SEPARABLE INJECTIVITY AND C " TENSOR PRODUCTS

Let A and B be C*-algebras and let D be a C*-subalgebra of B. We show that if D is separably injective then the triple (A,B,D) verifies the slice map conjecture. As an application, we prove that the minimal C*-tensor product A⊗B is separably injective if and only if both A and B are separably injective and either A or B is finite-dimensional.


PRELIMINARIES AND NOTATION.
Let A and B be C*-algebras and let A Q B denote their minimal (i.e., spatial) tensor product.Let L, denote the (',*-algebra of all, x 7t complex matrices for a positive integer n.If A B is a linear map then , (R) id,, A (R) M, B (R) M, is defined by (j(R)id,)(a,i) (i(a,)). is said to be completely positive if each (R) id, is positive.
A C*-algebra D is said to be injective if given C*-algebras E C_ F, any contractive completely positive map E D has a contractive conpletely positive extension V F D. We say that a C*-algebra D is separably injective if given separable C*- algebras E C_ F, any contractive completely positive map E D has a contractive completely positive extension F D. The separable injectivity in this paper is weaker than one in ( [7], [8]) and both coincide for commutative C*-algebras.A compact ltausdorff space is said to be substonean if every two disjoint co-zero sets have disjoint closures.A compact Hausdorff space X is substonean if and only if C(X) is separably injective [7, Theorem 4.6].
A C*-algebra D is said to be subhomogeneous if every irreducible representation is finite-dimensional with bounded dimension.In particulr, it is said to be t-homogeneous if every irreducible representation is n-dimensional.If D is subhomogeneous then we identify the spectrum D with the primitive ideal space [2, Chapters 3 and 4].maps satisfying R,,(zt (R) z) h(zt)z and L(z (R) z) h(z)z [10].For C*- subalgebras Ai of Di, we define the Fubini product 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 }.
For fixed C*-algebras A1 and A, F(Ai, A, D (R) D2) depends on D (R) D2.But they are all isomorhic and are the largest among them if Dt and D are injective.We denote by A (R)F A: any one of these isomorphic Fubini products of A and A [4].Let A and B be C*-algebras and let D be a C*-subalgebra.The triple (A, B, D) is said to verify the slice map conjecture if F(A,D,A (R) B) A (R) D [21.

