Table Of ContentAdvanced Courses in Mathematics CRM Barcelona
Thierry Giordano
David Kerr
N. Christopher Phillips
Andrew Toms
Crossed Products
of C*-Algebras,
Topological Dynamics,
and Classification
Advanced Courses in Mathematics
CRM Barcelona
Centre de Recerca Matemàtica
Managing Editor:
Enric Ventura
More information about this series at http://www.springer.com/series/5038
Thierry Giordano • David Kerr
N. Christopher Phillips • Andrew Toms
Crossed Products
of C*-Algebras, Topological
Dynamics, and Classification
Editor for this volume:
Francesc Perera, Universitat Autònoma de Barcelona
Thierry Giordano David Kerr
Department of Mathematics and Statistics Department of Mathematics
University of Ottawa Texas A&M University
Ottawa, ON, Canada College Station, TX, USA
N. Christopher Phillips Andrew Toms
Department of Mathematics Department of Mathematics
University of Oregon Purdue University
Eugene, OR, USA West Lafayette, IN, USA
ISSN 2297-0304 ISSN 2297-0312 (electronic)
Advanced Courses in Mathematics - CRM Barcelona
ISBN 978-3-319-70868-3 ISBN 978-3-319-70869-0 (eBook)
https://doi.org/10.1007/978-3-319-70869-0
Library of Congress Control Number: 2018948737
Mathematics Subject Classification (2010): 46L05, 46L10, 46L35, 46L40, 46L55
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Foreword
The classi(cid:12)cation of C -algebras via K-theoretic invariants was invigorated in the
(cid:3)
years prior to 2011, in part due to the discovery that the Cuntz semigroup (which
is,roughlyspeaking,thesemigroupofisomorphismclassesofcountablygenerated
Hilbert modules) contains an enormous amount of information. Explicit calcula-
tionsofthissemigroupledtoclassi(cid:12)cationtheoremsforC -algebrasofminimaldy-
(cid:3)
namical systems, among others. Connections between classi(cid:12)cation, C -algebras,
(cid:3)
and dynamics were strengthened by progresses on proving the hyper(cid:12)niteness of
amenablegroupactionsontheCantorset.Sincethen,theapplicationsoftheCuntz
semigrouptoclassi(cid:12)cationproblemshavebecomeanactiveandrapidlydeveloping
branch of functional analysis.
Motivatedbythesedevelopments,anintensiveresearchprogramwassetupat
theCentredeRecerca Matema(cid:18)tica(CRM)Barcelonafrom January to July 2011,
centered around the classi(cid:12)cation program of C -algebras.Inthe(cid:12)nalpartofthe
(cid:3)
semester,aconferenceandanintensiveadvancedcourseondynamicalsystemsand
operatoralgebraswereorganized.Theoverallaimofthisresearchprogramwasto
bringrenownedinternationalexpertstogetherinordertosharetheirinsights,and
provide the means for young mathematicians with research interests in this area
to learnaboutthe latest developments, to interact with researchers fromother
countries, and to initiate future collaborations.
The advanced course, entitled Crossed Products of C -Algebras, Topologi-
(cid:3)
cal Dynamics, and Classi(cid:12)cation, took place from June 14 to June 23, 2011. It
consisted of four series of lectures, delivered by Thierry Giordano (University
of Ottawa) on Minimal Topological Systems and Orbit Equivalence, David Kerr
(TexasA&M)onC -AlgebrasandTopologicalDynamics:FiniteApproximationand
(cid:3)
Paradoxicality,N.ChristopherPhillips(UniversityofOregon)onAnIntroductionto
Crossed Product C -AlgebrasandMinimalDynamics,andAndrewS.Toms(Purdue
(cid:3)
University) on The Cuntz Semigroup and the Classi(cid:12)cation of C -Algebras. This
(cid:3)
bookisbasedonnotesthattheauthorsusedfortheselectures,whichhaveevolved
to a considerable extent over the last few years. Thus, this text provides a very
solid introduction to the theory of topological dynamics and crossed products,
blended with the role of the Cuntz semigroup { an invariant that is intimately
related to the Elliott invariant { in the classi(cid:12)cation program.
Sincethedi(cid:11)erentaspectsofthebookarepresentedinalogicalsequence,the
de(cid:12)nition and main technical features of the Cuntz semigroup appear in the (cid:12)rst
part. (The so-called Cuntz comparison between positive elements in a C -algebra
(cid:3)
isanimportantnotionthatisusedthroughoutthebook.)Thissemigroup(orrather,
itsorder)constitutesthelinkbetweentwoseeminglyunrelatedregularityproperties,
namely (cid:12)nite nuclear dimension and absorption by the so-called Jiang{Su algebra
Z. This link is speci(cid:12)ed in the statement of the Toms{Winter conjecture.
v
vi Foreword
Group actions and crossed products of C -algebras are the main theme of
(cid:3)
the second part, which contains a large collection of examples and explicit com-
putations. This part focuses on methods and ideas rather than technical aspects,
although the methods do sometimes get quite technical. A thorough list of refer-
ences points the interested reader to the relevant literature where full proofs can
be found. This second part summarizes the basic structure of crossed products,
crossed products of simple C -algebras by (cid:12)nite groups (including a discussion of
(cid:3)
the Rokhlin property), crossed products by minimal homeomorphisms, and the
theory of large subalgebras with applications to crossed products.
