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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

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 $X$.

Authors

  • Mohammad B. AsadiSchool of Mathematics, Statistics and Computer Science
    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
  • Maria JoiţaDepartment of Mathematics
    Faculty of Applied Sciences
    University Politehnica of Bucharest
    313 Spl. Independentei
    060042 Bucureşti, Romania
    http://sites.google.com/a/g.unibuc.ro/maria-joita/
    e-mail

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