K-Theory for Group C*-Algebras and Semigroup C*-Algebras

325 Pages·2017·3.28 MB·English

by  Joachim Cuntz

Oberwolfach Seminars
47
K-Theory for Group
C*-Algebras
and Semigroup
C*-Algebras
Joachim Cuntz
Siegfried Echterhoff
Xin Li
Guoliang Yu

Oberwolfach Seminars
Volume 47
More information about this series at http://www.springer.com/series/4812
Joachim Cuntz • Siegfried Echterhoff • Xin Li
Guoliang Yu
K-Theory for Group C*-Algebras
and Semigroup C*-Algebras
Joachim Cuntz Siegfried Echterhoff
Mathematisches Institut Mathematisches Institut
Universität Münster Universität Münster
Münster, Nordrhein-Westfalen, Germany Münster, Nordrhein-Westfalen, Germany
Xin Li Guoliang Yu
Queen Mary University of London Department of Mathematics
School of Mathematical Sciences Texas A&M University
London, United Kingdom College Station, TX, USA
ISSN 1661-237X ISSN 2296-5041 (electronic)
Oberwolfach Seminars
ISBN 978-3-319-59914-4 ISBN 978-3-319-59915-1 (eBook)
DOI 10.1007/978-3-319-59915-1
Library of Congress Control Number: 2017955698
Mathematics Subject Classification (2010): 19-02, 19Kxx, 46Lxx
© Springer International Publishing AG 2017
This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the
material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broad-
casting, reproduction on microfilms or in any other physical way, and transmission or information storage and
retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or
hereafter developed.
The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication
does not imply, even in the absence of a specific statement, that such names are exempt from the relevant
protective laws and regulations and therefore free for general use.
The publisher, the authors and the editors are safe to assume that the advice and information in this book are
believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors
give a warranty, express or implied, with respect to the material contained herein or for any errors or
omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in
published maps and institutional affiliations.

