Strongly outer actions of amenable groups on Z-stable C^*-algebras

2018/11/01 by Gardella, Eusebio, Hirshberg, Ilan
#FOS: Mathematics #Functional Analysis (math.FA) #Operator Algebras (math.OA)

paper · doi:10.48550/arxiv.1811.00447

Abstract

Let A be a separable, unital, simple, Z-stable, nuclear C^*-algebra, and let α\colon G→ Aut(A) be an action of a countable amenable group G. If the trace space T(A) is a Bauer simplex and the action of G on ∂_eT(A) has finite orbits and Hausdorff orbit space, we show that α is strongly outer if and only if α\otimesid_Z has the weak tracial Rokhlin property. If G is moreover residually finite, then these conditions are also equivalent to α\otimesid_Z having finite Rokhlin dimension (in fact, at most 2). When the covering dimension of ∂_eT(A) is finite, we prove that α is cocycle conjugate to α\otimesid_Z. In particular, the equivalences above hold for α in place of α\otimesid_Z.

Related

• Strongly outer actions of amenable groups on Z-stable nuclear C^*-algebras
• Actions of nonamenable groups on Z-stable C^*-algebras
• Rokhlin-type properties, approximate innerness and Z-stability
• Equivariant Z-stability for single automorphisms on simple C^*-algebras with tractable trace simplices
• Dynamical comparison and Z-stability for crossed products of simple C^*-algebras