The third part o(cid:11)ers a treatment of crossed products via internal and exter-
nal approximation ideas, for groups (e.g., amenability and so(cid:12)city), C -algebras
(cid:3)
(e.g., nuclearity and being MF), and groups acting on spaces.Roughlyspeaking,
internal approximation is achieved by analysing subobjectsof the given object,
whileinthecaseofexternalapproximationthisisdonebystudyingotherobjects into
whichthegivenobjectmaps.Thispointofviewisrelated to certain aspectsofthe
classi(cid:12)cationprogram,whicharealsoexploredinthisthirdpart,particularlyincon-
nection with regularity properties that re(cid:13)ect a(noncommutative)zero-dimensional
behaviour.
The (cid:12)nal part of the book is devoted to developments in topological orbit
equivalence,inparticulartheclassi(cid:12)cationofminimalactionsof(cid:12)nitelygenerated
Abelian groups on the Cantor set. For this purpose, an invariant for topological
dynamical systems is provided, and the class of AF equivalence relations on the
Cantorsetiscompletelyclassi(cid:12)edbythisinvariant.Minimalactionsareshownto
be orbit equivalent toAF relations. Furthermore, areviewon topological fullgroups
associated to dynamicalsystems is provided, as well as results on the realization of
ergodic minimal systems by Cantor minimal systems.
The scientific committee of the saidintensiveresearchprogramonTheCuntz
Semigroup and Classi(cid:12)cation of C -Algebras was held at the CRM fromJanuary
(cid:3)
toJuly2011.Thescienti(cid:12)ccommitteeoftheintensiveresearchprogramconsistedof
George A. Elliott (University of Toronto), Andrew S. Toms (Purdue University),
Nathanial P. Brown (Pennsylvania State University), Joachim Cuntz (Universit(cid:127)at
Mu(cid:127)nster), Marius Dadarlat (Purdue University), and Mikael R(cid:28)rdam (University
of Copenhagen). The local organising committee consisted of Ramon Antoine,
Pere Ara, Joan Bosa, and Francesc Perera (Universitat Aut(cid:18)onoma de Barcelona).
The intensive research program was made possible not only by the support
of the CRM, but also of the National Science Foundation, AGAUR (2010ARCS
1-00098), the Ingenio Mathematica Programme (PMII-C5-0332), the Ministerio
de Ciencia e Innovaci(cid:19)on (MTM2010-10838-E, as well as MTM2008-06201-C02-
01/MTM), and the Comissionat per Universitats i Recerca de la Generalitat de
Catalunya (2009 SGR 1389).
Bellaterra, May 2017 Francesc Perera
Contents
I The Cuntz Semigroup and the Classification of C∗-Algebras 1
By Andrew S. Toms
1 Introduction 3
2 The Cuntz Semigroup 5
2.1 The Category Cu. . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2 Functionals and Cu . . . . . . . . . . . . . . . . . . . . . . . . . . 9
3 Structure of the Cuntz Semigroup 11
4 Elliott’s Program 17
4.1 Nuclearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
4.2 Counterexamples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
5 A New Conjecture 21
5.1 Z-Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
5.2 Strict Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
5.3 Nuclear Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
5.4 The Conjecture: Principle and Progress . . . . . . . . . . . . . . . 24
6 Nuts and Bolts: Proof Sketches 29
6.1 Proof Sketch for Theorem 5.4.7 . . . . . . . . . . . . . . . . . . . . 29
6.2 Proof Sketch for (a Special Case of) Theorem 5.4.8 . . . . . . . . . 31
Bibliography 35
II An Introduction to Crossed Product C*-Algebras
and Minimal Dynamics 39
By N. Christopher Phillips
7 Introduction and Motivation 41
8 Group Actions 51
8.1 Examples of Group Actions on Locally Compact Spaces . . . . . . 51
8.2 Examples of Group Actions on Noncommutative C∗-Algebras . . . 65
8.3 Additional Examples of Generalized Gauge Actions . . . . . . . . . 83
vii
viii Contents
9 Group C∗-algebras and Crossed Products 93
9.1 C∗-Algebras of Discrete Groups . . . . . . . . . . . . . . . . . . . . 93
9.2 Simplicity of the Reduced C∗-Algebra of a Free Group . . . . . . . 115
9.3 C∗-Algebras of Locally Compact Groups . . . . . . . . . . . . . . . 119
9.4 Crossed Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
