On the comparison of positive elements of a C-algebra by lower semicontinuous traces

arXiv:0806.1570v1 [math.OA] 10 Jun 2008ONTHECOMPARISONOFPOSITIVEELEMENTSOFA

C*-ALGEBRABYLOWERSEMICONTINUOUSTRACES

LEONELROBERT

Abstract.ItisshowninthispaperthattwopositiveelementsofaC*-

algebraagreeonalllowersemicontinuoustracesifandonlyiftheyare

equivalentinthesenseofCuntzandPedersen.Asimilarresultisalso

obtainedinthemoregeneralcasewherethetwoelementsarecomparable

bytheirvaluesonthelowersemicontinuoustraces.Thisresultisusedto

giveacharacterizationofthefunctionsontheconeoflowersemicontinuous

tracesofastableC*-algebrathatarisefrompositiveelementsofthealgebra.

1.Introduction

In[1],CuntzandPedersenconsideredtheproblemofcomparingpositive

elementsofaC*-algebrabytheirvaluesonthelowersemicontinuoustraces

onthealgebra.Theydefinedanequivalencerelationonthepositiveelements—

theCuntz-Pedersenrelation—andshowed,amongotherresults,thatiftheC*-

algebraissimple,thentwopositiveelementsareCuntz-Pedersenequivalentif

andonlyiftheyagreeonallthelowersemicontinuoustraces.Thequestion

whetherthiswastrueforanarbitraryC*-algebrawasleftunsettledintheir

paper,andisansweredaffirmativelyinTheorem1below.

LetAbeC*-algebra.RecallthatatraceonAisamapτ:A+→[0,∞]

thatisadditive,homogeneous,andsatisfiestheidentityτ(xx∗)=τ(x∗x).We

willbemostlyinterestedinthelowersemicontinuoustracesonA,thesetof

whichweshalldenotebyT(A).

Fora,b∈A+letussaythataisCuntz-Pedersenequivalenttob,denoted

bya∼b,ifa=󰀃

∞

i=1x

ix∗

iandb=󰀃

∞

i=1x∗

ix

iforsomex

i∈A.Letussaythat

aisCuntz-Pedersensmallerthanb,anddenotethisbya󰀎b,ifa∼a′≤bfor

somea′∈A+.Itwasshownin[1]thattherelations∼and󰀎aretransitive

(so∼isanequivalencerelation).

Theorem1.LetaandbbepositiveelementsofA.Thefollowingpropositions

aretrue.

(i)τ(a)≤τ(b)forallτ∈T(A)ifandonlyifforeveryǫ>0thereisδ>0

suchthat(a−ǫ)

+󰀎(b−δ)

+.

(ii)τ(a)=τ(b)forallτ∈T(A)ifandonlyifa∼b.

Remark.WeshallseeinSection3anexamplewhereτ(a)≤τ(b)forall

τ∈T(A),butitisnottruethata󰀎b.2.ProofofTheorem1TheproofofTheorem1isprecededbyanumberofpreliminarydefinitions

andresults.

Leta,b∈A+.Letuswritea󰀎

CPbifforeveryǫ>0thereisδ>0such

that(a−ǫ)

+󰀎(b−δ)

+.Since󰀎istransitive,therelation󰀎

CPisclearly

transitiveaswell.Letuswritea∼

CPbifa󰀎

CPbandb󰀎

CPa.Thisdefines

anequivalencerelationinA+.(ItwillbeshownintheproofofTheorem1(ii)

that∼

CPisthesameas∼.)

Proposition1.Leta,b,c,d∈A+.Thefollowingpropositionsaretrue.

(i)Ifa󰀎

CPbandc󰀎

CPdthena+c󰀎

CPb+d.

(ii)Ifa󰀎

CPbthenαa󰀎

CPαbforallα∈R+.

(iii)Ifa󰀎bthena󰀎

CPb.

Proof.Itwasshownin[2,Proposition2.3]thatforalla,b∈A+andǫ>0

thereisδ>0suchthat

(a−ǫ)

++(b−ǫ)

+󰀎(a+b−δ)

+,(1)

(a+b−ǫ)

+󰀎(a−δ)

++(b−δ)

+.(2)

Theseinequalitiesimply(i).

(ii)Thisisclear.

