On compact completely bounded maps of \(C^*\)-algebras
type: Item
instance of: scholarly article
title: On compact completely bounded maps of \(C^*\)-algebras
zbMATH Open document ID: 0533.46037
author: Tadasi Huruya
published in: Michigan Mathematical Journal
publication date: 1983-01-01T00:00:00Z
review text: Let \(A\) and \(B\) be \(C^*\)-algebras. For a linear map \(\phi\) of \(A\) into \(B\), it is possible to define associated maps \[ \phi\otimes\text{id}_ n :: A\otimes M_ n \to B\otimes M_ n, \] and \(\phi\) is said to be completely positive if each \(\phi\otimes\text{id}_ n\) is positive, and completely bounded if \(\sup_{n} \|\phi\otimes\text{id}_ n\|<\infty.\) This quantity is denoted \(\| \phi \|_{cb}\) when it exists. The main theorem says that the following four statements are equivalent. (1) Either A is finite dimensional or B satisfies that all its irreducible representations are on Hilbert spaces at most some positive integer. (2) Every bounded linear map of A into B is completely bounded. (3) Every compact linear map of A into B is completely bounded. (4) Every compact linear map of A into B is, in the norm \(\|.\|_{cb}\), a limit of maps with finite rank. The proof is based on [\textit{R. R. Smith}, J. Lond. Math. Soc., II. Ser. 27, 157-166 (1983; Zbl 0527.46052)] and [\textit{J. Tomiyama} and the author, J. Oper. Theory 10, 141-152 (1983; Zbl 0517.46043)]. Further related results are included. As an application of the above result of Smith, it is shown that if, on the algebraic tensor product \(A\odot B\), the greatest cross norm is equivalent to the projective \(C^*\)-cross norm, then A or B is finite dimensional.
Mathematics Subject Classification ID: 46L05
Mathematics Subject Classification ID: 46M05
zbMATH DE Number: 3846241
zbMATH Keywords: compact linear map
zbMATH Keywords: completely bounded map
zbMATH Keywords: completely positive map
zbMATH Keywords: cross norm
zbMATH Keywords: irreducible representations
zbMATH Keywords: projective \(C^*\)-cross norm
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1307/mmj/1029002852
OpenAlex ID: W1980488541
DOI: 10.1307/MMJ/1029002852
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description: scientific article; zbMATH DE number 3846241