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A BIVARIANT CHERN CHARACTER FOR FAMILIES OF SPECTRAL TRIPLES

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53 pages
A BIVARIANT CHERN CHARACTER FOR FAMILIES OF SPECTRAL TRIPLES Denis PERROT SISSA, via Beirut 2-4, 34014 Trieste, Italy September 7, 2004 Abstract In this paper we construct a bivariant Chern character defined on “families of spectral triples”. Such families should be viewed as a version of unbounded Kasparov bimodules adapted to the category of bornological algebras. The Chern character then takes its values in the bivariant entire cyclic cohomology of Meyer. The basic idea is to work within Quillen's algebra cochains formalism, and construct the Chern character from the exponential of the curvature of a superconnection, leading to a heat kernel regularization of traces. The obtained formula is a bivariant generalization of the JLO cocycle. Keywords: Bivariant entire cyclic cohomology, bornological algebras. 1 Introduction Recall that according to Connes [6], a noncommutative space is described by a spectral triple (A,H, D), where H is a separable Hilbert space, A an asso- ciative algebra represented by bounded operators on H, and D is a self-adjoint unbounded (Dirac) operator with compact resolvent, such that the commutator [D, a] is densely defined for any a ? A and extends to a bounded operator. The triple (A,H, D) carries a nontrivial homological information as a K-homology class of A.

  • unbounded bimodules

  • vector space

  • triples over

  • algebras

  • any bounded

  • bivariant entire

  • cyclic cohomology

  • algebras provide

  • cohomology theory


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ABIVARIANTCHERNCHARACTERFOR
FAMILIESOFSPECTRALTRIPLES

DenisPERROT
SISSA,viaBeirut2-4,34014Trieste,Italy
perrot@fm.sissa.it
September7,2004

Abstract
InthispaperweconstructabivariantCherncharacterdefinedon
“familiesofspectraltriples”.Suchfamiliesshouldbeviewedasaversion
ofunboundedKasparovbimodulesadaptedtothecategoryofbornological
algebras.TheCherncharacterthentakesitsvaluesinthebivariantentire
cycliccohomologyofMeyer.ThebasicideaistoworkwithinQuillen’s
algebracochainsformalism,andconstructtheCherncharacterfromthe
exponentialofthecurvatureofasuperconnection,leadingtoaheatkernel
regularizationoftraces.Theobtainedformulaisabivariantgeneralization
oftheJLOcocycle.
Keywords:
Bivariantentirecycliccohomology,bornologicalalgebras.

1Introduction
RecallthataccordingtoConnes[6],anoncommutativespaceisdescribedby
aspectraltriple(
A
,
H
,D
),where
H
isaseparableHilbertspace,
A
anasso-
ciativealgebrarepresentedbyboundedoperatorson
H
,and
D
isaself-adjoint
unbounded(Dirac)operatorwithcompactresolvent,suchthatthecommutator
[
D,a
]isdenselydefinedforany
a
∈A
andextendstoaboundedoperator.The
triple(
A
,
H
,D
)carriesanontrivialhomologicalinformationasa
K
-homology
classof
A
.ThemajormotivationleadingConnestointroduceperiodiccyclic
cohomology[5]isthatthelatteristhenaturalreceptacleforaCherncharacter
definedon
finitelysummable
representativesof
K
-homology.Thisfinitenesscon-
ditionwasremovedlaterandreplacedbytheweakerconditionof
θ
-summability,
i.e.theheatkernelexp(

tD
2
)associatedtothelaplacianoftheDiracoperator
hastobetrace-classforany
t>
0[7].Inthatcase,thealgebra
A
hastobe
endowedwithanormandtheCherncharacterofthespectraltripleisexpressed
asan
infinite-dimensionalcocycle
intheentirecycliccohomology
HE

(
A
).Ex-
cepttheoriginalconstructionofConnes,oneoftheinterestingexplicitformulas
forsuchaCherncharacterisprovidedbytheso-calledJLOcocycle[21].Here
theheatkernelplaystheroleofa
regulator
inthealgebraofoperatorson
H
,and
theJLOformulaincorporatesthedataofthespectraltripleinarathersimple
way.ThisledConnesandMoscovicitousethepowerfulmachineryofasymp-
toticexpansionsoftheheatkernel,givingriseto
local
expressionsextendingthe
1

classicalindextheoremsofAtiyah-Singertoveryinterestingnon-commutative
situations[9,10].
InthispaperwewanttogeneralizetheconstructionofaCherncharacterto
familiesofspectraltriples“overanoncommutativespace”describedbyasecond
associativealgebra
B
.Inthecontextof
C

