$K^ n$-positive maps in $C^ \ast$-algebras

Abstract:

Let ${K^n}$ be the set of $n$-positive maps of $B(H)$ to $B(H)$. A ${K^n}$-positive map of a ${C^ * }$-algebra $A$ to $B(H)$ is a positive linear map $\phi$ such that $\sum {{\text {Tr(}}\phi {\text {(}}} {a_i})b_i^t) \geq 0$ for any $\sum {{a_i} \otimes {b_i} \in \{ x \in A{ \otimes _\gamma }T(H)|{K^n}} { \ni ^\forall }\alpha ,({\text {id}} \otimes \alpha {\text {)(}}x{\text {)}} \geq {\text {0}}\}$. It is shown that the following three statements are equivalent. (1) Every ${K^n}$-positive map of $A$ to $B(H)$ is ${K^{n + 1}}$-positive. (2) Every ${K^n}$-positive map of $A$ to $B(H)$ is completely positive. (3) $A$ is an $n$-subhomogeneous ${C^ * }$-algebra.