0 ) + a2 (w − w0 )2 + · · · ,

.

(8.3)

where .aj ∈ B for each .j ≥ 0, and the series converges uniformly on compact subsets in .D(w0 , r). Likewise, since .f −1 (w) has a simple pole at .w0 , we can write f −1 (w) =

.

b−1 + b0 + b1 (w − w0 ) + b2 (w − w0 )2 + · · · , w − w0

(8.4)

where .bj ∈ B for each .j ≥ −1. Then the principle part of .f −1 (w)df (w) b−1 a1 is . w−w dw, and hence .q := R(f ; w0 ) = b−1 a1 . Furthermore, the fact 0 −1 (w) = f −1 (w)f (w) = I, w /= w , implies .f (w)f 0 .

a0 b−1 = b−1 a0 = 0, .

(8.5)

a0 b0 + a1 b−1 = b0 a0 + b−1 a1 = I, .

(8.6)

a0 b1 + a1 b0 + a2 b−1 = b1 a0 + b0 a1 + b−1 a2 = 0.

(8.7)

Therefore, q 2 = b−1 a1 b−1 a1 = b−1 (I − a0 b0 )a1 = q − b−1 a0 b0 a1 = q.

.

If q were equal to 0, then (8.6) would imply .b0 a0 = I . Multiplying the first equation in (8.5) on the left by .b0 , we would obtain .b−1 = 0, which contradicts u n the assumption that .f −1 (w) has a simple pole at .w0 . It is worth noting that if we define the left logarithmic residue p by the integral of .(df (w))f −1 (w) instead of .ωf in Definition 8.3, then p is an idempotent that is

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8 Compact Operators and Kernel Bundles

similar to q. In fact, if we set .s = b0 a0 b0 + b−1 a1 b−1 , then s is invertible with s −1 = a0 b0 a0 + a1 b−1 a1 and .sp = qs. If f is normal on domain .o and .f −1 (w) has poles at normal points .w1 , . . . , wm , then the Cauchy integral theorem implies

.

R(f ; o) :=

.

1 2π i

f ωf (w) = R(f ; w1 ) + · · · + R(f ; wm ).

(8.8)

∂o

The following corollary partially answers question 1. Corollary 8.5 If f is normal on domain .o and .f −1 (w) has simple poles at normal points, then .R(f ; o) is a sum of finitely many idempotents.

8.1.2 The Trace of Residues In the sequel, we assume .B = B(H) and .f (w) is everywhere Fredholm on .o. In this case, more details can be said about the singularity of .f −1 (w). The following lemma due to [101] is instrumental. Lemma 8.6 Assume f is normal on .D(w0 , r) with a unique normal point at .w0 where f is Fredholm. Then .f −1 (w) has a pole at 0 with Laurent series f −1 (w) =

E

.

bj (w − w0 )j ,

(8.9)

j ≥−s

where s is the order of the pole and .b−s , . . . , b−1 are finite rank operators. For a projection q on .H, we denote .I − q by .q ⊥ . Corollary 8.7 Let f be as in Lemma 8.6. Then there exist finitely many finite rank projections .q1 , . . . , qk such that ) ( ) ( f (w) = q1⊥ + (w − w0 )q1 · · · qk⊥ + (w − w0 )qk h(w),

.

where .h(w) is analytic and invertible everywhere on .D(w0 , r). Proof Clearly, if .s = 0 in (8.9), then .k = 0 and .h = f . The corollary holds trivially in this case. Suppose .f −1 has a pole of order .s ≥ 1. If .q1 denotes the projection from ⊥ .H onto .HOker b−s , then .b−s q 1 = 0. Since .b−s is of finite rank, so is .q1 . Moreover, assuming f is expanded as in (8.3), the fact .f −1 (w)f (w) = I for .w /= w0 implies .b−s a0 = 0, i.e., .rana0 ⊂ ker b−s , and it follows that .q1 a0 = 0. We set

f1 (w) =

.

{( ⎪ ⎨ q1⊥ + ⎪ ⎩

) 1 q1 f (w), w ∈ D(w0 , r), w /= w0 , w − w0

q1⊥ a0 + q1 a1 ,

w = w0 .

8.1 The Fredholm Determinant and Logarithmic Integral

191

Then a direct computation verifies that .f1 is analytic on .D(w0 , r) and f1−1 (w) = f −1 (w)(q1⊥ + (w − w0 )q1 )

.

is meromorphic on .D(w0 , r) with a possible pole only at .w0 of order strictly less than s. Furthermore, since .q1 is finite rank and .a0 = f (w0 ) is Fredholm, the operator .f1 (w0 ) = q1⊥ a0 + q1 a1 is Fredholm. In addition, we have the decomposition ) ( f (w) = q1⊥ + (w − w0 )q1 f1 (w),

.

(8.10)

where .f1 is normal on .D(w0 , r). Repeating this process, we may define .fj recursively as in (8.10) such that ) ( fj −1 (w) = qj⊥ + (w − w0 )qj fj (w), j = 2, . . . , k,

.

until .fk−1 (w) no longer has a pole at .w0 . The proof is thus completed by setting .h = fk . u n If one uses the fact .f (w)f −1 (w) = I and goes through some parallel steps, then it can be shown that ) ( ) ( f (w) = g(w) p1⊥ + (w − w0 )p1 · · · pk⊥ + (w − w0 )pk ,

.

(8.11)

for some finite rank projections .p1 , . . . , pk and an analytic function .g(w) that is invertible everywhere on .D(w0 , r). We leave the details as an exercise. In (8.11), observe that .pk is the projection onto .ker f (w0 ). Regarding the Maurer–Cartan form −1 df , (8.10) implies .ωf = f ( ωf (w) = f1−1 (w) q1⊥ +

.

1 q1 w − w0

= f1−1 (w)q1 f1 (w)

)( ) q1 f1 (w) + (q1⊥ + (w − w0 )q1 )f1, (w) dw

dw + ωf1 (w). w − w0

Thus by induction, we can write .ωf in the form ( ) −1 −1 .ωf = f1 (w)q1 f1 (w) + · · · + fk (w)qk fk (w)

dw + h−1 (w)h, (w)dw, w − w0 (8.12)

where .fk = h. Since .h−1 is analytic on the entire .D(w, r), the Cauchy integral theorem implies

192

8 Compact Operators and Kernel Bundles

f

1 .R(f ; w0 ) = 2π i

|w−w0 |=r

( ) −1 −1 f1 (w)q1 f1 (w) + · · · + fk (w)qk fk (w)

dw . w − w0

Note in particular that each summand in the integral is a function that takes value in finite rank operators. Thus the integrand is in .S1 for each w. It follows that TrR(f ; w0 ) = Trq1 + · · · + Trqk ,

.

(8.13)

which is always nonnegative. In particular, the residue is .R(f ; w0 ) = 0 if and only if .f −1 has no singularity at .w0 . If f is normal on domain .o and .f −1 (w) has poles at normal points .w1 , . . . , wm , then (8.8) implies TrR(f ; o) = TrR(f ; w1 ) + · · · + TrR(f ; wm ).

.

Therefore, .TrR(f ; o) = 0 if and only if .TrR(f ; wj ) = 0 for each j . Combining all the above observations, we have the following theorem which answers question 2. Theorem 8.8 Assume f is normal on .o and .f (w) is Fredholm at each normal point. Then .f (w) is invertible for every .w ∈ o if and only if .R(f ; o) = 0. If .g : o → S1 is analytic, then .f := I + g is clearly normal, and .f (w) is Fredholm with .indf (w) = 0 for each .w ∈ o. Moreover, Proposition 8.2 implies that .det f (w) is well-defined and analytic on .o. Moreover, if .det f (w) vanishes at some point .w0 ∈ o, then .f (w0 ) is not invertible, and it follows from the fact .indf (w0 ) = 0 that .f (w0 ) has a nontrivial kernel. Denote by .N0 the number of zeros of .det f (w) in .o, counting multiplicity. Theorem 8.9 Assume .g : o → S1 is analytic and .f = I + g. Then .TrR(f ; o) = N0 . In particular, if .det f (w) has a simple zero at .w0 , then .dim ker f (w0 ) = 1. Proof Since f is normal on .o, formula (8.1) and the argument principle lead to the equations 1 .TrR(f ; o) = 2π i =

1 2π i

1 = 2π i

f Trωf ∂o

f

( ) Tr (I + g(w))−1 dg(w)

∂o

f

d log det f (w) = N0 . ∂o

If .det f (w) has a simple zero at .w0 , we choose a small closed disc .D(w0 , r) ⊂ o that contains no other zeros of .det f (w). Then (8.13) and the above argument imply 1 = N0 = TrR(f ; w0 ) = Trq1 + · · · + Trqk

.

8.2 The Projective Spectrum of Compact Operators

193

for some .k ≥ 1. Since each .qj is a nontrivial projection (see the proof of Corollary 8.7), we must have .Trqj ≥ 1. It follows that .k = 1 and .Trq1 = 1. Using the factorization (8.11) and the similarity of .p1 and .q1 (exercise), we obtain .Trp1 = 1 which shows .dim ker f (w0 ) = 1. u n Exercise 8.10 1. Prove the factorization (8.11). 2. Prove equation (8.13). 3. Given the two factorizations of f in Corollary 8.7 and (8.11), show that after a possible reordering .qj is similar to .pj for each .1 ≤ j ≤ k. (Hint: Prove by induction, and use the remarks after the proof of Proposition 8.4).

8.2 The Projective Spectrum of Compact Operators This section first gives an analytic description of the projective spectrum of general compact operators. It lays a ground work for our later discussion on vector bundles over the spectrum. Then it will show that the commutativity of normal compact operators is determined by the geometry of their projective spectrum, giving a generalization of Theorem 1.19.

8.2.1 Thin Set A subset .S ⊂ Cn is called a thin set if for every point .λ ∈ S there exist a neighborhood D of .λ and a holomorphic function f on D such that .D ∩ S = Zf , where .Zf stands for the zero set of f . In other words, S is a thin set if it is locally the zero set of a holomorphic function. Evidently, such f is not unique. The point .λ is said to be regular if there exists such an f with gradient .∇f (λ) /= 0 or, equivalently, the differential .df (λ) /= 0. In this case, the tangent plane to .Zf at .λ is given by the equation . = 0, z ∈ Cn . And the implicit function theorem guarantees the existence of a neighborhood .D , of .λ such that .D , ∩ S is biholomorphic to the open unit ball of .Cn−1 . A thin set S is said to be smooth if every point of S is regular, in which case S is a complex manifold of dimension .n − 1. The following example illuminates the nuances. Example 8.11 Consider the thin set .S := {(z1 − z2 )2 = 0} ⊂ C2 . The differential d(z1 −z2 )2 = 2(z1 −z2 )(dz1 −dz2 ) vanishes everywhere on S. However, S is smooth because it is also the zero set of .z1 − z2 whose differential vanishes nowhere.

.

For compact operators .A1 , . . . , An ∈ K(H), we consider the pencil .A∗ (z) = I + zj Aj written with the summation convention. The following fact is observed in [213].

194

8 Compact Operators and Kernel Bundles

Proposition 8.12 If .A1 , . . . , An are compact operators on .H, then .P (A∗ ) is a thin set. Proof Since the set of finite rank operators is dense in .K(H), for any fixed λ ∈ P (A∗ ) there exist a small neighborhood D of .λ and the finite rank operators .F1 , . . . , Fn such that .|zj |||Aj −Fj || < 1 for every point .z ∈ D. Then .I +zj (Aj −Fj ) is invertible on D. Write .

A∗ (z) = I + zj (Aj − Fj ) + zj Fj ) ( ( ) −1 = I + zj (Aj − Fj ) I + (I + zj (Aj − Fj )) (zj Fj )

.

( ) =: I + zj (Aj − Fj ) Fλ (z), z ∈ D.

(8.14)

Observe that since .zj Fj is of finite rank, .Fλ is a holomorphic function from D into .I + S1 . Thus, the pencil .A∗ (z) is not invertible if and only if .Fλ (z) is not invertible. In light of Proposition 8.2(d), this occurs if and only if .det Fλ (z) = 0. Hence .D ∩ P (A∗ ) is the zero set of .det Fλ (z). u n Therefore, a point .λ ∈ P (A∗ ) is regular if the finite rank operators .F1 , . . . , Fn in the proof above can be chosen such that .∇ det Fλ (λ) /= 0.

8.2.2 Normal Compact Operators If .A1 , . . . , An are commuting operators, then .P (A∗ ) is a union of hyperplanes (Proposition 4.29). Surprisingly, the converse also holds for compact normal operators satisfying a mild condition [54, 168]. Thus, Theorem 1.19 has a generalization. The proof here is a simplification of that in the two papers. Recall that the point spectrum .σp (T ) is the set of eigenvalues of operator T . Definition 8.13 An operator .T ∈ B(H) is said to satisfy Condition A if there is an e > 0 such that the set .∩λ∈σp (T ) {w ∈ C | |1 − λw| ≥ e} is unbounded.

.

Lemma 8.14 Let .A1 and .A2 be compact operators on .H and assume .A1 is normal and satisfies Condition A. If there exists a nonzero complex number .μ with .|μ| = ||A2 || such that the complex line .{z ∈ C2 | z2 = 1/μ} ⊂ P (A∗ ), then there exists a nonzero vector .v ∈ H such that .A1 v = 0 and .A2 v = −μv. Proof Since .A1 satisfies Condition A, there exist .0 < e < 1 and a sequence .wn → ∞ such that .|1 + λwn | ≥ e for every .λ ∈ σp (A1 ). The assumption .{z ∈ C2 | z2 = 1/μ} ⊂ P (A∗ ) implies that .I + wA1 + A2 /μ is not invertible for every .w ∈ C. Since it is Fredholm with index 0, .ker(I + wn A1 + A2 /μ) is nontrivial and contains a unit vector .vn . The weak compactness of the closed unit ball .(H)1 gives rise to a subsequence of .{vn } that converges weakly to some .v ∈ (H)1 . Without loss of generality, we assume the sequence .{vn } converges weakly to v. Then, the

8.2 The Projective Spectrum of Compact Operators

195

compactness of .A1 and .A2 implies that .Ai vn → Ai v in norm for .i = 1, 2. We check that .v /= 0. Observe that the equality .(I + wn A1 + A2 /μ)vn = 0 implies .

1 1 A2 vn = − A2 v. n→∞ μ μ

lim (I + wn A1 )vn = − lim

n→∞

(8.15)

It is thus sufficient to check that the norm of the left-hand side is bounded below by a positive constant. In fact, let .P : H → ker A1 be the orthogonal projection and let .λ1 , λ2 , . . . be the nonzero eigenvalues of .A1 , counting multiplicity, with correE sponding orthonormal eigenvectors .e1 , e2 , . . .. Then .vn −P vn = j ej , and it follows that E 2 .||(I + wn A1 )vn || = ||P vn + (1 + λj wn )ej ||2 j

= ||P vn ||2 +

E

|1 + λj wn |2 ||2

j

≥ ||P vn ||2 + e 2 ||(I − P )vn ||2 ≥ e 2 ||vn ||2 = e 2 . Since .wn → ∞, dividing (8.15) by .wn , we also obtain .A1 v = 0. Moreover, since P A1 = 0, applying P to (8.15) we obtain

.

.

1 lim P vn = − P A2 v, n→∞ μ

which shows that the sequence .{P vn } is in fact norm convergent. Since .{vn } converges weakly to v, we must have .limn→∞ P vn = P v = v, which gives .P A2 v = −μv. Now, the assumption .|μ| = ||A2 || yields ||(I − P )A2 v||2 = ||A2 v||2 − ||μv||2 ≤ 0,

.

which shows that .A2 v = P A2 v = −μv, i.e., v is a common eigenvector of .A1 and A2 . u n

.

Lemma 8.15 Assume .A1 , A2 ∈ B(H) are normal and compact with .A1 satisfying Condition A. If .P (A∗ ) is a union of affine hyperplanes, then they have a common eigenvector. Proof Since .A2 is normal, it has an eigenvalue .μ such that .|μ| = ||A2 ||. In particular, .(0, −1/μ) ∈ P (A∗ ). Since the latter is a union of affine hyperplanes, one of them, say .H := {1 + λz1 + μz2 = 0}, contains .(0, −1/μ). By Lemma 1.14 the coefficient .λ is an eigenvalue of .A1 . If .λ = 0, then Lemma 8.14 gives what we need to prove. If .λ /= 0, we pick a sequence of nonzero scalars .{es | s = 1, 2, . . .} convergent to 0 such that .λs := λ + es μ /= 0 for every s. Define .Aes = A1 + es A2 . Then .(−1/λs , −es /λs ) ∈ H , and hence .λs I − A1 − es A2 is not invertible. Pick a unit vector .vs from .ker(λs I − A1 − es A2 ). Since the closed unit ball .(H)1 is weakly

196

8 Compact Operators and Kernel Bundles

compact, without loss of generality, we assume the sequence .{vs } converges weakly to v. Then the compactness of .A1 and .A2 implies .Ai vs → Ai v in norm for .i = 1, 2. Since .(λs I −A1 −es A2 )vs = 0, it follows that .{vs } in fact converges in norm to v. In particular, .||v|| = 1 and .A1 v = λv. We let .P : H → ker(A1 − λ) be the orthogonal projection. Then, since .A1 is normal, we have .P (A1 − λ) = 0 which implies 0 = P (λs I − A1 − es A2 )vs = es P (μ − A2 )vs , s = 1, 2, . . . .

.

Since .es /= 0 for each s, letting .s → ∞ we obtain .P (μ − A2 )v = 0, or equivalently, μv = P A2 v. As in the proof of Lemma 8.14, the fact that .|μ| = ||A2 || then gives .A2 v = μv, which establishes the lemma. u n .

We are now in a position to establish the following theorem. Theorem 8.16 Let .A1 , ..., An be normal compact operators satisfying Condition A. Then they commute if and only if .P (A∗ ) is a union of affine hyperplanes in .Cn . Proof The proof is a minor modification to the ending part of the proof to Theorem 1.19. The necessity follows from Proposition 4.29. For the sufficiency, to show that the operators pairwise commute, without loss of generality we assume .n = 2. Let .v1 be a common eigenvector of .A1 and .A2 guaranteed by Lemmas 8.14 and 8.15 such that .A1 v1 = λv and .A2 v1 = μv1 with .|μ| = ||A2 ||. Then with respect to the decomposition .H = Cv1 ⊕ H, , .A1 and .A2 are of the forms ( ) λ 0 . 0 A,1

and

( ) μ 0 , 0 A,2

respectively, where .A,1 and .A,2 are normal compact operators on .H, satisfying Condition A. Then .P (A∗ ) = P (A,∗ ) ∪ H , where .H = {1 + λz1 + μz2 = 0}. If , , , .A = A = 0, then there is nothing more to prove. Otherwise, since .P (A∗ ) is also 1 2 , a union of affine hyperplanes, we may repeat the above arguments for .A1 and .A,2 to obtain a common eigenvector .v2 ∈ H, . Continuing with this process, we obtain a set of orthogonal vectors .{vn | n = 1, . . . , m ≤ ∞} and a closed subspace .Hm (possibly equal to .H) spanned by it such that .A1 = A2 = 0 on .H⊥ m . With respect to the decomposition .H = Hm ⊕ H⊥ , the operators . A and . A are of the forms 1 2 m .

