Skew products of relations and the structure of simple \(C^ *\)-algebras
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. Pedersen}, J. Oper. Theory 4, 289-296 (1980; Zbl 0495.46040)], concerning the supply of projections in simple \(C^*\)-algebras.
projections in simple \(C^ *\)- algebras
scientific article; zbMATH DE number 3992193
scientific article