Complete positivity of elementary operators

Abstract:

In this paper, we prove that if $\mathcal {S}$ is an $n$-dimensional subspace of $L(H)$, then $\mathcal {S}$ is $([\frac {n}{2}]+1)$-reflexive, where $[\frac {n}{2}]$ denotes the greatest integer not larger than $\frac {n}{2}$. By the result, we show that if $\Phi ( \cdot )= \sum \limits _{i=1} \limits ^{n} A_{i}( \cdot )B_{i}$ is an elementary operator on a $C^{\ast }$-algebra $\mathcal {A}$, then $\Phi$ is completely positive if and only if $\Phi$ is $([\frac {n-1}{2}]+1)$-positive.