Cartan subalgebras and the UCT problem

TL;DR

This paper establishes that the presence of Cartan subalgebras in separable nuclear C*-algebras ensures they satisfy the UCT, and shows the UCT's stability under certain group actions respecting these subalgebras.

Contribution

It proves the UCT holds for C*-algebras with Cartan subalgebras and demonstrates its closure under specific crossed product constructions, linking the UCT problem to Cartan subalgebras and automorphisms of .

Findings

C*-algebras with Cartan subalgebras satisfy the UCT.

UCT is preserved under crossed products respecting Cartan subalgebras.

Conditions under which crossed products of  are UCT-satisfying are identified.

Abstract

We show that a separable, nuclear C*-algebra satisfies the UCT if it has a Cartan subalgebra. Furthermore, we prove that the UCT is closed under crossed products by group actions which respect Cartan subalgebras. This observation allows us to deduce, among other things, that a crossed product $\mathcal O_2\rtimes_\alpha \mathbb Z_p $ satisfies the UCT if there is some automorphism $\gamma$ of $\mathcal O_2$ with the property that $\gamma(\mathcal D_2)\subseteq \mathcal O_2\rtimes_\alpha \mathbb Z_p$ is regular, where $\mathcal D_2$ denotes the canonical masa of $\mathcal O_2$. We prove that this condition is automatic if $\gamma(\mathcal D_2)\subseteq \mathcal O_2\rtimes_\alpha \mathbb Z_p$ is not a masa or $\alpha(\gamma(\mathcal D_2))$ is inner conjugate to $\gamma(\mathcal D_2)$. Finally, we relate the UCT problem for separable, nuclear, $M_{2^\infty}$-absorbing C*-algebras to Cartan subalgebras and order two automorphisms of $\mathcal O_2$.