( ) D1 0 0 0

and, respectively,

( ) D2 0 , 0 0

where .D1 and .D2 are diagonals. Hence .A1 and .A2 commute.

u n

Since every self-adjoint operator satisfies Condition A, the following is immediate. Corollary 8.17 If .A1 , . . . , An ∈ K(H) are self-adjoint, then they commute if and only if .P (A∗ ) is a union of affine hyperplanes.

8.3 Kernel Bundles

197

Furthermore, since every matrix clearly satisfies Condition A, Theorem 1.19 is a consequence of Theorem 8.16. However, not every compact normal operator satisfies Condition A [168]. Example 8.18 Assume .{ek,i | k ∈ N, 1 ≤ i ≤ 2k } is an orthonormal basis for .H. k Denote by .αk,i the ith root of the equation .ξ 2 = 1,E set .βk = 1 + 1/2 + · · · + 1/k, αk,i and define .λk,i = βk . Then the operator .A = k,i λk,i ek,i ⊗ ek,i is compact and normal, and it fails Condition A. We leave the verification of this claim as an exercise. Exercise 8.19 1. Given .A1 , . . . , An ∈ K(H) and a regular point .λ ∈ P (A∗ ), show that the finite rank operators .F1 , . . . , Fn in the proof of Proposition 8.12 can be chosen such that .∇(det Fλ )(λ) /= 0. (Hint: Try to reduce it to a one-variable problem.) 2. Given compact operators .A1 , . . . , An on .H, show that if .H ⊂ Cn is an affine hyperplane contained in .P (A∗ ), then H is defined by the equation .1 + λ1 z1 + · · · + λn zn = 0, where .λj ∈ σ (Aj ), 1 ≤ j ≤ n. 3. For compact operators .A1 , . . . , An , prove that if .P (A∗ ) is a union of affine hyperplanes, then it is a countable union. 4. An operator .A ∈ B(H) is said to satisfy the strong Agmon condition, if there exist .θ ∈ (−π, π ) and .0 < δ < π such that the slice .{w ∈ C | θ − δ < Argw < θ + δ} contains no eigenvalues of A. Show that the strong Agmon condition implies Condition A. 5. Verify the claims in Example 8.18. 6. Prove that a compact operator K on .H is normal if and only if the projective spectrum of the pencil .I + z1 K + z2 K ∗ is a union of affine hyperplanes in .C2 . Project 8.20 1. Investigate whether Condition A is necessary for Lemma 8.14. If not, then Theorem 8.16 holds for all normal compact operators.

8.3 Kernel Bundles Consider a holomorphic function f on a complex domain .o ⊂ Cn+1 . A point .λ ∈ o ∂f is said to be regular for f if at least one of the partial derivatives . ∂z , 0 ≤ j ≤ n, j does not vanish at .λ or equivalently the differential .df (λ) /= 0. For matrices .A1 , . . . , An with characteristic polynomial .QA and eigensurface .P (A) = {z ∈ Cn+1 | QA (z) = 0}, the following holds. Lemma 8.21 If .λ is a regular point of .P (A), then .dim ker A(λ) = 1.

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Proof We show that .rankA(λ) = k − 1. As in Sect. 1.2, we let .CA (z) stand for the cofactor matrix of the linear pencil .A(z) = z0 I + z1 A1 + · · · + zn An . Then (1.10) gives ( ) dQA (z) = tr CAT (z)dA(z) .

.

If .λ is a regular point, then .dQA (λ) /= 0, implying .CA (λ) /= 0, i.e., at least one of the .(k − 1) × (k − 1) minors of .A(λ) is invertible. This shows that rank.A(λ) ≥ k − 1. On the other hand, since .λ ∈ P (A), we must have rank.A(λ) ≤ k − 1. u n The eigensurface .P (A) is said to be smooth if every point of it is regular. In this case .P (A) is a submanifold of .Cn+1 . Theorem 8.22 If .k × k matrices .A1 , . . . , An have a smooth eigensurface .P (A), then EA := {(λ, v) ∈ P (A) × Ck | v ∈ ker A(λ)}

.

is a line bundle over .P (A) with the canonical quotient map .π(λ, v) = λ. We shall call .EA the kernel bundle of the tuple .(A1 , . . . , An ). In this section, we shall prove the same fact for compact operators and show that .EA is a holomorphic line bundle.

8.3.1 On the Equivalence of Holomorphic Bundles One way to construct a Hermitian rank-k vector bundle over a complex manifold o is by means of a smooth function .f : o → Gr(k, H), where .Gr(k, H) is the Grassmannian manifold of all k-dimensional subspaces of a Hilbert space .H. We say that f is holomorphic if for every .λ ∈ o, there exist a neighborhood .A ⊆ o of .λ and holomorphic .H-valued functions .γ1 (z), . . . , γk (z) defined on .A such that .{γ1 (z), . . . , γk (z)} form a basis for .f (z). In this case, a Hermitian holomorphic rank-k bundle .

Ef = {(z, x) ∈ o × H | x ∈ f (z)}

.

(8.16)

is well-defined with the canonical quotient map .π(z, x) = z and the Hermitian metric given by the inner product of .H on each fiber .f (z). Recall that Theorem 7.8 asserts the existence of a unique connection on .Ef (the Chern connection) that is compatible with both the Hermitian metric and the holomorphic structure. Two ˜ are said to be Hermitian rank-k holomorphic bundles E over .o and .E˜ over .o ˜ such that equivalent if there exist biholomorphic maps .o : E → E˜ and .g : o → o the diagram

8.3 Kernel Bundles

199

.

commutes, where .oλ : Eλ → E˜ g(λ) is an isometric isomorphism for each .λ ∈ o. Furthermore, the bundles E and .E˜ are said to be locally equivalent if there exists ˜ g(A) are equivalent. In the a relatively open subset .A ⊆ o, such that .E|A and .E| ˜ and g is the identity sequel, we shall only concern ourselves with the case .o = o map. Thus, two holomorphic bundles .Ef and .Ef˜ defined in (8.16) are equivalent if and only if there exists a holomorphic function .U : o → Uk such that .f˜(λ) = U (λ)f (λ), λ ∈ o. In particular, if U is a constant function, then we say .Ef and .E ˜ are congruent. Obviously, congruent vector bundles are equivalent. In a short f while, we will see that, for holomorphic bundles with a certain “fullness” condition, equivalence also implies congruence. This is the Rigidity Theorem. V For a family of subspaces .{Vj | j ∈ J } in .H, we let . j ∈J Vj denote the closure of the linear span of .∪j ∈J Vj in .H. The following definition is to simplify the statement of some facts in the sequel. Definition 8.23 Given a smooth function .f : o → Gr(k, H), we say that the V bundle .Ef over .o is full if . λ∈o f (λ) = H. And we say it is locally full if .Ef |A is full for every open subset .A ⊂ o. Proposition 8.24 Let .f : o → Gr(k, H) be holomorphic. Then .Ef is full if and only if it is locally full. Proof It is clear that if .Ef is locally full, then it is full. Hence we only need to proveVthe other direction. Let .A be an open subset of .o and .x ∈ H be such that .x ⊥ z∈A f (z). We will prove by contradiction that .x = 0. ˚, denote the interior of .o, . Then .o ˚, is Let .o, = {z ∈ o | x ⊥ f (z)}, and let .o , clearly nonempty because it contains .A as an open subset. Suppose .o /= o and let ˚, . Then for any path-connected neighborhood .A0 of .λ, .λ be a boundary point of .o , ˚ the intersection .A0 ∩ o is nonempty. Since bundle .Ef is holomorphic, we can pick .A0 small enough so that there is a local holomorphic frame .{γ1 , . . . , γk } on .A0 . ˚, , we have .f (z) ⊥ x, that is, . = 0 for each j . Since For any .z ∈ A0 ∩ o each function . is holomorphic in z, the fact that it vanishes on the open set ˚, implies that it vanishes on the whole .A0 . This shows .λ ∈ o ˚, , which is a .A0 ∩ o , contradiction. This proves V that .o = o. By the assumption that .Ef is full, we must u n have .x = 0. Therefore, . z∈A f (z) = H, i.e., .Ef is locally full. This proposition has the following immediate but surprising consequence. Corollary 8.25 Let .V f : o → Gr(k, H) be holomorphic. If there exists a countable set .S ⊂ o such that . λ∈S f (λ) = H, then .Ef is locally full. It is striking because it implies V that, even for an open subset .A ⊂ o that is far away from S, we still have . z∈A f (z) = H. Furthermore, the bundle .Ef has the

200

8 Compact Operators and Kernel Bundles

following “rigidity” property which was first observed in Calabi [29] for the case k = 1.

.

Theorem 8.26 (Rigidity Theorem) Assume .f, f˜ : o → Gr(k, H) are holomorphic. If .Ef and .Ef˜ are full and locally equivalent, then they are congruent. Proof Suppose .Ef and .Ef˜ are locally equivalent. Then there is a relatively open | | set .A ⊆ o, and a biholomorphic bundle map .oA : Ef |A → Ef˜ |A which is an isometry on each fiber. One can choose a smaller .A if necessary so that (1) both .Ef and .Ef˜ have trivializations on .A, (2) .A lies inside a local chart of .o. Let .{γ1 , . . . , γk } be a holomorphic frame of .Ef on .A. If we define .γ˜j = oA (γj ), j = 1, . . . , k, then .γ˜1 , . . . , γ˜k form a holomorphic frame of .Ef˜ on .A, and = ,

.

z ∈ A,

due to the fact that .oA is an isometry on each fiber. Furthermore, since each .γj is holomorphic on .A, we have (p)

=

∂ p+q , ∂zp ∂ z¯ q

where .p = (p1 , p2 , . . . , pn ) and .q = (q1 , q2 , . . . , qn ) are multi-indices in .Zn+ , .

∂ p+q ∂ p1 +···+pn +q1 +···+qn ∂p (p) = , and γ (z) = γj (z). p p q q j ∂zp ∂ z¯ q ∂zp ∂z1 1 · · · ∂zn n ∂ z¯ 11 · · · ∂ z¯ nn

Therefore, (p)

= =

∂ p+q

∂zp ∂ z¯ q

(8.17)

∂ p+q (p) (q) = . ∂zp ∂ z¯ q

For any fixed .λ ∈ A, we set Sf,λ :=

V

.

(p)

(p)

{γ1 (λ), . . . , γk (λ)}

p∈Zn+

and define .Sf˜,λ likewise. Define a map .Uλ : Sf,λ → Sf˜,λ by (p)

(p)

Uλ (γj (λ)) = γ˜j (λ), 1 ≤ j ≤ k, p ∈ Zn+ .

.

(8.18)

8.3 Kernel Bundles

201 (p)

Note that .Uλ is first defined on .γj (λ) for each .1 ≤ j ≤ n and .p ∈ Zn+ , and then it is extended linearly to a map from .Sf,λ to .Sf˜,λ . Hence .Uλ is a linear operator for each .λ. Moreover, (8.17) implies that .Uλ is an isometry. Next, we show that .Sf,λ , Sf˜,λ , and .Uλ in fact do not depend on the choice of .λ in .A. Choose .A small enough so that every .γj has a Taylor series expansion γj (z) =

.

E (z − λ)r (r) γj (λ), z ∈ A, 1 ≤ j ≤ n, r! n

(8.19)

r∈Z+

|| || where .r = (r1 , r2 , . . . , rn ), .(z − λ)r = nj=1 (zj − λj )rj , and .r! = nj=1 rj !. Since .A is a relatively open subset in .o and .Ef and .E ˜ are full, they are locally full by f Proposition 8.24, i.e., V .

f (z) =

z∈A

V

f˜(z) = H.

z∈A

Therefore, H=

V

.

f (z) =

z∈A

V

{γ1 (z), . . . , γk (z)} ⊆ Sf,λ ⊆ H,

z∈A

which concludes that .Sf,λ = H. A parallel argument shows that .Sf˜,λ = H. It follows that .Uλ is a unitary operator on .H for each .λ ∈ A. In order to show that .Uλ is independent of .λ, we use the Taylor series expansion (8.19) to write for each j (p)

γj (z) =

.

E (z − λ)r (r+p) γj (λ), z ∈ A. r! n

r∈Z+

It follows that (E ) ( ) (z − λ)r (r+p) (p) γj (λ) Uλ γj (z) = Uλ r! n

.

r∈Z+

( ) E (z − λ)r (r+p) Uλ γ j = (λ) r! n r∈Z+

=

( ) E (z − λ)r (r+p) (p) (p) γ˜j (λ) = γ˜j (z) = Uz γj (z) . r! n

r∈Z+

This means for all .z ∈ A it holds that .Uz = Uλ , indicating that the function .Uz is locally constant on .o, and hence it is constant on .o since .o is path-connected. In conclusion, the equality .f˜(z) = Uf (z) holds for all .z ∈ o. u n

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8 Compact Operators and Kernel Bundles

8.3.2 Cowen–Douglas Operators The idea of kernel bundle arose in the work of Cowen–Douglas [32], which aims to find a complete unitary invariant for bounded linear operators. Let .o be a bounded domain in .C and .k ∈ N. An operator .T ∈ B(H) is said to be in the Cowen–Douglas class .Bk (o) provided that (1) (2) (3) (4)

o ⊂ σ (T ), ran(T − w) = H for every .w ∈ o, V . w∈o ker(T − w) = H, .dim ker(T − w) = k for every .w ∈ o. . .

One motivating example of such operator is the backward shift .S ∗ on .l2 (Z+ ) defined by .S ∗ (a0 , a1 , a2 , . . .) = (a1 , a2 , . . .), which is the adjoint of the unilateral shift S mentioned in Sect. 7.5. It is a good exercise to verify that .S ∗ ∈ B1 (D). For every .T ∈ Bk (o), there is a natural rank-k vector bundle over .o defined by ET = {(w, v) ∈ o × H | w ∈ o, v ∈ ker(T − w)}

.

with the canonical quotient map .π(w, v) = w. Lemma 8.27 Given .T ∈ Bk (o), its kernel bundle .ET is holomorphic. Proof We show that the function .f : o → Gr(k, H) defined by .f (w) = ker(T − w) is holomorphic, i.e., for every .w0 ∈ o there exist a neighborhood .A of .w0 and analytic .H-valued functions .γ1 , . . . , γk defined on .A such that .{γ1 (w), . . . , γk (w)} is a basis for .ker(T −w). For simplicity, we assume .w0 = 0. Observe that conditions (2) and (4) above indicate that T is Fredholm with index k. Then there exists .R ∈ B(H) such that .RT = I − P , where .P : H → ker T is the orthogonal projection. We define R(w) = (I − wR)−1 R, and P (w) = (I − wR)−1 P

.

for .w ∈ A := {w ∈ C | |w| < 1/||R||}. Then it is easy to see that .P (w) is an analytic rank-k function. Moreover, we have .

( ) ( ) ker(T − w) ⊂ ker R(w)(T − w) = ker I − P (w) ⊂ ranP (w).

Since .dim ker(T − w) = k = dim ranP (w), we must have .ker(T − w) = ranP (w). Hence, if .{e1 , . . . , ek } is a basis for .ker T , then the functions .γj (w) = P (w)ej , j = 1, . . . , k, have the required property. u n Thus, condition (3) above indicates that .ET is full and hence locally full by Proposition 8.24. The kernel bundle turns out to be a complete unitary invariant for operators in .Bk (o).

8.3 Kernel Bundles

203

Theorem 8.28 Two operators T and .T˜ in .Bk (o) are unitarily equivalent if and only if their kernel bundles .ET and .ET˜ are equivalent. Proof If T and .T˜ are unitarily equivalent, then there is a unitary U such that .U T = T˜ U . This implies .U (T − w) = (T˜ − w)U and consequently .U ker(T − w) = ker(T˜ − w) for every .w ∈ o. Therefore, .ET and .ET˜ are congruent and hence equivalent. On the other hand, since the kernel bundles .ET and .ET˜ are full, if they are equivalent, then the Rigidity Theorem (Theorem 8.26) asserts that they are congruent. Hence there exists a unitary .U ∈ B(H) such that .U ker(T − w) = ker(T˜ − w), w ∈ o. Thus, for every .v ∈ ker(T − w), we have .U (T − w)v = 0 = (T˜ − w)U v and consequently .U T v = T˜ U v. The fullness condition (3) above implies that .U T = T˜ U on .H. u n Observe that the condition (2) is not required for this theorem. In the case .T ∈ B1 (o), .ET is a holomorphic line bundle. If .γ is a local frame of .ET , then we have .(T − w)γ (w) = 0. Differentiating this equation, we obtain (T − w)γ , (w) − γ (w) = 0,

.

which implies .(T − w)2 γ , (w) = 0. With condition (2), one sees that .T − w is Fredholm with .ind(T − w) = 1 for every .w ∈ o. It follows that .ker(T − w)2 has dimension 2, and it is spanned by .{γ (w), γ , (w)}. An orthonormal basis can be obtained through the Gram–Schmidt process: e1 (w) =

γ (w) , and ||γ (w)||

e2 (w) =

||γ (w)||2 γ , (w) − γ (w) )1/2 . ( 2 , 2 , 2 ||γ (w)|| ||γ (w)|| ||γ (w)|| − ||

.

The function hT (w) = = ( )−1/2 2 2 , 2 , 2 = ||γ (w)|| ||γ (w)|| ||γ (w)|| − ||

.

is positive, and it reflects the local property of T . Indeed, with respect to the basis {e1 , e2 }, the restriction .(T − w)|ker(T −w)2 has the matrix representation

.

Nw :=

.

