Tracially Quasidiagonal Extensions

Lin, Huaxin

Canadian mathematical bulletin, Tome 46 (2003) no. 3, pp. 388-399

Résumé

It is known that a unital simple ${{C}^{*}}$ -algebra $A$ with tracial topological rank zero has real rank zero. We show in this note that, in general, there are unital ${{C}^{*}}$ -algebras with tracial topological rank zero that have real rank other than zero.Let $0\,\to \,J\,\to \,E\,\to A\,\to \,0$ be a short exact sequence of ${{C}^{*}}$ -algebras. Suppose that $J$ and $A$ have tracial topological rank zero. It is known that $E$ has tracial topological rank zero as a ${{C}^{*}}$ -algebra if and only if $E$ is tracially quasidiagonal as an extension. We present an example of a tracially quasidiagonal extension which is not quasidiagonal.

Détail

DOI : 10.4153/CMB-2003-040-1

Mots-clés : 46L05, 46L80, tracially quasidiagonal extensions, tracial rank