-algebras,suchobjectscorrespond
totheunboundedversionofKasparov’sbivariant
K
-theory[4].Inthispicture,
anelementofthegroup
KK
(
A
,
B
)isrepresentedbyatriple(
E
,ρ,D
),where
E
isanHilbert
B
-module.
D
shouldbeviewedasafamilyofDiracoperatorsover
B
,actingbyunboundedendomorphismson
E
,and
ρ
isarepresentationof
A
asboundedendomorphismsof
E
commutingwith
D
moduloboundedendomor-
phisms.Intheparticularcase
B
=
C
,thisdescriptionjustreducestospectral
triplesover
A
.TheconstructionofageneralbivariantCherncharacterasa
transformationfromanalgebraicversionof
KK
(
A
,
B
)(for
A
and
B
notneces-
sarily
C

-algebras)toabivariantcycliccohomologyhasalreadybeenconsidered
byseveralauthors.ForexampleNistor[25,26]constructedabivariantChern
characterfor
p
-summablequasihomomorphisms[12],withvaluesintheJones-
Kasselbivariantcycliccohomologygroups.CuntzandQuillenalsoconstructed
abivariantCherncharacterundersomesummabilityassumptions,withvalues
intheirowndescriptionofthebivariantperiodiccyclictheory[15,16].Onthe
otherhand,Puschniggconstructedawell-behavedcycliccohomologytheoryfor
C

-algebras,namelythe
localcycliccohomology
[29,30].Upongeneralizationof
apreviousworkofCuntz[13],thelocalcyclictheoryappearstobethesuitable
targetforacompletelygeneralbivariantCherncharacter(withoutsummabil-
ityassumptions)definedofKasparov’s
K
-theory.However,theexistenceand
propertiesofsuchconstructionsareoftenbasedonexcisionincycliccohomology
andtheuniversalpropertiesofbivariant
K
-theory.Byconsideringunbounded
bimoduleswewillfollowadifferentway,involvingheatkernelregularization
inthespiritoftheJLOcocycle,keepinginmindthatweareinterestedin
ex-
plicitformulas
forabivariantCherncharacterincorporatingthedata
ρ
and
D
.
Ourmotivationmainlycomesfromthepotentialapplicationstomathematical
physics,especiallyquantumfieldtheoryandstring/branetheory,wheresuch
objectsarisenaturally:

Theheatkernelmethodadmitsafunctionalintegralrepresentation.The
quantitiesunderinvestigationthencorrespondtoexpectationvaluesofobserv-
ablescorrespondingtosomequantum-mechanicalsystem.Thiswasfirstused
byAlvarez-Gaume´andWittenintheirstudyofmixed-gravitationalanomalies
[1],andledtotheasymptoticsymbolcalculusofGetzler[2].

ThebasicideaofintroducingaheatkernelregularizationofCherncharacters
inclassicaldifferentialgeometryisduetoQuillen[31].Bismutthensuccesfully
appliedthismethodinhisapproachoftheAtiyah-Singerindextheoremfor
familiesofellipticoperatorsonsubmersions[3].Itisworthmentioningthat
Bismutalsousesastochasticrepresentationoftheheatkernel.

TheBismut-Quillenapproachisessentialfortheanalyticandtopological
understandingofanomalies(bothchiralandgravitational)inquantumfield
theory[27,28].AbivariantCherncharacterdesignedinanequivariantsetting
mayshedsomelightontheinterplaybetweenBRScohomologyandtherecently
discoveredcycliccohomologyofHopfalgebras[10,11].

Twisted
K
-theoryand
K
-homologyrecentlyappearedinthephysicsliterature
2

throughtheclassificationof
D
-branes[23,34].Thisalsofallsintothescopeof
abivariantCherncharacter.
Firstwehavetoconsidertherightcategoryofalgebras.Forourpurpose,
itturnsoutthat
bornologicalassociativealgebras
areexactlywhatweneed.
Theseareassociativealgebrasendowedwithanadditionalstructuredescribing
thenotionofa
boundedsubset
.Completebornologicalalgebrasprovidethe
generalframeworkforentirecycliccohomology.Thistheoryhasbeendeveloped
indetailbyMeyerin[24].Theinterestingfeatureofthebivariantentirecyclic
cohomologyisthatitcontainsinfinite-dimensionalcocyclesandthuscanbeused
asthereceptacleofabivariantCherncharacterforourfamiliesofspectraltriples
carryingsomepropertiesof
θ
-summability.Giventwocompletebornological
algebras
A
and
B
,wewillconsiderthe
Z
2
-gradedsemigroupΨ

(
A
,
B
),

=
0
,
1,of
unbounded
A
-
B
-bimodules
.ThelatterisanadaptationofKasparov’s
unboundedbimodulestotherealmofbornologicalalgebras.Inourgeometric
picture,suchabimodulerepresentsafamilyofspectraltriplesoverthenon-
commutativespace
B
.OuraimistoconstructanexplicitformulaforaChern
characterdefinedonthesubsemigroupof
θ
-summablebimodules,
ch:Ψ

θ
(
A
,
B
e
)

HE

(
A
,
B
)
,

=0
,
1
,
(1)
carryingsuitablepropertiesofadditivity,differentiablehomotopyinvariance
andfunctoriality.Here
HE

