Z-stability, finite dimensional tracial boundaries and continuous rank functions

TL;DR

This paper extends the understanding of Z-stability in certain nonunital C*-algebras, showing they tensorially absorb the Jiang-Su algebra under specific conditions involving trace boundaries and rank functions.

Contribution

It generalizes recent theorems to nonunital C*-algebras with particular trace and rank properties, expanding the class of algebras known to be Z-stable.

Findings

Z-stability holds for specified nonunital C*-algebras

Conditions involve Choquet simplices with finite dimensional extreme boundary

Tensorial absorption of Jiang-Su algebra Z proven under these conditions

Abstract

We observe that a recent theorem of Sato, Toms-White-Winter and Kirchberg-Rordam also holds for certain nonunital C*-algebras. Namely, we show that an algebraically simple, separable, nuclear, nonelementary C*-algebra with strict comparison, whose cone of densely finite traces has as a base a Choquet simplex with compact, finite dimensional extreme boundary, and which admits a continuous rank function, tensorially absorbs the Jiang-Su algebra Z.