Uniqueness of the Injective III1 Factor
Based on lectures delivered to the Seminar on Operator Algebras at Oakland University during the Winter semesters of 1985 and 1986, these notes are a detailed exposition of recent work of A. Connes and U. Haagerup which together constitute a proof that al
- PDF / 6,153,988 Bytes
- 112 Pages / 468 x 684 pts Page_size
- 53 Downloads / 170 Views
1413 Steve Wright
Uniqueness of the Injective 111 1 Factor
Springer-Verlag Berlin HeidelbergNewYork London Paris Tokyo Hong Kong
Author
Steve Wright Department of Mathematics, Oakland University Rochester, MI, 48309-4401, USA
Mathematics Subject Classification (1980): 46L35, 46L 10 ISBN 3-540-52130-5 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-52130-5 Springer-Verlag New York Berlin Heidelberg
This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. Duplication of this publication or parts thereof is only permitted under the provisions of the German Copyright Law of September 9, 1965, in its version of June 24, 1985, and a copyright fee must always be paid. Violations fall under the prosecution act of the German Copyright Law.
© Springer-Verlag Berlin Heidelberg 1989 Printed in Germany Printing and binding: Druckhaus Beltz, Hemsbach/Bergstr. 2146/3140-543210 - Printed on acid-free paper
Contents page 1. Introduction
1
2. Part 1. Connes' Reduction of the Uniqueness Proof to the Bicentralizer Problem Chapter 1. Connes' Argument: outline and preliminary lemmas
9
Chapter 2. Araki's Property
25
Chapter 3. A Characterization of Int(M) for Type III Factors
41
Chapter 4. Trivial Bicentralizers and IIII Factors
61
Notes on Part I
79
3. Part II. Haagerup's Solution of the Bicentralizer Problem
82
4. References
106
5. Index
108
Introduction
These notes are based on the content of lectures delivered to the Seminar on Operator Algebras at Oakland University during the Winter semesters of 1985 and 1986. They are a detailed exposition of [10] and Section 2 of [16], which together constitute a proof of the uniqueness of the (separably acting) injective I III factor. The exposition contains nothing that is not already in [10] and [16], but merely fills in details in some of the arguments appearing there. Our hope is that the notes will contribute in some small way to an understanding and appreciation of these profound and beautiful results of Connes and Haagerup. The following rather informal discussion is intended to define terms and fix some notation relevant for the sequel and to historically orient the results with which it deals. We concentrate exclusively on only selected developments that focus directly on the classification of factors, and apologize here for the many serious omissions of developments in the general theory which consequently result. The classification of von Neumann algebras to within isomorphism has been the fundamental problem in their study and has motivated much of the work in the subject. (We will only consider von Neumann algebras acting in a separable Hilbert space.) Indeed, this was the underlying theme of the initial work in the 1930's of the founding fathers Murray and von Neumann [20], [30]. They isolated the factors, the von Ne
Data Loading...