(
A
,
B
)isthebivariantentirecycliccohomology
of
A
and
B
,and
B
e
istheunitalizationof
B
.Onthetechnicalside,wewill
useboththe
X
-complexdescriptionofcycliccohomologyduetoCuntz-Quillen
[15,16],andtheusual(
b,B
)-complexofConnes.The
X
-complexisusefulfor
someconceptualexplanationsoftheabstractpropertiesofcyclic(co)homology.
Givenacompletebornologicalalgebra
A
,itsentirecyclichomologyiscomputed
bythesupercomplex[24]
X
(
TA
):
TA

Ω
1
TA

,
(2)
where
TA
isthe
analytictensoralgebraof
A
,obtainedbyacertainbornological
completionofthetensoralgebraover
A
,andΩ
1
TA


1
TA
/
[
TA
,
Ω
1
TA
]
isthecommutatorquotientspaceoftheuniversalone-formsover
TA
.This
meansthattheentirecyclichomologyof
A
iscompletelydescribedthrough
thehomologicalpropertiesofitsanalytictensoralgebraindimension0and
1.Furthermore,takingtheanalytictensoralgebraof
TA
isharmless:indeed
X
(
TA
)and
X
(
TTA
)arehomotopicallyequivalentcomplexes.Inotherwords,
entirecyclichomologydoesnotdistinguishbetweenacompletebornological
algebraanditssuccessivenestedanalytictensoralgebras.Thisisaparticular
caseoftheanalyticversion[24]ofGoodwillie’stheorem[18].Thisresultisa
keypointofourbivariantCherncharacter.Theconstructionof(1)willfollow
twosteps:
a)UsingtheGoodwillietheorem,wefirstconstructaninvertiblebivariantclass
[
γ
]

HE
0
(
A
,
TA
)realizingtheequivalencebetweentheentirecyclichomologies
of
A
and
TA
.
b)WeconsiderabimoduleinΨ

(
A
,
B
e
).Thenundercertain
θ
-summability
conditions,weconstructanelement[
χ
]

HE

(
TA
,
B
)involvingtheexpo-
nentialofthecurvatureofasuperconnection,whichautomaticallyincorporates
3

thedesiredheatkernelregularization.ThisstepusesQuillen’stheoryofalgebra
cochainsasanessentialtool[32,33].Thenthecomposition[
γ
]

[
χ
]

HE

(
A
,
B
)
isthebivariantCherncharacter(1).
Thepaperisorganizedasfollows.Insections2,3werecallthebasic
definitionsandpropertiesofbornologicalspacesandentirecycliccohomol-
ogy.Insection4wepresentourconstructionoftheGoodwillieequivalence
[
γ
]

HE
0
(
A
,
TA
).ThesemigroupofunboundedbimodulesΨ

(
A
,
B
)isintro-
ducedinsection5.Sections6and7aredevotedtothefundamentalconstruction
oftheelement
χ

HE

(
TA
,
B
).Finally,weendthepaperwithanapplication
ofourCherncharactertothenon-bivariantcases,namelyordinary
K
-theory
and
K
-homologyinsection8.Inparticularwecheckthatthecomposition
producton
HE
describescorrectlytheindexpairingbetweenidempotentsand
spectraltriples.Besides,thestudyoftheBottclassallowstonormalizethebi-
variantCherncharacter.Theappendixcontainsastraightforwardadaptation,
tobornologicalalgebras,ofQuillen’salgebracochainsformalism.
Wewouldliketomentionalastpoint.Thereisapriorinoobviousinter-
sectionproductΨ(
A
,
B
)
×
Ψ(
B
,
C
)

Ψ(
A
,
C
)asinKasparovtheory.Alsowe
willneveraskifourconstructioniscompatiblewithsuchaproduct.Infact,it
ispossibletoshowthatthebivariantCherncharacteriscompatiblewiththe
Kasparovproducton
p
-summablequasihomomorphisms,butthisinvolvesare-
tractionofourentirecocyclesontoperiodicones.Thesematterswillbetreated
elsewhere.
Allalgebrasaresupposedtobebasedonthegroundfield
C
.Weworkin
thenon-unitalgradedcategory,i.e.homomorphismsbetweenalgebrasdonot
necessarilypreserveunits,andalloperationslikecommutators,tensorproducts,
etc...involvinggradedobjectsareautomaticallygraded.
2Bornology
Thissectionisintendedtogiveashortintroductionto
bornologicalvectorspaces
[20].Thesearevectorspaceswithanadditionalstructuredescribingabstractly
thenotionofboundedness.Concreteexamplesofbornologicalspacesarepro-
videdbynormedorlocallyconvexspaces.Bornologyisthecorrectframework
allowingthedevelopmentof
entire
cycliccohomologyinfullgenerality;thishas
beendonebyMeyerin[24].Sincethistopicisnotsofamiliartomathematical
physicists,wefeeltheneedtorecallthedefinitionsandbasicproperties.Our
sketchisbynomeanssupposedtogiveasufficientknowledgeaboutbornology;
wereferto[20,24]fordetails.
Let
V
beavectorspaceover
C
.Asubset
S
⊂V
isadiskiffitiscircledand
convex.Givenanysubset
S
,wedenoteby
S