( ) 0 hT (w) , w ∈ o. 0 0

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8 Compact Operators and Kernel Bundles

These matrices are called the local operators of T . Likewise, for an operator .T ∈ Bk (o), the local operator of T at .w ∈ o is defined as .Nw = (T − w)|ker(T −w)k+1 , and it is obviously nilpotent of order .k + 1. Deriving global invariants from local properties has always been a tempting challenge in mathematics. Thus, can T be recovered from its local operators? The following theorem answers the question, and it is the main result of [32]. Theorem 8.29 Two .Bk (o)-class operators T and .T˜ are unitarily equivalent if and only if their local operators .Nw and .N˜ w are unitarily equivalent at every .w ∈ o. For .T ∈ B1 (o), the Ricci curvature of the Chern connection on the line bundle 2 ||γ (w)||2 ET is .RT (w) = − ∂ log∂w∂ , where .γ is any holomorphic frame of .ET (see w¯ Example 7.9). It leads us to the following captivating observation.

.

Corollary 8.30 Two .B1 (o)-class operators are unitarily equivalent if and only if their kernel bundles have the same Ricci curvature everywhere on .o. Proof The necessity is easy to see. For the sufficiency, given an operator .T ∈ B1 (o), one verifies by direct computation that .RT (w)h2T (w) = −1, w ∈ o. Thus, if .T˜ ∈ B1 (o) is such that .RT (w) = RT˜ (w), then .hT (w) = hT˜ (w), indicating that ˜ w are unitarily equivalent at every .w ∈ o. The corollary then follows .Nw and .N from Theorem 8.29. u n Given .T ∈ Bk (o), a holomorphic cross-section .γ of .ET is said to be spanning if .span{γ (w) | w ∈ o} is dense in .H. It is shown in Zhu [248] that such crosssections always exist, and they give rise to an analytic model for T . We set .Ωˆ = {w¯ | w ∈ Ω}. Theorem 8.31 Every operator .T ∈ Bk (o) is unitarily equivalent to the adjoint of ˆ multiplication by w on a Hilbert space of holomorphic functions on .o. Proof We fix a spanning holomorphic cross-section .γ of .ET . For each .x ∈ H, we ˆ = {xˆ | x ∈ H}. It is ˆ and set .H ˆ = , ¯ w ∈ o, define the function .x(w) ˆ is a space of holomorphic functions on .o. Moreover, it is a Hilbert clear that .H space with the inner product defined by . ˆ = , x, y ∈ H. Thus, the map ˆ .U : H → H which sends x to .x ˆ is unitary. Setting .S = U T U ∗ , we check that the ∗ ˆ Indeed, for any .x ∈ H and .w ∈ o, ˆ we adjoint .S is the multiplication by w on .H. have (S ∗ x)(w) ˆ = (U T ∗ U ∗ x)(w) ˆ = (U T ∗ x)(w)

.

∗ x(w) = = T¯

= ¯ = ¯ = w x(w). ˆ u n Literature Note In addition to the study of unitary equivalence, there is an extensive work on the similarity of two Cowen–Douglas operators. For details, we

8.3 Kernel Bundles

205

refer the reader to [68, 137]. Efforts have also been made to extend the Cowen– Douglas theory to tuples of commuting operators. This development can be found in Curto [46], Misra [161], Wang [234], as well as the references therein.

8.3.3 The Kernel Bundle of Compact Operators For .A1 , . . . , An ∈ K(H), Proposition 8.12 indicates that the projective spectrum P (A∗ ) is a thin set. Hence, if .P (A∗ ) is smooth, then it is a complex manifold. The goal of this subsection is to show that in this case the kernel bundle over .P (A∗ ) is a Hermitian holomorphic line bundle, and it is a unitary invariant for the tuple .(A1 , . . . , An ). In view of Theorem 8.28, this is an extension of the Cowen–Douglas theory to noncommuting operators. For more details about the discussion here, we refer the reader to [130, 131]. Recall from Sect. 8.2 that given any .λ ∈ P (A) there exist finite rank operators .F1 , . . . , Fn and a small neighborhood D of .λ such that .|zj |||(Aj −Fj )|| < 1, z ∈ D. Hence the operator-valued function .

Fλ (z) = I + (I + zj (Aj − Fj ))−1 (zj Fj )

.

is holomorphic and Fredholm on D. Lemma 8.32 If a point .λ ∈ P (A∗ ) is regular, then .dim ker A∗ (λ) = 1. Proof For a regular point .λ ∈ P (A∗ ), the finite rank operators .F1 , . . . , Fn in the proof of Proposition 8.12 can be chosen such that .∇ det Fλ (λ) /= 0 (Sect. 8.3 exercise 1). Moreover, observe that .Fλ (z) is holomorphic and Fredholm at every z in a small neighborhood D of .λ. Consider the linear function .zλ (w) = λ + w∇(det Fλ )(λ), w ∈ C. We pick a small .r > 0 so that the analytic disk .zλ (rD) lies inside D. Furthermore, since the vector .∇(det Fλ )(λ) is nonzero and is normal to the tangent plane of .P (A∗ ) at .λ, the disk .zλ (rD) intersects .P (A∗ ) transversally at .λ, and hence the zeros of the analytic function .φ(w) := det Fλ (zλ (w)) are discrete. So we may assume further that r is small enough such that .zλ (rD) ∩ P (A∗ ) = {λ}. Therefore, Proposition 8.2(d) implies that .Fλ (zλ (w)) is invertible for each .0 < |w| < r, and it is Fredholm at .w = 0. Moreover, since .φ(0) = 0 and φ , (0) =

.

= = ||∇(det Fλ )(λ)||2 > 0, the function .φ has a simple zero at .w = 0. Proposition 8.9 then implies .

dim ker A∗ (λ) = dim ker Fλ (λ) = 1. u n

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8 Compact Operators and Kernel Bundles

Therefore, if .P (A∗ ) is smooth, Lemma 8.32 gives rise to the kernel bundle EA := {(z, x) ∈ P (A∗ ) × H | x ∈ ker A∗ (z)},

.

with the canonical quotient map .π(z, x) = z. However, a more delicate analysis is needed to show that .EA is holomorphic. Theorem 8.33 If .P (A∗ ) is smooth, then .EA is a holomorphic line bundle over P (A∗ ).

.

Proof It only remains to show that .EA has a holomorphic local frame .γ at every point .λ ∈ P (A∗ ). We let D, .Fλ , and .zλ (w) be defined as in the proof of Lemma 8.32. For .τ ∈ P (A∗ ), we consider the analytic disc .zτ (rD). As noted before, the disc .zλ (rD) intersects .P (A∗ ) transversally at .λ, and .det Fλ (zλ (w)) has a simple zero at .w = 0. Since all involved functions are analytic, there exist a small .r > 0 and a small neighborhood .V ⊂ P (A∗ ) of .λ that meet the following requirements: (1) For every .τ ∈ V , the disc .zτ (rD) intersects .P (A∗ ) transversally at .τ . (2) .V + rBn ⊂ D, and .zτ (rD) ∩ P (A∗ ) = {τ } for every .τ ∈ V . (3) For each fixed .τ ∈ V , the function .det Fλ (zτ (w)) has a simple zero at .w = 0. Then .Fλ (zτ (w)) is a normal Fredholm operator-valued function on .rD with a unique normal point .w = 0. Corollary 8.7 and (8.11) give the factorization ) ( ) ( Fλ (zτ (w)) = g(w) p1⊥ + (w − w0 )p1 · · · pk⊥ + (w − w0 )pk , |w| < r.

.

Since .Fλ (zτ (w)) has a simple zero at .w = 0, the proof of Theorem 8.9 shows that .k = 1. Therefore, we can write .Fλ (zτ (w)) = g(w)(p⊥ + wp), where .g(w) is analytic and invertible everywhere in .rD, and p is a rank-1 projection onto .ker Fλ (τ ) = ker A∗ (τ ). It is worth noting that, although g and p both depend on .τ , we shall see that this dependence presents no challenge to the proof. Now we can write with the summation convention that ( ) A∗ (zτ (w)) = I + zτ,j (w)(Aj − Fj ) Fλ (zτ (w)) ( ) = I + zτ,j (w)(Aj − Fj ) g(w)(p ⊥ + wp).

.

( ) ˆ and recall ωA∗ (z) = A−1 Denote . I + zτ,j (w)(Aj − Fj ) g(w) by .g(w) ∗ (z)dA∗ (z), c .z ∈ P (A∗ ). Then for .|w| ≤ r, we have ( ) 1 ⊥ ˆ + wp) p)gˆ −1 (w)d g(w)(p w ( ) 1 = (p⊥ + p)gˆ −1 (w) gˆ , (w)(p⊥ + wp) + g(w)p ˆ dw w ( ) = p⊥ gˆ −1 (w)gˆ , (w)p⊥ + wp⊥ gˆ −1 (w)gˆ , (w)p dw

ωA∗ (zτ (w)) = (p⊥ +

8.3 Kernel Bundles

207

( + pgˆ

−1

) 1 −1 1 , ⊥ (w)gˆ (w)p + pgˆ (w)gˆ (w)p + p dw. w w ,

Since the first three summands are analytic in w, the Cauchy integral theorem implies Q(τ ) :=

f

1 2π i

|w|=r

(8.20)

ωA∗ (zτ (w))

) ( f 1 1 −1 1 pgˆ (w)gˆ , (w)p⊥ + p dw 2π i |w|=r w w ) ( ) ( f f 1 −1 1 1 1 , ⊥ gˆ (w)gˆ (w)dw p + dw p =p 2π i |w|=r w 2π i |w|=r w

=

= pgˆ −1 (0)gˆ , (0)p⊥ + p. A direct calculation verifies that Q2 (τ ) = Q(τ ), Q(τ )p = p, p ⊥ Q(τ ) = 0.

.

In other words, .Q(τ ) is a rank-1 idempotent that maps .H onto .ker A∗ (τ ). Moreover, since .zτ (w) is holomorphic in .τ ∈ V , the integral expression (8.20) implies that .Q(τ ) is analytic in .τ as well. Pick any nonzero vector .e ∈ ker A∗ (λ) and define .γ (τ ) = Q(τ )e. Then .γ (λ) = e /= 0, and hence .γ is a holomorphic local frame of .EA in a neighborhood of .λ ∈ P (A∗ ). u n

8.3.4 An Example with Rank-1 Projections We use a simple example to illustrate the construction of the kernel bundle for compact operators as well as the computation of its curvature. Let .e1 and .e2 be linearly independent unit vectors in .H and consider the rank-1 projections .Ai = ei ⊗ ei , i = 1, 2. Since multiplying .e1 by a unimodular scalar does not change .A1 , without loss of generality we assume . ≥ 0 and write it as .cos α for some .α ∈ (0, π/2]. Then, for .A∗ (z) = I + z1 e1 ⊗ e1 + z2 e2 ⊗ e2 , we have A∗ (z)ej = (I + z1 e1 ⊗ e1 + z2 e2 ⊗ e2 )ej = ej +

2 E

.

zi ei .

i=1

Let .H = span{e1 , e2 }. With respect to the decomposition .H = H ⊕ H ⊥ , the pencil .A∗ (z) is similar to .W (z) ⊕ IH ⊥ , where

208

8 Compact Operators and Kernel Bundles

( W (z) =

.

1 + z1 z1 cos α z2 cos α 1 + z2

) = I2 + diag(z1 , z2 )G

( ) and G is the Gramian matrix . . It follows that P (A∗ ) = {z ∈ C2 | 1 + z1 + z2 + z1 z2 sin2 α = 0},

.

(8.21)

which is a special case of Corollary 5.7. It is not hard to check that .P (A∗ ) is smooth if and only if . /= 0, in which case .P (A∗ ) is a complex manifold of dimension 1, and the kernel bundle .EA is a Hermitian holomorphic line bundle on .P (A∗ ) due to Theorem 8.33. Moreover, we have .ker A∗ (z) = C((1 + z2 )e1 − z2 cos αe2 ), and thus .γ (z) = (1 + z2 )e1 − z2 cos αe2 is a global holomorphic frame of .EA . Furthermore, ||γ (z)||2 = |1 + z2 |2 sin2 α + cos2 α, z ∈ P (A∗ ),

.

and the curvature form of .EA is O(z) = ∂∂ log ||γ (z)||2 ( ) = ∂∂ log |1 + z2 |2 sin2 α + cos2 α

.

=∂

− cot2 αdz2 ∧ dz2 (1 + z2 )dz2 = ( )2 , z2 ∈ C. |1 + z2 |2 + cot2 α |1 + z2 |2 + cot2 α

For n linearly independent unit vectors( .e1 , . . .), en ∈ H, their relative position is described by the Gramian matrix .G = . We set .Hn = span{e1 , . . . , en } and consider the projections .Aj = ej ⊗ ej , 1 ≤ j ≤ n. Then with respect to the decomposition .H = Hn ⊕ Hn⊥ , the pencil .A∗ (z) can be written as ( A∗ (z) =

.

Wn (z) 0 0 I

) ,

(8.22)

where .Wn (z) is the matrix representation of the restriction .A∗ (z)|Hn . Thus .P (A∗ ) = {z ∈ Cn | det Wn (z) = 0}, and its smoothness is characterized by the Gramian matrix G of the vectors .e1 , . . . , en . The following theorem was conjectured in [130] and proved in [158]. Theorem 8.34 For linearly independent rank-1 projections .Aj = ej ⊗ ej , 1 ≤ j ≤ ( ) n, the spectrum .P (A∗ ) is smooth if and only if .rank G−1 + diag(z1 , . . . , zn ) ≥ n − 1 for all .z ∈ Cn . However, this theorem is not entirely satisfactory because ideally the smoothness of .P (A∗ ) should be described by the relative position of the n vectors, or the entries of G, without resorting to z. At this time, such a description only exists

8.3 Kernel Bundles

209

for .n ≤ 3 [130]. Polynomial .det Wn (z) mentioned above is in fact the characteristic polynomial of the projections .A1 , . . . , An . The following is shown recently [131]. Theorem 8.35 Two tuples of rank-1 projections are unitarily equivalent if and only if they have the same characteristic polynomial.

8.3.5 A Criterion for the Unitary Equivalence In the spirit of Theorem 8.28, this subsection aims to describe the unitary equivalence of two compact operator tuples .A = (A1 , . . . , An ) and .B = (B1 , . . . , Bn ) on .H by means of the kernel bundle. Throughout this section, we assume .M := P (A∗ ) = P (B∗ ), and it is smooth and path-connected. First, in the special case that A and B are pairs of finite rank projections or tuples of rank-1 projections, Theorem 1.24 and Theorem 8.35 already imply that they are unitarily equivalent. The fullness condition of kernel bundle also makes sense for a single operator. Definition 8.36 For .T ∈ B(H), we say that its spectrum .σ (T ) is full if V .

ker(T − λ) = H.

(8.23)

λ∈σ (T )

Thus, every Cowen–Douglas operator has a full spectrum, and so does every normal compact operator. If T is compact and .dim H = ∞, then .0 ∈ σ (T ). To give more weight to T ’s nonzero eigenvalues, we say that .σ (T ) is strongly full if only nonzero .λ is considered in (8.23). Lemma 8.37 Given .A1 , . . . , An ∈ K(H) with smooth projective spectrum .P (A∗ ), if .σ (Aj ) is strongly full for some j , then .EA is locally full. Proof The smoothness condition implies that .P (A∗ ) is a complex manifold of dimension .n − 1, and moreover .EA is a Hermitian holomorphic line bundles in light of Theorem 8.33. Without loss of generality, we assume .σ (A1 ) is strongly full. Clearly, a nonzero scalar .α is in .σ (A1 ) if and only if .ξα := (− α1 , 0, · · · , 0) ∈ P (A∗ ). Since .σ (A1 ) is strongly full, we have H=

V

.

0/=λ∈σ (A1 )

ker(A1 − λ) =

V

ker A∗ (ξλ ),

0/=λ∈σ (A1 )

which implies that .EA is locally full by Corollary 8.25.

u n

This tantalizing fact shows that, although defined locally, the smoothness of .P (A∗ ) has some global implications. With some additional spectral conditions, one can derive the unitary equivalence of A and B from the equivalence of their kernel bundles.

210

8 Compact Operators and Kernel Bundles

Lemma 8.38 If every operator in the two tuples A and B has strongly full spectrum, then the equivalence of .EA and .EB implies the unitary equivalence of A and B. Proof The strong fullness of the spectra .σ (A1 ) and .σ (B1 ), together with Corollary 8.37, indicates that .EA and .EB are both full. If they are equivalent, then Theorem 8.26 ensures the existence of a unitary operator U such that .fB (λ) = UfA (λ) for all .λ ∈ M. It follows that .U A∗ (λ) = B∗ (λ)U holds on .ker A∗ (λ). For each i and .0 /= a ∈ σ (Ai ), it is obvious that .λ = (0, . . . , 0, −1 a , 0, . . . , 0) ∈ P (A∗ ), −1 where . a is at the ith coordinate. Substituting this .λ into the equation .U A∗ (λ) = B∗ (λ)U , we obtain that .U Ai = Bi U holds on .ker(aI − Ai ). The strong fullness u n assumption on .σ (Ai ) then implies .U Ai = Bi U on .H for each i. The strong fullness condition in Lemma 8.38, though it appears to be a lot to ask for, is necessary because the pencil .A∗ (z) and the associated kernel bundle .EA do not capture the information about .ker Ai . The following example helps to illuminate this point. Example 8.39 Let .A = (A1 , A2 ), where ⎛

0 ⎜0 .A1 = ⎜ ⎝0 0

0 0 0 0

0 1 0 0

⎞ 0 0⎟ ⎟, 1⎠ 0

⎛

0 ⎜1 A2 = ⎜ ⎝0 0

0 0 0 0

0 0 0 1

⎞ 1 0⎟ ⎟, 0⎠ 0

and .B = (A2 , A1 ). Thus, ⎛

1 ⎜ z2 .A∗ (z) = ⎜ ⎝0 0

0 0 1 z1 0 1 0 z2

⎞ z2 0⎟ ⎟, z1 ⎠ 1

⎛

1 ⎜ z1 B∗ (z) = ⎜ ⎝0 0

0 0 1 z2 0 1 0 z1

⎞ z1 0⎟ ⎟, z2 ⎠ 1

and they have the same smooth projective spectrum .M = {z ∈ C2 | 1 − z1 z2 = 0}. For any point .z = (z1 , z2 ) ∈ o, since .z1 z2 = 1, we have .

ker A∗ (z) = C(−z2 , z12 + z22 , −z1 , 1)T , ker B∗ (z) = C(−z1 , z12 + z22 , −z2 , 1)T .

It can be checked that .EA and .EB are both full, and the map defined by ( .

) ( ) z, η(−z2 , z12 + z22 , −z1 , 1) → z, η(−z1 , z12 + z22 , −z2 , 1) , z ∈ M, η ∈ C,

establishes the unitary equivalence of the two kernel bundles. However, A and B are evidently not unitarily equivalent because .rankA1 /= rankA2 . Since injective normal compact operators have strongly full spectra, we have the following consequence of Lemma 8.38.

