TEN LECTURES ON OPERATOR ALGEBRAS

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Conference Board of the Mathematical Sciences REGIONAL CONFERENCE SERIES IN Ì Á THEMA TICS

supported by the National Science Foundation

Number 55

TEN LECTURES ON OPERATOR ALGEBRAS

by William Arveson

Published for the Conference Board of the Mathematical Sciences

bythe American Mathematical Society

Providence, Rhode Island

http://dx.doi.org/10.1090/cbms/055

Expository Lectures

from the CBMS Regional Conference

held at Texas Tech University

August 1-5, 1983

Research supported in part by National Science Foundation Grant MCS83-02061.

1980 Mathematics Subject Classifications. Primary 47D25; Secondary 47C05.

Library of Congress Cataloging in Publication Data

Arveson, William.

Ten lectures on Operator algebras.

(Regional Conference series in mathematics; no. 55)

Bibliography: p.

1. Operator algebras-Addresses, essays, lectures. I. Conference Board of the

Mathematical Sciences. II. Title. III. Title: 10 lectures on Operator algebras. IV. Series.

QAl.R33no. 55 [QA326] 510s [512'.55] 84-9222

ISBN 0-8218-0705-6

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Contents

Introduction vii Lecture 1. Origins in Single Operator theory 1 Lecture 2. Triangulär and quasitriangular Operator algebras 9 Lecture 3. Brief on quantum mechanics and Schrödinger Operators 17 Lecture 4. Nonselfadjoint Operator algebras and the Feynman-Kac formula . . .29 Lecture 5. Commutative subspace lattices 41 Lecture 6. Reflexive Operator algebras 51 Lectures 7 and 8. Hyperreflexive Operator algebras 61 Lecture 9. The absorption principle 73 Lecture 10. Similarity of reflexive Operator algebras 85 References 91

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Introduction

This book contains somewhat expanded versions of ten lectures delivered at Texas Tech University during the summer of 1983. The Operator algebras of the title are nonselfadjoint algebras of Operators on Hubert space.

This subject is new, and has shown remarkable growth in the last twenty years. Indeed, when I was finishing my graduate studies in 1964 I knew of only three papers that addressed themselves seriously to nonselfadjoint Operator algebras ([42, 61], and a paper of John Schue, The structure of hyperreducible triangulär algebras, Proc. Amer. Math. Soc. 15 (1964), 766-772). Á few of us believed in the sixties that this was a promising way to approach the theory of Single Operators, but we certainly did not see how and were not even sure if that would be accomplished. What actually happened was that the subject developed in several directions, and was pursued entirely on its own merits. When the applications to Single Operator theory did come, they came unexpectedly and in surprising ways (see Lecture 10). These results are deep and, looking back on it now, I must say that I cannot conceive of any way that the methodology of single Operator theory could have produced them.

The subject matter for these lectures has been selected using subjective criteria. Some of it has historical interest, some of it seems timely or important (at least to me), some of it seems to suggest new directions, and some of it is just fun to communicate. I have had to omit several of my favorite topics on which there has been significant progress, including noncommutative Silov boundaries, abstract dilation theory, and algebras defined by group actions [7, 8, 73, 74, 50,12, 49].

Some of the material is expository and is presented largely without proofs (Lectures 1, 2, 5, 6). Some is expository but with complete proofs or complete ideas of proofs (Lectures 7, 8, 10). Lectures 7 and 8 expand on some notes distributed to the participants in a seminar at Berkeley during the spring quarter 1983; Lecture 9 is based on a lecture delivered in Busteni, Romania, in September 1983. Lecture 4 contains new material relating to the Feynman-Kac formula. I have taken some care to present complete proofs, and to develop the background material from classical mechanics and quantum mechanics in Lecture 3.

Finally, the references are by no means complete. I have referenced only those items I know about that relate to the subject matter of these lectures. The reader

Vll

vm INTRODUCTION

will find a more complete bibliography in the survey of John Erdos [29], which also contains a discussion of several topics not mentioned in these lectures.

I would like to thank the National Science Foundation for granting financial support to this lecture series, the Conference Organizer, Gareth Ashton, for his efforts to make the Conference work, and the Mathematics Department of Texas Tech University for the hospitality they extended to all the participants.

Note added December 5, 1983, concerning Lectures 4 and 5. Barry Simon has pointed out that connections between the Feynman-Kac formula and dilation theory have been observed previously in the mathematical physics literature. We particularly want to call attention to the work of Abel Klein [75, 76, 77, 78] on Osterwalder-Schrader positivity, and of Klein and Landau [79, 80] on the path Space approach to perturbation theory and KMS Systems.

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References

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