A covariant Stinespring type theorem for $\tau$-maps

TL;DR

This paper extends Stinespring's theorem to covariant $ au$-maps between Hilbert modules over $C^*$-algebras and von Neumann algebras, establishing a bijective correspondence with crossed product modules under covariance.

Contribution

It introduces a covariant Stinespring type theorem for $ au$-maps and demonstrates a bijective correspondence with crossed product modules in the covariant setting.

Findings

Established a covariant Stinespring type theorem for $ au$-maps.

Proved a bijective correspondence between covariant $ au$-maps and crossed product modules.

Extended the theory to the case where $ au$ is completely positive.

Abstract

Let $\tau$ 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 $\tau$-map if $$\langle T(x),T(y)\rangle=\tau(\langle x, y\rangle)~\mbox{for all}~x,y\in E.$$ A Stinespring type theorem for $\tau$-maps and its covariant version are obtained when $\tau$ is completely positive. We show that there is a bijective correspondence between the set of all $\tau$-maps from $E$ to $F$ which are $(u',u)$-covariant with respect to a dynamical system $(G,\eta,E)$ and the set of all $(u',u)$-covariant $\widetilde{\tau}$-maps from the crossed product $E\times_{\eta} G$ to $F$, where $\tau$ and $\widetilde{\tau}$ are completely positive.