8.3 Kernel Bundles

211

Proposition 8.40 If A and B are tuples of injective normal compact operators with smooth and identical projective spectra, then they are unitarily equivalent if and only if .EA and .EB are equivalent. It is not clear whether the same holds for general injective compact operators. We leave it as a project. Exercise 8.41 1. Show that .S ∗ ∈ B1 (o), where .S ∗ is the backward shift. 2. Show that every .B1 (o)-class operator is irreducible. 3. Given .T ∈ B2 (o), find a matrix representation for its local operator .Nw , w ∈ o. 4. Check that .P (A∗ ) in (8.21) is smooth if and only if .sin2 α /= 1. 5. Consider matrices ( ) ( ) 01 00 .A1 = , A2 = . 00 10 (a) Show that .P (A∗ ) is smooth. (b) If .B1 and .B2 are .2 × 2 matrices such that .P (B∗ ) = P (A∗ ) and .EA is equivalent to .EB , show that A is unitarily equivalent to B. 6*. Consider the decomposition (8.22) for the case .n = 3. Compute .det W3 (z) and find a condition on the vectors .e1 , e2 , and .e3 with which the polynomial is irreducible. 7. Prove Theorem 8.35 for the case .n = 3. Project 8.42 1. Consider matrices .A1 , . . . , An ∈ Mk (C) and let .A(z) = z0 I +z1 A1 +· · ·+zn An . (a) Investigate the smoothness of .p(A) ⊂ Pn . (b) If .p(A) is smooth, describe the Chern character of .EA and the Kähler form on .p(A). 2. Given n linearly independent unit vectors .e1 , . . . , en ∈ H, we consider the projections .pj = ej ⊗ ej , 1 ≤ j ≤ n, and the hyperplane arrangement n ran(I −p ). Study the connection between the following two subjects: .S := ∪ j j =1 (a) The factorization of the characteristic polynomial .Qp (b) The de Rham cohomology of .Sc 3. Consider the pencil .A∗ (z) = I + z1 Vh + z2 Vh∗ , where .Vh is the Volterra operator on .H 2 (D) defined in Sect. 7.6.2. Compute .P (A∗ ) and determine whether it is smooth. 4. Suppose A and B are tuples of injective compact operators with smooth and identical projective spectrum, and assume their kernel bundles .EA and .EB are both full. Does the equivalence of .EA and .EB imply the unitarily equivalence of the tuples A and B?

Chapter 9

Weak Containment and Amenability

This chapter extends the discussion in Chap. 2 to finitely generated groups. We assume throughout this chapter that the Hilbert space .H has infinite dimension. In this case, Kuiper’s theorem [148] asserts that the group of unitaries .U (H) is contractible. Recall that given two unitary representations .(π, H) and .(ρ, H) of a group G, we say that .π contains .ρ and write .ρ < π if .ρ is equivalent to a subrepresentation of .π . Observe that .1G < π if and only if there exists a nonzero vector .v ∈ H such that .π(g)v = v for all .g ∈ G. In other words, v is a common eigenvector for all .π(g) corresponding to the common eigenvalue 1. Such vector v is called an invariant vector of .π . Containment is a partial order on the set of unitary representations of G. However, it is too rigid to be useful in the study of noncompact groups. In this chapter, we will discuss the notion of weak containment and use it to describe the amenability of finitely generated groups. We shall also see how weak containment and amenability are reflected through the projective spectrum of the group. For more details about the topics in this chapter, we refer the reader to [18, 62, 244].

9.1 Locally Compact Groups A topological group G is said to be locally compact if its unit 1 has an open neighborhood D whose closure is compact. Thus, in this case every .x ∈ G has a neighborhood xD with compact closure. The following theorem due to Haar is fundamental, and its proof can be found in most real analysis books, for instance [116]. Proposition 9.1 Every locally compact group G has a positive regular Borel measure .μ, unique up to a positive scalar multiple, such that: (a) For every Borel subset .Q ⊂ G and .g ∈ G, .μ(gQ) = μ(Q). © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 R. Yang, A Spectral Theory Of Noncommuting Operators, https://doi.org/10.1007/978-3-031-51605-4_9

213

214

9 Weak Containment and Amenability

(b) For every open set .D ⊂ G, .μ(D) > 0. (c) For every compact subset .K ⊂ G, .μ(K) < ∞. Such measure u is called a Haar measure on G. The Banach spaces .Lp (G, μ), p ≥ 1, are thus well-defined. Example 9.2 The following are some well-known examples of Haar measure: (1) The Lebesgue measure dm on .Rn (2) The counting measure on a countable group (3) As an open subset in .R2n , the general linear group .GLn (R) has Haar measure dμ(x) =

.

dm , x ∈ GLn (R). | det x|n

(4) Consider the Heisenberg group G consisting of matrices ⎛ ⎞ 1x z . ⎝0 1 y ⎠ , x, y, z ∈ R. 001 Then the Lebesgue measure .dm = dxdydz is a Haar measure on G.

9.1.1 Convolution Given .f ∈ L1 (G) and .g ∈ Lp (G), 1 ≤ p ≤ ∞, their convolution is defined as f

f (x)g(x −1 y)dμ(x).

f ∗ g(y) =

.

(9.1)

G

Proposition 9.3 For .f ∈ L1 (G) and .g ∈ Lp (G), it holds .||f ∗ g||p ≤ ||f ||1 ||g||p . Proof The case .p = ∞ is obvious, so we only prove the case .1 ≤ p < ∞. Since f ∈ L1 (G), it gives rise to a measure .μf on G such that .dμf (x) = |f (x)|dμ(x). Without loss of generality, we assume .||f ||1 = 1, and hence .μf is a probability measure on G. Thus, Hölder’s inequality gives

.

||f

.

p ∗ g||p

|p f |f | | −1 | | = | f (x)g(x y)dμ(x)| dμ(y) G G )p f (f −1 ≤ |f (x)||g(x y)|dμ(x) dμ(y) G

f f ≤ G G

G

|g(x −1 y)|p dμf (x)dμ(y)

9.1 Locally Compact Groups

215

f f =

|g(x −1 y)|p dμ(y)dμf (x) = ||g||p . p

G G

u n An important case of Proposition 9.3 is when .p = 1, in which the space .(L1 (G), ∗) becomes a Banach algebra. Observe that the convolution .∗ is a natural extension of the multiplication in .C[G]. Moreover, the adjoint on .C[G] defined in (2.1) can be extended to .L1 (G) such that .f ∗ (x) = f (x −1 ). Using the G-invariance of .μ, one verifies that .||f ∗ ||1 = ||f ||1 . Therefore, the algebra .L1 (G) is also equipped with an adjoint operation. However, it is not a .C ∗ -algebra because .||f ∗ ∗ f ||1 is in general not equal to .||f ||21 . A representation .(π, H) of G naturally extends to a representation of .L1 (G) such that f .π(f ) := f (x)π(x)dμ(x), f ∈ L1 (G), (9.2) G

where the convergence of the integral is with respect to the weak topology on .B(H). It is a good exercise to check the following properties for .f, g ∈ L1 (G): (b) π(f ∗ ) = π(f )∗ ;

(a) ||π(f )|| ≤ ||f ||1 ;

.

(c) π(f ∗ h) = π(f )π(h).

(9.3)

Thus, the map .π : L1 (G) → B(H) is a .∗ -homomorphism. We denote by .Cπ∗ (G) the norm closure of the set .π(L1 (G)) in .B(H) and call it the .C ∗ -algebra of G with respect to .π . If G has a finite generating set S, then .Cπ∗ (G) is the .C ∗ -algebra generated by the unitaries .π(x), x ∈ S.

9.1.2 Revisit the Regular Representation For each fixed .x ∈ G, there is a natural isometric action .h → Lp (G), 1 ≤ p ≤ ∞, defined by

xh

of G on

.

.x

h(y) = h(x −1 y), h ∈ Lp (G), y ∈ G.

(9.4)

Due to the G-invariance of .μ, it is clear that .||x h||p = ||h||p . In the case .p = 2, we have .x h = λG (x)h, where .λG is the (left) regular representation of G. Observe that for every .f ∈ L1 (G), definition (9.2) gives λG (f )h(y) = f ∗ h(y), h ∈ L2 (G).

.

(9.5)

The norm closure of .λG (L1 (G)) in .B(L2 (G)), denoted by .Cr∗ (G), is the reduced group .C ∗ -algebra of G. In the case G is a discrete group, the canonical trace .tr on

216

9 Weak Containment and Amenability

C[G] extends to be a tracial state on .Cr∗ (G) defined by .tr a = , where .δ1 is the characteristic function of .{1} ⊂ G. Furthermore, for .f, h ∈ L2 (G), we have

.

f ∗ h∗ (y) =

f

.

f (x)h∗ (x −1 y)dμ(x) =

G

f f (x)h(y −1 x)dμ(x) = . G

(9.6) Hence, functions of the form .f ∗ h∗ on G are often called the matrix coefficients of .λG . Proposition 9.4 For a locally compact group G with Haar measure .μ, the following are equivalent: (a) G is compact. (b) .1G < λG . (c) .μ(G) < ∞. Proof To see .(a) ⇒ (b), we notice that if G is compact, then the constant function 1 ∈ L2 (G) and it is an invariant vector with respect to .λG . Thus .1G < λG . For .(b) ⇒ (c), if .1G is contained in .λG , then .λG has an invariant vector .η ∈ L2 (G), i.e., .η(x) = λG (x −1 )η(1) = η(1) for every .x ∈ G. Thus .η is a constant. It follows that f 2 .|η(1)| μ(G) = |η(x)|2 dμ(x) = ||η||2 < ∞.

.

G

The fact .(c) ⇒ (a) can be proved by contradiction. Since G is locally compact, there exists a neighborhood .D ' of .1 ∈ G such that its closure .D ' is compact. Suppose G was not compact. There exists .g1 ∈ G \ D ' . Likewise, there exists gn ∈ G \ (D ' ∪ g1 D ' ∪ · · · ∪ gn−1 D ' ), n ≥ 2.

.

Choose a neighborhood D of .1 ∈ G such that .D = D −1 and .D 2 ⊂ D ' . If there exists .x ∈ gm D ∩ gn D, where .m > n, then .x = gm y = gn y ' for some .y, y ' ∈ D, and it follows that .gm = gn y ' y −1 ∈ gn V which would contradict our selection of .gn . This implies that .gn D, n = 1, 2, . . . , are disjoint open subsets in G. Therefore, μ(G) ≥

∞ E

.

n=1

which contradicts (c).

μ(gn D) =

∞ E

μ(D) = ∞,

n=1

u n

Definition 9.5 A continuous function f on a topological space X is said to vanish at .∞ if . ∀e > 0, the set .{x ∈ X | |f (x)| ≥ e} is compact. We let .C0 (X) denote the set of all continuous functions on X that vanish at .∞ and let .Cc (X) denote the set of all continuous functions on X with compact support. Clearly, if X is compact, then .Cc (X) = C0 (X) = C(X). In general, the space .Cc (X) is not closed in .C(X), but .C0 (X) is.

9.2 Weak Containment

217

Lemma 9.6 Let G be a locally compact group. Then .f ∗ g ∈ C0 (G) for .f, g ∈ L2 (G). Since .Cc (G) is dense in .L2 (G), it is sufficient to prove the lemma for .f, g ∈ Cc (G). But this is clear in view of the fact that .supp(f ∗ g) ⊂ suppf · suppg (exercise). Corollary 9.7 Let G be a locally compact but noncompact group. Then .λG contains no finite dimensional subrepresentation. Proof We prove this by contradiction. Suppose .(π, K) is a representation of G such that .dim K = n < ∞, and .{v1 , . .(. , vn } is an )orthonormal basis of .K. Then for each .x ∈ G, the .n × n matrix . is unitary. In particular, En 2 . j =1 || = 1. Suppose .π is a subrepresentation of .λG . Then without loss of generality we can assume .K ⊂ L2 (G) and .π = λG |K . Then (9.6) would imply n E .

||2 =

j =1

n E

|v1 ∗ vj∗ (x)|2 ≥ 1, x ∈ G.

j =1

But this would contradict Lemma 9.6.

u n

For a noncompact abelian group G, every irreducible representation is onedimensional (Corollary 2.3). Hence, Corollary 9.7 in this case implies that .λG contains no irreducible representation of G. This fact indicates that the containment relation of representations is too rigid for noncompact groups. A weaker notion of containment is needed to compare the representations. Exercise 9.8 1. Verify the claim of Example 9.2 (3). 2. Consider a locally compact group G and .f, g ∈ Cc (G). (a) Show that .supp(f ∗ g) ⊂ suppf · suppg. (b) Prove Lemma 9.6. 3. Prove the three properties in (9.3). 4. For a finite group G, show that .λG contains every irreducible representation of G. 5. Let G be a discrete group. Verify that the canonical trace on .Cr∗ (G) defined by .tr a = is a tracial state.

9.2 Weak Containment Group representation makes linear functional an important tool in the study of groups. Given a representation .(π, H) of group G and vectors .u, v ∈ H, the function

218

9 Weak Containment and Amenability π fu,v (x) = , x ∈ G,

.

(9.7)

is called a matrix coefficient of .π . Clearly, it extends to be a linear functional on π simply as .f Cπ∗ (G). For convenience, we shall write .fu,v u,v whenever there is only one representation in concern. In the case .u = v, the notation .fv,v will be further reduced to .fv , and we shall call .fv a function of positive type associated with .π . Moreover, we say .fv is normalized if .||v|| = 1. A normalized function of positive type extends to be a vector state .φv on .Cπ∗ (G) defined by .φv (a) = . Apparently, if v is a unit invariant vector for .π , then .φv : Cπ∗ (G) → C is a onedimensional representation of .Cπ∗ (G). In other words, the state .φv is a multiplicative linear functional on .Cπ∗ (G). A good reference for the discussion here is [62]. A weaker notion of containment is defined by functions of positive type.

.

Definition 9.9 Let .(π, H) and .(ρ, K) be representations of a locally compact group G. We say that .π is weakly contained in .ρ (denoted by .π ≺ ρ) if for every .u ∈ H, every compact subset .Q ⊂ G, and every .e > 0, there exist .v1 , . . . , vn ∈ K such that | | n E | π | ρ |. f (x) − fvi (x)|| < e, x ∈ Q. | u

(9.8)

i=1

In short, this rather technical definition says that .π ≺ ρ if every function of positive type associated with .π can be approximated on any compact subset by a finite sum of functions of positive type associated with .ρ. If .π ≺ ρ and .ρ ≺ π , then .π and .ρ are said to be weakly equivalent and we write .π ∼ ρ.

9.2.1 Some General Properties Weak containment has the following properties: (1) .π < ρ implies .π ≺ ρ. (2) If .π ≺ ρ and .π ' and .ρ ' are unitarily equivalent to .π and .ρ, respectively, then ' ' .π ≺ ρ . (3) For unitary representations .π, ρ, and .σ of G, if .π ≺ ρ and .ρ ≺ σ , then .π ≺ σ . (4) Let .mπ denote the direct sum of m copies of .π . Then .mπ ∼ π . It follows from (2) and (3) that weak containment is a partial order on the equivalence classes of unitary representations of G. We consider a normalized function of positive type .fuπ . If .π ≺ ρ, then for any compact subset .Q ⊂ G containing 1 and .e > 0, there exist vectors .v1 , . . . , vn ∈ K such that (9.8) holds. Evaluating at .x = 1 ∈ G, we obtain

9.2 Weak Containment

219

| | n E | | 2| |1 − ||v || i | < e. |

(9.9)

.

i=1

Setting .αj = ||vj ||2 /

En

2 i=1 ||vi || , j

n E

fvρi (x) =

.

i=1

= 1, . . . , n, we can rewrite

(E n

||vi ||2

i=1

)( E n i=1

αi f

ρ vi ||vi ||

) (x) .

Then (9.8) and (9.9) show that .fuπ can be approximated arbitrarily close by a convex sum of normalized functions of positive type associated with .ρ. We summarize the observation as follows. Remark 9.10 .π ≺ ρ if and only if for every compact subset .Q ⊂ G containing 1, every unit vector .u ∈ H, and .e > 0, there exist unit vectors .v1 , . . . , vn and nonnegative numbers .α1 , . . . , αn with .α1 + · · · + αn = 1 such that | | n E | | π ρ | < e, x ∈ Q. |f (x) − α f (x) i vi | | u

.

i=1

Weak containment is best described by means of the group .C ∗ -algebras. We first recall a well-known fact regarding .∗ -homomorphisms. Lemma 9.11 Let .A and .B be unital .C ∗ -algebras. Then an injective homomorphism .φ : A → B is isometric.

∗-

.

Proof Since the .C ∗ -algebra norm has the property .||a ∗ a|| = ||a||2 , it suffices to show that .||φ(a)|| = ||a|| for all .a ∈ A such that .a ≥ 0. First, since .0 ≤ ||a||I − a and .∗ -homomorphism maps positive elements to positive elements, we have 0 ≤ φ(||a||I − a) = ||a||I − φ(a),

.

which shows .||φ(a)|| ≤ ||a||. In particular, .φ is continuous. On the other hand, setting .b = ||φ(a)||I − a, we have .φ(b) = ||φ(a)||I − φ(a) ≥ 0. Then by the uniqueness of positive square root and the continuity of .φ, we have φ(b) =

.

(√ ) / / φ(b)∗ φ(b) = φ(b∗ b) = φ b∗ b .

The injectivity of .φ then implies .b =

√ b∗ b ≥ 0, which shows .||a|| ≤ ||φ(a)||.

u n

Proposition 9.12 Let .(ρ, K) and .(π, H) be two unitary representations of a discrete group G. Then the following statements are equivalent: (a) .π ≺ ρ. (b) .||π(h)|| ≤ ||ρ(h)|| for each .h ∈ C[G].

220

9 Weak Containment and Amenability

(c) The map .φ : Cρ∗ (G) −→ Cπ∗ (G) induced by .φ(ρ(x)) = π(x), x ∈ G, extends to a surjective .∗ -homeomorphism. (d) .σ (π(h)) ⊂ σ (ρ(h)) for each .h ∈ L1 (G). Proof (a).⇒(b). For any nonzero .h ∈ C[G], the set .Q := supp(h∗ ∗ h) is a finite subset containing 1. For any unit vector .u ∈ H, the function .fuπ extends to be a vector state .φuπ on .Cπ∗ (G). Remark 9.10 ensures the existence of unit vectors .v1 , . . . , vn and .α1 , . . . , αn ≥ 0 with .α1 + · · · + αn = 1 such that | | n E | π ∗ | ρ ∗ |φ (h ∗ h) − | < e, α φ (h ∗ h) i vi | u |

.

i=1

which, by the properties in (9.3), is equivalent to | | n E | | 2| |. ||π(h)u||2 − αi ||ρ(h)vi || | < e. | i=1

It follows that ||π(h)u||2
0 there exists a unit vector .ξ ∈ H such that .

sup ||π(x)ξ − ξ || < e.

