Room 217

On a tensor-analogue of the Schur product

V. S. Sunder

IMSc

We consider the {\em tensorial Schur product} $R \circ^\otimes S = [r_{ij} \otimes s_{ij}]$ for $R \in M_n(\CA), S\in M_n(\CB),$ with $\CA, \CB ~\mbox{unital}~ C^*$-algebras, prove that such a `tensorial Schur product' of positive operators is again positive, and then use this fact to prove (an apparently marginally more general version of) the classical result of Chi that a linear map $\phi:M_n \rar \M_d$ is completely positive if and only if $[\phi(E_{ij}] \in M_n(M_d)^+$, where of course $\{E_{ij}:1 \leq i.j \leq n\}$ denotes the usual system of matrix units in $M_n (:= M_n(\C))$. We also discuss some other corollaries of the main result.