Ranks of operators in simple C*-algebras

Marius Dadarlat; Andrew S. Toms

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arXiv · 0912.0675

Ranks of operators in simple C*-algebras

Marius Dadarlat·Andrew S. Toms

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Abstract

Let A be a unital simple separable C*-algebra with strict comparison of positive elements. We prove that the Cuntz semigroup of A is recovered functorially from the Murray-von Neumann semigroup and the tracial state space T(A) whenever the extreme boundary of T(A) is compact and of finite covering dimension. Combined with a result of Winter, we obtain Z \otimes A isomorphic to A whenever A moreover has locally finite decomposition rank. As a corollary, we confirm Elliott's classification conjecture under reasonably general hypotheses which, notably, do not require any inductive limit structure. These results all stem from our investigation of a basic question: what are the possible ranks of operators in a unital simple C*-algebra?

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Marius Dadarlat, Andrew S. Toms. 2009-12-03. Ranks of operators in simple C*-algebras. https://arxiv.org/abs/0912.0675

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