ON TENSOR PRODUCTS OF GROUP C ALGEBRAS AND RELATED TOPICS

Description

ON TENSOR PRODUCTS OF GROUP C?-ALGEBRAS AND RELATED TOPICS CLAIRE ANANTHARAMAN-DELAROCHE Abstract. We discuss properties and examples of discrete groups in connec- tion with their operator algebras and related tensor products. Contents 1. Introduction 2 2. Preliminary background and notations 3 2.1. Maximal and minimal tensor products 3 2.2. Nuclear C?-algebras 4 2.3. The weak expectation property 6 2.4. Exact C?-algebras 7 2.5. Groups and operator algebras 8 2.6. Crossed products 10 3. Amenable groups and their C?-algebras 11 4. Kirchberg factorization property 14 5. Amenable dynamical systems 18 5.1. Definition and basic results 18 5.2. Boundary amenability 19 5.3. Amenable actions on universal compactifications 22 6. The Akemann-Ostrand property (AO) and related notions 24 6.1. Property (S) 24 6.2. Property (H) 26 6.3. The C?-algebra of the biregular representation 28 7. Applications of property (AO) 32 7.1. Non-nuclearity in K-theory 32 7.2. Solid von Neumann algebras 33 References 33 The author acknowledges the hospitality and support of the Centre Bernoulli at the EPFL in Lausanne, where she began to write this exposition in January 2007, and of the Fields Institute in Toronto where it was completed in October 2007. 1

• c?


• qq qq


• basic results


• amenable groups


• calkin algebra


• positive maps


• tt tt


• every homomorphism

Sujets

• Toronto
• Lausanne
• Calkin algebra

Informations

• Publié par: pefav
• Nombre de lectures: 17
• Langue: English

Extrait

ONTENSORPRODUCTSOFGROUP
C
∗
-ALGEBRASAND
RELATEDTOPICS

CLAIREANANTHARAMAN-DELAROCHE

Abstract.
Wediscusspropertiesandexamplesofdiscretegroupsinconnec-
tionwiththeiroperatoralgebrasandrelatedtensorproducts.

Contents
1.
Introduction
2
2.
Preliminarybackgroundandnotations
3
2.1.Maximalandminimaltensorproducts3
2.2.Nuclear
C
∗
-algebras4
2.3.Theweakexpectationproperty6
2.4.Exact
C
∗
-algebras7
2.5.Groupsandoperatoralgebras8
2.6.Crossedproducts10
3.
Amenablegroupsandtheir
C
∗
-algebras
11
4.
Kirchbergfactorizationproperty
14
5.
Amenabledynamicalsystems
18
5.1.Definitionandbasicresults18
5.2.Boundaryamenability19
5.3.Amenableactionsonuniversalcompactifications22
6.
TheAkemann-Ostrandproperty
(
AO
)
andrelatednotions
24
6.1.Property(
S
)24
6.2.Property(
H
)26
6.3.The
C
∗
-algebraofthebiregularrepresentation28
7.
Applicationsofproperty
(
AO
)32
7.1.Non-nuclearityin
K
-theory32
7.2.SolidvonNeumannalgebras33
References33

TheauthoracknowledgesthehospitalityandsupportoftheCentreBernoulliattheEPFLin
Lausanne,whereshebegantowritethisexpositioninJanuary2007,andoftheFieldsInstitute
inTorontowhereitwascompletedinOctober2007.
1

2

CLAIREANANTHARAMAN-DELAROCHE

QQQ((1.
Introduction
Thisexpositionisanextendedversionofnotesforalecturepresentedatthe
CentreBernoulliinJanuary2007.Ourpurposeistoreviewthreefactorization
problemsrelatedwiththebiregularrepresentationofaninfinitecountablediscrete
group
1
Γ.
Let
λ
and
ρ
berespectivelytheleftandrightregularrepresentationsofΓ.
Thebiregularrepresentationistherepresentation
λ
∙
ρ
:(
s,t
)
7→
λ
(
s
)
ρ
(
t
)ofΓ
×
ΓintheHilbertspace
`
2
(Γ).If
C
∗
(Γ),
C
λ
∗
(Γ),denoterespectivelythefulland
reduced
C
∗
-algebrasassociatedwithΓ,thebiregularrepresentationextendsto
homomorphisms
λ
∙
ρ
:
C
∗
(Γ)
⊗
max
C
∗
(Γ)
→B
(
`
2
(Γ))
,
2∗∗(
λ
∙
ρ
)
r
:
C
λ
(Γ)
⊗
max
C
λ
(Γ)
→B
(
`
(Γ))
,
thankstotheuniversalpropertyofthemaximaltensorproduct.Itisnaturalto
askforwhichgroupsthehomomorphisms(
λ
∙
ρ
)
r
or
λ
∙
ρ
canbefactorizedthrough
minimaltensorproducts:
C
λ
∗
(Γ)
⊗
max
C
Q
λ
∗
(Γ)
C
∗
(Γ)
⊗
max
C
Q
∗
(Γ)
QQQQ
(
Q
λ
Q
∙
ρ
)
r
QQQQQQ
λ
Q
∙
ρ
P
r


