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

zbMATH Open1399.46085arXiv1410.4491MaRDI QIDQ4603330

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Publication date: 19 February 2018

Abstract: Let $au$ be a linear map from a unital $C^{\ast }$-algebra $CMcalA$ to a von Neumann algebra $mathematicalB$ and let $CMcalC$ be a unital $C^{\ast }$-algebra. A map $T$ from a Hilbert $CMcalA$-module $E$ to a von Neumann $CMcalC$-$CMcalB$ module $F$ is called a $au$-map if ${\displaystyle }$langle T(x),T(y) angle= au(langle x, y angle)~mbox{for all}~x,yin E.${\displaystyle }$ A Stinespring type theorem for $au$-maps and its covariant version are obtained when $au$ is completely positive. We show that there is a bijective correspondence between the set of all $au$-maps from $E$ to $F$ which are $(u^{\prime },u)$-covariant with respect to a dynamical system $(G,eta,E)$ and the set of all $(u^{\prime },u)$-covariant $widetildeau$-maps from the crossed product $Eime{s}_{eta}G$ to $F$, where $au$ and $widetildeau$ are completely positive.

Full work available at URL: https://arxiv.org/abs/1410.4491

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