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

Volume 255 / 2020

Mohammad B. Asadi, Reza Behmani, Maria Joiţa Studia Mathematica 255 (2020), 27-53 MSC: Primary 46L08; Secondary 46L07. DOI: 10.4064/sm190220-4-9 Published online: 4 May 2020


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 $X$.


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

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