QQQQQ
(
(
P


QQ
C
λ
∗
(Γ)
⊗
min
C
λ
∗
(Γ)
/
/
B
(
`
2
(Γ))
,C
∗
(Γ)
⊗
min
C
∗
(Γ)
/
/
B
(
`
2
(Γ))
.
Inthereducedsettingtheanswerissimple.Thisisrecalledinsection2.
The
threefollowingconditionsareequivalent:
(i)Γ
isamenable;
(ii)
P
r
isanisomorphism;
(iii)(
λ
∙
ρ
)
r
passestoquotient.
Asalways,itismuchmoredifficulttodealwithfull
C
∗
-algebras.Section3is
devotedtothissituationwhichhasbeenstudiedbyKirchberg.HedefinedΓto
havethefactorizationproperty(
F
)if
λ
∙
ρ
passestoquotient.Amonghismany
deepresultsarethefollowingones:
•
Everyresiduallyfinite
2
grouphasproperty
(
F
)
.
•
TheconverseistrueforgroupshavingtheKazhdanproperty
(
T
)
.
Thefamilyofgroupssuchthat
P
isanisomorphismiscontainedintheclass
ofproperty(
F
)groups.Itisstillmysterious.Kirchbergprovedthat
P
isan
isomorphismifandonlyif
C
∗
(Γ)
hasLance’sweakexpectationproperty
(WEP).
Oneofthemostimportantopenproblemis
whetherthefull
C
∗
-algebraofthefree
1
Unlessstatedotherwise,inthispaperallgroupsareassumedtobeinfinitecountable.
2
Wereferwiththenextsectionsfornotionsusedandnotdefinedintheintroduction.

TENSORPRODUCTSOFGROUP
C
∗
-ALGEBRAS

TTT**3

groupwithinfinitelymanygeneratorshasthe
WEP.Theanswertothisquestion
wouldhaveimportantconsequencesinoperatoralgebrastheory.
Anotherimportantopenproblemconcerningfull
C
∗
-algebrasis
whetherthe
exactnessof
C
∗
(Γ)
impliestheamenabilityof
Γ.Kirchberggavea
positiveanswer
whenassumingthat
Γ
hasproperty
(
F
).
Forthisquestionalso,thesituationisnicerforreduced
C
∗
-algebras:
C
λ
∗
(Γ)
is
exactifandonlyif
Γ
isboundaryamenable
.Weintroducethislatternotionin
Section4.OurmainpurposeistoprovidetoolsforthestudyoftheAkemann-
OstrandpropertyinSection5.
LetΓbethefreegroupwith
n
≥
2generators.AkemannandOstrandproved
that,although(
λ
∙
ρ
)
r
doesnotfactorizevia
C
λ
∗
(Γ)
⊗
min
C
λ
∗
(Γ),thisishowever
thecaseforitscompositionwiththeprojection
Q
from
B
(
`
2
(Γ))ontotheCalkin
algebra
Q
(
`
2
(Γ)):
C
λ
∗
(Γ)
⊗
max
C
λ
∗
(Γ)
TTTP
TTT
Q
T
◦
T
(
T
λ
T
∙
ρ
)
r
rTTTC
λ
∗
(Γ)
⊗
min
C
λ
∗
(Γ)
/
/
Q
(
`
2
(Γ))
,
Whensuchaphenomenonoccurs,onesaysthatΓhastheAkemann-Ostrand
property(
AO
).Weintroducetwoboundaryamenabilitypropertieswhichimply
the(
AO
)property:property(
H
)duetoHigsonandtheweakercondition(
S
)due
toSkandalis.Amongthegroupswithproperty(
AO
)aretheamenablegroupsand
Gromovhyperbolicgroups.
Non-amenablegroupswithproperty(
AO
)haveveryremarkablefeatures.We
mentiontwooftheminSection6.FirsttheapplicationduetoSkandalisgiving
examplesof
C
∗
-algebrasnotnuclearin
K
-theory
.Second,themorerecentresult
ofOzawawhotookadvantageofproperty(
AO
)togiveasimpleproofthat
the
vonNeumannalgebraofthefreegroup
F
n
,
n
≥
2
,(andofmanyothergroups)is
aprimefactor
.
Section2isdevotedtobasicpreliminaryresultsanddefinitionsthatwehave
triedtomakeascondensedandcompleteaspossible,forthereader’sconvenience.
Allalongthetext,ourobjectiveistogiveprecisedefinitions,sufficientlymany
examples,andachoiceofproofswethinktoberepresentativeandnottootech-
nical.Wehavetriedtoprovidereferencesforassertionsstatedwithoutproof.
2.
Preliminarybackgroundandnotations
Forfundamentalsof
C
∗
-algebraswereferto[20,17,49,66].
2.1.
Maximalandminimaltensorproducts.
Let
A
and
B
betwo
C
∗
-algebras,
anddenoteby
A

B
theiralgebraictensorproduct.A
C
∗
-norm
on
A

B
isa
normofinvolutivealgebrasuchthat
k
x
∗
x
k
=
k
x
k
2
for
x
∈
A

B
.Thereare

4

CLAIREANANTHARAMAN-DELAROCHE

twonaturalwaystodefine
C
∗
-normsonthe
∗
-algebra
A

B
.Recallfirstthata
representationofa
C
∗
-algebrainaHilbertspace
H
isahomomorphism
3
from
A
intothe
C
∗
-algebra
B
(
H
)ofallboundedoperatorson
H
.Animportantfactto
keepinmindisthathomomorphismsof
C
∗
-algebrasareautomaticallycontractive
.spamThe
maximal
C
∗
-norm
isdefinedby
k
x
k
max
=sup
k
π
(
x
)
k
where
π
runsoverallhomomorphismsfrom
A

B
intosome
B
(
H
)
4
.The
maximal
tensorproduct
isthecompletion
A
⊗
max
B
of
A

B
forthis
C
∗
-norm.
The
minimaltensorproduct
isdefinedbytakingspecific