itscircledconvexhull:itisthe
smallestdiskcontaining
S
.If
S
isadisk,itslinearspan
V
S
isendowedwitha
semi-norm
||||
S
whoseunitballistheclosureof
S
.
S
iscalledcompletantiff
V
S
isaBanachspace.
Definition2.1
Let
V
beavectorspace.A(convex)bornology
S
(
V
)
isacol-
lectionofsubsetsof
V
verifyingthefollowingaxioms:
4

•{
x
}∈
S
(
V
)
foranyvector
x
∈V
.

S
1
+
S
2

S
(
V
)
forany
S
1
,S
2

S
(
V
)
.

If
S

S
(
V
)
,then
T

S
(
V
)
forany
T

S
.

S


S
(
V
)
forany
S

S
(
V
)
.
Any
S

S
(
V
)
iscalleda
smallsubset
ofthebornologicalspace
V
.
Thebornology
S
(
V
)iscalledcompletantiffanysmallsubset
S

S
(
V
)is
containedinacompletantsmalldisk.Inthatcase,(
V
,
S
(
V
))isa
complete
bornologicalvectorspace
.
Example2.2
If
V
isalocallyconvexspace,thenthe
boundedbornology
Bound
(
V
)isthecollectionofsubsets
S
boundedforallseminormson
V
.If
V
iscompleteforthelocallyconvextopology,thenitiscompleteasabornolog-
icalspace.Fre´chetspacesendowedwiththeboundedbornologyareimportant
examplesofcompletebornologicalspaces.
Example2.3
If
V
isanyvectorspace,thefinebornology
Fine
(
V
)isthesmall-
estadmissiblebornology:asubsetissmalliffitiscontainedinthediskedhull
ofafinitenumberofpointsof
V
.Inparticular,anysmallsubsetiscontainedin
afinite-dimensionalsubspaceof
V
.Abornologicalspacewithfinebornologyis
alwayscompletebecausefinite-dimensionalspacesarecomplete.
Example2.4
Ausefulwaytoconstructabornologyon
V
istostartfrom
acollection
U
ofsubsetsnotsatisfyingtheaxiomsofabornology,andthen
toconsiderthesmallestbornology
S
(
V
)containing
U
.Wesaythat
S
(
V
)is
generatedby
U
.
Bornologicalconvergence:
Asequence
{
x
n
}
n

N
ofpointsinabornological
space
V
issaidtoconvergebornologicallytothelimit
x

∈V
iffthereisasmall
disk
S

S
(
V
)suchthat
x
n

x


S
forany
n
andlim
n
→∞
||
x
n

x

||
S
=0.A
setissaidtobeclosedforthebornologyiffitissequentiallyclosedforbornolog-
icallyconvergentsequences.Theclosedsetsforthebornologyfulfilltheaxioms
ofatopology,henceabornologicalspacehasalsoatopology(thoughingeneral
notavectorspacetopology).
Boundedmaps:
Let
V
and
W
betwobornologicalvectorspaces.Alinear
map
l
:
V→W
is
bounded
iff
l
(
S
)

S
(
W
)foranysmall
S

S
(
V
).Anarbi-
traryset
{
l
j
}
j

J
oflinearmapsis
equibounded
iff
{
l
j
(
x
)
|
j

J,x

S
}
isasmall
subsetof
W
forany
S

S
(
V
).WedenotebyHom(
V
,
W
)thevectorspaceof
boundedlinearmapsbetween
V
and
W
.Thesetsofequiboundedmapsforma
bornologycalledthe
equiboundedbornology
onHom(
V
,
W
).Itiscompleteif
W
iscomplete.Wewillalwaysendowthespacesofboundedlinearmapswiththe
equiboundedbornology.
Completions:
Let
V
beabornologicalvectorspace.Its
bornologicalcom-
pletion
V
c
isthecompletebornologicalvectorspacedefinedasthesolutionof
thefollowinguniversalproblem:thereisaboundedlinearmap
u
:
V→V
c
suchthat,foranycompletebornologicalspace
W
andanyboundedlinearmap
5

l
:
V→W
,thereisauniqueboundedlinearmapfrom
V
c
to
W
factorizing
l
.Thecompletionalwaysexists,andcanbeexplicitlyrealizedastheinductive
limitofasystemofBanachspaces(see[20]andtheappendixof[24]).Itisof
courseuniquebyuniversality.If
V
isanormedspaceendowedwiththebounded
bornology,thenitsbornologicalcompletioncoincideswithitsHausdorffcom-
pletion.However,itshouldbestressedthattheuniversalmap
V→V
c
mayfail
tobeinjectiveforanarbitrarybornologicalspace
V
.
Multilinearmaps:
An
n
-linearmap
l
:
V
1
×
...
×V
n
→W
betweenbornologi-
calspacesisboundediff
l
(
S
1
,...,S
n
)

