On a Rokhlin property for abelian group actions on C$^*$-algebras
Johannes Christensen, Robert Neagu, Gábor Szabó
TL;DR
The paper develops a unified framework for abelian Rokhlin-type properties of locally compact abelian group actions on C*-algebras and their duals. It introduces the abelian Rokhlin property and a rational variant, and proves a central duality: abelian Rokhlin property is dual to pointwise strong approximate innerness (and vice versa). This duality is formulated via equivariantly sequentially split maps and extended to encompass Z-stability and Rokhlin-dimension considerations, recovering known results for flows and finite/compact groups. The authors further show that the abelian Rokhlin property yields a complete description of densely defined lower semicontinuous traces on crossed products as traces induced from invariant traces on the coefficient algebra, thereby unifying many trace-structure results in the literature.
Abstract
In this article, we study the so-called abelian Rokhlin property for actions of locally compact, abelian groups on C$^*$-algebras. We propose a unifying framework for obtaining various duality results related to this property. The abelian Rokhlin property coincides with the known Rokhlin property for actions by the reals (i.e., flows), but is not identical to the known Rokhlin property in general. The main duality result we obtain is a generalisation of a duality for flows proved by Kishimoto in the case of Kirchberg algebras. We consider also a slight weakening of the abelian Rokhlin property, which allows us to show that all traces on the crossed product C$^*$-algebra are canonically induced from invariant traces on the the coefficient C$^*$-algebra.