Crossed Products With Continuous Trace

149 Pages·1996·15.76 MB·English

by  Siegfried Echterhoff

Selected Titles in This Series
586 Siegfried Echterhoff, Crossed products with continuous trace, 1996
585 Gilles Pisier, The operator Hilbert space OH, complex interpolation and tensor norms,
1996
584 Wayne W. Barrett, Charles R. Johnson, and Raphael Loewy, The real positive definite
completion problem: Cycle completability,, 1996
583 Jin Nakagawa, Orders of a quartic field, 1996
582 Darryl McCollough and Andy Miller, Symmetric automorphisms of free products, 1996
581 Martin U. Schmidt, Integrable systems and Riemann surfaces of infinite genus, 1996
580 Martin W. Liebeck and Gary M. Seitz, Reductive subgroups of exceptional algebraic
groups, 1996
579 Samuel Kaplan, Lebesgue theory in the bidual of C(X), 1996
578 Ale Jan Homburg, Global aspects of homoclinic bifurcations of vector fields, 1996
577 Freddy Dumortier and Robert Roussarie, Canard cycles and center manifolds, 1996
576 Grahame Bennett, Factorizing the classical inequalities, 1996
575 Dieter Heppel, Idun Reiten, and Sverre O. Smal0, Tilting in Abelian categories and
quasitilted algebras, 1996
574 Michael Field, Symmetry breaking for compact Lie groups, 1996
573 Wayne Aitken, An arithmetic Riemann-Roch theorem for singular arithmetic surfaces,
1996
572 Ole H. Hald and Joyce R. McLaughlin, Inverse nodal problems: Finding the potential
from nodal lines, 1996
571 Henry L. Kurland, Intersection pairings on Conley indices, 1996
570 Bernold Fiedler and Jiirgen Scheurle, Discretization of homoclinic orbits, rapid forcing
and "invisible" chaos, 1996
569 Eldar Straume, Compact connected Lie transformation groups on spheres with low
cohomogeneity, I, 1996
568 Raul E. Curto and Lawrence A. Fialkow, Solution of the truncated complex moment
problem for flat data, 1996
567 Ran Levi, On finite groups and homotopy theory, 1995
566 Neil Robertson, Paul Seymour, and Robin Thomas, Excluding infinite clique minors, 1995
565 Huaxin Lin and N. Christopher Phillips, Classification of direct limits of even Cuntz-circle
algebras, 1995
564 Wensheng Liu and Hector J. Sussmann, Shortest paths for sub-Riemannian metrics on
rank-two distributions, 1995
563 Fritz Gesztesy and Roman Svirsky, (m)KdV solitons on the background of quasi-periodic
finite-gap solutions, 1995
562 John Lindsay Orr, Triangular algebras and ideals of nest algebras, 1995
561 Jane Gilman, Two-generator discrete subgroups of PSL(2,R), 1995
560 F. Tomi and A. J. Tromba, The index theorem for minimal surfaces of higher genus, 1995
559 Paul S. Muhly and Baruch Solel, Hilbert modules over operator algebras, 1995
558 R. Gordon, A. J. Power, and Ross Street, Coherence for tricategories, 1995
557 Kenji Matsuki, Weyl groups and birational transformations among minimal models,
1995
556 G. Nebe and W. Plesken, Finite rational matrix groups, 1995
555 Tomas Feder, Stable networks and product graphs, 1995
554 Mauro C. Beltrametti, Michael Schneider, and Andrew J. Sommese, Some special
properties of the adjunction theory for 3-folds in P5, 1995
553 Carlos Andradas and Jesus M. Ruiz, Algebraic and analytic geometry of fans, 1995
(Continued in the back of this publication)
This page intentionally left blank
MEMOIRS
- L * A of the
American Mathematical Society
Number 586
Crossed Products with
Continuous Trace
Siegfried Echterhoff
September 1996 • Volume 123 • Number 586 (first of 4 numbers) • ISSN 0065-9266
American Mathematical Society
Providence, Rhode Island
1991 Mathematics Subject Classification.
Primary 46L40; Secondary 22D25, 22D30.