(9.10)

x∈Q

Proof For every .x ∈ G and every unit vector .ξ ∈ H, we have ||π(x)ξ −ξ ||2 =

.

= ||π(x)ξ ||2 +||ξ ||2 −− = 2−2Re ()=2Re (1−) ≤ 2|1−|. On the other hand, |1 − | = || ≤ ||π(x)ξ − ξ ||.

.

The proposition then follows from the definition of weak containment.

u n

For any fixed .x ∈ G, we may let .Q = {x} and set .e = 1/n, n = 1, 2, . . ., in the above lemma to obtain a sequence of unit vectors .ξn such that .||π(x)ξn −ξn || < 1/n. The next fact is then immediate. It also follows from Proposition 9.12.

222

9 Weak Containment and Amenability

Corollary 9.16 Let .(π, H) be a unitary representation of G, and assume .1G ≺ π . Then .1 ∈ σ (π(x)) for every .x ∈ G. Moreover, if G is a finite group and .dim H < ∞, we may let .Q = G. The sequence ξn above then converges to an invariant vector for .π . Thus, .1G ≺ π implies .1G < π . A locally compact group G is said U to ben compactly generated if there exists a compact set .Q ⊂ G such that .G = ∞ n=1 Q . Recall that Baire’s Category Theorem asserts that for a locally compact Hausdorff space X, if .{Fj } is a sequence of closed U F also has empty interior. For a compactly subsets with empty interior, then . ∞ j =1 j generated group, the compact set Q in Lemma 9.15 can be a fixed one.

.

Theorem 9.17 Let G be a locally compact group that is generated by a compact set Q, and assume .(π, H) is a unitary representation of G. Then .1G ≺ π if and only if for every .e > 0, there exists a unit vector .ξ ∈ H such that .supx∈Q ||π(x)ξ − ξ || < e. Proof Without loss of generality, assume .1 ∈ Q. If .1G ≺ π , then Lemma 9.15 yields the “only if” part. To prove the other direction, for every .n ∈ N the assumption ensures the existence of a unit vector .ξ ∈ H such that .supx∈Q ||π(x)ξ − ξ || < e/n. Hence, for .x1 , . . . , xn ∈ Q, we estimate that ||π(x1 x2 . . . xn )ξ − ξ ||

.

=||π(x1 x2 . . . xn )ξ −π(x1 x2 · · · xn−1 )ξ + . . . +π(x1 x2 )ξ −π(x1 )ξ +π(x1 )ξ −ξ || ≤||π(x1 · · · xn−1 )(π(xn )ξ − ξ )|| + · · · + ||π(x1 )(π(x2 )ξ − ξ )|| + ||π(x1 )ξ − ξ || 0,Uthere exists a .ξ such that (9.10) holds for .Qn . n Since G is a Baire space and .G = ∞ n=1 Q , Baire’s Category Theorem ensures the n 0 existence of .n0 ∈ N such that .Q contains an interior point, say .g0 . This means that there exists an open neighborhood U of the unit .1 ∈ G such that .g0 U ⊆ Qn0 . ' For any Uncompact subset .Q ⊂ G, U∞there nexist elements .y1 , . . . , yn ∈ G such that ' .Q ⊆ j =1 yj U . The fact .G = n=1 Q implies the existence of .N ∈ N such that yj U ⊆

N U

.

Qk , j = 1, . . . , n,

k=1

U k n from which it follows that .Q' ⊆ N k=1 Q . Since .Q , n = 1, 2, . . ., is an increasing sequence of subsets due to the assumption .1 ∈ Q, we have .Q' ⊆ QN for some large integer N, and therefore (9.10) holds for .Q' . This shows .1G ≺ π . u n Consider a countable group G generated by a finite set .S = {g1 , . . . , gn }. U (S ∪ S −1 )n , where .S −1 = {g1−1 , . . . , gn−1 }. Recall Apparently, we have .G = ∞ n=1 that, given a representation .(π, H) of G, the associated Markov operator is defined as

9.3 Amenability

223

Mπ =

.

1 (π(g1 ) + · · · + π(gn )), n

and the triangle inequality implies .||Mπ || ≤ 1. Theorem 9.18 Suppose group G is finitely generated and .π is a unitary representation of G. Then .1G ≺ π if and only if .1 ∈ σ (Mπ ). Proof Assume G is generated by .S = {g1 , . . . , gn }. If .1G ≺ π , then Proposition 9.12(d) indicates that .{1} = σ (M1G ) ⊂ σ (Mπ ). For the other direction, since ∗ .1 ∈ σ (Mπ ), either .Mπ − I or .Mπ − I is not bounded below. Without loss of generality, we assume the former occurs. Then there exists a sequence of unit vectors .ξk ∈ H such that .||Mπ ξk − ξk || → 0 (and hence .||Mπ ξk || → 1). It follows that n E .

||π(gi )ξk − ξk ||2 = n(2 − 2Re )

i=1

= n(||Mπ ξk − ξk ||2 + 1 − ||Mπ ξk ||2 ) → 0, which implies .||π(gi )ξk − ξk || = ||ξk − π(gi−1 )ξk || → 0 for each i. Applying Theorem 9.17 for the case .Q = S ∪ S −1 , we obtain .1G ≺ π . u n Exercise 9.19 1. Show that the function of positive type .fuπ on group G (see (9.7)) extends to be a bounded positive linear functional on .Cπ∗ (G). 2. Let .φ be a positive linear functional on a unital .C ∗ -algebra .B. Prove (a) .φ(a ∗ ) = φ(a) and .|φ(a ∗ b)|2 ≤ φ(a ∗ a)φ(b∗ b). (b) .||φ|| = φ(I ). (c) The set .{b ∈ B | φ(b∗ b) = 0} is a closed left ideal in .B. 3. Suppose .(π, H) and .(ρ, K) are representations of a compact group G. Prove that .π ≺ ρ if and only if .π < ρ. 4. Show that the two representations .ρ+ and .ρ− of .GL3 (Z/3Z) in Sect. 2.2.2.2 are not weakly equivalent. 5. In Example 9.14, verify that . → einθ0 for every n. 6*. Complete the proof of Proposition 9.12 by showing that (d) implies (a). (Hint: First show (d) implies (b), which gives .ker ρ ⊂ ker π . Then consult [62, Theorem 3.4.4]).

9.3 Amenability The Banach–Tarski paradox claims that there exists a decomposition of a solid ball in .R3 into several disjoint pieces such that they can be reassembled by means of

224

9 Weak Containment and Amenability

rotation and translation to form two solid balls whose sizes are identical to that of the original one. Lying at the core of this bizarre phenomenon is the Axiom of Choice which allows the construction of nonmeasurable sets. The main tool used to make such a decomposition is the free group .F2 . The notion of amenable group was introduced by von Neumann [175] to describe groups that do not give rise to the Banach–Tarski paradox.

9.3.1 Invariant Mean Consider a set X with a positive measure .μ. A bounded linear functional .φ on L∞ (X, μ) naturally gives rise to a finitely additive measure .νφ defined by

.

νφ (E) := φ(χE ),

(9.11)

where E is any measurable subset in X, and χE is its characteristic function. The norm .||φ|| is equal to the total variation of .νφ . In particular, if .φ is a state on .L∞ (X), then .νφ is a finitely additive probability measure (called a mean) on X. However, such .νφ is usually not countably additive because the infinite series χE + χE + · · · may not converge in .L∞ (X) for disjoint sets .E1 , E2 , . . . with positive measure. Indeed, the dual space .(L∞ (X))∗ can be identified via (9.11) with the space of finitely additive measures on X that are absolutely continuous with respect to .μ and have finite total variation (Dunford-Schwartz [71], p.296). Furthermore, it is wellknown that .(L1 (X))∗ = L∞ (X), and every .f ∈ L1 (X) gives rise to a functional ∞ ∗ .fˆ ∈ (L (X)) defined by 1

fˆ(h) =

f

h(x)f (x)dμ(x), h ∈ L∞ (X).

.

2

(9.12)

X

Thus, for any .μ-measurable set E, we have νfˆ (E) = fˆ(χE ) =

f f (x)dμ(x), E

and this defines a countably additive measure on X that is absolutely continuous with respect to .μ and has the total variation equal to .||f ||1 . On the other hand, the Radon–Nikodym theorem asserts that every such measure .ν on X is of the form 1 .ν ˆ , where .f = dν/dμ ∈ L (X). It is easy to verify that .||fˆ|| = ||f ||1 . Thus, f we may simply regard .L1 (X) as a closed subspace of .(L∞ (X))∗ with respect to the norm topology. Denote by .L1 (X)1,+ the set of functions .f ∈ L1 (X) such that 1 ∞ .f ≥ 0 and .||f || = 1. Then every .f ∈ L (X)1,+ gives a state .fˆ on .L (X). The ∞ set of all states on .L (X), denoted by .M, is a closed convex subset in the closed unit ball of .(L∞ (X))∗ , and hence it is compact with respect to the weak.∗ topology (w.∗ -topology for short) according to the Banach–Alaoglu theorem.

9.3 Amenability

225

In the case X is countable with counting measure, it is customary to write .L1 (X) as .l1 . The following facts are useful in the sequel. Details can be found in Rudin [196, pg.111, 1(a)] and Conway [41, pg.135, Prop. 5.2]. Proposition 9.20 The following facts hold: (a) .L1 (X) is w.∗ -dense in .(L1 (X))∗∗ = (L∞ (X))∗ . (b) If X is countable, a sequence in .l1 converging weakly to 0 converges in norm to 0. For a locally compact group G with Haar measure .μ, a state .φ ∈ (L∞ (G))∗ is said to be an invariant mean if .φ(x h) = φ(h), h ∈ L∞ (G), x ∈ G. In this case, .νφ (xE) = νφ (E) for every .x ∈ G and Borel subset .E ⊂ G. Hence .νφ is an invariant finitely additive probability measure. Definition 9.21 A locally compact group G is amenable if it has an invariant mean. ˆ Apparently, if G is compact, then .1 ∈ L1 (G)1,+ and .1/μ(G) is an invariant mean on G. Thus every compact group is amenable. For noncompact groups, even for abelian ones such as .R and .Z, proving the existence of an invariant mean is a nontrivial matter. As intended by von Neumann, the free group .F2 is a distinguished example of non-amenable groups. Example 9.22 Consider the free group .F2 with generators a and b. We let .Sa be the set of elements in .F2 whose reduced form (see Sect. 2.3) starts with a. Likewise, we have .Sb , Sa −1 , and .Sb−1 . Apparently, these four subsets are disjoint, and F2 = {1} ∪ Sa ∪ Sb ∪ Sa −1 ∪ Sb−1 .

.

Moreover, since all words in .Sa are in reduced form, we have .a −1 Sa = {1} ∪ Sa ∪ Sb ∪ Sb−1 . If there were an invariant finitely additive probability measure .μ on .F2 , then we would have μ(a −1 Sa ) = μ({1}) + μ(Sa ) + μ(Sb ) + μ(Sb−1 ) > μ(Sa ).

.

This contradiction shows that .F2 is non-amenable.

9.3.2 The Markov–Kakutani Theorem Consider a locally convex topological vector space X and a compact convex subset K ⊂ X. Brouwer’s fixed-point theorem asserts that every continuous self-map of K has a fixed point. A map .T : K → K is said to be affine if

.

T (tv + (1 − t)u) = tT (v) + (1 − t)T (u), u, v ∈ K, 0 ≤ t ≤ 1.

.

226

9 Weak Containment and Amenability

The Markov–Kakutani theorem states that every commuting family .{Tj | j ∈ J } of continuous affine self-maps of K has a common fixed point, i.e., there exists .v0 ∈ K such that .Ti (v0 ) = v0 for every .j ∈ J . Given a locally compact group G, its action on .L∞ (G) induces an affine action ∞ ∗ .g → Tg of G on the dual .(L (G)) defined by Tg (m)f = m(g f ), f ∈ L∞ (G), m ∈ (L∞ (G))∗ .

.

If .{mk } is a sequence in .(L∞ (G))∗ that w.∗ -converges to m, then Tg (mk )f = mk (g f ) → m(g f ) = Tg (m)f, f ∈ L∞ (G),

.

which shows that .Tg is continuous with respect to the w.∗ -topology on .(L∞ (G))∗ . Theorem 9.23 Every locally compact abelian group G is amenable. Proof Recall that .M stands for the set of states on .L∞ (G). It is convex and compact with respect to the w.∗ -topology on .(L∞ (G))∗ . Since G is abelian, the set .{Tg | g ∈ G} is a commuting family of continuous affine self-maps on .M. The Markov– Kakutani theorem then implies the existence of a common fixed point .m0 ∈ M, giving an invariant mean on G. u n The invariant mean .m0 above can be constructed explicitly for some abelian groups. Example 9.24 Consider group .Z. For each .n ∈ N, we let .En = {−n, . . . , 0, . . . , n} and define a mean .νn on .Z by νn (A) =

.

|A ∩ En | , A ⊂ Z. 2n + 1

Observe that |νn (1 + A) − νn (A)| ≤

.

1 . 2n + 1

(9.13)

Let .mn be the state on .l∞ (Z) induced by .mn (χA ) = νn (A). Then the sequence .{mn } contains a subsequence w.∗ -converging to some state .m ∈ M. It follows from (9.13) that m is invariant.

9.3.3 A Spectral Description of Amenability The following theorem describes amenability by weak containment and spectral theory. It is a combination of results in [124, 143, 144, 190]. Theorem 9.25 For a finitely generated group G, the following are equivalent:

9.3 Amenability

227

(a) G is amenable. (b) .1G ≺ λG . (c) .1 ∈ σ (MλG ), where .MλG is the Markov operator. Proof We assume G is generated by a finite set .S = {x1 , . . . , xn }. The equivalence of (b) and (c) follows from Theorem 9.18. We now prove (b).⇒(a). Since .1G ≺ λG , Lemma 9.17 implies the existence of unit vectors .fn ∈ L2 (G) such that ||y fn − fn ||2
||z||2 }, j = 0, 1, . . . , n.

.

Then the vector .(n, −1, . . . , −1) ∈ H0 ∩ o0 , showing that .Fn is not amenable by Proposition 9.38 (a). However, if we let .ξ0 , . . . , ξn be the distinct .(n + 1)th roots of unity, then the vector .(ξ0 , . . . , ξn ) ∈ H0 ∩ P (Aλ ). Hence .H0 intersects nontrivially with both .P (Aλ ) and its complement. Literature Note Exercise 3 in Sect. 2.1 indicates that if a group G contains an element of finite order .n > 1, then .C[G] contains a nontrivial idempotent. Recall that a countable group G is said to be torsion free if it contains no element of finite order greater than 1. Evidently, every finite group has torsion, and so do the dihedral group .D∞ and the Grigorchuk group .G. The free abelian group .Zn , n ≥ 1, and the free groups .Fn , n ≥ 2, are torsion free. Kaplansky’s idempotent conjecture asserts that if G is torsion free, then .C[G] contains no nontrivial idempotent. The

9.4 Haagerup Groups and Kazhdan’s Property (T)

233

conjecture was later posed for the reduced group .C ∗ -algebra .Cr∗ (G) by Kaplansky and Kadison. It is shown in [120] that the conjecture holds for torsion-free Haagerup groups. But it is not known whether the conjecture holds for the Kazhdan group .SL3 (Z). There are several other related conjectures concerning the group algebra, for example, the Baum–Connes conjecture, the Borel conjecture, the Farrel–Jones conjecture, and the Novikov conjecture. For more details, we refer the reader to [227, 236]. Project 9.41 1. Characterize finitely generated groups for which the converse of Proposition 9.36 holds. 2. In view of Example 9.40 and the fact that free groups are not Kazhdan, is it true that for a noncompact finitely generated Kazhdan group G, we have .P (Aπ ) ∩ H0 = {0} for every representation .π that does not weakly contain .1G ?

Chapter 10

Self-similarity and Julia Sets

The symmetry of a measure space .(X, μ) is described by measure-preserving group actions it admits. Such group actions naturally give rise to a unitary representation, called the Koopman representation, of the group on .L2 (X, μ). On the other hand, the symmetry of .(X, μ) also enables the construction of groups with certain desired properties. Of particular interest is the case when .X = [0, 1] with Lebesgue measure .μ, through which the first example of a group of intermediate growth, the Grigorchuk group .G, was constructed [103]. Subsequently, it was discovered that .G displays a self-similarity property. Such property is not unique to .G, and it is shared by many other groups, such as the infinite dihedral group .D∞ , the lamplighter group .L, and the basilica group considered in this chapter. Remarkably, the self-similarity gives rise to a rational map on the projective space whose Julia set is linked with the projective spectrum of the group. This final chapter is devoted to a description of this connection.

10.1 Some Basics of Complex Dynamics Given a domain .o ⊂ Cn , a sequence of holomorphic functions .{fk } on .o is said to converge normally to f if the convergence is uniform on every compact subset of .o. The Cauchy integral formula then implies that f is also holomorphic. A sequence of vector-valued holomorphic maps .{Fk : o → Cm } is said to converge normally if each component converges normally. Definition 10.1 A family of holomorphic maps .{Fλ : o → Cm | λ ∈ A} is called a normal family if every sequence in the family contains a normally convergent subsequence. For a holomorphic map .H = (H1 , . . . , Hn ) : o → o, we shall use the notation H k to denote the kth iteration of H . For instance, the notation .H 2 (z) stands for

.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 R. Yang, A Spectral Theory Of Noncommuting Operators, https://doi.org/10.1007/978-3-031-51605-4_10

235

236

10 Self-similarity and Julia Sets

H (H (z)). Complex dynamics studies various issues concerning the convergence of n the iteration sequence .{H k }∞ 1=1 . The interest is primarily on the case .o = C with H being a polynomial or rational map. If H is such a map with degree d, then it is not hard to see that .H k is of degree .d k . It is often conducive to consider the map ˆ n = Cn ∪ {∞}. A point .z ∈ Cn is called a fixed on the extended complex space .C point for a map H if .H (z) = z. Fixed points of a map H are classified according to ) ( i the moduli of the eigenvalues .μ1 , . . . , μn of its Jacobian matrix .H ' (z) := ∂H ∂zj . A fixed point z is said to be

.