9.5 Reduced Crossed Products. . . . . . . . . . . . . . . . . . . . . . . 139
9.6 Computation of Some Examples of Crossed Products . . . . . . . . 151
10 Some Structure Theory for Crossed Products by Finite Groups 169
10.1 Introductory Remarks on the Structure of C∗-Algebras . . . . . . . 169
10.2 Crossed Products by Finite Groups . . . . . . . . . . . . . . . . . . 183
10.3 The Rokhlin Property for Actions of Finite Groups . . . . . . . . . 188
10.4 The Tracial Rokhlin Property for Actions of Finite Groups . . . . 204
11 Crossed Products by Minimal Homeomorphisms 221
11.1 Minimal Actions and Their Crossed Products . . . . . . . . . . . . 221
11.2 Classifiability: Introduction and a Special Case . . . . . . . . . . . 232
11.3 Minimal Homeomorphisms of Finite-Dimensional Spaces . . . . . . 247
12 Large Subalgebras and Applications to Crossed Products 275
12.1 The Cuntz Semigroup . . . . . . . . . . . . . . . . . . . . . . . . . 275
12.2 Large Subalgebras . . . . . . . . . . . . . . . . . . . . . . . . . . . 282
12.3 Basic Properties of Large Subalgebras . . . . . . . . . . . . . . . . 290
12.4 Large Subalgebras and the Radius of Comparison . . . . . . . . . . 298
12.5 Large Subalgebras in Crossed Products by Z . . . . . . . . . . . . 308
12.6 Application to the Radius of Comparison of Crossed Products . . 315
12.7 Open Problems on Large Subalgebras . . . . . . . . . . . . . . . . 324
Bibliography 329
III C∗-Algebras and Topological Dynamics:
Finite Approximation and Paradoxicality 349
By David Kerr
13 Introduction 351
14 Internal Measure-Theoretic Phenomena 361
14.1 Amenable Groups and Nuclearity . . . . . . . . . . . . . . . . . . . 361
14.2 Amenable Actions, Nuclearity, and Exactness . . . . . . . . . . . . 365
14.3 The Type Semigroup, Invariant Measures, and Pure Infiniteness. . 368
14.4 The Universal Minimal System . . . . . . . . . . . . . . . . . . . . 376
14.5 Minimal Actions, Pure Infiniteness, and Nuclearity . . . . . . . . . 379
Contents ix
15 External Measure-Theoretic Phenomena 381
15.1 Sofic Groups, Sofic Actions, and Hyperlinearity . . . . . . . . . . . 381
15.2 Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383
15.3 Combinatorial Independence. . . . . . . . . . . . . . . . . . . . . . 388
15.4 Mean Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . 390
16 Internal Topological Phenomena 393
16.1 Locally Finite Groups and AF Algebras . . . . . . . . . . . . . . . 393
16.2 Dimension and K-Theoretic Classification . . . . . . . . . . . . . . 394
16.3 Minimal Homeomorphisms of Zero-Dimensional Spaces . . . . . . . 399
16.4 Minimal Homeomorphisms of Finite-Dimensional Spaces . . . . . . 402
16.5 Mean Dimension and Comparison in the Cuntz Semigroup . . . . . 406
17 External Topological Phenomena 413
17.1 Groups Which Are Locally Embeddable into Finite Groups . . . . 413
17.2 Chain Recurrence, Residually Finite Actions, and MF Algebras . . 416
Bibliography 425
IV Minimal Topological Systems and Orbit Equivalence 433
By Thierry Giordano
18 Introduction 435
19 Cantor Dynamics 437
19.1 Definitions of Some Dynamical Concepts . . . . . . . . . . . . . . . 437
19.2 Cantor Minimal Systems . . . . . . . . . . . . . . . . . . . . . . . . 438
19.3 Dynamic and Ordered Bratteli Diagrams. . . . . . . . . . . . . . . 438
19.3.1 Bratteli Diagrams . . . . . . . . . . . . . . . . . . . . . . . 438
19.3.2 Ordered Bratteli Diagrams and the Bratteli–Vershik Model 440
20 E´tale Equivalence Relations 445
20.1 E´tale Equivalence Relations . . . . . . . . . . . . . . . . . . . . . . 445
20.2 Isomorphism and Orbit Equivalence . . . . . . . . . . . . . . . . . 448
20.3 AF-Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449
20.4 Invariants of E´tale Equivalence Relations . . . . . . . . . . . . . . 452
20.4.1 Pre-Ordered and Ordered Groups. . . . . . . . . . . . . . . 452
20.4.2 Dimension Groups . . . . . . . . . . . . . . . . . . . . . . . 457
20.4.3 Pre-Ordered Groups Associated to an E´tale Equivalence
Relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 458
20.5 The Bratteli–Elliott–Krieger Theorem . . . . . . . . . . . . . . . . 461
21 The Absorption Theorem 463
