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## Studia Mathematica

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## On the ternary domain of a completely positive map on a Hilbert $C^{\ast }$-module

### Tom 255 / 2020

Studia Mathematica 255 (2020), 27-53 MSC: Primary 46L08; Secondary 46L07. DOI: 10.4064/sm190220-4-9 Opublikowany online: 4 May 2020

#### Streszczenie

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 $X$.

#### Autorzy

College of Science
University of Tehran
Tehran, Iran
and
School of Mathematics
Institute for Research in Fundamental Sciences (IPM)
P.O. Box 19395-5746
Tehran, Iran
e-mail
• Reza BehmaniDepartment of Mathematics
Kharazmi University
50, Taleghani Ave.
15618 Tehran, Iran
e-mail
Faculty of Applied Sciences
University Politehnica of Bucharest
313 Spl. Independentei
060042 Bucureşti, Romania 