Cartan subalgebras and the UCT problem, II

Abstract

We show that outer approximately represenbtable actions of a finite cyclic group on UCT Kirchberg algebras satisfy a certain quasi-freeness type property if the corresponding crossed products satisfy the UCT and absorb a suitable UHF algebra tensorially. More concretely, we prove that for such an action there exists an inverse semigroup of homogeneous partial isometries that generates the ambient C*-algebra and whose idempotent semilattice generates a Cartan subalgebra. We prove a similar result for actions of finite cyclic groups with the Rokhlin property on UCT Kirchberg algebras absorbing a suitable UHF algebra. These results rely on a new construction of Cartan subalgebras in certain inductive limits of Cartan pairs. We also provide a characterisation of the UCT problem in terms of finite order automorphisms, Cartan subalgebras and inverse semigroups of partial isometries of the Cuntz algebra ${\mathcal{O}}_2$. This generalizes earlier work of the authors.

minor revisions; final version, accepted for publication in Math. Ann.; 26 pages

References in corpus (7)

• Classification of finite simple amenable $\mathcal{Z}$-stable $C^{\ast }$-algebras
• Renault's Equivalence Theorem for Groupoid Crossed Products
• Sequentially split $\ast$-homomorphisms between ${\mathrm{C}}^{\ast }$-algebras
• Strongly self-absorbing C*-dynamical systems, III
• Topological higher-rank graphs and the $C^{\ast }$-algebras of topological 1-graphs
• Some classifiable groupoid C*-algebras with prescribed K-theory
• Graph-Based Models for Kirchberg Algebras

Cited by in corpus (7)

• The inner structure of boundary quotients of right LCM semigroups
• Cartan subalgebras in C*-algebras. Existence and uniqueness
• Every classifiable simple C*-algebra has a Cartan subalgebra
• Classifiable ${\mathrm{C}}^{\ast }$-algebras from minimal $\mathbb{Z}$-actions and their orbit-breaking subalgebras
• Constructing Menger manifold C*-diagonals in classifiable C*-algebras
• On diagonal quasi-free automorphisms of simple Cuntz-Krieger algebras
• On Conjugacy of MASAs in Graph $C^{\ast }$-Algebras