Finite rank approximation and semidiscreteness for linear operators
type: Item
instance of: scholarly article
title: Finite rank approximation and semidiscreteness for linear operators
zbMATH Open document ID: 0989.46033
author: Christian Le Merdy
publication date: 1999-11-24T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/75405
review text: A \(C^*\)-algebra \(B\) is called to be nuclear if \(A\otimes_{\min}B= A\otimes_{\max}B\) holds true for all \(C^*\)-algebras \(A\). This is equivalent to the existence of a net \(B@>\alpha_i>> M_{n_i} @>\beta_i>> B\) of completely positive contractions \(\alpha_i\), \(\beta_i\) with \(\lim\beta_i\alpha_i= I_B\) pointwise. G. Pisier has introduced an operator version of this concept: A completely bounded map \(u: Y\to B\), \(B\) any \(C^*\)-algebra and \(Y\) any operator space, is called to be \(c\)-nuclear, if for every \(C^*\)-algebra \(A\) the mapping \(I_A\otimes u: A\otimes_{\min} Y\to A\otimes_{max}B\) is bounded with a norm less than \(c\). There is again an equivalent characterization by finite rank approximation \(Y@> \alpha_i>> M_{n_i} @>\beta_i>> B\) of \(u\). For the case of von Neumann algebras the notion of nuclearity has to be replaced by that of semidiscreteness: a von Neumann algebra \(M\) is called to be semidiscrete, if \(A\otimes_{\min} M= A\otimes_{\text{nor}}M\) for each \(C^*\)-algebra \(A\). (\(A\otimes_{\text{nor}}M\) denotes the normal tensor product introduced by E. Effros.) Again in this case there is an equivalent characterization of nets of finite rank operators \(M@> \alpha_i>> M_{n_i}@> \beta_i>> M\). In the paper under review the author introduces an operator version of semidiscreteness as follows: a map \(u: Z\to M\), \(Z\) an operator space, is called to be \(c\)-semidiscrete, if for any \(C^*\)-algebra \(A\) the tensor map \(I_A\otimes u: A\otimes_{\min} Z\to A\otimes_{\text{nor}}M\) is bounded with a norm less than \(c\). As a main result, an equivalent characterization of the \(c\)-discreteness by finite rank approximation of \(u\) in the above sense is given, and the relationship to the \(c\)-nuclearity is studied. Since a \(C^*\)-algebra \(B\) is nuclear iff \(B^{**}\) is semidiscrete, one has to compare \(c\)-nuclear maps \(u: Y\to B\) with properties of \(u^{**}\). The author shows that for completely bounded mappings \(u: Y…
reviewed by: Heinz Junek
Mathematics Subject Classification ID: 46L06
Mathematics Subject Classification ID: 46L99
Mathematics Subject Classification ID: 46B28
Mathematics Subject Classification ID: 47L20
Mathematics Subject Classification ID: 46L10
Mathematics Subject Classification ID: 47L30
zbMATH DE Number: 1368752
zbMATH Keywords: approximation
zbMATH Keywords: tensor products
zbMATH Keywords: von Neumann algebras
zbMATH Keywords: operators algebras
zbMATH Keywords: nuclear \(C^*\)-algebras
zbMATH Keywords: completely bounded map
zbMATH Keywords: nuclearity
zbMATH Keywords: semidiscreteness
zbMATH Keywords: nets of finite rank operators
zbMATH Keywords: tensor map
zbMATH Keywords: \(c\)-semidiscreteness
zbMATH Keywords: \(c\)-nuclearity
MaRDI profile type: Publication
OpenAlex ID: W2319104208
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DOI: 10.5802/AIF.1741
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published in: Annales de l’institut Fourier
description: scientific article; zbMATH DE number 1368752