Complete positivity of elementary operators

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.