Nuclear \(C^ *\)-algebras and related questions
type: Item
instance of: scholarly article
title: Nuclear \(C^ *\)-algebras and related questions
zbMATH Open document ID: 0622.46041
DOI: 10.1007/BF01088202
published in: Journal of Soviet Mathematics
publication date: 1987-01-01T00:00:00Z
review text: If \({\mathfrak A}_ 1\) and \({\mathfrak A}_ 2\) are \(C^*\)-algebras due to their exact representations \(\rho_ j: {\mathfrak A}_ j\to B(H_ J)\), where B(H) denotes the algebra of all bounded operators in the Hilbert space H, the minimal norm \(\| \|_{\min}\) can be defined on their tensor product \({\mathfrak A}_ 1\otimes {\mathfrak A}_ 2\). On the other hand there exists the maximal norm \(\| a\|_{\max}=\sup_{\gamma}\| a\|_{\gamma}\) where the supremum is taken over all possible \(C^*\)- norms. The \(C^*\)-algebra \({\mathfrak A}\) is called nuclear, if for any \(C^*\)-algebra \({\mathfrak B}\) the norms \(\| a\|_{\min}=\| a\|_{\max}\) coincide on the tensor product \({\mathfrak A}\otimes {\mathfrak B}\) (i.e. the \(C^*\)-norm is defined there uniquely). The example of such nuclear algebra is \(M_ n({\mathbb{C}})\)-the \(n\times n\) matrices over the complex field. The results obtained by the different authors in the last 10 years on nuclear \(C^*\)-algebras are exposed (73 references).
Mathematics Subject Classification ID: 46L05
Mathematics Subject Classification ID: 46M05
zbMATH DE Number: 4009103
zbMATH Keywords: cross-norm
zbMATH Keywords: minimal norm
zbMATH Keywords: tensor product
zbMATH Keywords: maximal norm
zbMATH Keywords: nuclear \(C^ *\)- algebras
author: A. I. Shtern
MaRDI profile type: Publication
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