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*The Tracial Topological Rank of C-Algebras**

Authors:	Lin Huaxin

Institution:	Department of Mathematics, University of Oregon Eugene, Oregon 97403-1222, USA and East China Normal University Shanghai 200062, China, hxlin{at}noether.uoregon.edu

Abstract:	We introduce the notion of tracial topological rank for C-algebras.In the commutative case, this notion coincides with the coveringdimension. 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 aprojection in M_n(C(X)), have tracial topological rank no morethan k. Non-nuclear C-algebras can have small tracial topologicalrank. It is shown that if A is a simple unital C-algebra withtracial topological rank k (< ${infty}$ ), then (i) A is quasidiagonal, (ii) A has stable rank 1, (iii) A has weakly unperforatedK₀(A), (iv) A has the following Fundamental Comparabilityof Blackadar:if p, q $isin$ A are two projections with ${tau}$ (p) < ${tau}$ (q)for all tracialstates ${tau}$ on A, then p ${cupre}$ q . 2000 MathematicsSubject Classification: 46L05, 46L35.

Keywords:	tracial topological rank simple C*-algebras
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