Operato r Theor y Operato r Algebra s

an d Application s

http://dx.doi.org/10.1090/pspum/051.1

PROCEEDING S O F SYMPOSI A IN PUR E MATHEMATIC S

Volum e 51 , Par t 1

Operato r Theor y Operato r Algebra s

an d Application s

Willia m B. Arveso n an d Ronal d G . Douglas , Editor s

AMERICA N MATHEMATICA L SOCIET Y

PROVIDENCE , RHOD E ISLAN D

PROCEEDINGS OF THE SUMMER RESEARCH INSTITUTE ON OPERATOR THEORY/OPERATOR ALGEBRAS AND APPLICATIONS

HELD AT UNIVERSITY OF NEW HAMPSHIRE DURHAM, NEW HAMPSHIRE

JULY 3-23, 1988

with support from the National Science Foundation, Grant DMS-8714162

1980 Mathematics Subject Classification (1985 Revision). Primary 46L, 47A, 47B, 58G.

Library of Congress Cataloging-in-Publication Data

Operator theory: operator algebras and applications/William B. Arveson and Ronald G. Dou­glas, editors. p. cm.—(Proceedings of symposia in pure mathematics; v. 51) ISBN 0-8218-1486-9 I. Operator algebras — Congresses. 2. Operator theory — Congresses. I. Arveson, William. II. Douglas, Ronald G. III. Series. QA326.067 1990 90-33771 512'.55-dc20 CIP

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Contents Preface xi

PART 1

The spectral C*-algebra of an ^-semigroup WILLIAM ARVESON 1

Two-sided Lagrange-Sylvester interpolation problems for rational matrix functions J. A. BALL, I. C. GOHBERG, AND L. RODMAN 17

Nonstationary inertia theorems, dichotomy, and applications A. BEN-ARTZI AND I. GOHBERG 85

Toeplitz operators, quantum mechanics, and mean oscillation in the Bergman metric LEWIS COBURN 97

Towards a functional calculus for subnormal tuples: The minimal nor­mal extension and approximation in several complex variables JOHN B. CONWAY 105

On the technique of comparison algebra for elliptic boundary problems on noncompact manifolds H. O. CORDES 113

Composition operators on Hilbert spaces of analytic functions: A status report CARL C. COWEN 131

Random Toeplitz operators R. E. CURTO, P. S. MUHLY, AND J. XlA 147

Isomorphisms of nest algebras and their quotients KENNETH R. DAVIDSON 171

Invariants for Hilbert modules RONALD G. DOUGLAS 179

Multivariate multipliers for groups and their operator algebras E. G. EFFROS AND Z.-J. RUAN 197

Beyond commutant lifting J. WILLIAM HELTON 219

vi CONTENTS

Similarity and approximation of operators DOMINGO A. HERRERO 225

A primer on A^-theory N I G E L D. HIGSON 239

Complete distributivity ALAN HOPENWASSER 285

Operator algebraic invariants for elliptic operators JERRY KAMINKER 307

The supersymmetric replica trick and smoothness of the density of states for random Schrodinger operators ABEL KLEIN 315

Some recent progress in nest algebras DAVID R. LARSON 333

Rigidity theorems in spaces of analytic functions VERN I. PAULSEN 347

Hankel operators and multivariate stationary processes VLADIMIR V. PELLER 357

The principal index JOEL PINCUS 373

Refinement theory for nonselfadjoint operator algebras STEPHEN C. POWER 395

Some remarks on the index theory for semigroups of endomorphisms of Q3(£) ROBERT T. POWERS 405

Deformation quantization and operator algebras M A R C A. RIEFFEL 411

Toeplitz and Hankel operators, wavelets, NWO sequences, and almost diagonalization of operators RICHARD ROCHBERG 425

K and KK : Topology and operator algebras JONATHAN ROSENBERG 445

Applications of C*-algebras and operator theory to proper holomorphic mappings NORBERTO SALINAS 481

Function theory and de Branges's spaces DONALD E. SARASON 495

Algebraic quantization and stability IRVING SEGAL 503

Analytic operator algebras BARUCH SOLEL 519

CONTENTS vii

Automorphisms which preserve unitary equivalence classes of normal states

U. HAAGERUP AND E. ST0RMER 531

Cocycle conjugacy of group actions on factors

MASAMICHI TAKESAKI 539

Pseudodifferential operators and ^-homology

MICHAEL E. TAYLOR 561

Toeplitz operators and index theory in several complex variables

HARALD UPMEIER 585

Szego expansions for operators with smooth or nonsmooth symbol

HAROLD WIDOM 599

Analytic theory of subnormal n -tuple of operators

DAOXING XIA 617

PART 2

Reduced Hankel operators and radial measures GREGORY T. ADAMS 1

From algebras of normal operators to intersecting hyperplanes

E. A. AZOFF AND H. A. SHEHADA 11

Random walks, A^-theory for AF algebras, and classical statistical mechanics

B. M. BAKER AND D. E. HANDELMAN 17

Some C4 and C$ norm inequalities related to the paving problem

K. BERMAN, H. HALPERN, V. KAFTAL, AND G. WEISS 29

Cyclic composition operators on H2

P. S. BOURDON AND J. H. SHAPIRO 43

Representation theory of U(oo)

ROBERT P. BOYER 55

Trace class integral kernels

CHRISTOPHER M. BRISLAWN 61

Products of operators with singular continuous spectra

GEON HO CHOE 65

viii CONTENTS

Joint hyponormality: A bridge between hyponormality and subnor-mality R A U L E. CURTO 69

