Topologizing UCTs for unital extensions
type: Item
instance of: scholarly article
title: Topologizing UCTs for unital extensions
zbMATH Open document ID: 1455.46059
author: Shudong Liu
author: Junping Liu
author: Changguo Wei
published in: Banach Journal of Mathematical Analysis
publication date: 2020-08-21T00:00:00Z
review text: Let \(A\) and \(B\) be C*-algebras and let \(K(H)\) be the C*-algebra of compact operators on a separable Hilbert space. Kasparov has identified his odd KK-groups \(\mathrm{KK}_1(A,B)\) with a certain group of unitary equivalence classes of extensions of \(B\otimes K(H)\) by \(A\). If \(A\) is unital, we may restrict to extensions that are unital. Depending on whether we use unitaries in the multiplier algebra of \(B\otimes K(H)\) or in the Calkin algebra, we get the relations of weak and strong unitary equivalence of such unital extensions. The Universal Coefficient Theorem computes KK-groups in terms of homomorphisms between K-theory groups and extensions. If \(A\) is unital and satisfies the Universal Coefficient Theorem, then the two unital Ext-groups may also be described in such terms, using the class of the unit in \(\mathrm{K}_0(A)\) as an extra ingredient. The KK-groups have been equipped with a natural, possibly non-Hausdorff topology. And it has been shown that the exact sequence in the Universal Coefficient Theorem is an exact sequence of topological groups when it applies. In the present article, the unital Ext-groups are equipped with suitable topologies, making them pseudo-Polish groups. It is shown that the exact sequences that relate the unital Ext-groups to K-theory are exact as sequences of topological groups.
reviewed by: Ralf Meyer
Mathematics Subject Classification ID: 46L05
Mathematics Subject Classification ID: 22A05
Mathematics Subject Classification ID: 19K33
Mathematics Subject Classification ID: 46L80
zbMATH DE Number: 7238119
zbMATH Keywords: C*-algebra
zbMATH Keywords: Kasparov theory
zbMATH Keywords: Ext-theory
zbMATH Keywords: universal coefficient theorem
zbMATH Keywords: unital extension
MaRDI profile type: Publication
full work available at URL: s43037-020-00075-w
OpenAlex ID: W3036661010
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DOI: 10.1007/S43037-020-00075-W
description: scientific article