Separable injectivity and C* tensor products

VOL. 14 NO. (1991) 139-148

SEPARABLE INJECTIVITY

AND

C" TENSOR

PRODUCTS

TADASIHURUYA

Facultyo[Education

NiigataUniversity

Niigata, 950-21

JAPAN

and

SEUNG-HYEOKKYE

Department

of Mathematics

Song

Sim Collegefor

Women

Bucheon,

Seoul422-743,

KOREA

(Received January 26, 1990)

ABSTRACT. Let

A

and

B

be

C*-

algebras and let

D

beaC*-subalgebraofB. We

show that if

D

is separably injective then the triple

(A, B,

D)

verifies the slice map conjecture. Asanapplication, weprove that theminimal

C*-tensor

product

A

(R)

B

is

separablyinjective if andonlyifboth

A

and Bareseparablyinjective andeither

A

or

B

isfinite-dimensional.

KEY WORDS

AND

PHRASES.

Ca-algebra, C*-tensor

product, injective

Ca-algebra,

separablyinjective C*-algebra, slice map.

1980

AMS SUBJECT

CLASSIFICATION CODE.

46L05.

1.

INTRODUCTION.

Smith and Williams introduced thenotionofseparableinjectivity inconnection with thestudyofcompletelybounded maps

([7],

[8]).

Asstatedintheintroduction of

[8],

it is weaker than the related concept of injectivity and yet is appropriate for certain desirable extensionproblems. Fromthispointof view,we study

C*-tensor

products.

Let

A

and

B

be

C*-algebras

and D be a C*-subalgebra ofB. If

D

is injective, then thetriple

(A, B,

D)

verifiesthe slice mapconjecturein thesenseofWassermann

2. PRELIMINARIES AND NOTATION.

LetAand BbeC*-algebrasand let AQBdenote their minimal(i.e., spatial) tensor

product. Let L, denote the

(’,*-algebra

of all, x 7tcomplex matrices for a positive integern. If A B isalinear map then

,

(R)id,, A(R)M, B(R)M, isdefinedby

(j(R)id,)(a,i)

(i(a,)).

is saidtobe completely positiveifeach(R) id, is positive.

A

C*-algebra Dis said tobe injective if given

C*-algebras E

C_

F,

any contractive

completely positive map E D has a contractiveconpletely positive extension

V

F

D.

We

say that a

C*-algebra D

is separablyinjective ifgiven separable

C*-algebras EC_

F,

any contractivecompletelypositivemap E Dhasacontractive completely positive extension

F

D. The separable injectivity in this paper is weakerthanonein

([7],

[8])

andbothcoincideforcommutative

C*-algebras. A

compact ltausdorffspace is saidtobesubstoneanif every twodisjointco-zerosets havedisjoint closures. Acompact Hausdorff spaceXis substoneanif andonly if

C(X)

isseparably injective

[7,

Theorem

4.6].

A C*-algebra D

issaid to besubhomogeneous ifevery irreducible representation is finite-dimensional withboundeddimension.

In

particulr,it is said tobe t-homogeneous if every irreduciblerepresentation is n-dimensional. If

D

is subhomogeneousthen we

identify the spectrum

D

with the primitiveidealspace

[2,

Chapters 3 and

4].

For 1, 2 let

Di

be a

C*-algebra

and let

hi

D.

The right slice map

R,,

D

(R)

D

D2

andthe leftslice map

L,

D

(R)

D

D

areunique boundedlinear mapssatisfying

R,,(zt

(R)

z)

h(zt)z

and

L(z

(R)

z)

h(z)z

[10].

For

C*-subalgebras Ai

of Di, wedefinetheFubiniproduct

F(At,A2,

Dt

(R)

D2)

of

At

and

A2

with respect to

D

(R)

D

[11]

by

F(A,

A, Di

(R)

D)

{z

q

Dt

(R)

D2-

R,,

(z)

q

A2

and

L,(z)

q

A1

for all

ht

q

D’

and

h

D

}.

Forfixed

C*-algebras

A1

and

A,

F(Ai, A,

D

(R)

D2)

dependson

D

(R)

D2.

Butthey

areall isomorhic andarethelargestamong them if

Dt

and

D

areinjective. Wedenote by

A

(R)F

A:

anyoneoftheseisomorphicFubiniproductsof

A

and

A

[4].

Let Aand

B beC*-algebrasand let D beaC*-subalgebra. Thetriple

(A,

B,

D)

is said to verify the slice map conjecture if

F(A,D,A

(R)

B)

A(R)D

[21.

