The Tracial Topological Rank of C*‐Algebras

We introduce the notion of tracial topological rank for C*‐algebras. In the commutative case, this notion coincides with the covering dimension. Inductive limits of C*‐algebrasof the form PM n ( C ( X )) P , where X is a compact metric space with dim X ⩽ k , and P is a projection in M n ( C ( X )), have tracial topological rank no more than k . Non‐nuclear C*‐algebras can have small tracial topological rank. It is shown that if A is a simple unital C*‐algebra with tracial topological rank k (< ∞), then (i) A is quasidiagonal, (ii) A has stable rank 1, (iii) A has weakly unperforated K 0 ( A ), (iv) A has the following Fundamental Comparability of Blackadar: if p , q ∈ A are two projections with τ( p ) < τ( q ) for all tracial states τ on A , then p ≼ q . 2000 Mathematics Subject Classification : 46L05, 46L35.