Commun. Korean Math. Soc. 1997 Vol. 12, No. 3, 571-577
Dong-Yun Shin Seoul City University
Abstract : We first show that if $\psi{}:{}M_n(B(H))\rightarrow M_n(B(H))$ is a $D_n \otimes F(H)$-bimodule map, then there is a matrix $A \in M_n$ such that $\psi{}={}S_A.$ Secondly, we show that for an operator space $\Cal E,$ $A\in M_n,$ the Schur product map $S_A{}:{} M_n( \Cal E) \rightarrow M_n(\Cal E)$ and $\phi_A : M_n(\Cal E )\rightarrow \Cal E , $ defined by $\phi_A ([x_{ij}])=\sum _{i,j=1} ^n a_{ij}x_{ij},$ we have $\Vert S_A \Vert {}={}\Vert S_A \Vert_{cb}{}={}\Vert A \Vert _S,$ $\Vert \phi_A \Vert {}={}\Vert \phi_A \Vert_{cb}{}={}\Vert A \Vert _1 $ and obtain some characterizations of $A$ for which $S_A$ is contractive.
