Ranks of operators in simple C*-algebras

TL;DR

This paper investigates the possible ranks of operators in unital simple C*-algebras, establishing conditions under which the Cuntz semigroup can be reconstructed and confirming a classification conjecture without inductive limit assumptions.

Contribution

It provides a functorial reconstruction of the Cuntz semigroup from K-theoretic data and tracial states, and confirms Elliott's classification conjecture under broad conditions.

Findings

Cuntz semigroup recovered from K-theory and traces

Z nsor A is isomorphic to A under certain conditions

Classification conjecture verified without inductive limit assumptions

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?