3.

THE

SLICE

MAP

PROBLEM.

A C*-algebra A

is said tohave property

(S)

if

(A, B,

D)

verifiestheslice map conjec-turefor every

C*-algebra

Band every

C*-subalgebra

DofB

[12].

Wenow considera

D. SubhomogeneousorinjectiveC*-algebrashave property

(S’)

[11].

THEOREM 1. Let

D

bea

C*-algebra.

If

D

isserarably injective, then

D

hasproperty

(s’).

PROOF: Let

A

bea

C*-algebra

and

B

a

C*-algebra

containing D. Letr

F(A,

D,A(R)

B).

Then thereexistsasequence

{rn}

such thata’,

n)

a(i, n)(R)b(i, ..),

lim,,,,

where eacha(i,

n)

A

and each

b(i, n)

B. Let

Bo

be theC*-subalgebra generated by

{b(i,n)

1,...,m(n),n

1, 2,...

}

and let

Do

be the C*-subalgebra ofD generated by

{Rn

(x):

h A*

}.

Thenwehave

Do

C_

Bo,

x

F(A,

Do, A

(R)

Bo)

by a sinila.r argument of

[4,

Lemma

5].

By

hypothesis, there exists a contractive

completelypositivemap

b"

B0

Dwhichextendsthe identityembeddingof

Do

into D. Then

Rh((IA

(R)

b)(:))=

dp(Rh(:))

Rh(:r.)

(h

A*).

Since

{Rh

h

A*}

is total

[10,

Theorem

1],

z

(IA

(R)

)(x)

A

(R)

D

and so

F(A,D,A

(R)

B)

C_

A

(R)D.

The oppositeinclusion is immediate.

Itisknown that thedirectsumof two

C*-algebras

having property

(S’)

hasproperty

(S’).

Inordertoshow that Theorem 1 givesanewexample having property

(S’),

two results will beneeded.

In

the proofof

[7,

Theorem

4.6],

Smith and Williamsobtained thefollowinglemma.

LEMMA

2. Let

B

and

D

be

C*-algebras.

Then thereexistsaonetoonecorrespondence

between completelypositivemapsqb

B

D

(R)M, andcompletely positivemaps

B

(R)

M,,

D

foranypositiveintegern.

Weremark thatOis notnecessarily normpreserving and that 0satisfiesthat

O()IA(R)M,

0(lA

fora

C

-subalgebra

A

of

B,

where

O(gP)la(R)M"

and

0(IA

denote the restrictions of

0(b)

and

b

to

A

(R)

M,,

and

A,

respectively.

The proof of thefollowingproposition isbasedon anideaof

[6,

Theorem

2.1].

PROPOSITION 3. Let D bea

C*-algebra. I

D isseparablyinjective, then D(R)M,, is

separablyinjective for any positiveintegern.

PROOF: Let

A

be a separable

C*-algebra

and let

b

A

D

(R)

M,

be acontractive

completelypositive map. Let

B

beaseparable

C*-algebra

containing A. Wewillshow that

b

hasanormpreserving, completely positiveextension B

D

(R)

M,,.

that

(A)

C_

Do

(R)11,1,. Let

A

and

D

denote the C*- algebrasobtained by adjoining

identities to

A

and

Do,

respectively. Then the unital map

A

Dx

defined by

ff

(a

+

al)

(a)

+

al iscompletelypositiveby

[1,

Lemma

3.9].

By

hypothesis, there exists a contractivecompletelypositive map r

D

D which extends the identity

embedding of

Do

intoD. Definethe map

if2

A1

D

(R)

M,

by

2(a)

7r((a)).

Then

b

is acontractivecompletely positivemap from

A1

to D(R)M, whichextends

b.

Hencewelnay assumethat

A

has the identity u.

Using thesame notations as in Lemma 2, we have the map

0(b)

A

(R)

M,

D associated with

.

Since

D

is separably injective, thereexists a completely positive extension

bl

B(R)M,

D

of

0().

Again by Lemma 2 we have the completely

positive map

a

B

D

(R)

M,.,

such that

0(fin)

b.

By

the remark aboutLemma 2,

ffa

extends

.

Definethecompletelypositivemap

B

D(R)

Mn

by

(b)

da(ubu).

Then

b

is anextensionof

.

If b E

B

with

II

b

II_<

1, then

II

,(a)I1=11

.,.(,,a,,)I1<_11

.,.

IIII

,,,,

I1_<11

;.,,.(,,)I1=11

(,,)I1=11

II.

