On the ternary domain of a completely positive map on a Hilbert $C^{\ast }$-module
Volume 255 / 2020
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 the ‘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. 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 associated 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 particular sense to be defined later) 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 $X$.
