Almost finiteness, comparison, and tracial $\mathcal{Z}$-stability

Abstract

Inspired by Kerr's work on topological dynamics, we define tracial $\mathcal{Z}$-stability for sub-$C^{\ast }$-algebras. We prove that for a countable discrete amenable group $G$ acting freely and minimally on a compact metrizable space $X$, tracial $\mathcal{Z}$-stability for the sub-$C^{\ast }$-algebra $(C(X)\subseteq C(X)⋊G)$ implies that the action has dynamical comparison. Consequently, tracial $\mathcal{Z}$-stability is equivalent to almost finiteness of the action, provided that the action has the small boundary property.

Final submitted version, to appear in JFA. 24 pages

References in corpus (4)

• Completely positive maps of order zero
• Nuclear dimension of simple C*-algebras
• Étale groupoids and their $C^{\ast }$-algebras
• Uniform property Gamma

Cited by in corpus (1)

• The Symbolic Extension Theory in Topological Dynamics