• • • • •

Attracting if .|μj | < 1 for all j . Super-attracting if .|μj | = 0 for all j . Repelling if .|μj | > 1 for all j . Parabolic if .|μj | = 1 for some j . A saddle point if there exist i and j with .|μi | < 1 and .|μj | > 1.

Moreover, H is said to be degenerate if there exists .z ∈ o such that .H (z) = 0. Otherwise, we say H is nondegenerate. ˆn → C ˆ n , its Fatou set Definition 10.2 Given a non-constant rational map .H : C ˆ n on which the sequence .{H k }∞ is a normal F(H ) is the maximal open subset of .C k=1 ˆ n \ F(H ). family. The Julia set .J(H ) is the complement .C

.

Evidently, the Julia set is closed, and both .J(H ) and .F(H ) are invariant under the map H . If a is an attracting fixed point for H , then there exists a scalar .0 < s ' < 1 such that .||H ' (a)v|| ≤ s ' ||v|| for every .v ∈ Cn , and it follows from Taylor expansion that there exists a small open ball U centered at a and a scalar .s ' < s < 1 such that ||H (z) − H (a)|| = ||H (z) − a|| ≤ s||z − a||, z ∈ U.

.

Thus .H (U ) ⊂ U , and .{H n } converges uniformly on U to the constant map .z → a. Definition 10.3 Suppose a is an attracting fixed point for H , then the basin of attraction around a is the maximal open domain .Ba ⊂ Cn containing a such that for every .w ∈ Ba one has .H k (w) → a. It is clear that .Ba ⊆ F(H ). However, a fixed point may have empty basin of attraction. We collect some relevant facts for the classical case .n = 1 from Gamelin [95]. Theorem 10.4 (Montel) A family of meromorphic functions on a domain .o ⊂ C that omits three values is normal. ˆ →C ˆ be a polynomial map of degree .d ≥ 2. Theorem 10.5 Let .f : C ˆ (a) .B∞ is an open connected subset of .C. (b) .J(f ) coincides with the boundary of .B∞ , and it is nonempty and compact. (c) Each bounded component of the Fatou set .F(f ) is simply connected.

10.1 Some Basics of Complex Dynamics

237

Fig. 10.1 The Julia set of = z2 + (0.285 + 0.01i), Wikipedia

.f (z)

Fig. 10.2 The Julia set of .f (z) =

1+2z3 , 3z2

Wikipedia

Even for very simple maps, the Julia set can be rather sophisticated and strikingly beautiful (Figs. 10.1 and 10.2). Observe that .B∞ = F(f ) in Fig. 10.1, and it has only one connected component. The Fatou set in Fig. 10.2 has infinitely many components. However, the Julia set can also be very simple. The following example is instrumental for later discussions. Its proof is a good exercise [12]. ˆ its Julia set Example 10.6 For the Tchebyshev polynomial .T (z) = 2z2 − 1 on .C, k ˆ \ [−1, 1]. is .J(T ) = [−1, 1]. The sequence .{T } converges normally on .B∞ = C

238

10 Self-similarity and Julia Sets

10.1.1 Complex Dynamics in Cn A helpful method of studying the dynamical properties of a map is to reduce it to a simpler form via conjugation. Let .o1 and .o2 be domains in .Cn . Two holomorphic self-maps .F : o1 → o1 and .H : o2 → o2 are said to be semi-conjugate if there exists a holomorphic map .o : o1 → o2 such that .H ◦ o = o ◦ F . In the case such .o is biholomorphic, we say F and H are conjugate. Since the semi-conjugacy implies .H k ◦ o = o ◦ F k , k = 1, 2, . . ., the two maps have similar dynamical properties. We let .Aut(Cn ) denote the group of polynomial automorphisms of .Cn . For the case .n = 2, the group contains two simple types of maps: (1) Affine maps .(z1 , z2 ) :→ (az1 + bz2 + c, a ' z1 + b' z2 + c' ), where .ab' − a ' b /= 0. (2) Elementary maps .(z1 , z2 ) :→ (az1 + b, p(z1 ) + b' z2 ), where .ab' /= 0, and p is a one-variable polynomial. The following classical theorem is due to Jung [139]. Theorem 10.7 The group .Aut(C2 ) is generated by the affine and elementary maps. However, despite much effort since then, little is known about the structure of Aut(Cn ) for .n ≥ 3. The problem is closely related to the well-known Jacobian conjecture which claims that every polynomial map .F : Cn → Cn with a nonzero constant Jacobian determinant is in Aut.(Cn ), i.e., it has a polynomial inverse, for instance, see [151, 226]. For .n = 2, a particular composition of affine and elementary maps turns out to be very important.

.

Definition 10.8 The generalized Hénon map is defined by (z1 , z2 ) :→ (z2 , p(z2 ) − αz1 ), α /= 0,

.

where p is a polynomial. The following theorem is due to Friedland–Milnor [83]. Theorem 10.9 Every automorphism in .Aut(C2 ) is conjugate to either an affine map or an elementary map or a Hénon map. Thus, the dynamical properties of a polynomial self-map of .C2 can be traced back to that of the three types of maps. For .p ≥ 1, the p-norm on .Cn is defined by .||z||p = (|z1 |p + · · · + |zn |p )1/p . We shall denote .|| · ||2 simply by .|| · ||. A polynomial map .H : Cn → Cn is said to be homogeneous if all the components of H are homogeneous polynomials of the same degree. For such a map, the following fact can be proved using the compactness of the closed unit sphere. Lemma 10.10 Assume .H : Cn → Cn is a homogeneous map of degree .d ≥ 0. (a) There exists a constant .M > 0 such that .||H (z)|| ≤ M||z||d , z ∈ Cn .

10.1 Some Basics of Complex Dynamics

239

(b) If H is nondegenerate, then there exists .m > 0 such that .||H (z)|| ≥ m||z||d , z ∈ Cn . In particular, part (a) asserts that .H /||z||d is bounded on .Cn \ {0}. When .d ≥ 2, the lemma implies the existence of positive numbers r and R such that (a ' ) ||H (z)|| < ||z||/2 if ||z|| < r;

.

(b' ) ||H (z)|| > 2||z|| if ||z|| > R. (10.1)

Observe that (a.' ) implies .||H k (z)|| < 2−k ||z||, k = 1, 2, . . . , and hence the origin 0 is a fixed point with a nonempty basin of attraction. Proposition 10.11 Assume .H : Cn → Cn is a homogeneous map of degree .d ≥ 2. (a) The basin of attraction .B0 is nonempty and Stein. (b) .B0 is a circular domain, i.e., .z ∈ B0 implies .cz ∈ B0 for every .c ∈ C with .|c| ≤ 1. (c) .B0 is bounded if and only if H is nondegenerate. Proof Statements (b) and (c) are easy to see. To prove (a), we let r be as in (10.1) and set o=

∞ U

.

H −k (rBn ).

k=1

It is clear that .B0 ⊂ o. On the other hand, if .z ∈ o, then there exists integer .m ≥ 1 such that .H m (z) ∈ rBn . It follows that .||H m+k (z)|| < 2−k ||H m (z)||, k = 1, 2, . . ., which implies .z ∈ B0 . This shows that .B0 = o. The ball .rBn is obviously Stein. An application of Proposition 4.34 shows that .o is Stein. u n

10.1.2 The Green Function Proposition 10.11 prepares the ground for introducing the Green function. Definition 10.12 Assume .H : Cn → Cn is a homogeneous map of degree .d ≥ 2 and ρ(z) = sup{a > 0 | az ∈ B0 }, z ∈ Cn .

.

The Green function of H is defined as .h(z) = − log ρ(z). The Green function plays an important role in complex dynamics. The treatment here is mostly given in Ueda [240]. Observe that h is a real function and .h(z) = −∞ for every .z ∈ H −1 (0). The following lemma summarizes some basic properties, and its proof is left as an exercise.

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10 Self-similarity and Julia Sets

Lemma 10.13 Let H be as in Definition 10.12 with Green function h. Then (a) .h(z) − log ||z|| is a homogeneous function of degree 0. (b) .h(H (z)) = dh(z). (c) .B0 = {z ∈ Cn | h(z) < 0}. In particular, if z is a nonzero fixed point of H , then (b) implies .h(z) = dh(z), and hence .h(z) is either 0 or .−∞. Theorem 10.14 Let H be as in Definition 10.12 with Green function h. Then the n sequence .{d −k log ||H k (z)||}∞ k=1 converges pointwise to .h(z) on .C \ {0}. And the convergence is uniform if H is nondegenerate. Proof When H is nondegenerate, the function .γ (z) = log ||H (z)|| − d log ||z|| is well-defined and bounded on .Cn \ {0} by Lemma 10.10 (a). Writing .

1 1 log ||H (z)|| = log ||z|| + γ (z), d d

substituting z by .H k−1 (z), and multiplying both sides by .d 1−k , we obtain the recursion relation d −k log ||H k (z)|| = d 1−k log ||H k−1 (z)|| + d −k γ (H k−1 (z)).

.

It follows that d −k log ||H k (z)|| = log ||z|| + d −1 γ (z) + · · · + d −k γ (H k−1 (z)).

.

(10.2)

Since .d ≥ 2 and .γ is bounded, the right-hand side of (10.2) converges uniformly as k → ∞. Set

.

˜ h(z) := lim d −k log ||H k (z)||.

.

k→∞

It remains to show that .h˜ coincides with the Green function h. To this end, we note that Lemma 10.13 (b) implies .h(H k (z)) = d k h(z). Therefore, we have ( ) h − h˜ = lim d −k h(H k (z)) − log ||H k (z)|| .

.

k→∞

(10.3)

Since .h(z) − log ||z|| is homogeneous of degree 0 by Lemma 10.13 (a), it is bounded on .Cn \ {0}. Therefore, the limit in (10.3) is 0. In the case H is degenerate, since .||H (z)||/||z||d is bounded on .Cn \ {0}, there exists a scalar .M > 0 such that γ (z) := log ||H (z)|| − d log ||z|| − M < 0

.

on .Cn \ {0}. Thus, using the recursion relation repeatedly, we have

10.1 Some Basics of Complex Dynamics

241

d −k log ||H k (z)|| = d 1−k log ||H k−1 (z)|| + d −k M + d −k γ (H k−1 (z))

.

( =

log ||z|| + d

−1

γ (z) + · · · + d

−k

) γ (H

k−1

(z)) + M

k E

d −j .

j =1

Since the partial sum inside the parenthesis is decreasing, it is convergent (possibly to .−∞). The remaining part of the proof is similar to that in the nondegenerate case. u n It is worth pointing out that if .h0 (z)( is any function on .Cn \ {0}) such that .h0 (z) − log ||z|| is bounded, then the fact .d −k h0 (H k (z)) − log ||H k (z)|| → 0 implies that .

lim d −k h0 (H k (z)) = lim d −k log ||H k (z)|| = h(z), z ∈ Cn \ {0}.

k→∞

k→∞

(10.4)

Corollary 10.15 Let H be as in Definition 10.12. Then its Green function h is plurisubharmonic on .Cn \ {0} and pluriharmonic on .F(H ). Proof If H is nondegenerate, Lemma 10.14 asserts that the sequence .{d −k log || k n k .H (z)||} converges uniformly to the Green function h on .C \{0}. Since .log ||H (z)|| n is plurisubharmonic for each k, h is plurisubharmonic on .C \{0}. If H is degenerate, then the proof of Lemma 10.14 shows that .{d −k log ||H k (z)||} is a decreasing sequence of plurisubharmonic functions that converges pointwise to h. Hence h is plurisubharmonic on .Cn \ {0}. For any .p ∈ F(H ), there exists a neighborhood V of p with its compact closure .V ⊂ F(H ) such that the family .{H k } is normal on V . Then there exists a subsequence .{H kj } converging normally to some holomorphic map .H∗ on V . Choose a hyperplane L in .Cn that does not contain .H∗ (p). Up to a change of coordinates, we may assume without loss of generality that .L = {z ∈ Cn | z1 = 0}. Pick a small .e > 0 and consider an .e-neighborhood of L in .Cn defined by n | |z | < e||z||}. Due to the normal convergence of .{H kj }, by .Ne = {z ∈ C 1 possibly shrinking V we may assume .H kj (V ) ∩ Ne = ∅ for all sufficiently large j , and .{H kj } converges uniformly on V . Define { h0 (z) =

.

log ||z||

for z ∈ Ne ,

log(|z1 |/e)

for z ∈ Cn \ Ne .

(10.5)

Then .0 ≤ h0 (z) − log ||z|| ≤ log(1/e), z ∈ Cn , and it follows from (10.4) that the sequence .{d −kj h0 (H kj (z))} converges uniformly to the Green function h on V . Since .h0 (H kj (z)) is pluriharmonic on V for all sufficiently large j , h is u n pluriharmonic on V . Exercise 4 in Sect. 4.2 leads us to the following fact. Theorem 10.16 Assume .H : Cn → Cn is a homogeneous map of degree .d ≥ 2. Then every path-connected component of the Fatou set .F(H ) is Stein.

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10 Self-similarity and Julia Sets

10.1.3 Complex Dynamics in Pn and the Indeterminacy Sets Given a map .H = (H0 , . . . , Hn ) : Cn+1 → Cn+1 , where .H0 , . . . , Hn are homogeneous polynomials of the same degree .d ≥ 2, it induces a map .H ' : Pn → Pn through the commuting diagram Cn+1

H

φ .

Pn

Cn+1 φ

H

Pn ,

(10.6)

where .φ is the canonical projection. Conversely, if .H ' : Pn → Pn is a holomorphic map, then there exists a nondegenerate homogeneous polynomial map .H : Cn+1 → Cn+1 such that the above diagram commutes. In the case H is degenerate, the map ' n .H is not well-defined at the common zeros of .H0 , . . . , Hn because .P contains no origin. Thus the set .φ(H −1 (0)) consists of singular points for .H ' . If there are only a ' finite number of such singular ( −kpoints, ) the map .H is called a meromorphic map. For ' .k ≥ 1, we set .Ik (H ) = φ H (0) and call it the indeterminacy set of .H ' k . Clearly, ∞ I (H ' ). For convenience, in ' ' ' .Ik (H ) ⊂ Ik+1 (H ) for each k. We set .I∞ (H ) = ∪ k=1 n the sequel we shall denote .H ' by H , respectively, .Ik (H ' ) by .Ik , and .I∞ (H ' ) by .I∞ whenever there is no risk of confusion. Definition 10.17 Given a meromorphic map .H : Pn → Pn , its extended indeterminacy set is defined as .EH = I∞ . We shall also write .EH as E for convenience. It is often rather technical to determine E, and this fact can add many (and often unnecessary) complications to the study of dynamics in the projective space. It is natural to consider E as part of the Julia set .J(H ). Thus, a point .p ∈ Pn \ E is a Fatou point if it has a neighborhood n k .U ⊂ P \ E on which the family .{H | k = 1, 2, . . .} is normal. It is worth pointing out that there exists meromorphic map .H : P2 → P2 whose Julia set is equal to .P2 [86], for example, the following map: H ([z0 : z1 : z2 ]) = [(z0 − z2 )2 : (z0 − 2z2 )2 : z02 ].

.

Complex dynamics is an active and extensive field. For more information, we refer the reader to [11, 141] for the study in one variable, and to [86–88, 167] for treatises in several variables. Exercise 10.18 1. Prove the claims in Example 10.6. (Hint: Write .x = cos t and discuss the cases t is real or complex.) 2. Consider the Hénon map .H (z1 , z2 ) = (z2 , z22 + 2(1 − z1 )) in .C2 .

10.2 Self-similar Group Representations

243

(a) Find and classify its fixed points. (b) Lift the map H to a quadratic homogeneous map in .P2 . (Hint: Use the chart .z0 /= 0.) 3. Suppose a is an attracting fixed point of a holomorphic map H on .Cn . Prove that .Ba is simply connected. 4. Verify (10.4). 5. Prove Lemma 10.13. 6. Consider the holomorphic map .H (z1 , z2 ) = (z1 , z1 z2 ) on .C2 . Show that its range is not open, and hence the open mapping theorem fails on .C2 .

10.2 Self-similar Group Representations If a locally compact group G has a measure-preserving ( action on ) a measure space (X, μ), then the Koopman representation .π : G → U L2 (X, μ) is defined by

.

π(g)f (x) = f (g −1 x), x ∈ X, g ∈ G.

.

Clearly, if .X = G and .μ is the Haar measure, then the Koopman representation is the left regular representation of G. In general, the measure space X can be a planar figure, a manifold, or a rooted tree. Self-similar group representation first appeared in [103] which constructed the Grigorchuk group .G (Sect. 5.4) through a measure-preserving action on dyadic subintervals of .[0, 1]. Self-similar representation can also be defined abstractly, without referring to any measure-preserving group actions. Definition 10.19 A unitary representation .(π, H) of a group G is said to be selfsimilar (or d-similar) if there exist an integer .d ≥ 2 and a unitary operator .W : H → Hd such that every entry of the .d × d block matrices .πˆ (g) = W π(g)W ∗ , g ∈ G, is either equal to 0 or of the form .π(x) for some .x ∈ G. In this case, since .πˆ (g) is a unitary operator on .Hd and each of its nonzero entries is a unitary operator on .H, every row or column of .πˆ (g) has precisely one nonzero entry. Not every group is self-similar, and it is not clear whether there exists a characterization of groups with a self-similar representation.

10.2.1 Reshuffling the Dyatic Intervals An important example of Koopman representation occurs when .X = [0, 1) and .μ is the Lebesgue measure. Consider the set of dyadic numbers .D := {j/2k | k ≥ 1, 0 ≤ j < 2k }. Dyatic subintervals of .[0, 1) are intervals .[a, b) with .a, b ∈ D and .a < b. It is not hard to see that .D is dense in .[0, 1], despite the fact that it

244

10 Self-similarity and Julia Sets

is a small subset of rationals. We can express every number in .[0, 1) by a binary sequence, omitting the dot, i.e., a sequence (denoted by w in the sequel) of 0s and 1s. To make this expression unique, we assume that a dyatic number ends with a sequence of 0s instead of 1s. For example, .0.5 is expressed as .100 · · · rather than .0111 · · · . Any measure-preserving action g on .[0, 1) lifts canonically to a unitary map .π(g) : L2 [0, 1) → L2 [0, 1) such that .π(g)f (x) = f (g −1 x). Many self-similar group representations are constructed through an action of the group on .[0, 1) that reshuffles the dyadic subintervals. Since the numbers in .[0, 1) are represented by strings w of 0s and 1s, we can describe the group action on such strings. For simplicity, we let .H stand for .L2 [0, 1). For any .f ∈ H, we denote its restrictions to .[0, 1/2) and .[1/2, 1) by .f1 and .f2 , respectively. Then it is not hard to check that the map .W : H → H2 defined by (x ) 1 1 ⊕ √ f2 Wf (x) = √ f1 2 2 2

(

.