Library of Congress Cataloging-in-Publication Data
Echterhoff, Siegfried, 1960-
Crossed products with continuous trace / Siegfried Echterhoff.
p. cm. - (Memoirs of the American Mathematical Society, ISSN 0065-9266; no. 586)
"September 1996, volume 123, number 586 (first of 4 numbers)."
Includes bibliographical references.
ISBN 0-8218-0563-0 (alk. paper)
1. C*-algebras. 2. Locally compact groups. 3. Representations of groups. 4. Crossed prod
ucts. I. Title. II. Series.
QA3.A57 no. 586
[QA326]
510s-dc20
[512'.55] 96-21893
CIP
Memoirs of the American Mathematical Society
This journal is devoted entirely to research in pure and applied mathematics.
Subscription information. The 1996 subscription begins with Number 568 and consists of
six mailings, each containing one or more numbers. Subscription prices for 1996 are $391 list,
$313 institutional member. A late charge of 10% of the subscription price will be imposed on
orders received from nonmembers after January 1 of the subscription year. Subscribers outside
the United States and India must pay a postage surcharge of $25; subscribers in India must pay
a postage surcharge of $43. Expedited delivery to destinations in North America $30; elsewhere
$92. Each number may be ordered separately; please specify number when ordering an individual
number. For prices and titles of recently released numbers, see the New Publications sections of
the Notices of the American Mathematical Society.
Back number information. For back issues see the AMS Catalog of Publications.
Subscriptions and orders should be addressed to the American Mathematical Society, P. O. Box
5904, Boston, MA 02206-5904. All orders must be accompanied by payment. Other correspondence
should be addressed to Box 6248, Providence, RI 02940-6248.
Copying and reprinting. Individual readers of this publication, and nonprofit libraries
acting for them, are permitted to make fair use of the material, such as to copy a chapter for use
in teaching or research. Permission is granted to quote brief passages from this publication in
reviews, provided the customary acknowledgement of the source is given.
Republication, systematic copying, or multiple reproduction of any material in this publication
(including abstracts) is permitted only under license from the American Mathematical Society.
Requests for such permission should be addressed to the Assistant to the Publisher, American
Mathematical Society, P.O. Box 6248, Providence, RI 02940-6248. Requests can also be made by
e-mail to [email protected].
Memoirs of the American Mathematical Society is published bimonthly (each volume consisting
usually of more than one number) by the American Mathematical Society at 201 Charles Street,
Providence, RI 02904-2213. Periodicals postage paid at Providence, Rhode Island. Postmaster:
Send address changes to Memoirs, American Mathematical Society, P. O. Box 6248, Providence,
RI 02940-6248.
© Copyright 1996, American Mathematical Society. All rights reserved.
This publication is indexed in Science Citation Index®, SciSearch®, Research Alert®,
CompuMath Citation Index®, Current Contents®/Physical, Chemical & Earth Sciences.
Printed in the United States of America.
©The paper used in this book is acid-free and falls within the guidelines
established to ensure permanence and durability.
&«$ Printed on recycled paper.
