Operator Valued Maps on Hilbert $C^{\ast }$-Modules

Abstract

We provide a characterization for operator valued completely bounded linear maps on Hilbert $C^{\ast }$-modules in terms of $\phi$-maps. Also, we show that for every operator valued completely positive map $\phi$ on a $C^{\ast }$-algebra $\mathcal{A}$, there is a unique (up to multiplication by a unitary operator) non-degenerate $\phi$-map on each Hilbert $\mathcal{A}$-module.