S
(
W
)foranysmallsets
S
i

S
(
V
i
).If
W
iscomplete,thenthereisauniquebounded
n
-linearmap
V
1
c
×
...
×V
nc
→W
factorizing
l
.
Completedtensorproducts:
Let
V
1
and
V
2
betwobornologicalvector
spaces.Weendowtheiralgebraictensorproduct
V
1
⊗V
2
withthebornol-
ogygeneratedbythesubsets
S
1

S
2
,forany
S
i

S
(
V
i
).Thecompletionof
V
1
⊗V
2
withrespecttothisbornologyisthe
completedtensorproduct
V
1

ˆ
V
2
.
Thecompletedtensorproductisassociative,whencethedefinitionofthe
n
-fold
tensorproduct
V
1

ˆ
...

ˆ
V
n
of
n
bornologicalspaces.Thelatterisuniversalfor
thebounded
n
-linearmaps
V
1
×
...
×V
n
→W
withcompleterange
W
.
Algebras:
Abornologicalalgebraisabornologicalspace
A
endowedwitha
boundedbilinearmap(product)
A×A→A
.Thealgebra
A
iscompleteiff
itiscompleteasavectorspace.Inthispaperwewillbeconcernedonlywith
associativebornologicalalgebras.
Subspaces,quotients:
Let
V
beabornologicalvectorspaceand
W⊂V
a
vectorsubspace.Thereisacanonicalbornologyon
W
:asubset
S
∈W
issmall
iffitissmallfor
V
.Ontheotherhand,thequotientspace
V
/
W
hasalsoa
bornology:
S

S
(
V
/
W
)iffthereisasmall
T

S
(
V
)suchthat
S
=
T
mod
W
.
When
V
iscomplete,thenthesubspace
W⊂V
andquotient
V
/
W
arecomplete
iff
W
isbornologicallyclosedin
V
.
Bornologicalcomplexes:
Abornologicalspace
V
withaboundedlinearmap

:
V→V
satisfying

2
=0isabornologicalcomplex.Itshomologyisasusual
thebornologicalvectorspace
H

(
V
)=Ker
∂/
Im

.
3Entirecycliccohomology
Herewerecalltheformulationofcyclic(co)homologywithinthe
X
-complex
frameworkofCuntzandQuillen[15].Theanalyticadaptationofthattheory
presentedbyMeyerin[24]allowstodefineelegantlythe
entire
cyclichomology,
cohomologyandbivariantcohomologyfor
bornological
algebras.Therearein
facttwoequivalentwaystodescribetheentirecycliccohomologyofacomplete
bornologicalalgebra
A
.ThefirstoneistouseConnes’(
b,B
)complexofnon-
commutativeformscompletedwithrespecttoacertainbornology;wecallthis
completionΩ
ǫ
A
.Thesecondoneisthe
X
-complexofthecompletedtensoral-
gebra
TA
.Thesecomplexesarehomotopyequivalent[24],andgiverisetothe
definitionofentirecycliccohomology.TheconstructionofthebivariantChern

6

characterproposedinourpaperusessimultaneouslythe(
b,B
)-complexand
X
-
complexapproaches.Also,athirdcomplexwillbeneededasanintermediate
step;wecallitthecompleteddeRham-KaroubicomplexΩ
δ
A

.

3.1Non-commutativedifferentialforms
ferentialformsover
A
isthedirectsumΩ
A
=
n

0
Ω
n
A
ofthe
n
-dimensional
Let
A
beacompletebornologicalalgebra.Thea
L
lgebraofnon-commutativedif-
subspacesΩ
n
A
=
A
e

ˆ
A

ˆ
n
for
n

1andΩ
0
A
=
A
,where
A
e
=
A⊕
C
isthe
unitalizationof
A
.Itiscustomarytousethedifferentialnotation
a
0
da
1
...da
n
(resp.
da
1
...da
n
)forthestring
a
0

a
1
...

a
n
(resp.1

a
1
...

a
n
).The
differential
d

n
A→
Ω
n
+1
A
isuniquelyspecifiedby
d
(
a
0
da
1
...da
n
)=
da
0
da
1
...da
n
and
d
2
=0.ThemultiplicationinΩ
A
isdefinedasusualand
fulfillstheLeibnizrule
d
(
ω
1
ω
2
)=

