Certain \(C^*\)-algebras with real rank zero and their corona and multiplier algebras. II
type: Item
instance of: scholarly article
title: Certain \(C^*\)-algebras with real rank zero and their corona and multiplier algebras. II
zbMATH Open document ID: 0816.46058
DOI: 10.1007/BF00961332
author: Shuang Zhang
published in: \(K\)-Theory
publication date: 1993-04-01T00:00:00Z
review text: The author investivates the structures of projections and \(K\)-theory of purely infinite simple \(C^*\)-algebras, their multiplier and corona algebra. For a \(\sigma\)-unital purely infinite \(C^*\)-algebra \(\mathcal A\) the author shows that the homotopy classes \(D_ h(M({\mathcal A}\otimes {\mathcal K}))\) of nontrivial projections in the multiplier algebra \(M({\mathcal A}\otimes {\mathcal K})\) can be identified with the disjoint union \(K_ 0({\mathcal A})\dot{\cup} K_ 0({\mathcal A})\dot{\cup}\{\infty\}\), where \(\{\infty\}\) denotes the path component of projections in \(M({\mathcal A}\otimes {\mathcal K})\) whose images in the generalized Calkin algebra \({\mathcal C}:= C({\mathcal A}\otimes {\mathcal K})= M({\mathcal A}\otimes {\mathcal K})/{\mathcal A}\otimes {\mathcal K}\) are nontrivial projections. Furthermore, there is a bijection between \(D_ h({\mathcal C})\) and \(K_ 1({\mathcal A})\). As consequences of these results the author proves for the Cuntz algebras \({\mathcal O}_ n\) \((2\leq n< \infty)\) that the number of homotopy classes of nontrivial projection in \(\mathcal A\), in \(M({\mathcal A}\otimes {\mathcal K})\) and in \(\mathcal C\) are \(n-1\), \(2n-1\) and 1 respectively. If \(\mathcal A\) is a type III factor then the number of path components of nontrivial projections in \(M({\mathcal A}\otimes {\mathcal K})\) and in \(M({\mathcal C})\) are 3 and 1 respectively. In the second section the author shows that the generalized Fredholm index map defines a bijection between the set of path components of unitaries in the generalized Calkin algebra \({\mathcal C}\) and \(D_ h({\mathcal A})\). In the third section the author considers matroid algebras. Let \(\mathcal A\) be a separable infinite matroid algebra; then it is well-known [\textit{G. A. Elliott}, Ann. Math., II. Ser. 100, 407-422 (1974; Zbl 0296.46062)] that there exists a unique proper closed ideal \(J\) of the multiplier algebra \(M({\mathcal A})\) which contains \(\mathcal A\) str…
Mathematics Subject Classification ID: 46L05
Mathematics Subject Classification ID: 46L80
zbMATH DE Number: 149763
zbMATH Keywords: structures of projections
zbMATH Keywords: \(K\)-theory of purely infinite simple \(C^*\)- algebras
zbMATH Keywords: multiplier and corona algebra
zbMATH Keywords: Cuntz algebras
zbMATH Keywords: homotopy classes
zbMATH Keywords: generalized Fredholm index map
zbMATH Keywords: generalized Calkin algebra
zbMATH Keywords: separable nonunital finite matroid algebra
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf00961332
OpenAlex ID: W2074304494
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description: scientific article; zbMATH DE number 149763