Skew products of relations and the structure of simple \(C^ *\)-algebras
type: Item
instance of: scholarly article
title: Skew products of relations and the structure of simple \(C^ *\)-algebras
zbMATH Open document ID: 0613.46049
DOI: 10.1007/BF01246943
author: Alex Kumjian
author: Bruce Blackadar
published in: Mathematische Zeitschrift
publication date: 1985-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/173566
review text: The authors present two types of constructions for \(C^*\)-algebras, and three interesting, specific examples of simple \(C^*\)-algebras, with ''few'' projections, which can be constructed in either fashion. The first construction is an inductive limit. Let \(\rho\) be an automorphism of a \(C^*\)-algebra B and \(\theta_{\rho}\) the ''twisted double embedding'' of B into \(M_ 2(B)\) defined by \(\theta_{\rho}(b)=\left[ \begin{matrix} b\\ 0\end{matrix} \begin{matrix} 0\\ \rho (b)\end{matrix} \right]\). Then \(M^{\rho}_{\infty}(B)\) is the inductive limit of \((M_{2^ n}(B)\), \(\theta_{\rho}\otimes id)\). For the second construction, let R be a principal discrete groupoid with unit space X (abbreviated, ''R is a relation on X''). For R a relation on X and Q a relation on Y, the authors define the notion of R-Q cocycle \(\phi\) and a corresponding skew-product relation \(R\times_{\phi}Q\) on \(X\times Y\). The construction is then that of a groupoid \(C^*\)-algebra [\textit{J. Renault}, A groupoid approach to \(C^*\)-algebras, Lecture Notes Math. 793 (1980; Zbl 0433.46049)], where the groupoid is a skew-product relation. Concerning properties of the specific examples so constructed, one of the algebras is stable simple with no non-zero projections (and neither K- theory nor the theory of AF algebras are required for the construction or verification), while another, although it has projections, has the property that any matrix algebra over it has distinct traces which agree on all projections [unlike the projectionless algebra constructed by \textit{B. Blackader}, J. Oper. Theory 5, 63-71 (1981; Zbl 0494.46056)]. The authors also study some of their constructions, as well as tensor products of the compacts with the Bunce-Deddens algebras [\textit{J. Bunce} and \textit{J. Deddens}, J. Funct. Anal. 19, 13-24 (1975; Zbl 0313.46047)], in relation to various properties, considered in [\textit{B. Blackadar}, Semesterbericht Funktionalanalysis, Tübingen, W83] and [\textit{G. P…
Mathematics Subject Classification ID: 46L05
Mathematics Subject Classification ID: 46L55
zbMATH DE Number: 3992193
zbMATH Keywords: skew products of relations
zbMATH Keywords: inductive limit
zbMATH Keywords: twisted double embedding
zbMATH Keywords: principal discrete groupoid
zbMATH Keywords: cocycle
zbMATH Keywords: groupoid \(C^ *\)-algebra
zbMATH Keywords: traces
zbMATH Keywords: tensor products
zbMATH Keywords: Bunce-Deddens algebras
zbMATH Keywords: projections in simple \(C^ *\)- algebras
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3992193