The tracial topological rank of \(C^*\)-algebras (Q2766415)

There have been two notions of rank in \(C^*\)-algebra theory, namely (topological) stable rank and real rank, as a noncommutative analogue of topological dimension. The author of the paper under review introduces another notion, i.e., tracial (topological) rank, which has turned out to be useful for the classification of (simple) nuclear \(C^*\)-algebras; see \textit{H. Lin} [Trans. Am. Math. Soc. 353, 693-722 (2001; Zbl 0964.46044)] and also Chapter 3 of \textit{H. Lin} [An introduction to the classification of amenable \(C^*\)-algebras, Singapore, World Scientific (2001; Zbl 1013.46055)]. NEWLINENEWLINENEWLINEIn the present paper, the author, apart from several nice examples, gives a deep analysis of the topic, establishes that a separable \(C^*\)-algebra \(C(X)\) has tracial rank if and only if \(\dim X=n\), and shows that a unital simple \(C^*\)-algebra \(A\) with tracial rank \(k\) has stable rank \(1\), real rank no more than \(1\), weakly unperforated \(K_\circ (A)\) with the Riesz property, and Blackadar's Fundamental Comparison Property. He also proves that if a unital \(C^*\)-algebra \(A\) has tracial rank \(k\), then every unital hereditary \(C^*\)-subalgebra of \(A\) and every quotient of \(A\) by a closed ideal has tracial rank no more than \(k\) . NEWLINENEWLINENEWLINEIt is proved that a unital simple \(C^*\)-algebra with tracial rank \(k\) and with a unique normalized trace has tracial rank zero.