(iii)Supposethata=󰀃

∞

i=1x

ix∗

iand󰀃

∞

i=1x∗

ix

i≤b.Letǫ>0.By

thelemmaofKirchbergandRørdam[3,Lemma2.2](seealsotheremark

after[2,Lemma2.2]),therearen∈Nandǫ

1>0suchthat(a−ǫ)

+󰀎

(󰀃

n

i=1x

ix∗

i−ǫ

1)

+.Wehave

(a−ǫ)

+󰀎(n󰀂

i=1x

ix∗

i−ǫ

1)

+󰀎n󰀂

i=1(x

ix∗

i−ǫ

2)

+∼n󰀂

i=1(x∗

ix

i−ǫ

2)

+

󰀎(n󰀂

i=1x

ix∗

i−ǫ

3)

+󰀎(b−ǫ

4)

+.

Intheabovechainofinequalitieswehaveapplied(1),(2)andthat(xx∗−ǫ)

+∼

(x∗x−ǫ)

+forallǫ>0(see[2,Proposition2.3]).󰀁

LetusdenotebyA

CPthequotientA+/∼

CP.WeconsiderA

CPordered

bytheorder󰀌a󰀍≤󰀌b󰀍ifa󰀎

CPb,where󰀌a󰀍and󰀌b󰀍denotetheequivalence

classesofthepositiveelementsaandb.WealsoendowA

CPwiththeaddition

operation󰀌a󰀍+󰀌b󰀍:=󰀌a+b󰀍.TheorderofA

CPiscompatiblewiththeaddition

operation,i.e.,󰀌a󰀍≤󰀌a󰀍+󰀌b󰀍.Thus,A

CPisanorderedsemigroupwith0.

InordertoproveTheorem1(i)wewillapplythefollowingpropositionto

theorderedsemigroupA

CP.

Proposition2.([4,Proposition3.2])LetSbeanorderedsemigroupwith0

andwiththepropertythatif(k+1)x≤kyforsomex,y∈Sandk∈N,then

x≤y(i.e.,Sisalmostunperforated).ThefollowingimplicationholdsinS:

Ifx≤MyforsomeM∈N,andλ(x)

isadditive,order-preserving,andsatisfiesλ(y)=1,thenx≤y.

2NoticethatthesemigroupA

CPsatisfiesthehypothesesofthepreceding

proposition.Infact,inA

CPwehavethatk󰀌x󰀍≤k󰀌y󰀍implies󰀌x󰀍≤󰀌y󰀍,

becausewecanmultiplytheelementsofA

CPbypositiverealscalars.

Noticealsothatforeveryadditivemapλ:A

CP→[0,∞],themaponA+

definedbyτ

λ(a):=λ(󰀌a󰀍)isatrace,becauseitisadditiveandsatisfiesthe

traceidentity(homogeneityholdsautomaticallyforanyadditivemapwithvaluesin[0,∞]).Thistracemaynotbelowersemicontinuous.ToTheorem1

(i)wewillthenneedthefollowinglemma.

Lemma1.([2,Lemma3.1])Letτ:A+→[0,∞]beatraceontheC*-algebra

A.Then˜τ(a)=sup

ǫ>0τ((a−ǫ)

+)isalowersemicontinuoustrace.

ProofofTheorem1(i).Itisclearthatifa󰀎

CPbthenτ(a)≤τ(b)forevery

τ∈T(A).

Letusassumethatτ(a)≤τ(b)forallτ∈T(A).Foreveryclosedtwo-sided

idealIofA,themapdefinedbyτ

I(x)=0ifx∈I+andτ

I(x)=∞otherwise,

isalowersemicontinuoustraceonA.Sinceτ

I(a)≤τ

I(b)foranysuchtraceit

followsthatIdeal(a)⊆Ideal(b).Letǫ>0.Wehave(a−ǫ)

+∈Ped(Ideal(b))+;

hence(a−ǫ)

+=󰀃

m

i=1y

iby∗

iforsomey

i∈A.Thisimpliesthat(a−ǫ)

+󰀎Mb

forsomeM∈N,andso󰀌(a−ǫ)

+󰀍≤M󰀌b󰀍.Letusshowthatwealsohave

λ(󰀌(a−ǫ)

+󰀍)

CP→[0,∞]

suchthatλ(󰀌b󰀍)=1.ByProposition2,thiswillimplythat󰀌(a−ǫ)