Algebras generated by classical pseudodifferential operators on open Riemannian manifolds ALEXANDER DYNIN 93

A similarity invariant C. K. FONG, E. A. NORDGREN, H. RADJAVI, AND P. ROSENTHAL 99

Two themes in index theory on singular varieties J. S. Fox, P. HASKELL, AND W. PARDON 103

CP-Duality for C - and W*-algebras ICHIRO FUJIMOTO 117

A note on special CSL algebras with nontrivial cohomology FRANK GILFEATHER 121

Actions of compact groups on C*-algebras E. C. GOOTMAN AND A. J. LAZAR 127

Parallel sums of operators W. L. G R E E N AND T. D. MORLEY 129

The zero-two law and the peripheral spectrum of positive contractions in Banach lattice algebras J. J. GROBLER 135

Finite weight projections in von Neumann factors H. HALPERN, V. KAFTAL, AND L. ZSIDO 143

Toeplitz operators on noncommutative tori and their real valued index RONGHUI Ji 153

Close C*-algebras are A^-equivalent M A H M O O D KHOSHKAM 159

On derivations of operator algebras implemented by bounded operators EDWARD V. KISSIN 163

Composition operators and weighted polynomial approximation T. L. KRIETE, III AND B. D. M A C C L U E R 175

Seminormality for measure-theoretic composition operators ALAN LAMBERT 18 3

Local derivations and local automorphisms oi3B{X) D. R. LARSON AND A. R. SOUROUR 187

How to use primeness to describe properties of elementary operators MARTIN MATHIEU 195

Shifts on Krein spaces BRIAN W. M C E N N I S 201

CONTENTS ix

Inner completely positive maps on von Neumann algebras JAMES A. M I N G O 213

Isometries of nest algebras R. L. MOORE AND T. T. TRENT 219

Chern character for proper T-manifolds HITOSHI MORIYOSHI 221

Selfadjointness of quantum Hamiltonians of relativistic and nonrela-tivistic particles in electromagnetic fields MICHIRO NAGASE AND TOMIO U M E D A 235

Actions of compact groups on C*-algebras COSTEL PELIGRAD 243

Invariants for triangular AF algebras J. R. PETERS AND Y. T. POON 247

A>theory and noncommutative homotopy theory N. CHRISTOPHER PHILLIPS 255

Equivalence bimodules in the representation theory of reductive groups R. J. PLYMEN 267

Spectral theory and sheaf theory IV MIHAI PUTINAR 273

On factorization of self adjoint operator polynomials LEIBA RODMAN 295

The Kasparov groups for commutative C*-algebras and Spanier-White­head duality CLAUDE SCHOCHET 307

How are the Toeplitz C* -algebras of Reinhardt domains affected by taking pseudoconvex hulls? ALBERT J.-L. SHEU 323

Factor state extensions on 0 2

JOHN S. SPIELBERG 333

On the Dauns-Hofmann theorem SILVIU TELEMAN 337

Subnormal operators with a common invariant subspace T. T. T R E N T AND W. W O G E N 343

C*-algebras and the classification of finite groups M A R T I N E. WALTER 345

Bivariant Chern character I XIAOLU W A N G 355

Composition operators acting on spaces of holomorphic functions on domains in Cn

WARREN R. W O G E N 361

x CONTENTS

Rigidity theorem and Buerling theorem KEREN YAN 367

On certain unitary operators and composition operators KEHEZHU 371

Preface

During the last twenty years operator theory has come of age. The sub­ject has developed in several directions, using new and powerful methods that have led to the solution of basic problems thought to be inaccessible in the sixties. Some of these developments have made mutually enriching con­tact with other areas of mathematics, including algebraic topology and index theory, complex analysis in one and several variables, and probability theory.

This period has seen the full characterization of quasitriangular opera­tors in terms of the Fredholm index, the classification of families of essen­tially normal operators via C*-algebraic extensions, and the consequences of the latter subject culminating in the unification of C*-algebraic ^-homology and AT-cohomology in the Kasparov ATAT-bifunctor. The invariant subspace problem has been solved for subnormal operators and related classes. The classical Weyl-von Neumann theorem has been vastly generalized to sepa­rable C*-algebras. The Ringrose problem on the multiplicity of nests of subspaces has been solved, using algebraic methods that have provided strik­ing insight into the behavior of operators under similarity transforms. The classical perturbation theory for Schrodinger operators has been transformed and simplified by the use of path integrals and the Feynman-Kac formula. A rich theory of completely positive and completely bounded maps of C*-algebras has emerged, and this has had significant implications for operator theory, including dilation theory, the characterization of operators having annular spectral sets, and the partial solution of Sz.-Nagy's problem. The C*-algebras of Toeplitz operators associated with a large class of domains in Cn are now clearly understood. Finally, the structure of broad classes of reflexive operator algebras has been penetrated and put into the context of "noncommutative" spectral synthesis.

These algebraic methods are diverse, and they touch upon a broad area of mathematics. In addition to the interrelations alluded to above, there have been direct applications to systems theory, complex variables, and sta­tistical mechanics. Moreover, significant problems and motivations have arisen from the subject's traditional underpinnings for partial differential equations. While it would not be possible or perhaps even desirable to at­tempt to unify these results and methods, it seemed an appropriate time to

XI

Xll PREFACE

summarize progress and examine the common points of view that now run through the subject. Thus, we organized the 1988 AMS Summer Institute on Operator Theory/Operator Algebras and Applications. The present two vol­umes contain papers representing most of the invited talks, as well as many of the seminar talks that were given there. We think that the points made above are amply borne out in these Proceedings.

William B. Arveson Ronald G. Douglas