English

Cartan subalgebras and the UCT problem

Operator Algebras 2017-06-22 v3

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 O2αZp\mathcal O_2\rtimes_\alpha \mathbb Z_p satisfies the UCT if there is some automorphism γ\gamma of O2\mathcal O_2 with the property that γ(D2)O2αZp\gamma(\mathcal D_2)\subseteq \mathcal O_2\rtimes_\alpha \mathbb Z_p is regular, where D2\mathcal D_2 denotes the canonical masa of O2\mathcal O_2. We prove that this condition is automatic if γ(D2)O2αZp\gamma(\mathcal D_2)\subseteq \mathcal O_2\rtimes_\alpha \mathbb Z_p is not a masa or α(γ(D2))\alpha(\gamma(\mathcal D_2)) is inner conjugate to γ(D2)\gamma(\mathcal D_2). Finally, we relate the UCT problem for separable, nuclear, M2M_{2^\infty}-absorbing C*-algebras to Cartan subalgebras and order two automorphisms of O2\mathcal O_2.

Keywords

Cite

@article{arxiv.1511.02697,
  title  = {Cartan subalgebras and the UCT problem},
  author = {Selçuk Barlak and Xin Li},
  journal= {arXiv preprint arXiv:1511.02697},
  year   = {2017}
}

Comments

v3: 18 pages; minor modifcations, updated reference list, one new reference added. To appear in Adv. Math. v2: 18 pages; minor modifications mostly in Abstract, Introduction and Section 3. More significant changes in Section 4. Main results are unchanged. v1: 17 pages

R2 v1 2026-06-22T11:40:31.541Z