Thiscompletes the proof.

EXAMPLE

4. Let

N

be the

Stone-ech

compactitication of the set N ofpositive

integersandN* thecoronaset

/N

N. For

each positiveintegeruput

D,

C(N*)

(R)

M,. Let

Doo

denote the

C*-algebra

ofboundedsequences

{x,

}

such that

,

D,

for eachn. Then

Doo

isseparablyinjective and hasnodecomposition

Doo

Do

Di

such that

D

is

subhomogeneous

and

Di

isinjective.

Prtoo’" Foreach nthere existsaprojection ofnorm onefrom

D,

onto

C(N*)

(R)1,,

where1,, denotesthe identityof

M,.

Thealgebra

C(N*)

isseparablyinjective by

[7,

Theorem

4.6],

but isnot injective. Then

D,

is separably injectiveby Proposition 3,

butis not injective. Hence

Doo

isseparably injective, but isnot injective.

Suppose

that

Doo

has a decomposition

Doo

Do

Di.

Then there exist central projectionsp andqof

Do

such thatp q 1, where 1 denotes the identity of

Doo.

We

have thesequence

{p,

)

of projections of

C(N*)

such thatp

{p,

(R)

1,}.

Hence

D,

{z

e

Doo’

{Zn}

with

n e (C(N*)p,.,)(R)Mr,

forall

n}.

Ifp, 0, thereexists an irreducible representation of

D,

with dimensionn. Since

D0

is subhomogeneous,

wehavean integer

no

such thatpn 0 for alln

> no.

Put

Di.no

{x

Do,:,

x,

{z,}

with z, 0 forall n

< no}.

Then

Di.,

is not injective.

On

the other hand,

thereexistsaprojection ofnorm onefrom

Di

onto

Di.no

and hence

Di.no

isinjective. This isacontradiction andcompletestheproof.

4.

C*-TENSOR

PRODUCTS

OF SEPARABLY INJECTIVE C*-ALGEBRAS.

THEOREM 5. LetA andB be

C*-algebras.

Thefollowingtwo statementsare equiva-lent"

(i)

A(R)B isseparablyinjective.

(ii)

Both

A

and B areseparablyinjective and either

A

or

B

isfinite-dimensional.

Weneedseverallemmas.

LEMMA 6. Let

Ai

bea

C*-subalgebra

ofa

C*-algebra

D, for 1,2, 3. Then, under the obviousidentification, wehave

F(F(A,

(R)

A.,D,

(R)

D),Aa,(D,

(R)

D:)(R) Da)

F(A,, F(A,

Aa,

D

(R)

Da),

D,(R)

(D,.

(R)

Da)).

PROOF" Letz q.

F(F(A,A:,Di(R)D:),Az,(D(R)D:)(R)D3)

and

hi

q.

D’

for/=1,2,3. Then, wehave

Rh:(Rh,(Z))

Rh,(R)ha(Z)

A3,

Lh:(ni,(z))

Rh,(Lh(z))

q_

A:,

because

L,a(z)

F(A,A2,

D

(R)

D2)

bythe assumption. These imply that

R,,

(z) e F(A2, Aa, D2

(R)

D3).

Now,

wehave

L,=(R)la(z)

Ll,(Ll,

a(z))

_

A1,

because

Lha(Z)

_

F(AI,

A:,

DI

(R)D:)

by the assumption. Since the familyofall product functionals

h

(R)

ha

on

D

(R)

Da

istotal, weobtain

L(z)

At

forall h

e (D2

(R)

D3)*

byastandardapproximation argument

(see,

forexample,

[11,

Lemma

2.1]).

Hencewe

have

z q.

F(A, F(A:,

A3,

D2

(R)

D3), D1

(R)

(D2

(R)

D3)).

Thereverseinclusioncanbe shown similarly.

LEMMA

7. Let

A

beaninfinite-dimensional

C*-algebra

and

D

bea

non-subhomogeneous

C*-algebra.

Then

D

(R)

A

isnot separablyinjective.

PROOF: Let

B(H)

be theC*-algebraofallboundedlinearoperatorson aHilbert space

H

such that

B(H)

_D

D

(R)A.Since

A

isinfinite-dimensional, thereexistsan

orthogonal

sequence

{A,,

}

of commutative

C*-subalgebras

ofA. The

C*-subalgebra

generatedby

{A, }

maybe identitiedwiththec0-sum

n

A,, of

{An }.

B(H)(R)F(D(nAn))

F(B(H),D(R)(r,

An),B(H)(R) B(H))

C_

F(B(H),D(R) A,B(H)(R)

B(H))

B(tI)(R)(D(R)

A).