1+x 2

) , x ∈ [0, 1),

(10.7)

is a unitary operator. For the self-similar representations considered later in this chapter, this W shall play the role required in Definition 10.19. If we identify the dyadic intervals with the vertices in an infinite rooted binary tree, for example, .[0, 12 ) with .T0 and .[ 12 , 43 ) with .T10 , then every number in .[0, 1) corresponds to a unique infinite path down the tree. Hence the group action on .[0, 1) has an equivalent description on the tree. Since reshuffling of the dyadic subintervals preserves the length, the corresponding action on the tree preserves each layer. Figure 10.3 shows the first four layers of the tree. For treatises on selfsimilar representations, we refer the reader to [105, 174]. Fig. 10.3 Rooted binary tree

T

T0

T1

T000

T10

T01

T00

T001

T010

T011

T100

T11

T101

T110

T111

10.2 Self-similar Group Representations

245

10.2.2 Three Illuminating Examples This subsection describes the self-similar representations for the infinite dihedral group .D∞ , the lamplighter group .L, and the Grigorchuk group .G. 1. The Koopman representation .π of .D∞ = on .H = 2 L [0, 1) is realized by the following action of .D∞ on .[0, 1): a(0w) = 1w, a(1w) = 0w; t (0w) = 0a(w), t (1w) = 1t (w).

.

Thus, it can be verified that the Koopman representation of .D∞ on .H is 2-similar: [ ] 0I ∼ ˆ = .π(a) = π(a) , I 0

[ ] π(a) 0 ∼ π(t) = πˆ (t) = . 0 π(t)

(10.8)

Moreover, the following is known [108]. Theorem 10.20 The Koopman representation .π of .D∞ is weakly equivalent to the regular representation .λD∞ . 2. The lamplighter group .L, generated by a shift a and an involution c, is presented as L = .

.

Alternatively, it can be viewed as the semidirect product .Z x (⊕j ∈Z Z2 ) defined by (k, X) · (m, Y ) = (k + m, k(Y ) + X), k, m ∈ Z, X, Y ∈ ⊕j ∈Z Z2 ,

.

where .k(Y ) is the action of .Z on .⊕j ∈Z Z2 by the bilateral shift, i.e., if .k > 0, then k(Y ) shifts Y by k units to the right, and if .k < 0, then shift Y to the left by .|k| units. Another description of .L is given by considering the ring .R = Z2 [t, t −1 ] and the automorphisms

.

(a · p)(t) = tp(t), c · p = p + 1, p ∈ Z2 [t, t −1 ].

.

Using the generators a and .b = ac, the action of .L on .[0, 1) is as follows: a(0w) = 1b(w), a(1w) = 0a(w); b(0w) = 0a(w), b(1w) = 1b(w),

.

c(0w) = 1w,

c(1w) = 0w.

Thus, the Koopman representation .π of .L on .H is 2-similar: ( πˆ (a) =

.

) ( ) ( ) 0 π(a) π(a) 0 0I , πˆ (b) = , π(c) ˆ = . π(b) 0 0 π(b) I 0

(10.9)

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10 Self-similarity and Julia Sets

3. Recall from Sect. 5.4 that the Grigorchuk group .G is generated by four involutions .a, b, c, and .d with the following infinite set of algebraic equations: a 2 = b2 = c2 = d 2 = bcd = 1,

.

σ k ((ad)4 ) = σ k ((adacac)4 ) = 1, k = 0, 1, 2, · · · , where .σ is the substitution:. a → aca, b → d, c → b, d → c. Although the presentation is rather complicated, its Koopman representation .π on .H is relatively simple, and it is realized by the following action on .[0, 1): a(0w) = 1w,

a(1w) = 0w;

b(0w) = 0a(w),

b(1w) = 1c(w);

c(0w) = 0a(w),

c(1w) = 1d(w);

d(0w) = 0w,

d(1w) = 1b(w).

.

Hence, it is also 2-similar: (

) 0I .π ˆ (a) = , I 0 ( ) π(a) 0 πˆ (c) = , 0 π(d)

( ) π(a) 0 πˆ (b) = , 0 π(c) ( ) I 0 πˆ (d) = . 0 π(b)

(10.10)

10.3 Renormalization Maps If .(π, H) is a d-similar representation of a finitely generated group .G = , then it is of great interest to investigate how the self-similarity is reflected by the projective spectrum .p(Aπ ), where .Aπ (z) = z0 I + z1 π(g1 ) + · · · + zn π(gn ). Due to the unitary equivalence of .π and .πˆ , the pencil .Aπ (z) is unitarily equivalent to the .d ×d block matrix pencil .Aπˆ (z). Thus, in the case .d = 2, the Schur complement leads us to a canonical map in the projective space .Pn that preserves .p(Aπ ). Given the explicit relations between .π and .π ˆ in the previous section, it becomes quite appealing to derive such maps for the groups .D∞ , .L, and .G. For linear operators .A, B, C, and D on a Hilbert space .H, when A is invertible, we have ( )( ) ) ( I 0 AB A B . . = −CA−1 I CD 0 D − CA−1 B Operator .D − CA−1 B is called the Schur complement of the block matrix. The following consequence is immediate. Lemma 10.21 Suppose A is invertible. Then the block matrix

10.3 Renormalization Maps

247

( .

AB CD

) (10.11)

is invertible if and only if .D − CA−1 B is invertible. Moreover, if .AC = CA, then the block matrix is invertible if and only if .AD − CB is invertible.

10.3.1 The Infinite Dihedral Group Let .π be the 2-similar representation of .D∞ given in (10.8). Then ) ( z1 z0 + z2 π(a) ∼ . .Aπ (z) = z0 + z1 π(a) + z2 π(t) = Aπ ˆ (z) = z1 z0 + z2 π(t) (10.12) Therefore, the pencil .Aπ (z) is invertible if and only if the .2 × 2 block matrix on the right-hand side of (10.12) is invertible. For convenience, we shall frequently write 2 2 .π(g) simply as g in the subsequent computations. In the case .z /= z , the pencil 0 2 2 2 −1 . In this case, the block .z0 + z2 a is invertible and its inverse is .(z0 − z2 a)(z − z ) 0 2 matrix .Aπˆ (z) is invertible if and only if the Schur complement z0 + z2 t − z12 (z0 − z2 a)(z02 − z22 )−1 =

.

z0 (z02 − z12 − z22 ) z02 − z22

+

z12 z2 z02 − z22

a + z2 t

is invertible. This leads us to define the following so-called renormalization map: ( F (z0 , z1 , z2 ) :=

.

z0 (z02

− z12

− z22 ), z12 z2 , z2 (z02

− z22 )

) .

(10.13)

Therefore, in the case .z02 /= z22 , a point .z ∈ P (Aπ ) if and only if .F (z) ∈ P (Aπ ). In the case .z02 = z22 , we have .Aπ (F (z)) = −z12 (z0 I − z2 π(a)), which is non-invertible because the classical spectrum .σ (π(a)) = {±1}. Corollary 10.22 The projective spectrum .P (Aπ ) is invariant under the map F .

10.3.2 The Lamplighter Group For the lamplighter group .L and its Koopman representation .π , the linear pencil z0 + z1 π(a) + z2 π(b) + z3 π(c) does not lead to a simple renormalization map. The reader should try and find where the hurdle lies. Therefore, Zu–Yang–Lu [249] considered the linear pencil

.

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10 Self-similarity and Julia Sets

Aπ (z) = z0 + z1 π(c) + z2 (π(a) + π(b)) + z3 (π(a −1 ) + π(b−1 )).

.

Then by (10.9), we have ) ( z0 + z2 a + z3 a −1 z1 + z2 a + z3 b−1 . Aπ (z) ∼ = Aπˆ (z) = z1 + z2 b + z3 a −1 z0 + z2 b + z3 b−1

.

(10.14)

A simple matrix multiplication shows that .

( ( ) ) I −I I 0 Aπˆ (z) 0 I −I I ) ( z1 − z0 + z2 (a − b) 2(z0 − z1 ) . = z1 − z0 + z3 (a −1 − b−1 ) z0 + z2 b + z3 b−1

Hence, by Lemma 10.21, when .z0 /= z1 , the pencil .Aπ (z) is invertible if and only if 2(z0 − z1 )(z0 + z2 b + z3 b−1 ) − (z1 − z0 + z3 (a −1 − b−1 ))(z1 − z0 + z2 (a − b))

.

= (z02 − z12 − 2z2 z3 ) + 2z2 z3 c + z2 (z0 − z1 )(a + b) + z3 (z0 − z1 )(a −1 + b−1 ) is invertible. This gives rise to the following renormalization map: ( ) L(z0 , z1 , z2 , z3 ) := z02 − z12 − 2z2 z3 , 2z2 z3 , z2 (z0 − z1 ), z3 (z0 − z1 ) .

.

(10.15)

For any point .z ∈ P (Aπ ), if .z0 /= z1 , then the above observation shows that Aπ (L(z)) is not invertible. If .z0 = z1 , then .L(z) = 2z2 z3 (−1, 1, 0, 0). Since .σ (π(c)) = {±1}, the pencil .Aπ (L(z)) = 2z2 z3 (π(c) − I ) is also not invertible. .

Corollary 10.23 The projective spectrum .P (Aπ ) is invariant under the map L.

10.3.3 The Grigorchuk Group Interestingly, although the Grigorchuk group .G is much more complicated than .D∞ , its renormalization map can be derived in much the same way. Let .π be the 2-similar representation of .G given in (10.10). We shall consider the pencil Bπ (z) = z0 I + z1 π(a) + z2 π(b) + z3 π(c) + z4 π(d), z ∈ C5 .

.

Then the 2-similarity relation (10.10) implies

10.3 Renormalization Maps

249

) ( z1 z0 + (z2 + z3 )a + z4 ∼ . .Bπ (z) = Bπ ˆ (z) = z1 z 0 + z 2 c + z3 d + z4 b

(10.16)

Observe that .σ (π(a)) = {±1}. If .(z0 +z4 )2 /= (z2 +z3 )2 , then .(z0 +z4 )+(z2 +z3 )a is invertible, and its inverse is .

(z0 + z4 ) − (z2 + z3 )a . (z0 + z4 )2 − (z2 + z3 )2

Hence, in this case .Bπ (z) is invertible if and only if the Schur complement z0 + z2 c + z3 d + z4 b − z12 (z0 + z4 + (z2 + z3 )a)−1

.

(10.17)

is invertible. Therefore, rewriting (10.17) as z 0 + z 2 c + z3 d + z4 b −

.

( ) z12 z0 + z4 − (z2 + z3 )a 2 2 (z0 + z4 ) − (z2 + z3 )

and then regrouping the terms by the generators, we see that .Bπ (z) is invertible if and only if the rational pencil ( . z0 −

z12 (z0 + z4 ) (z0 + z4 )2 − (z2 + z3 )2

) +

z12 (z2 + z3 ) a + z4 b + z2 c + z3 d (z0 + z4 )2 − (z2 + z3 )2 (10.18)

is invertible. In order to obtain a polynomial map, we multiply the above rational pencil by the function .α(z) := (z0 + z4 )2 − (z2 + z3 )2 , and this produces the map ( ) R(z) := z0 α − z12 (z0 + z4 ), z12 (z2 + z3 ), z4 α, z2 α, z3 α , z ∈ C5 .

.

(10.19)

Thus, in the case .α(z) /= 0, pencil .Bπ (z) is (invertible if and only if .Bπ (R(z)) is ) invertible. If .α(z) = 0, then .Bπ (R(z)) = −z12 (z0 + z4 )I − (z2 + z3 )π(a) , which is not invertible because .σ (π(a)) = {±1}. Corollary 10.24 The projective spectrum .P (Bπ ) is invariant under the map R. Corollaries 10.22, 10.23, and 10.24 suggest that something general may be true for groups with a 2-similar representation. However, one hurdle to obtaining such a general result is the lack of a canonical form of linear pencil which permits a renormalization map for every 2-similar group representation. This is what we have encountered in the case with the lamplighter group. For more discussion on the maps F , L, and R, we refer the reader to [59, 106, 107, 110, 113]. Exercise 10.25 1. Consider the operator W defined in (10.7).

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10 Self-similarity and Julia Sets

(a) Check that W is unitary. (b) Verify the identities in (10.8). 2. The map F in (10.13) is derived through the assumption that upper-left block .z0 + z2 π(a) in (10.12) is invertible. Derive a similar map by assuming instead that the lower-right block .z0 + z2 π(t) is invertible. 3. Determine and classify the fixed points of the map F in (10.13). 4. The basilica group .B = acts on .[0, 1) by the following rules: .a(0w) = 0w, a(1w) = 1b(w); b(0w) = 1w, and .b(1w) = 0a(w). (a) Express .πˆ (a) and .π(b), ˆ where .π is the Koopman representation. (b) Derive a renormalization map F for the pencil .Aπ (z) = z0 I + z1 π(a) + z2 π(b). (c) Does it hold that .F (P (Aπ )) ⊂ P (Aπ )? (d) Find the set of fixed points of F , if there is any, and classify them. 5. Consider the map .F : C3 → C3 defined in (10.13) and the map .R : C5 → C5 defined in (10.19). Show that (a) .||F (z)||2 ≤ ||z||32 and (b) .||R(z)||1 ≤ ||z||31 . 6*. Consider the map R defined in (10.19) associated with the Grigorchuk group .G. (a) Find the set of fixed points of R. (b) Show that the origin .0 is the only attracting fixed point of the map R. (c) Show that the point .[−1 : −2 : 1 : 1 : 1] is in the projective spectrum .p(Bπ ). Project 10.26 1. Using the generating set .{a, b} or .{a, b, c} to derive a renormalization map for the lamplighter group .L. (Note that a is not an involution in this case.) 2. Investigate whether phenomenon such as Corollary 10.24 occurs for other groups with a 2-similar representation. 3. Study the projective spectra for the basilica group B with respect to the regular representation and the Koopman representation .π . 4. Compute the Green functions for the maps .F, L, and R defined in (10.13), (10.15), and (10.19), respectively.

10.4 The Julia Set of D∞ The renormalization maps .F, L, and R above reflect the self-similarity of the Koopman representations of the groups .D∞ , L, and .G, respectively. Since the maps are homogeneous, they induce the corresponding maps in the projective spaces. For convenience, we shall denote these maps also by .F, L, and R. The compactness of the projective spaces offers an advantage for the study of their dynamical properties. In this section, we shall take a closer look at the map F associated with .D∞ .

10.4 The Julia Set of D∞

251

10.4.1 Determining the Indeterminacy Set In the projective space, the map F defined in (10.13) is expressed as F ([z0 : z1 : z2 ]) = [z0 (z02 − z12 − z22 ) : z12 z2 : z2 (z02 − z22 )], z ∈ P2 .

.

(10.20)

It is a pleasure to check that the indeterminacy set I1 (F ) = {[±1 : 1 : 0], [0 : 1 : 0], [±1 : 0 : 1]}.

.

(10.21)

Thus, map F is meromorphic. Moreover, the set .I1 reflects the spectral property σ (π(a)) = σ (π(t)) = {±1}. With a bit more effort, the set .I2 can be determined as well. However, it becomes increasingly cumbersome to determine .Ik when k gets larger, not mentioning .I∞ or the extended indeterminacy set E. In light of Theorem 5.2, Proposition 9.36, and Theorem 10.20, the projective spectrum

.

U

p(Aπ ) =

{z ∈ P2 | z02 − z12 − z22 − 2z1 z2 x = 0},

.

(10.22)

−1≤x≤1

and Corollary 10.22 shows .F (p(Aπ )) ⊂ p(Aπ ). A very helpful tool for the study ˆ defined by of the iteration sequence .{F n } is the map .τ : P2 → C

τ (z) =

.

⎧ ⎪ ⎨0,

if z02 − z12 − z22 = 0;

z − z1 − z2 ⎪ ⎩ 0 , 2z1 z2 2

2

(10.23)

2

otherwise.

Thus, in view of (10.22), we have .τ (z) ∈ [−1, 1] if and only if .z ∈ p(Aπ ). The following connection between the map F and the Tchebyshev polynomial .T (x) = 2x 2 − 1 is pivotal for discussions in the sequel. Lemma 10.27 .T n (τ (z)) = τ (F n (z)), z ∈ P2 \ In (F ), n = 1, 2, . . . Its proof is left as an exercise. To simplify the computation of .F n , we use the function .τ (z) to write F (z) = [2τ (z)z0 z1 z2 : z12 z2 : z1 z2 (2τ (z)z2 + z1 )], z ∈ P2 \ I1 .

.

This suggests that the complications of the indeterminacy sets .Ik (F ) are largely due to the common factor .z1 z2 . Therefore, in order to avoid this unnecessary complication, we make another renormalization and consider the map Fπ (z) := [2τ (z)z0 : z1 : 2τ (z)z2 + z1 ].

.

(10.24)

252

10 Self-similarity and Julia Sets

Clearly, we have .Fπ (z) = F (z) if .z1 z2 /= 0. It is important to observe that, since .τ is homogeneous of degree 0, the map .Fπ on .P2 is a homogeneous rational map of degree 1. This lends great convenience to the study. Lemma 10.28 For the map .Fπ , we have .E = I1 = {[±1 : 0 : 1]}. Observe that .E ⊂ p(Aπ ). Furthermore, Lemma 10.27 gives rise to the following connection. Proposition 10.29 For .n ≥ 1, the following diagram is commutative: P2 \ E

F πn

τ .

Cˆ

P2 \ E τ

Tn

ˆ C.

This connection is crucial because the iterations of .Fπ can now be studied through that of the Tchebyshev polynomial T . Example 10.6 can therefore assist.

10.4.2 Projective Spectrum and the Julia Set To simplify subsequent calculations, define functions pn (z) = 2n

n−1 ||

.

T k (τ (z)), z ∈ P2 , n = 1, 2, . . .

k=0

We take a closer look at the case .z ∈ pc (Aπ ), in which .τ (z) ∈ / [−1, 1]. Example 10.6 then implies .T k (τ (z)) /= 0 for every .k ≥ 0. Moreover, if .z1 z2 = 0, then .z02 − z12 − z22 /= 0, and hence .τ (z) = ∞, which implies .pn (z) = ∞ for each n. We let fn (z) =

n E

.

j =1

1 , n ≥ 2, z ∈ pc (Aπ ). pj (z)

(10.25)

Then each .fn is holomorphic on .pc (Aπ ) and vanishes on the subset .{z1 z2 = 0} ∩ pc (Aπ ). Proposition 10.29 enables us to rewrite .Fπn . Lemma 10.30 For .n ≥ 2, we have [ Fπn (z) = z0 :

.