Printed on acid-free paper
This book is published under the trade name Birkhäuser, www.birkhauser-science.com
The registered company is Springer International Publishing AG
The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland
Contents
Preface ix
1 Introduction 1
2 Crossed products and the Mackey–Rieffel–Green machine
Siegfried Echterhoff 5
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 Some preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2.1 C∗-algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2.2 Multiplier algebras . . . . . . . . . . . . . . . . . . . . . . . 7
2.2.3 Commutative C∗-algebras and functional calculus . . . . . 7
2.2.4 Representation and ideal spaces of C∗-algebras . . . . . . . 9
2.2.5 Tensor products . . . . . . . . . . . . . . . . . . . . . . . . 10
2.3 Actions and their crossed products . . . . . . . . . . . . . . . . . . 11
2.3.1 Haar measure and vector-valued integration on groups . . . 11
2.3.2 C∗-dynamical systems and their crossed products . . . . . . 12
2.4 Crossed products versus tensor products . . . . . . . . . . . . . . . 16
2.5 The correspondence categories . . . . . . . . . . . . . . . . . . . . 19
2.5.1 Hilbert modules . . . . . . . . . . . . . . . . . . . . . . . . 20
2.5.2 Morita equivalences . . . . . . . . . . . . . . . . . . . . . . 21
2.5.3 The correspondence categories . . . . . . . . . . . . . . . . 24
2.5.4 The equivariant correspondence categories . . . . . . . . . . 25
2.5.5 Induced representations and ideals . . . . . . . . . . . . . . 26
2.5.6 The Fell topologies and weak containment . . . . . . . . . . 30
2.6 Green’s imprimitivity theorem and applications . . . . . . . . . . . 31
2.6.1 The imprimitivity theorem . . . . . . . . . . . . . . . . . . 31
2.6.2 The Takesaki–Takai duality theorem.. . . . . . . . . . . . . 39
2.6.3 Permanence properties of exact groups . . . . . . . . . . . . 40
2.7 Induced representations and the ideal structure of crossed products 43
2.7.1 Induced representations of groups and crossed products . . 43
2.7.2 The ideal structure of crossed products . . . . . . . . . . . 52
2.7.3 The Mackey machine for transformation groups . . . . . . . 61
v
vi Contents
2.8 The Mackey–Rieffel–Green machine for twisted crossed products . 64
2.8.1 Twisted actions and twisted crossed products . . . . . . . . 64
2.8.2 The twisted equivariant correspondence category and the
stabilisation trick . . . . . . . . . . . . . . . . . . . . . . . . 68
2.8.3 Twisted Takesaki–Takai duality . . . . . . . . . . . . . . . . 71
2.8.4 Stability of exactness under group extensions . . . . . . . . 71
2.8.5 Induced representations of twisted crossed products . . . . 73
2.8.6 Twisted group algebras, actions on K and Mackey’s little
group method . . . . . . . . . . . . . . . . . . . . . . . . . . 74
3 Bivariant KK-Theory and the Baum–Connes conjecure
Siegfried Echterhoff 81
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
3.2 Operator K-Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 83
3.3 Kasparov’s equivariant KK-theory . . . . . . . . . . . . . . . . . . 88
3.3.1 Graded C∗-algebras and Hilbert modules . . . . . . . . . . 88
3.3.2 Kasparov’s bivariant K-groups . . . . . . . . . . . . . . . . 90
3.3.3 The Kasparov product . . . . . . . . . . . . . . . . . . . . . 93
3.3.4 Higher KK-groups and Bott-periodicity . . . . . . . . . . . 98
3.3.5 Excision in KK-theory . . . . . . . . . . . . . . . . . . . . 106
3.4 The Baum–Connes conjecture . . . . . . . . . . . . . . . . . . . . . 107
3.4.1 The universal proper G-space . . . . . . . . . . . . . . . . . 107
3.4.2 The Baum–Connes assembly map . . . . . . . . . . . . . . 110
3.4.3 Proper G-algebras and the Dirac dual-Dirac method . . . . 115
3.4.4 The Baum–Connes conjecture for group extensions . . . . . 128
3.5 The going-down (or restriction) principle and applications . . . . . 129
3.5.1 The going-down principle . . . . . . . . . . . . . . . . . . . 129
3.5.2 Applications of the going-down principle . . . . . . . . . . . 136
3.5.3 Crossed products by actions on totally disconnected spaces 139
4 Quantitative K-theory for geometric operator algebras
Guoliang Yu 149
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
4.2 Geometric C∗-algebras . . . . . . . . . . . . . . . . . . . . . . . . 151
4.3 Quantitative K-theory for C∗-algebras . . . . . . . . . . . . . . . . 153
4.4 A quantitative Mayer–Vietoris sequence . . . . . . . . . . . . . . . 154
4.5 Dynamic asymptotic dimension and K-theory of crossed product
C∗-algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
4.6 Asymptotic dimension for geometric C∗-algebras and the Ku¨nneth
formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
4.7 Quantitative K-theory for Banach algebras . . . . . . . . . . . . . 161
Contents vii
5 Semigroup C∗-algebras
Xin Li 167
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
5.2 C∗-algebras generated by left regular representations . . . . . . . . 168
5.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
5.3.1 The natural numbers . . . . . . . . . . . . . . . . . . . . . . 169
5.3.2 Positive cones in totally ordered groups . . . . . . . . . . . 170
5.3.3 Monoids given by presentations . . . . . . . . . . . . . . . . 170
5.3.4 Examples from rings in general, and number theory in
particular . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
5.3.5 Finitely generated abelian cancellative semigroups . . . . . 174
5.4 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