1
ω
2
+(

)
|
ω
1
|
ω
1

2
,where
|
ω
1
|
isthede-
greeof
ω
1
.EachΩ
n
A
isacompletebornologicalspacebyconstruction,and
weendowΩ
A
withthedirectsumbornology.ThisturnsΩ
A
intoacomplete
bornologicaldifferentialgraded(DG)algebra,i.e.themultiplicationmapand
d
arebounded.ItistheuniversalcompletebornologicalDGalgebragenerated
.ybAOnΩ
A
aredefinedvariousoperators.Firstofall,theHochschildboundary
b

n
+1
A→
Ω
n
A
is
b
(
ωda
)=(

)
n
[
ω,a
]for
ω

Ω
n
A
,and
b
=0onΩ
0
A
=
A
.
Oneeasilyshowsthat
b
isboundedand
b
2
=0.ThentheKaroubioperator
κ

n
A→
Ω
n
A
isconstructedoutof
b
and
d
:
1

κ
=
db
+
bd.
(3)
Therefore
κ
isboundedandcommuteswith
b
and
d
.ThelastoperatorisConnes’
B

n
A→
Ω
n
+1
A
,
B
=(1+
κ
+
...
+
κ
n
)
d
onΩ
n
A
,
(4)
whichisboundedandverifies
B
2
=0=
Bb
+
bB
and

=
κB
=
B
.
WenowdefinethreeotherbornologiesonΩ
A
,leadingtothenotionof
entire
cycliccohomology:

The
entirebornology
S
ǫ

A
)isgeneratedbythesets
[[
n/
2]!
S
e
(
dS
)
n
,S

S
(
A
)
,
(5)
0≥nwhere[
n/
2]=
k
if
n
=2
k
or
n
=2
k
+1,and
S
e
=
S
+
C
.Thatis,asubset
ofΩ
A
issmalliffitiscontainedinthecircledconvexhullofasetlike(5).We
writeΩ
ǫ
A
forthecompletionofΩ
A
withrespecttothisbornology.Ω
ǫ
A
will
bethecomplexofentirechains.
S•
The
analyticbornology
S
an

A
)isgeneratedbythesets
n

0
S
e
(
dS
)
n
,
S

S
(
A
).ThecorrespondingcompletionofΩ
A
isΩ
an
A
.Itisrelatedtothe
X
-complexdescriptionofentirecyclichomology(seebelow).

7


The
deRham-Karoubibornology
S
δ

A
)isgeneratedbythecollection
n1Sofsets
n

0[
n/
2]!
S
e
(
dS
),
S

S
(
A
),withcompletionΩ
δ
A
.Thiswillgiverise
tothedeRham-Karoubicomplex.
ThemultiplicationinΩ
A
isboundedforthethreebornologiesabove,as
wellasalltheoperators
d,b,κ,B
.Moreover,the
Z
2
-graduationofΩ
A
givenby
evenandoddformsispreservedbythecompletionprocess,sothatΩ
ǫ
A
,
Ω
an
A
andΩ
δ
A
are
Z
2
-gradeddifferentialalgebras,endowedwiththeoperators
b,κ,B
fulfillingtheusualrelations.Inparticular,Ω
ǫ
A
iscalledthe(
b,B
)-complex
of
entirechains
.Notealsothatthemultiplicationordivisionof
n
-formsby
[
n/
2]!obviouslyprovidelinearbornologicalisomorphismsbetweenΩ
ǫ
A
,
Ω
an
A
andΩ
δ
A
.
3.2Theanalytictensoralgebra
Let
A
beacompletebornologicalalgebra,Ω
A

+
A⊕
Ω

A
the
Z
2
-graded
algebraofdifferentialforms.TheevenpartΩ
+
A
isatrivialygradedsubalgebra.
WeendowΩ
+
A
withanewassociativeproduct,the
Fedosovproduct
[15]
ω
1

ω
2
=
ω
1
ω
2


1

2

1
,
2

Ω
+
A
.
(6)
Associativityiseasytocheck.Infactthealgebra(Ω
+
A
,

)isisomorphictothe
ˆnon-unitaltensoralgebra
T
A
=
n

1
A

n
,underthecorrespondence
L+Ω
A∋
a
0
da
1
...da
2
n
←→
a
0

ω
(
a
1
,a
2
)

...

ω
(
a
2
n

1
,a
2
n
)

T
A
,
(7)
where
ω
(
a
i
,a
j
):=
a
i
a
j

a
i

a
j
∈A⊕A

ˆ2
isthe
curvature
of(
a
i
,a
j
).Itturns
outthattheFedosovproduct

isboundedforthebornology
S
an
restrictedto
Ω
+
A
[24],andthusextendstotheanalyticcompletionΩ
a+n
A
.Thecomplete
bornologicalalgebra(Ω
a+n
A
,

)isalsodenotedby
TA
andcalledthe
analytic
tensoralgebra
of
A
in[24].
3.3
X
-complex
The
X
-complexfirstappearedinQuillen’sworkonalgebracochains[32],and
thenwasusedbyCuntz-Quillenintheirformulationofcyclichomology[15,16].
Herewerecallthe
X
-complexconstructionforbornologicalalgebras,following
Meyer[24].
Let
A
beacompletebornologicalalgebra.The
X
-complexof
A
isthe
Z
2
-graded
complex
d♮X
(
A
):
A
o
o/
/
Ω
1
A

