THE TRACIAL TOPOLOGICAL RANK OF C^*-ALGEBRAS

Abstract

We introduce the notion of tracial topological rank for ${\rm C}^*$-algebras. In the commutative case, this notion coincides with the covering dimension. Inductive limits of ${\rm C}^*$-algebras of the form $PM_n(C(X))P$, where $X$ is a compact metric space with ${\rm dim\,} X\le k$, and $P$ is a projection in $M_n(C(X))$, have tracial topological rank no more than $k$. Non-nuclear ${\rm C}^*$-algebras can have small tracial topological rank. It is shown that if $A$ is a simple unital ${\rm C}^*$-algebra with tracial topological rank $k$ ($<\infty$), then \begin{enumerate} \item[(i)] $A$ is quasidiagonal, \item[(ii)] $A$ has stable rank $1$, \item[(iii)] $A$ has weakly unperforated $K_0(A)$, \item[(iv)] $A$ has the following Fundamental Comparability of Blackadar: if $p,q\in A$ are two projections with $\tau(p)<\tau(q)$ for all tracial states $\tau$ on $A$, then $p\preceq q$. 2000 Mathematics Subject Classification: 46L05, 46L35.