Operations on continuous bundles of \(C^*\)-algebras
type: Item
instance of: scholarly article
title: Operations on continuous bundles of \(C^*\)-algebras
zbMATH Open document ID: 0835.46057
DOI: 10.1007/BF01461011
author: Eberhard Kirchberg
author: Simon Wassermann
published in: Mathematische Annalen
publication date: 1996-04-16T00:00:00Z
full work available at URL: https://eudml.org/doc/165387
review text: We give examples of separable continuous \(C^*\)-algebra bundles \({\mathcal A}_1\), \({\mathcal A}_2\) over \(\widehat{\mathbb{N}}\) and separable \(C^*\)- algebras \(B\), \(C\) such that the fibrewise minimal \(C^*\)-algebra tensor product \({\mathcal A}_1\otimes_{\min} B\) and the fibrewise maximal \(C^*\)-algebra tensor product \({\mathcal A}_2\otimes_{\max} C\) are not continuous \(C^*\)-bundles. It follows that \({\mathcal A}_1\) is not a subbundle of a trivial bundle. We also prove the general results (i) that a \(C^*\)-algebra \(A\) is exact (respectively nuclear) if and only if for every continuous bundle of \(C^*\)-algebras the fibrewise minimal (respectively maximal) \(C^*\)-algebra tensor product with \(A\) is a continuous bundle of \(C^*\)-algebras and (ii) that a continuous \(C^*\)-algebra bundle \(\mathcal A\) with exact fibre algebras has exact bundle algebra if and only if its fibrewise minimal tensor product with any fixed \(C^*\)-algebra is continuous.
Mathematics Subject Classification ID: 46L45
Mathematics Subject Classification ID: 46M20
Mathematics Subject Classification ID: 46L99
zbMATH DE Number: 840206
zbMATH Keywords: separable continuous \(C^*\)-algebra bundles
zbMATH Keywords: fibrewise minimal \(C^*\)- algebra tensor product
zbMATH Keywords: fibrewise maximal \(C^*\)-algebra tensor product
MaRDI profile type: Publication
cites work: C* -Tensor Norms and Slice Maps
cites work: Q4012533
cites work: A Simple C*-Algebra Generated by Two Finite-Order Unitaries
cites work: Lifting problems and local reflexivity for \(C^ *\)-algebras
cites work: Tensor products of operator algebras
cites work: The Heisenberg group and \(K\)-theory
cites work: Q3331747
cites work: On subalgebras of the CAR-algebra
cites work: On non-semisplit extensions, tensor products and exactness of group \(C^*\)-algebras
cites work: On nuclear C\(^*\)-algebras
cites work: Q3205815
cites work: Continuous fields of \(C^*\)-algebras coming from group cocycles and actions
cites work: On the cross-norm of the direct product of \(C^ *\)-algebras
cites work: On tensor products of certain group \(C^*\)-algebras
cites work: Tensor Products of Free-Group C* -Algebras
cites work: \(C^*\)-algebras associated with groups with Kazhdan's property T
description: scientific article; zbMATH DE number 840206