,
(8)
bwhereΩ
1
A

isthecompletionofthecommutatorquotientspaceΩ
1
A
/b
Ω
2
A
=
Ω
1
A
/
[
A
,
Ω
1
A
]endowedwiththequotientbornology.Theclassofthegenericel-
ement(
a
0
da
1
mod[
,
])

Ω
1
A

isusuallydenotedby
♮a
0
da
1
.Themap
♮d
:
A→
Ω
1
A

thussends
a
∈A
to
♮da
.Also,theHochschildboundary
b

1
A→A
vanishesonthecommutators[
A
,
Ω
1
A
],hencepassestoawell-definedmap
b

1
A

→A
.Explicitlytheimageof
♮a
0
da
1
by
b
isthecommutator[
a
0
,a
1
].
Thesemapsareboundedandsatisfy
♮d

b
=0and
b

♮d
=0,sothat
X
(
A
)

8

indeeddefinesacomplete
Z
2
-gradedbornologicalcomplex.
Wenowfocusonthe
X
-complexoftheanalytictensoralgebra
TA
.Inthat
case,Ω
1
TA


1
TA
/
[
TA
,
Ω
1
TA
]isalwayscomplete,andasabornological
vectorspace
X
(
TA
)iscanonicallyisomorphictotheanalyticcompletionΩ
an
A
.
Herewemusttakecareofanotationalproblem.Sincethesymbol
d
isalready
usedforthedifferentialonΩ
A
,wealwayschoosetheboldprint
d
forthe
differentialonΩ
TA
.Thenthecorrespondencebetween
X
(
TA
)andΩ
an
A
goes
asfollows[15,24]:first,onehasa
TA
-bimoduleisomorphism
Ω
1
TA≃T
e
A⊗
ˆ
A⊗
ˆ
T
e
A
(9)
x
d
ay

x

a

y
for
a
∈A
,x,y
∈T
e
A
,
where
T
e
A
:=
C
⊕TA
istheunitalizationof
TA
.Thisimpliesthatthebornolog-
icalspaceΩ
1
TA

isisomorphicto
T
e
A⊗
ˆ
A
,whichcanfurtherbeidentified
withtheanalyticcompletionofoddformsΩ
a

n
A
,throughthecorrespondence
x

a

xda
,

a
∈A
,x
∈T
e
A
.Thuscollectingtheevenpart
X
0
(
TA
)=
TA
and
theoddpart
X
1
(
TA
)=Ω
1
TA

together,yieldsalinearbornologicalisomor-
phism
X
(
TA
)

Ω
an
A
.Westilldenoteby(

d
,b
)theboundariesinducedon
Ω
an
A
throughthisisomorphism;CuntzandQuillenexplicitlycomputedthem
intermsoftheusualoperatorsondifferentialforms[15]:
b
=
b

(1+
κ
)
d
onΩ
2
n
+1
A
,
(10)

d
=
κ
i
d

κ
2
i
b
onΩ
2
n
A
.
2
X
nn
X

1
i
=0
i
=0
Thecrucialresult[15,24]isthatthecomplex(Ω
an
A
,♮
d
,b
)=
X
(
TA
)is
homotopyequivalenttothecomplexofentirechainsΩ
ǫ
A
endowedwiththe
differential(
b
+
B
).Letusrecallbrieflythejob[15,24].TheKaroubioperator
κ
verifiesthepolynomialidentity(
κ
n

1)(
κ
n
+1

1)=0onΩ
n
A
,hence
κ
2
alsoverifiesapolynomialidentity.Itfollowsthat
κ
2
hasadiscretespectrum
σ
,
andΩ
A
decomposesintothedirectsumofthegeneralizedeigenspaces
V
λ
for
any
λ

σ
.Oneoftheeigenvaluesof
κ
2
is1,withmultiplicity2.Let
P
bethe
projectionofΩ
A
onto
V
1
,vanishingontheothereigenspaces.Since
P
andits
orthogonalprojection
P

commutewithalloperatorscommutingwith
κ
,the
subspaces
P
Ω
A
and
P

Ω
A
arestablewithrespectto
d,b
and
B
.Oneshows[24]
that
P,P

areboundedforthebornologies
S
an

A
)and
S
ǫ

A
),henceextend
tothecompletionsΩ
an
A
andΩ
ǫ
A
.Moreover,thesubcomplex(
P

Ω
an
A
,♮
d
,b
)
iscontractible,henceΩ
an
A
retractson
P
Ω
an
A
forthedifferential(

d
,b
).Also,
(
P

Ω
ǫ
A
,b
+
B
)iscontractibleandΩ
ǫ
A
retractson
P
Ω
ǫ
A
forthedifferential
(
b
+
B
).Let
c

