On a characterization of positive maps

Abstract

Abstract. We extend the theory of decomposable maps by giving a detailed description of k-positive maps. A relation between transposition and modular theory is established. The structure of positive maps in terms of modular theory (the generalized Tomita-Takesaki scheme) is examined. 1. Definitions, notations and stating the problem For any C∗-algebra A let A + denote the set of all positive elements in A. A state on a unital C∗-algebra A is a linear functional ω: A → C such that ω(a) ≥ 0 for every a ∈ A + and ω(I) = 1 where I is the unit of A. By S(A) we will denote the set of all states on A. For any Hilbert space H we denote by B(H) the set of all bounded linear operators on H. A linear map ϕ: A → B between C∗-algebras is called positive if ϕ(A +) ⊂ B +.