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.
Citation
Sze-Kai Tsui. "HILBERT ${ C}^*$-MODULES : A USEFUL TOOL." Taiwanese J. Math. 1 (2) 111 - 126, 1997. https://doi.org/10.11650/twjm/1500405228
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