ON TENSOR PRODUCTS OF GROUP C ALGEBRAS AND RELATED TOPICS

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