Derivations, Dissipations and Group Actions on C*-algebras
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1229 Ola Bratteli
Derivations, Dissipations and Group Actions on C*-algebras
Springer-Verlag Berlin Heidelberg New York London Paris Tokyo
Author
Ola Bratteli Institute of Mathematics, University of Trondheim, N-7034 Trondheim-NTH, Norway
Mathematics Subject Classification (1980): 46L55, 22025, 20M20, 34C35, 43A45, 47B47, 47005, 54H20
ISBN 3-540-17199-1 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-17199-1 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 those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to "Verwertungsgesellschaft Wort", Munich. © Springer-Verlag Berlin Heidelberg 1986 Printed in Germany
Printing and binding: Druckhaus Beltz, Hemsbach/Bergstr. 2146/3140-543210
PREFACE
These lecture notes are based on a series of lectures given In a seminar at the Research Institute of Mathematical Sciences, Kyoto University, in 1984-85. I am greatly indebted to Huzihiro Araki for arranging the visit to Kyoto University, and thanks are due to the participants of the seminar for their interest. The following colleagues gave valuable critical remarks to, and pointed out mistakes in, parts of previous versions of these notes: Huzihiro Araki, Charles J.K. Batty, George A. Elliott, David E. Evans, Akio Ikunishi, Akitaka Kishimoto and Derek W. Robinson. It is also a pleasure to thank Toshie Ito and the other "international" secretaries at RIMS, as well as Monica Grund in Trondheim, for their typing of various parts of the manuscript. Trondheim, May 1986 Ola Bratteli
CONTENTS Introduction with historical remarks
1
1. General theory of derivations
4
1.1. Basic notions
4
1.2. Bounded *-derivations
5
1.3. Unbounded *-derivations
7
1.4. Closed and pre-closed derivations 1.5. Generators and pre-generators
9 15
1.6. Classification of all closed derivations on a given C*-algebra 1.6.1. Functional analysis of the domains of derivations
21 21
1.6.2. Classification of the closed derivations on the compact operators
24
1.6.3. Classification of the closed derivations of an AF-algebra
25
1.6.4. Classification of closed derivations on abelian C*-algebras
25
1.6.4.1. The O-dimensional case
27
1.6.4.2. The 1-dimensional case
29
1.6.4.3. The 2-dimensional case
32
2. Non-commutative vectorfields
34
2.1. General introduction and motivation
34
2.2. Classes of smooth elements and spectral theory for group actions
39
2.2.1. Differentiable elements
39
2.2.2. Spectral theory for abelian groups
41
2.2.3. Spectral theory for compact groups
44
2.2.4. The algebra of G-finite elements
45
2.3. Closability and automatic continuity of noncommutative vector fields
48
2.3.1. Automatic continuity of derivations of An' n = 1 , 2 , ... ,DO
48
2.3.2. Closability of der
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