THE SLICE MAP PROBLEM.
A C*-algebra A is said to have property (S) if (A, B, D) verifies the slice map conjec- ture for every C*-algebra B and every C*-subalgebra D of B [12].We now consider a property (S') as follows.A C*-algebra D is said to have property (S') if (A, B, D) veri- fies the slice map conjecture for every C*-algebra A and every C*-algebra B containing D. Subhomogeneous or injective C*-algebras have property (S') [11].
THEOREM 1.Let D be a C*-algebra.If D is serarably injective, then D has property (s').
PROOF: Let A be a C*-algebra and B a C*-algebra containing D. Let r F(A, D, A(R) B).Then there exists a sequence {rn} such that a', n) a(i, n)(R)b(i, ..), lim,,,, where each a(i, n) A and each b(i, n) B. Let Bo be the C*-subalgebra generated by {b(i,n) 1,...,m(n),n 1, 2,... } and let Do be the C*-subalgebra of D generated by {Rn (x): h A* }.Then we have Do C_ Bo, x F(A, Do, A (R) Bo) by a sinila.rargument of [4, Lemma 5].By hypothesis, there exists a contractive completely positive map b" B0 D which extends the identity embedding of Do into D. Then Rh((IA (R) b)(:))= dp(Rh(:)) Rh(:r.) The opposite inclusion is immediate.
It is known that the direct sum of two C*-algebras having property (S') has property (S').In order to show that Theorem 1 gives a new example having property (S'), two results will be needed.In the proof of [7, Theorem 4.6], Smith and Williams obtained the following lemma.LEMMA 2. Let B and D be C*-algebras.Then there exists a one to one correspondence between completely positive maps qb B D (R) M, and completely positive maps B (R) M,, D for any positive integer n.
We remark that O is not necessarily norm preserving and that 0 satisfies that O()IA(R)M, 0(lA for a 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 the following proposition is based on an idea of [6, Theorem 2.1].
PROPOSITION 3. Let D be a C*-algebra.I D is separably injective, then D (R) M,, is separably injective for any positive integer n.
PROOF: Let A be a separable C*-algebra and let b A D (R) M, be a contractive completely positive map.Let B be a separable C*-algebra containing A. We will show that b has a norm preserving, completely positive extension B D (R) M,,.
Since the image (A) is separable, there exists a separable C*-subalgebra Do such that (A) C_ Do (R) 11,1,.Let A and D denote the C*-algebras obtained by adjoining identities to A and Do, respectively.Then the unital map A Dx defined by ff (a + al) (a) + al is completely positive by [1, Lemma 3.9].By hypothesis, there exists a contractive completely positive map r D D which extends the identity embedding of Do into D. Define the map if2 A1 D (R) M, by 2(a) 7r((a)).
Then b is a contractive completely positive map from A1 to D (R) M, which extends b.Hence we lnay assume that A has the identity u.
This completes the proof.Prtoo'" For each n there exists a projection of norm one from D, onto C(N*) (R) 1,, where 1,, denotes the identity of M,.The algebra C(N*) is separably injective by [7, Theorem 4.6], but is not injective.Then D, is separably injective by Proposition 3, but is not injective.Hence Doo is separably injective, but is not injective.
Suppose that Doo has a decomposition Doo Do Di.Then there exist central projections p and q of Do such that p q 1, where 1 denotes the identity of Doo.We have the sequence {p, ) of projections of C(N*) such that p {p, (R) 1,}.Hence D, {z e Doo' {Zn} with n e (C(N*)p,.,)(R)Mr,for all n}.Ifp, 0, there exists an irreducible representation of D, with dimension n.Since D0 is subhomogeneous, we have an integer no such that pn 0 for all n > no.Put Di.no {x Do,:, x, {z,} with z, 0 for all n < no}.Then Di., is not injective.On the other hand, there exists a projection of norm one from Di onto Di.no and hence Di.no is injective.This is a contradiction and completes the proof.In this section we prove the following theorem.THEOREM 5. Let A and B be C*-algebras.The following two statements are equiva- lent" (i) A (R) B is separably injective.
(ii) Both A and B are separably injective and either A or B is finite-dimensional.
The reverse inclusion can be shown similarly.LEMMA 7. Let A be an infinite-dimensional C*-algebra and D be a non-subhomogeneous C*-algebra.Then D (R) A is not separably injective.PROOF: Let B(H) be the C*-algebra of all bounded linear operators on a Hilbert space H such that B(H) _D D (R) A. Since A is infinite-dimensional, there exists an orthogonal sequence {A,, } of commutative C*-subalgebras of A. The C*-subalgebra generated by {A, } may be identitied with the c0-sum n A,, of {An }.
LBMMA 8. Let A and B be Calgebr.I A B is parably injtive, then both A and B are separably injective.
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 has a contractive completely positive extension 1 F A (R) B. Let h be a state of B such that h(b) 1. Define 1 F A by bl(z) =/,((x)).Then I is the desired extension of b.This implies that A is separably injective.A similar argument shows that B is separably injective.
The following lemma is a slight modification of the proof of [8, Propositon 2.6].LEMMA 9. Let A and B be C*-algebras and let A and B denote the C*-algebras obtained by adjoining identities to A and B, respectively.I A (R) B is separably injective then A (R) B is separably injective.
Paoor-Let E C_ F be separable C*-algebras and let q E A (R) B be a contractive completely positive map.Choose Am and Bo be separable C*-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 of unit norm such that a,b, and c act as identities of Am, B0 and the C*-subalgebra generated by Am (R) Bo and a (R) b, respectively.We note that (a (R) b)(1 (R) d) a (R) bd (1 (R) d)(a (R) b) for each d ( Bo.Let h be the state of A which annihilates A. Define E A (R) B by (x) cqi(x)c and t? E B by t(z) R,((z)).By Lemma 8, B is separably injective.Then 0 has a contractive completely positive extension 61 F B. Define 6." F CI(R)B by 6(x)= l(R)61(x).
A symmetric argument shows that A B is separably injective.
LEMMA 10. t A be a unital innitdensional subhomogenus C*-algebra.Then there exist a homomorphism ofA and a norm one projection such that the image (x(A)) is imorhpic to the Calgebra of all continuous functions on me in,hire compact Hausdorff space X.
Paoor" By [2, 3.6.3Proposition] and the proof of [, Threm 3.2], we may sume that there exists a cled twsid ideal J such that A/J is finite-dimensional, J is n-homogeneous and J is an infinite set.
Suppose first that J h a limit point.By [2, 3.6.4Proposition] J is a locally compact Hausdo spe.Thus there exists a clo twsid ideal J0 such that (J/J0) an infinite compt Hauo spe.Let (J/J0) X.Then C(X) may be identified with the center of J/Jo.Let " A A/Jo be the quotient map.From [2, 3.6.4Proposition] for a $ A the map tr,(A(a)) is continuous on J, where tr, denotes the normaliz trace on M,.Note that ker A ker for eh A $ X. Define (A) C(X) by ((a))(A) tr,(A(a)) (a A, X).
It is ey to see that and are desired maps.
Suppose now that J h no limit point.Let T be a non-empty set.t (M,) be the C*-algebra of (zx) (zx) such that z M, for all fi T and sup I I ll< and let (M,) be the ideal of t(M,) such that for eh > 0 II 1 for all but a finite number of indic A.
(ii) := (i).Since a finite-dimensional C*-algebra is a finite direct sum of matrix algebras, Propositon 3 implies that A (R) B is separably injective.

For 1 , 2 let
Di be a C*-algebra and let hi D. The right slice map R,, D (R) D D2 and the left slice map L, D (R) D D are unique bounded linear

EXAMPLE 4 .
Let N be the Stone-ech compactitication of the set N of positive integers and N* the corona set /N N.For each positive integer u put D, C(N*)(R)    M,.Let Doo denote the C*-algebra of bounded sequences {x, } such that , D, for each n.Then Doo is separably injective and has no decomposition Doo Do Di such that D is subhomogeneous and Di is injective.