an
A→
Ω
ǫ
A
bethebornologicalvectorspaceisomorphism
c
(
a
0
da
1
...da
n
)=(

)
[
n/
2]
[
n/
2]!
a
0
da
1
...da
n

n

N
.
(11)
Then
c
mapsisomorphically
P
Ω
an
A
onto
P
Ω
ǫ
A
,andunderthiscorrespondence,
theboundaries(

d
,b
)and
b
+
B
coincide:
c

1
(
b
+
B
)
c
=(

d
,b
)on
P
Ω
an
A
.Itfol-
lowsthatthe
X
-complex
X
(
TA
)ishomotopyequivalenttothe(
b
+
B
)-complex
ofentirechainsΩ
ǫ
A
.Thisleadstothedefinitionofentirecyclic(co)homology:
Definition3.1
Let
A
beacompletebornologicalalgebra.
i)Theentirecyclichomologyof
A
isthehomologyofthe
X
-complexofthe
9

analytictensoralgebra
TA
:
HE

(
A
)=
H

(
X
(
TA
))
,

=0
,
1
,
(12)
orequivalently,the
(
b
+
B
)
-homologyofthe
Z
2
-gradedcomplexofentirechains
.ΩǫAii)Let
(
X
(
TA
))

bethe
Z
2
-gradedcomplexof
bounded
mapsfrom
X
(
TA
)
to
C
,withdifferentialthetransposedof
(

d
,b
)
.Thentheentirecycliccohomology
of
A
isthecohomologyofthisdualcomplex:
HE

(
A
)=
H

((
X
(
TA
))

)
,

=0
,
1
.
(13)
iii)If
A
and
B
arecompletebornologicalalgebras,then
Hom(
X
(
TA
)
,X
(
TB
))
denotesthespaceof
bounded
linearmapsfrom
X
(
TA
)
to
X
(
TB
)
.Itisnat-
urallyacomplete
Z
2
-gradedbornologicalcomplex,thedifferentialofamap
f
correspondingtothecommutator
(

d
,b
)

f

(

)
|
f
|
f

(

d
,b
)
.Thebivariant
entirecycliccohomologyof
A
and
B
isthenthecohomologyofthiscomplex:
HE

(
A
,
B
)=
H

(Hom(
X
(
TA
)
,X
(
TB
)))
,

=0
,
1
.
(14)
Inthecase
A
=
C
,oneshows[24]that
X
(
T
C
)ishomotopicallyequivalentto
X
(
C
):
C

0,thustheentirecyclichomologyof
C
issimply
HE
0
(
C
)=
C
and
HE
1
(
C
)=0.Thisimpliesthatforanycompletebornologicalalgebra
A
,we
gettheusualisomorphisms
HE

(
C
,
A
)

HE

(
A
)and
HE

(
A
,
C
)

HE

(
A
).
Furthermore,sincethecompositionofboundedmapsisbounded,thereisa
well-definedcompositionproductonbivariantentirecycliccohomology:
HE
i
(
A
,
B
)
×
HE
j
(
B
,
C
)

HE
i
+
j
+2
Z
(
A
,
C
)
,i,j
=0
,
1(15)
forcompletebornologicalalgebras
A
,
B
,
C
.Anyboundedhomomorphism
ρ
:
A→B
extendstoaboundedhomomorphism
ρ

:
TA→TB
bysetting
ρ

(
a
1

...

a
n
)=
ρ
(
a
1
)

...

ρ
(
a
n
).Theboundednessof
ρ

becomesobviousoncewe
+rewriteitusingtheisomorphism
TA≃

an
A
,

),since
ρ

(
a
0
da
1
...da
2
n
)=
ρ
(
a
1
)

(
a
1
)
...dρ
(
a
2
n
).Thehomomorphism
ρ

givesrisetoabounded
X
-
complexmorphism
X
(
ρ

):
X
(
TA
)

X
(
TB
):
x
7→
ρ

(
x
)(16)
♮x
d
y
7→
♮ρ

(
x
)
d
ρ

(
y
)

x,y
∈TA
.
Wewritech(
ρ
)fortheclassof
X
(
ρ

)in
HE
0
(
A
,
B
).Itisthesimplestexam-
pleofbivariantCherncharacter.Last,remarkthat
HE

(
A
,
A
)isa
Z
2
-graded
unitalring,theunitcorrespondingtotheCherncharacteroftheidentityho-
momorphismof
A
.
3.4TheentiredeRham-Karoubicomplex
Thereisstillanothercomplexrelatedtocyclichomology,namelythedeRham-
Karoubicomplex[22].Inourcontextofbornologicalalgebras,wehaveto
consideritscompletedversion.Solet
A
beacompletebornologicalalgebra.
Recallth
S
atthedeRham-Karoubibornology
S
δ
onΩ
A
isgeneratedbythe
subsets
n

0[
n/
12]!
S
e

S

n
,foranysmallset
S

S
(
A
).Thecompletionof
Ω
A
withrespecttothisbornologyisΩ
δ
A
.LetΩ
δ
A

bethe
completion
of
01