Publication date:
20 August 2017
Source:Advances in Mathematics, Volume 316
Author(s): Selçuk Barlak, Xin Li
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 ${\mathcal{O}}_2{⋊}_{\alpha }{\mathbb{Z}}_p$ satisfies the UCT if there is some automorphism γ of ${\mathcal{O}}_2$ with the property that $\gamma ({\mathcal{D}}_2)\subseteq {\mathcal{O}}_2{⋊}_{\alpha }{\mathbb{Z}}_p$ is regular, where ${\mathcal{D}}_2$ denotes the canonical masa of ${\mathcal{O}}_2$. We prove that this condition is automatic if $\gamma ({\mathcal{D}}_2)\subseteq {\mathcal{O}}_2{⋊}_{\alpha }{\mathbb{Z}}_p$ 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 ${\mathcal{O}}_2$.