Operator Algebra Seminar

Speaker: Hannes Thiel, Westfälische Wilhelms-Universität Münster

Title: Ranks of operators in simple C*-algebras with stable rank one

Abstract: Let A be a unital, separable, simple C*-algebra with stable rank one.
Every positive element in the stabilization of A induces a rank function on the space of tracial states of A. We show that all possible rank functions are realized this way.

This implies that the Cuntz semigroup of A is almost divisible as soon as it is almost unperforated. Assuming additionally that A has locally finite nuclear dimension (for
example, that A is an ASH-algebra), we obtain that strict comparison of positive elements in A implies that A tensorially absorbs the Jiang-Su algebra. In particular, this verifies the Toms-Winter conjecture for ASH-algebras with stable rank one.