5.4.1 Embedding semigroups into groups . . . . . . . . . . . . . . 175
5.4.2 Graph products . . . . . . . . . . . . . . . . . . . . . . . . . 176
5.4.3 Krull rings . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
5.5 C∗-algebras attached to inverse semigroups, partial dynamical
systems and groupoids . . . . . . . . . . . . . . . . . . . . . . . . . 182
5.5.1 Inverse semigroups . . . . . . . . . . . . . . . . . . . . . . . 182
5.5.2 Partial dynamical systems . . . . . . . . . . . . . . . . . . . 188
5.5.3 E´tale groupoids . . . . . . . . . . . . . . . . . . . . . . . . . 192
5.5.4 The universal groupoid of an inverse semigroup . . . . . . . 195
5.5.5 Inverse semigroup C∗-algebras as groupoid C∗-algebras . . 197
5.5.6 C∗-algebras of partial dynamical systems as C∗-algebras of
partial transformation groupoids . . . . . . . . . . . . . . . 200
5.5.7 The case of inverse semigroups admitting an idempotent
pure partial homomorphism to a group . . . . . . . . . . . 203
5.6 Amenability and nuclearity . . . . . . . . . . . . . . . . . . . . . . 204
5.6.1 Groups and groupoids . . . . . . . . . . . . . . . . . . . . . 205
5.6.2 Amenability for semigroups . . . . . . . . . . . . . . . . . . 208
5.6.3 Comparing reduced C∗-algebras for left cancellative
semigroups and their left inverse hulls . . . . . . . . . . . . 209
5.6.4 C∗-algebras generated by semigroups of projections . . . . 216
5.6.5 The independence condition . . . . . . . . . . . . . . . . . . 222
5.6.6 Construction of full semigroup C∗-algebras . . . . . . . . . 230
5.6.7 Crossed product and groupoid C∗-algebra descriptions of
reduced semigroup C∗-algebras . . . . . . . . . . . . . . . . 232
5.6.8 Amenability of semigroups in terms of C∗-algebras . . . . . 235
5.6.9 Nuclearity of semigroup C∗-algebras and the connection to
amenability . . . . . . . . . . . . . . . . . . . . . . . . . . . 238
5.7 Topological freeness, boundary quotients, and C∗-simplicity . . . . 239
5.8 The Toeplitz condition . . . . . . . . . . . . . . . . . . . . . . . . . 249
5.9 Graph products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256
5.9.1 Constructible right ideals . . . . . . . . . . . . . . . . . . . 257
5.9.2 The independence condition . . . . . . . . . . . . . . . . . . 261
viii Contents
5.9.3 The Toeplitz condition . . . . . . . . . . . . . . . . . . . . . 265
5.10 K-theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268
5.11 Further developments, outlook, and open questions . . . . . . . . . 270
6 Algebraic actions and their C∗-algebras
Joachim Cuntz 273
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273
6.2 Single algebraic endomorphisms . . . . . . . . . . . . . . . . . . . . 275
6.2.1 The K-theory of A[ϕ] . . . . . . . . . . . . . . . . . . . . . 278
6.2.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281
6.3 Actions by a family of endomorphisms, ring C∗-algebras . . . . . . 283
6.4 Regular C∗-algebras for ax+b -semigroups . . . . . . . . . . . . . 286
6.5 The K-theory for C∗(R(cid:2)R×) . . . . . . . . . . . . . . . . . . . . 289
λ
6.6 KMS-states. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292
7 Semigroup C∗-algebras and toric varieties
Joachim Cuntz 297
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297
7.2 Toric varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298
7.3 The regular C∗-algebra for a toric semigroup . . . . . . . . . . . . 301
Bibliography 307
Preface
This book grew out of the Oberwolfach Seminar “K-Theory for Group C∗-
Algebras and Semigroup C∗-Algebras” organized by the authors in October
2014. In this seminar, we discussed several classes of C∗-algebras of current
interest, including group C∗-algebras, crossed products, semigroup C∗-algebras
and C∗-algebras of algebraic actions. Since K-theory is the most important tool
to analyse and classify C∗-algebras, our main goal was to explain methods for
explicit K-theory computations, both in a general framework and in the case
of our particular classes of examples. Building on this, we showed how these
K-theory tools help to shed some light on the structure of our C∗-algebras. On
the one hand, we wanted to give a friendly introduction to these topics without
assuming too much background, and on the other hand, our intention was to
present an overview of recent developments.
This book essentially consists of expanded versions of our lecture notes for the
seminar, together with additional material and some new results. As most of the
topics of the seminar have not yet appeared in book form, this is an excellent
opportunity to give an overview as well as a unified and systematic treatment of
material which can only be found scattered across the literature.
We would like to thank the Mathematisches Forschungsinstitut Oberwolfach, its
director Professor Dr. Gerhard Huisken and its highly professional staff for pro-
viding an excellent environment in which to work. It is also a pleasure to thank
all the participants for their interest in the subject, their enthusiasm and many
interesting discussions, which altogether provided a very inspiring atmosphere.
The first named author has been supported by the DFG through CRC 878 and
by the ERC through AdG 267079. The second named author has been supported
by the DFG through CRC 878. The research of the third named author has been
supported by EPSRC grant EP/M009718/1. The work of the last author has
been partially supported by the US National Science Foundation and the Chinese
Science Foundation.
ix