On the ternary domain of a completely positive map on a Hilbert C*-module

TL;DR

This paper introduces the concept of the ternary domain of a completely positive map on a Hilbert C*-module, characterizes it, and explores its relationships with dilation triples and linking algebra structures.

Contribution

It defines the ternary domain for completely positive maps on Hilbert C*-modules and provides characterizations and connections to dilation theory and linking algebras.

Findings

The ternary domain is a Hilbert C*-module over the multiplicative domain.

The ternary domain of the algebra is a closed two-sided *-ideal.

Every CP map induces a unique CP map on the linking algebra.

Abstract

We associate to an operator valued completely positive linear map $\varphi$ on a $C^{\ast }$-algebra $A$ and a Hilbert $C^{\ast }$-module $X$ over $A$ a subset $X_{\varphi }$ of $X,$ called '\textit{ternary domain}' of $\varphi$ on $X,$ which is a Hilbert $C^{\ast }$-module over the multiplicative domain of $\varphi $ and every $\varphi $-map (i.e., associated quaternary map with $\varphi $) acts on it as a ternary map. We also provide several characterizations for this set. The ternary domain \ of $\varphi $ on $A\ $ is a closed two-sided $\ast $-ideal $T_{\varphi }$ of the multiplicative domain of $\varphi $. We show that $XT_{\varphi }=X_{\varphi }$ and give several characterizations of the set $X_{\varphi }.$ Furthermore, we establish some relationships between $X_{\varphi }$ and minimal Stinespring dilation triples associate to $\varphi $. Finally, we show that every operator valued completely positive linear map $\varphi $ on a $C^{\ast }$ -algebra $A$ induces a unique (in a some sense) completely positive linear map on the linking algebra of $X$ and we determine its multiplicative domain in terms of the multiplicative domain of $\varphi $ and the ternary domain of $\varphi $ on $X$.