Taiwanese Journal of Mathematics

HILBERT ${ C}^*$-MODULES : A USEFUL TOOL

Sze-Kai Tsui

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Abstract

In this article, we show how the concept of Hilbert $C^*$-module can be used to investigate completely positive linear maps. We show when two unital pure completely positive linear maps of a $C^*$-algebra into $M_n$ are unitarily equivalent. We also develop and characterize a concept of weak containment between two completely positive linear maps of a $C^*$-algebra into a von Neumann algebra. In preparation, we exhibit some basic known properties of Hilbert $C^*$-modules. In addition, we explore the norm of the standard Hilbert column $C^*$-modules and show it is the Haagerup tensor norm of two operator spaces.

Article information

Source
Taiwanese J. Math., Volume 1, Number 2 (1997), 111-126.

Dates
First available in Project Euclid: 18 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1500405228

Digital Object Identifier
doi:10.11650/twjm/1500405228

Mathematical Reviews number (MathSciNet)
MR1452088

Zentralblatt MATH identifier
0885.46050

Subjects
Primary: 46L99: None of the above, but in this section 46L05: General theory of $C^*$-algebras
Secondary: 46M99: None of the above, but in this section 46C50: Generalizations of inner products (semi-inner products, partial inner products, etc.)

Keywords
Hibert $C^*$-modules standard Hilbert $C^*$-modules completely positive linear maps GNS-Stinespring construction weak containment Haagerup tensor product

Citation

Tsui, Sze-Kai. HILBERT ${ C}^*$-MODULES : A USEFUL TOOL. Taiwanese J. Math. 1 (1997), no. 2, 111--126. doi:10.11650/twjm/1500405228. https://projecteuclid.org/euclid.twjm/1500405228


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