K n -Positive Maps in C ∗ -Algebras

Abstract

Let Kn be the set of n-positive maps of B(H) to B(H). A Knpositive map of a C*-algebra A to B(H) is a positive linear map q5 such that ETr(q5(ai)bt) > Oforany Ea,bi E {x E A0@T(H)jK n Dv a, (id0oa)(x) > 0}. It is shown that the following three statements are equivalent. (1) Every K -positive map of A to B(H) is Kf+'-positive. (2) Every Kn-positive map of A to B(H) is completely positive. (3) A is an n-subhomogeneous C*-algebra. Introduction. The concepts of mapping cones K and K-positive maps which have introduced by St0rmer in [4] are powerful tools when one considers the extension problem of positive maps in C*-algebras. In this paper, we investigate Kn-positive maps of a C*-algebra A to the algebra B(H) of all bounded operators on a Hilbert space H, which are induced by the intermediary of the set of all n-positive maps Kn (which is one of the mapping cones) in B2 (H)+, and which may have a close relation with the extension problem. The positivity of Kn-positive maps is stronger than the positivity of B2 (H)+positive maps and weaker than the positivity of CP(H)-positive maps, and becomes stricter as n grows large. For the definitions of mapping cones, B2 (H)+, and CP(H) we refer the reader to the paper [4]. According to Proposition 1 in the next section, a Kn-positive map is an npositive map. This is not true, however, in the reverse direction, as it is known that a 1-positive map is not in general K1-positive (that is B2 (H)+-positive). The author has recently shown in [2] that the cone which corresponds to the n-positive maps of A to B(H) is n=onv-Y{ ( ai 0b ) (Z, ai 0 bi) a. E Al bi T(H), i = 1 n} in (A0,,T(H))+, where T(H) is the set of all trace class operators on H. Combined with the results of [4], the relation among the cones in (A&,,T(H))+ is the following: cA c ... c cA C CA C ... C (A T (H))+ n n n 1 P(A, K') c ... c P(A, Kn) c P(A, Kn+1) c ... c P(A, CP(H)) When we study the extension problem of positive maps in C*-algebras, it is important whether the above inclusions are strict or not. Thus, we encounter a problem which is similar to the conjecture posed by M. D. Choi [1] in 1972. He Received by the editors January 21, 1986 and, in revised form, May 21, 1986. 1980 Mathematics Subject Classification (1985 Revision). Primary 46L05.