Complete positivity of elementary operators

Abstract

In this paper, we prove that if S is an n-dimensional subspace of L(H), then S is ([i ] + 1)-reflexive, where [n ] denotes the greatest integer not n larger than '. By the result, we show that if 1?( ) = E Ai( )Bi is an 2=1 elementary operator on a C*-algebra A, then 'D is completely positive if and only if 'D is ([n1 ] + 1)-positive. In this paper, let H denote a complex Hilbert space. Let H(') denote the direct sum of n copies of H. For T E L(H), we write T(') for the operator on H(') which is the direct sum of n copies of T; the notation is extended to a subset of L(H) by S(n) = {T(n) E L(H(n)): T E S}. If S is a subspace of L(H), S is called n-reflexive if S(n) = ref (S(n)) =_ {T(n) E L(H(n)): T(n)X E [S(n)X], for all x E H (n)}, where [] denotes norm closed linear span. By the definiton, we have that if S is m-reflexive, then S is n-reflexive for n > m. A separating vector for a subspace S of L(H) is a vector x E H such that T 4 Tx, T E S, is an invective map. For x, y E H, let x 0 y denote the rank-one operator u | 4 (u, x)y. Let A denote a C*-algebra. Then A is called primitive, if A has a faithful irreducible representation on some Hilbert space. An elementary operator AP on A n is a linear mapping of the form AP: T F4 AiTBi, where {Ai},nL1 and {Bi},nU1 are i=l subsets of A. In this paper, we assume that all elementary operators are nonzero. A linear map 4J on A is positive (resp. hermitian-preserving) if 4)(T) is positive (resp. hermitian) for all positive (resp. hermitian) T in A. We define 4'n = 4{I 0 In: Mn(A) -4 Mn(A) by 4) 0 In((Tij)nxn) = (4)(Ti ))nxn. 4) is said to be n-positive if 4J 0 In is positive. If 4J is n-positive for all n, then 4J is said to be completely positive. In [4], Magajna states the following problem: For each positive integer r determine the smallest k = k(r) such that all rdimensional subspaces of L(H) are k-reflexive. In [4], Magajna proves k < r. In this paper, we prove that if S is an n-dimensional subspace of L(H), then S is ([n] + 1)-reflexive. Also by this result, we study complete positivity of elementary operators on a C*-algebra A. We prove that if n 4 ) = Ai( )Bi is an elementary operator on a C*-algebra A, then 4) is i=l1 Received by the editors July 8, 1996 and, in revised form, May 14, 1997. 1991 Mathematics Subject Classification. Primary 47B47, 47B49; Secondary 46L05.