A covariant Stinespring type theorem for $τ$-maps

Abstract

Let $τ$ be a linear map from a unital $C^*$-algebra $\CMcal A$ to a von Neumann algebra $\mathematical B$ and let $\CMcal C$ be a unital $C^*$-algebra. A map $T$ from a Hilbert $\CMcal A$-module $E$ to a von Neumann $\CMcal C$-$\CMcal B$ module $F$ is called a $τ$-map if $$\langle T(x),T(y)\rangle=τ(\langle x, y\rangle)~\mbox{for all}~x,y\in E.$$ A Stinespring type theorem for $τ$-maps and its covariant version are obtained when $τ$ is completely positive. We show that there is a bijective correspondence between the set of all $τ$-maps from $E$ to $F$ which are $(u',u)$-covariant with respect to a dynamical system $(G,η,E)$ and the set of all $(u',u)$-covariant $\widetildeτ$-maps from the crossed product $E\times_η G$ to $F$, where $τ$ and $\widetildeτ$ are completely positive.

Final version, To appear in "Surveys in Mathematics and its Applications"