Approaching the UCT problem via crossed products of the Razak-Jacelon algebra

TL;DR

This paper links the UCT problem for separable, nuclear C*-algebras to the UCT property of crossed products involving the Razak-Jacelon algebra and finite cyclic group actions, offering a new perspective on the problem.

Contribution

It introduces a novel approach connecting the UCT problem to crossed products of the Razak-Jacelon algebra, leveraging recent classification results and characterizations involving tracially AF algebras.

Findings

UCT for separable, nuclear C*-algebras depends on crossed products with the Razak-Jacelon algebra.

Characterizes the UCT problem via finite cyclic group actions on the Razak-Jacelon algebra.

Recovers previous characterizations involving the Cuntz algebra O_2.

Abstract

We show that the UCT problem for separable, nuclear $\mathrm C^*$-algebras relies only on whether the UCT holds for crossed products of certain finite cyclic group actions on the Razak-Jacelon algebra. This observation is analogous to and in fact recovers a characterization of the UCT problem in terms of finite group actions on the Cuntz algebra $\mathcal O_2$ established in previous work by the authors. Although based on a similar approach, the new conceptual ingredients in the finite context are the recent advances in the classification of stably projectionless $\mathrm C^*$-algebras, as well as a known characterization of the UCT problem in terms of certain tracially AF $\mathrm C^*$-algebras due to Dadarlat.