Hence, itfollows that

B(H)(R)F(D(R)(nAn))

F(B(H),D(R)(O,An),B(H)(R)(D(R)A))

F(B(H),F(D,.A.,D(R)

A),B(H)(R)(D(R) A))

(by

[12,

Threm

4])

F(F(B(H),D,B(H)D),An,(B(H)D)A)

(by Lemma

6)

F(B(H)

@

D, .A.,

(B(H)

@D)@

A)

(B(H) D)

@

(,A,)

(by

[12,Th,’em

4])

=B(H)(D@(,A,)).

Since

D

(,An)

is canonically *-imorphic to

n(n

An),

this contradicts

[5,

Theorem

3.2].

Hence

F(B(H),

(n A),

B(H)

@

B(H))

containsproperly

B(H)

(D

A),

andso

D

A

isnotseparably injective by Theorem 1.

LBMMA

8. Let

A

andBbe

Calgebr. I

A Bisparablyinjtive, then both A

and

B

areseparablyinjective.

PaOOF" Let

E

C_

F

be separable

C*-algebras.

Let

b

E

A

be a contractive

completely positive map.

Let

b be a positive element of

B

with

II

b

I1=

1. Define

E

A(R)

B

by

(x)

(:)

(R)b. Then hasa contractive completely positive

extension

1

F

A

(R)B. Leth beastate ofBsuch that

h(b)

1.Define

1

F

A

by

bl(z) =/,((x)).

Then

I

is thedesired extensionof

b.

Thisimplies that A is

separablyinjective.

A

similar argumentshowsthat

B

isseparably injective. Thefollowinglemma isaslight modification of theproofof

[8,

Propositon

2.6].

LEMMA

9. Let

A

and

B

be

C*-algebras

and let

A

and

B

denote the

C*-algebras

obtainedby adjoiningidentities to

A

and

B,

respectively.

I A

(R)

B

isseparablyinjective then

A

(R)

B

isseparably injective.

Paoor- Let E C_

F

beseparable

C*-algebras

andlet

q

E

A

(R)

B

beacontractive completely positivemap. Choose

Am

and

Bo

be separableC*-subalgebras such that

q(E)

_C

Am

(R)

Bo

+

CI(R)

Bo.

By

[8,

Proposition

2.5]

there exist positive elements a E A,b E

B

and c q

A

(R)

B

ofunit normsuch that

a,b,

and c act as identities of

Am, B0

and the

C*-subalgebra generated

by

Am

(R)

Bo

anda (R)

b,

respectively.

We

note that

(a

(R)

b)(1

(R)

d)

a(R)bd

(1

(R)

d)(a

(R)

b)

foreach d (

Bo.

Let h be the state of

A

which annihilatesA. Define

E

A

(R)

B

by

(x)

cqi(x)c and t?

E

B

completelypositiveextension

61

F B.Define

6."

F CI(R)Bby

6(x)= l(R)61(x).

By hypothesis hasacontractivecompletelypositiveextension

g,

F

,4B. Define

Ct

F--

A

B

by

di(x)

(a

b)e(x)(a

b)

+

(I

(a

b)2)O2(x)(I

(a

b)2)

Since

(z)

tnay be written

(a@b,(l_(a@b),))

(,(x)

₀

O(x)

0

) (

(l-(ab)2)

ab

t

is acontractivecompletelypositivemap. Tos extends

,

letz Eand write

(z)

y+

z,y

AoBo,

z

CIBo.

Then

(z)

y+czc

y+zc and

O(x)

z. Thus

()

(,

)(

+

)(

)

+

(

(a

))

(

(

))

+

(

a) +

(

(a

))

().

Hence A B isseparablyinjective.

A

symmetricargumentshows that

A

B

isseparably injective.

LEMMA

10. t

A

beaunitalinnitdensionalsubhomogenus C*-algebra. Then thereexistahomomorphism of

A

andanorm oneprojection such that the image

(x(A))

isimorhpictothe

Calgebra

of allcontinuous functions on mein,hire

compact HausdorffspaceX.

Paoor"

By

[2,

3.6.3

Proposition]

and theproofof

[,

Threm

3.2],

wemaysume that thereexists a cled twsid ideal Jsuch that

A/J

is finite-dimensional, J is

n-homogeneous and

J

is aninfiniteset.

Suppose

first thatJhalimitpoint.

