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arxiv: 1511.02697 · v3 · pith:N6WTUXT4new · submitted 2015-11-09 · 🧮 math.OA

Cartan subalgebras and the UCT problem

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keywords mathcalgammaalphacartanmathbbrtimessubalgebrascrossed
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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 $\mathcal O_2$.

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