10 9 8 7 6 5 4 3 2 1 00 99 98 97 96
Contents
Abstract vii
Introduction 1
Chapter 1. Preliminaries and basic definitions 11
1.1. Twisted covariant systems and crossed products 11
1.2. Representation and ideal spaces of C*-algebras 15
1.3. Morita equivalence and imprimitivity bimodules 16
1.4. Induced representations and the Mackey-Green machine 18
Chapter 2. Morita equivalent twisted actions and duality 29
2.1. Morita equivalent twisted actions 29
2.2. Morita equivalence and duality for abelian twisted systems 34
Chapter 3. Representations of type I abelian twisted systems 37
3.1. Maximally pointwise unitary subgroups 37
3.2. Abelian twisted systems with type I crossed products 44
3.3. Maximally pointwise unitary subgroups for dual systems 48
Chapter 4. Subgroup crossed products 55
4.1. Subgroup algebras and regularizations 55
4.2. Open regularizations and subgroup actions 61
4.3. Decomposition of crossed products by subgroup algebras 64
4.4. Subgroup crossed products and duality 69
Chapter 5. Crossed products with continuous trace 77
5.1. Subgroup crossed products and cr-proper G-spaces 77
5.2. Pointwise unitary subgroup actions 82
5.3. Systems with continuous choices of maximally pointwise unitary
subgroups 91
5.4. Continuous trace for systems with Hausdorff quasi-orbit space 95
5.5. Systems with continuously varying stabilizers 100
Chapter 6. Applications and Examples 115
vi CONTENTS
6.1. Crossed products by cr-proper actions 115
6.2. Actions of R, Z and T 119
6.3. Examples and applications to certain group C*-algebras 121
Some concluding remarks 129
References 131
Abstract
We prove necessary and sufficient conditions for a separable twisted covariant
system (A, G, a, r) to have a continuous-trace crossed product A>^ G under
Q)T
the general assumptions that A has continuous trace and G/N is abelian. We
r
give a complete description of such systems if the G-quasi-orbit space of A is
HausdorfF or if the stability groups for the action of G on A vary continuously.
We also prove some relations for the Dixmier-Douady class 6(A x G) in terms
a?r
of the Dixmier-Douady class 6(A) of A and the twisted action (a, r) of G on A.
On our way to do this we introduce the notion of choices of maximally pointwise
unitary subgroups for (A, G, a, r) and give necessary and sufficient conditions
for A >4 G to be of type I in terms of such choices. In a final chapter we apply
a)T
our results to the study of group C*-algebras of some solvable and nilpotent Lie
groups.
Key words and phrases. C*-algebras, C*-dynamical systems, twisted covariant systems,
twisted crossed products, continuous trace, duality, induced representations, regularizations,
type I crossed products, group C*-algebras, nilpotent groups, solvable groups.
This work constitutes the author's Habilitationsschrift submitted to the University of Pader-
born in November 1993
Received by the editor August 5, 1994
This page intentionally left blank
Introduction
During the last fifteen years the study of covariant systems (or C*-dynamical
systems) (A, G, a) and their crossed products A y\ G has grown to its own
a
subject in the theory of operator algebras. In this work we are mainly inter
ested in the study of separable abelian covariant systems, i.e. covariant systems
(A, G, a) where A is a separable C*-algebra and G is a second countable abelian
locally compact group, such that A and A x G have continuous trace. The main
a
motivation for this study is the fact that the knowledge of such systems provides
important information about the structure of the group C*-algebras of many
solvable and nilpotent groups (see [65] for an excellent discussion of this the
ory).
If A is a C*-algebra, then let A denote the set of all equivalence classes of
irreducible ^representations of A, the dual space of A, equipped with the Jacob-
son topology. An element a of the positive cone A+ of A is called a continuous
trace element of A if the map
A -• RU {oo}; p K-> Tr p(a)
is finite and continuous on A, where Trp(a) denotes the usual trace of the op
erator p(a) on the Hilbert space H of p. A is said to be a C*-algebra with
p
continuous trace if the linear hull CA of all continuous trace elements of A forms
a dense ideal of A.
C*-algebras with continuous trace are very well understood by their represen
tation theory. If A has continuous trace, then A is a locally compact Hausdorff
space and A may be realized as a section algebra TQ(E) of a C*-bundle p : E —• A
such that each fiber A is isomorphic to the compact operators on the Hilbert
p
space H . If, in addition, A is separable and stable, i.e., if A = A (g> /C, where
p
/C denotes the algebra of compact operators on /2(N), then A is a section al
gebra of a locally trivial C*-bundle. This implies that each point in A has an
open neighborhood W such that Iw — n{kercr;cr e A \ W} is isomorphic to
Co(W,/C). Moreover, associated to each separable continuous trace C*-algebra
A, there is an element 8(A) of the third Cech cohomology group iJ3(A,Z), the
so-called Dixmier-Douady class of A. If A and B are two separable and stable
l