] z1 : z2 + z1 fn (z) , z ∈ pc (Aπ ). pn (z)

Proof For .z ∈ P2 \ E, we write

10.4 The Julia Set of D∞

253 (1)

(1)

(1)

Fπ (z) = [2τ (z)z0 : z1 : 2τ (z)z2 + z1 ] =: [z0 : z1 : z2 ] = z(1)

.

and use the fact that .τ (z(1) ) = τ (Fπ (z)) = T (τ (z)) to obtain (1)

(1)

(1)

(1)

Fπ2 (z) = [2τ (z(1) )z0 : z1 : 2τ (z(1) )z2 + z1 ]

.

= [22 τ (z)T (τ (z))z0 : z1 : 22 τ (z)T (τ (z))z2 + z1 (1 + 2T (τ (z)))]. (10.26) To prove the lemma by induction, we assume (n−1)

Fπn (z) = [2T n−1 (τ (z))z0

.

(n)

(n)

(n−1)

: z1 : 2T n−1 (τ (z))z2

+ z1 ]

(n)

=: [z0 : z1 : z2 ], n ≥ 2. Then (n)

(n)

(n)

(n)

Fπn+1 (z) = [2T (τ (z(n) ))z0 : z1 : 2T (τ (z(n) ))z2 + z1 ]

.

(n)

(n)

= [2T n (τ (z))z0 : z1 : 2T n (τ (z))z2 + z1 ]. (n)

(n)

Substituting .z0 and .z2 using the induction assumption, we can rewrite .Fπn+1 (z) as (n−1)

[22 T n (τ (z))T n−1 (τ (z))z0

.

( ) (n−1) : z1 : 2T n (τ (z)) 2T n−1 (τ (z))z2 + z1 + z1 ].

Applying the process iteratively, we obtain Fπn (z)

.

)] ( [ n−1 n−1 n−1 k || || E || T k (τ (z)) : z1 : 2n z2 T k (τ (z)) + z1 1 + 2k T n−i (τ (z)) = 2n z0 k=0

k=0

k=1

i=1

[ )] ( n−1 E pn (z) = z0 pn (z) : z1 : z2 pn (z) + z1 1 + . pk (z) k=1

Since .pn (z) does not vanish on .pc (Aπ ), the proof is completed by factoring out .pn . u n We are in position to prove the following inclusion which connects the resolvent set with the Fatou set. Lemma 10.31 .pc (Aπ ) ⊂ F(Fπ ). ˆ is a rational map, for every compact subset Proof Since .τ (z) : p c (Aπ ) → C c ˆ \ [−1, 1]. Thus .{T k (τ (z))} converges .K ⊂ p (Aπ ) the image .τ (K) is compact in .C

254

10 Self-similarity and Julia Sets

uniformly to .∞ on K according to Example 10.6. In particular, there exists an N ∈ N such that . |pn1(z)| ≤ 2−n for every .n ≥ N and .z ∈ K. This implies that the series

.

f (z) = lim fn (z) =

.

n→∞

1 1 + + ··· p1 (z) p2 (z)

(10.27)

converges uniformly on K. Lemma 10.30 then implies that the sequence .{Fπn (z)} converges normally to the map F∗ ([z0 : z1 : z2 ]) = [z0 : 0 : z2 + z1 f (z)], z ∈ p c (Aπ ).

(10.28)

.

u n On the other hand, if .z ∈ p(Aπ ), then .x := τ (z) ∈ [−1, 1], and hence we can rewrite .T (x) in terms of cosine by setting .x = cos θ , where .θ ∈ [0, π ]. Then .T (cos θ ) = cos(2θ ) and hence .T n (cos θ ) = cos(2n θ ) for all .n ∈ N. The kth Tchebyshev polynomial of the first kind .Tk (x) is defined by the equality .cos(kθ ) = Tk (cos θ ). Hence T n (cos(θ )) = T2n (cos(θ )).

(10.29)

.

Thus, when .θ /= 0 or .π, a simple computation gives n || .

T j (cos θ ) =

j =0

sin(2n+1 θ ) . 2n+1 sin θ

(10.30)

This fact leads to the main theorem of this section. Theorem 10.32 .J(Fπ ) = p(Aπ ). Proof First of all, Lemma 10.31 implies .J(Fπ ) ⊂ p(Aπ ). So it only remains to show the inclusion in the opposite direction. In the proof of Lemma 10.30, it is shown that for .z ∈ P2 \ E we have [ )] ( n−1 n−1 n−1 k || || E || Fπn (z) = 2n z0 T k (τ ) : z1 : 2n z2 T k (τ ) + z1 1 + 2k T n−i (τ ) .

.

k=0

k=0

k=1

i=1

If .ξ = [ξ0 : ξ1 : ξ2 ] ∈ p(Aπ ), then as before we write .τ (ξ ) = cos θ for some / Z for any integer .m ≥ 0. 0 ≤ θ ≤ π . Suppose . πθ ∈ [0, 1] is non-dyadic, i.e., .2m πθ ∈ A review of (10.22) and definition (10.23) indicates that .ξ1 /= 0 in this case. In light of (10.30), we have

.

)] [ ( n−1 E sin(2n θ ) sin(2n θ ) sin(2n θ ) , Fπn (ξ ) = ξ0 : ξ 1 : ξ2 + ξ1 1 + sin θ sin θ 2n−1 sin(2k+1 θ )

.

k=1

10.4 The Julia Set of D∞

255

[ ] n E sin θ k −1 = ξ0 : ξ1 + ξ sin θ (sin(2 θ )) : ξ , 2 1 sin(2n θ ) k=1

which does not converge as .n → ∞. If .ξ were a Fatou point, then there would exist n a path-connected neighborhood V of .ξ and a subsequence .{Fπ j } that converges normally to a holomorphic map .Fˆ∗ on V . Since .pc (Aπ ) is dense in .P2 , the map .Fˆ∗ must be a holomorphic extension of the limit function .F∗ in (10.28) from .V ∩ | sin θ | | ˆ p c (Aπ ) to V . But due to the fact .| sin(2 n θ) ≥ sin θ > 0 for all n, the map .F∗ is not continuous at .ξ . This contradiction shows .ξ ∈ J(Fπ ). The density of non-dyatic numbers in .[0, 1] and the closedness of .p(Aπ ) and .J(Fπ ) then give the inclusion .p(Aπ ) ⊂ J(Fπ ). u n The following corollary summarizes what we know about the Fatou set .F(Fπ ). Corollary 10.33 The following hold for the map .Fπ defined in (10.24): (a) .F(Fπ ) = p c (Aπ ). (b) The iteration sequence .{Fπn } converges normally on .pc (Aπ ) to the map F∗ ([z0 : z1 : z2 ]) = [z0 : 0 : z2 + z1 f (z)],

.

where f is as defined in (10.27).

10.4.3 The Limit of Iterations Interestingly, the function f in Corollary 10.33 can also be determined. First, the following lemma is easy to prove by induction. Lemma 10.34 If .w ∈ C is not a dyadic multiple of .π, then n E .

csc(2k w) = cot(w) − cot(2n w), n ≥ 2.

k=1

Theorem 10.35 .f (z) = τ (z) −

/ τ 2 (z) − 1, z ∈ pc (Aπ ).

Proof Since .cos w is an entire function, it has a local inverse function .cos−1 d whenever . dw cos w = − sin w /= 0, i.e., whenever w is not an integer multiple of .π . It is not hard to check that equality (10.30) holds for all complex numbers w for which .θ := cos−1 w is well-defined. Suppose .z ∈ pc (Aπ ) is such that θ(z) cos−1 (τ (z)) . is non-dyadic. Then the functions .fn defined in (10.25) can be π := π written as

256

10 Self-similarity and Julia Sets

fn (z) =

.

n E sin (θ (z)) ( ) , n ≥ 2. sin 2k θ (z) k=1

Hence, by Lemma 10.34 we have E fn (z) csc(2k θ (z)) = cot(θ (z)) − cot(2n θ (z)). = sin(θ (z)) n

.

(10.31)

k=1

In light of Lemma 10.31, the sequence .{cot(2n θ (z))} converges normally on c .p (Aπ ). To determine the limit, we use equality (10.30) to write .

sin θ (z) 1 ( ) = n+1 ||n , n ≥ 2. j 2 sin 2n+1 θ (z) j =0 T (τ (z))

ˆ \ [−1, 1], and consequently .T j (τ (z)) → ∞ Since .z ∈ pc (Aπ ), we have .τ (z) ∈ C ( ) as .j → ∞ by Example 10.6. It follows that .sin 2n+1 θ (z) → ∞ as .n → ∞, and hence / ± 1 − sin2 (2n θ (z)) n .c := lim cot(2 θ (z)) = lim n→∞ n→∞ sin(2n θ (z)) = ±i. To determine which value to choose for c, we let .n → ∞ in (10.31) to obtain f (z) = cos(θ (z)) − c sin(θ (z)) / / = τ (z) − c 1 − τ 2 (z) = τ (z) ± τ 2 (z) − 1.

.

On the other hand, since f (z) = lim fn (z) n→∞ ( ) 1 1 1 + = 1+ + ··· , 2T (τ (z)) 22 T (τ (z))T 2 (τ (z)) 2τ (z)

.

we must have .f (z) = 0 when .τ (z) = ∞. This proves the theorem.

u n

One observes that, since .τ is well-defined on the entire .P2 , the function f extends to .P2 . Investigation of its properties is left as an exercise. Concluding Remarks Theorem 10.32 suggests an intrinsic connection between self-similarity, Julia set, and projective spectrum. Even though .D∞ is a rather simple group, the associated dynamical map .Fπ defined in (10.24) is by no means trivial. Two main factors that contributed to the establishment of Theorem 10.32 are:

10.4 The Julia Set of D∞

257

1. An explicit description of the projective spectrum of .D∞ in (10.22) 2. The semi-conjugacy of .Fπ with the Tchebyshev polynomial (Proposition 10.29) No analogous facts are currently known for the lamplighter group .L or the Grigorchuk group .G. However, despite the complexity of the two groups, the 2similarity of their Koopman representations described in Sect. 10.3 seems simple enough to warrant further progress along this line. Indeed, some promising results about .L have been discovered in [249], and some observations are made about .G in the exercise. Exercise 10.36 1. 2. 3. 4. 5.

Verify Lemma 10.27 and show that the same relation holds for .Fπ in (10.24). Compute the indeterminacy sets .I1 and .I2 for the map .Fπ . Determine the indeterminacy set .I1 for the map L in (10.15). Verify Lemma 10.34. The function f in Theorem 10.35 can be extended to .P2 .

(a) Show that .f −1 (T) = p(Aπ ). (b) Determine the set of discontinuities for f . (c) Find the zero set of f . 6. Consider the element .u = 12 (b + c + d − 1) ∈ C[G] defined in Sect. 5.4. Let 5 3 .M = {z ∈ C | z2 = z3 = z4 } and define the isomorphism .X : C → M by ( ) (w0 , w1 , w2 ) → w0 − w2 /2, w1 , w2 /2, w2 /2, w2 /2 .

.

(a) For the pencils .A(z) = z0 1 + z1 a + z2 u and .B(z) = z0 1 + z1 a + z2 b + z3 c + z4 d in .C[G], show that .A(w) = B(X(w)), w ∈ C3 . (b) Let F be the map in (10.13) and R be the map in (10.19). Show that .X(F (w)) = R(X(w)), w ∈ C3 . (c) Show that .w ∈ F(F ) if and only if .X(w) ∈ F(R |M ). Project 10.37 1. Study the projective spectrum of the groups .L and .G with respect to their regular representations and Koopman representations. 2. Investigate the Julia set of the map L for .L and the map R for .G. In particular, is the projective spectrum .p(Bπ ) a subset of .J(R)?

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Index

Symbols .C0 -representation, .φ-singular,

229

kernel, 198 locally full, 199

105

A Affine map, 225 Algebra Calkin, 70 Cuntz, 83 differential graded, 140 irrational rotation, 85 Almost commuting tuple, 71 Amenable group, 225 Atkinson theorem, 71

B Backward shift, 202 Baire’s Category Theorem, 222 Banach–Tarski paradox, 223 Basin of attraction, 236 Betti number, 120 Beurling’s theorem, 176 Bianchi identity, 122 Blaschke product, 81 Borel subalgebra, 60 Braid arrangement, 133 Bundle congruent, 199 equivalence, 198 full, 199 holomorphic, 198

C Calabi–Yau manifold, 151 Cartan decomposition, 51 matrix, 54 ’s criterion, 45 subalgebra, 51 Character, 25 Characteristic polynomial of group, 26 of group representation, 26 of Lie algebra, 43 of Lie algebra representation, 43 of matrices, 2 Chern class, 126 connection, 149 form, 126 Chern–Weil homomorphism, 125 Class function, 25 Cohomology cyclic, 140 de Rham, 120 Hochschild, 139 Commutant, 24 Conjugate functions, 238 Connection on a vector bundle, 121 Convolution, 214 Curvature, 121 form, 122 Cyclic cocycle, 140

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 R. Yang, A Spectral Theory Of Noncommuting Operators, https://doi.org/10.1007/978-3-031-51605-4

269

270 D Degenerate map, 236 Derivation of a Lie algeba, 48 Determinant Fredholm, 187 Fuglede–Kadison, 104 Harpe–Skandalis, 106 Determinantal representation, 17 Differential form closed, 120 exact, 120 holomorphic, 120 Domain holomorphically convex, 79 of holomorphy, 79 pseudoconvex, 79 Stein, 79 Dual of a group, 25 Dynkin diagram, 54

E Eigensurface, 2 Essentially normal tuple, 72 Euler characteristic, 120 Exact complex, 67 Exceptional Lie algebras, 54 Extended indeterminacy set, 242 Exterior algebra, 66

F Factor, 104 finite, 104 type II.1 , 104 Fatou set, 236 Finite extension, 114 Fixed point, 236 Følner’s property, 228 Form Chern–Simons, 127, 129 Maurer–Cartan, 128 Fredholm tuple, 71 Function Beta, 128 Green, 239 growth, 113 holomorphic, 120 inner, 176 outer, 177 pluriharmonic, 145 of positive type, 218

Index Fundamental form of a metric, 150 of a pencil, 153

G Gelfand map, 66 Generic contraction, 103 form, 19 position, 98 Group basilica, 250 Coxeter, 36 determinant, 27 Grigorchuk, 115 Grothendieck, 82 Haagerup, 229 Heisenberg, 214 of intermediate growth, 115 Kazhdan, 230 Klein four-, 117 lamplighter, 245 linear, 115 locally compact, 213 nilpotent, 114 quaternion, 31 solvable, 114 torsion free, 81 von Neumann algebra, 25 Weyl, 53 Growth exponential, 114 polynomial, 114

H Haar measure, 214 unitary, 88 Hartogs extension theorem, 76 figure, 78 Heisenberg algebra, 50 Hénon map, 238 Hermitian structure, 148 Holomorphic function, 145 section, 147 vector bundle, 147 Holomorphically convex hull, 79 Hyperinvariant subspace, 186 Hyperplane arrangement, 133

Index I Indeterminacy set, 242 Index of Fredholm operator, 71 of Fredholm tuple, 71 of subgroup, 114 Inner automorphism, 35 derivation, 49 Invariant linear functional, 123 mean, 225 polynomial, 125 vector, 213 J Jacobian conjecture, 238 Jacobi’s formula, 7 Joint resolvent set, 2 Joint spectrum algebraic, 66 approximate point, 74 essential Taylor, 72 Harte, 73 projective, 75 Taylor, 69 Julia set, 236 K Kähler manifold, 150 metric, 150 potential, 150 Kazhdan’s property (T), 230 K-group, 82 Killing form, 45 Koszul complex, 66 L Lie algebra nilindex of, 42 nilpotent, 42 nilradical of, 42 radical of, 42 semisimple, 41 simple, 41 solvable, 42 Local equivalence, 199

271 M Mahler measure, 112 Markov–Kakutani theorem, 226 Markov operator, 91 Matrix Coxeter, 36 metric, 148 spectral, 61 Maurer–Cartan form, 106 of a pencil, 119 Maximal ideal space, 65 Mean, 224 Meromorphic map, 242 Metric Fubini-Study, 151 Hermitian, 150 Morita equivalence, 85 Multiplicative linear functional, 65

N Noncommutative probability space, 88 Normal convergence, 235 family, 235 function, 188 matrix, 12 pair of projections, 99 point, 188

O Operator compact, 70 composition, 87 Cowen–Douglas, 202 Dolbeault, 147 Fredholm, 70 Hilbert–Schmidt, 167 local, 204 model, 180 quasinilpotent, 105 Schatten p-class, 187 strictly cyclic, 184 subnormal, 181 Toeplitz, 180 trace class, 167 Volterra, 180 weighted shift, 184

272 P Partial isometry, 82 Pauli matrices, 3 Plemelj-Smithies formula, 5 Polarization identity, 125 Polynomial cubic, 17 monic, 17 Poincaré, 133 quadric, 17 real-zero, 21 Tchebyshev, 37, 237 Power index of simple Lie algebras, 57 set, 182 R R-diagonal operator, 89 Reduced form of words, 32 group C*-algebra, 25 Regular point, 193 Renormalization map, 247 Representation adjoint, 3, 42 irreducible, 24 Koopman, 243 reducible, 24 regular, 25 self-similar, 243 sub-, 24 Tits, 38 unitary, 24 Reproducing kernel, 175 Residue, 189 Ricci curvature tensor, 151 flat, 151 form, 151 Root long, 53 short, 53 of a simple Lie algebra, 51 Rooted binary tree, 244 Root system base, 52 exceptional, 54 irreducible, 53 isomorphic, 54 rank, 52 of a simple Lie algebra, 51 of vectors, 52 R-transform, 89

Index S Schur complement, 246 Semi-conjugacy, 238 Semidirect product of groups, 35 Semidirect sum, 49 Semisimple element, 51 Shift bilateral, 85 unilateral, 84 Similar tuples, 14 Simple root, 53 Smooth surface, 198 variety, 152 Spectral mapping theorem, 70 Spectrum full, 209 point, 194 strongly full, 209 Stably isomorphic C*-algebras, 85 State, 153 faithful, 156 tracial, 157 vector, 157 Structure constants, 42

T Thin set, 193 Trace canonical, 23, 25 graded, 141

U Uniform boundedness principle, 80 Unitary equivalence, 14

V Variational calculus, 172 Von Neumann–Wold decomposition, 84

W Weak containment, 218 equivalence, 218 Weakly bounded, 80 Wedge product, 66 Word length, 32 metric, 32