By

[2,

3.6.4

Proposition]

Jisalocally compact Hausdo spe. Thusthereexistsaclo twsid ideal

J0

such that

(J/J0)

an infinitecompt

Hauo

spe. Let

(J/J0)

X. Then

C(X)

maybe identifiedwith the center of

J/Jo.

Let

"

A

A/Jo

be the quotientmap. From

[2,

3.6.4

Proposition]

fora$

A

the map

tr,(A(a))

is continuouson

J,

wheretr,denotes the normaliz traceon

M,.

Notethat ker

A

ker foreh

A

$X. Define

(A)

C(X)

by

((a))(A)

tr,(A(a))

(a

A,

X).

Itis ey toseethat and aredesired maps.

Suppose

now that Jhnolimitpoint. Let Tbeanon-emptyset. t

(M,)

be

Let J Y. Definep" A

.(M,*)

by

p(a)(/)-

,(a)

(aE

A,,

Since

Y

isdiscrete, by

[2,

10.10.1]

wehave

p(J)=

c.(M,*).

Let./,"

.

(M,,)--(M,)/c.(M,*)

denotethequotient map. Since

(pp)-l(O)

_

J, pp(A) is

finit;e-dimensional. Hencethereexistsafinite set

{al,...a:}

ofAsuch that

{lp(al),...

ltp(a)}

spanspp(A). Thenwehave

p(A)

c.(M,) +

Cp(al)+-..

+

Cp(a:).

Let

X

be the one-point compactification ofthe set N of positive integers. Then

C(X)

(R) lr, may be identified with the

C*-algebra

of convergent sequences of ele-lnents ofM,*. Let

p(a,)

(m)

g,(M,).

Passing to convergent subsequences, there exists a sequence

{A,*}

of

Y

such that

(m.)

E

C(X)(R)

M,, for each i. Define

,"

e(/tl,*)--

e

{.}

(/l/l,,)

by

’((a,x))

(a.).

Thent,p(a)

e C(X)(R)

M,

for a EA. Let r ,p. Then

r(A)

C(X)(R)

M,,. Define

,

r(A)

C(X)

by

,(-(,,))(..)

Itiswell known that

b

has thedesiredproperty.

Using Choi-Effros lifting theorem

[1],

Smith andWilliamsshowedinthe proof of

[,

Lemma

3.3]

that every quotient

algebra

ofanuclear separably injective

C*-algebra

is

separably injective.

By

thisusefulresult wehave thefollowinglemma.

LEMMA

11. Let

A

beanuclearseparablyinjective

C*-algebra. Let

r bea

*-homomorpbism of

A

and let

B

be a commutative

C*-subalgebra

of

r(A).

I

there

existsanormoneprojection qi

r(A)

B

such that

(r(A))

B,

then

B

isseparably

injective.

Ptool"

Let E

(:::

F

beseparable

C*-algebras

and let E

B

be a contractive

completelypositive map.

By

the aboveremark,

r(A)

isseparablyinjective. Then has

acontractivecompletely positiveextension F

r(A).

Then

2

F

B

isacontractivecompletelypositiveextensionof

.

Hence

B

isseparably injective. Ploo" o" THEOREM 5:

(i)

=: (ii).

By Lemma 8, itsuffices toshow thateither Aor

Bisfinite-dimensional. Todo this,wemay assumethat

A

and

B

are unitalby Lemma

9.

Supposethat Aand

B

are infinite-dimensional. IfA orBisnon-subhomogeneous, itfollowsfromLemma7 that

A

(R)Bis notseparablyinjective. This isacontradiction.

rx,r2, infinitecompact llousdorffspaces

Xt,

X.

and norm oneprojections

0t

r(A)

C(Xt),

ck

r(B)

C(X:).

By

Le,nma 11 and

[7,

Theoren,

4.6]

.\’, a,,d

X.

are

substonean. %may identify

C(X)(R)

C(X,.)

with

C(X

x

X).

Then (34)

r

G

r,.(AQ

B)

C(Xt

X:)

isanormoneprojection suchthat

b

qi:(r

Or.(AO B))

C(X

x

X).

Again by Lemma 11 and

[7,

Theorem

4.6]

X

x

X

issubstonean. But

thiscontradicts

[3,

Proposition

1.7].

(ii)

:=

(i). Since a finite-dimensional C*-algebra is a finite direct sum of matrix

algebras,Propositon 3 implies that

A

(R)

B

isseparablyinjective.

ACKNOWLEDGMENT: The second authorwas partiallysupportedby Korea

Sci-enceandEngineering Foundation, 1989-90.

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