A set of maps from \(K\) to \(\mathrm{End}_ A(\ell _ 2(A))\) isomorphic to \(\mathrm{End}_{A(K)}(\ell _ 2(A(K)))\). Applications
type: Item
instance of: scholarly article
title: A set of maps from \(K\) to \(\mathrm{End}_ A(\ell _ 2(A))\) isomorphic to \(\mathrm{End}_{A(K)}(\ell _ 2(A(K)))\). Applications
zbMATH Open document ID: 0579.46033
DOI: 10.1007/BF01000337
author: M. Frank
published in: Annals of Global Analysis and Geometry
publication date: 1985-01-01T00:00:00Z
review text: Let \(A\) be a \(C^*\)-algebra, \(K\) be a compact space, \(A(K)\) be the \(C^*\)- algebra of all continuous maps from \(K\) into \(A\), \(\ell_ 2(A)\) be the Hilbert \(A\)-module being standard for all countably generated Hilbert \(A\)- modules in the sense of Kasparov's stabilization theorem. We investigate a set of maps from \(K\) into \(\mathrm{End}_ A(\ell_ 2(A))\) which is isomorphic to \(\mathrm{End}_{A(K)}(\ell_ 2(A(K))).\) We describe the subsets which are isomorphic to \(\mathrm{End}_{A(K)}(\ell_ 2(A(K))),\) \(\mathrm{GL}_{A(K)}(\ell_ 2(A(K)))\) and \(\mathrm{GL}_{A(K)}(\ell_ 2(A(K))),\) respectively. As an application we deduce a criterion for the self-duality of \(\ell_ 2(A)\) for \(A\) being commutative.
Mathematics Subject Classification ID: 46H25
Mathematics Subject Classification ID: 46L05
Mathematics Subject Classification ID: 46M20
zbMATH DE Number: 3927662
zbMATH Keywords: countably generated Hilbert \(A\)-modules
zbMATH Keywords: Kasparov stabilization
zbMATH Keywords: theorem
zbMATH Keywords: operators on Hilbert \(C^*\)-modules
zbMATH Keywords: compact space
MaRDI profile type: Publication
cites work: Q3854185
cites work: Q4744815
cites work: On the contractability of the unitary group of the Hilbert space over a C*-algebra
full work available at URL: https://doi.org/10.1007/bf01000337
OpenAlex ID: W2316197086
description: scientific article; zbMATH DE number 3927662