Cartan subalgebras and the UCT problem, II
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Mathematische Annalen
Cartan subalgebras and the UCT problem, II Selçuk Barlak1 · Xin Li2 Received: 2 February 2019 / Revised: 1 February 2020 © The Author(s) 2020
Abstract We study the connection between the UCT problem and Cartan subalgebras in C*algebras. The UCT problem asks whether every separable nuclear C*-algebra satisfies the UCT, i.e., a noncommutative analogue of the classical universal coefficient theorem from algebraic topology. This UCT problem is one of the remaining major open questions in the structure and classification theory of simple nuclear C*-algebras. Since the class of separable nuclear C*-algebras is closed under crossed products by finite groups, it is a natural and important task to understand the behaviour of the UCT under such crossed products. We make a contribution towards a better understanding by showing that for certain approximately inner actions of finite cyclic groups on UCT Kirchberg algebras, the crossed products satisfy the UCT if and only if we can find Cartan subalgebras which are invariant under the actions of our finite cyclic groups. We also show that the class of actions we are able to treat is big enough to characterize the UCT problem, in the sense that every such action (even on a particular Kirchberg algebra, namely the Cuntz algebra O2 ) leads to a crossed product satisfying the UCT if and only if every separable nuclear C*-algebra satisfies the UCT. Our results rely on a new construction of Cartan subalgebras in certain inductive limit C*-algebras. This new tool turns out to be of independent interest. For instance, among other things, the second author has used it to construct Cartan subalgebras in all classifiable unital stably finite C*-algebras.
Communicated by Andreas Thom. Selçuk Barlak is supported by the Villum Fonden project grant ‘Local and global structures of groups and their algebras’ (2014–2018). The Xin Li is supported by EPSRC Grant EP/M009718/1.
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Xin Li [email protected] Selçuk Barlak [email protected]
1
Aschaffenburg, Germany
2
School of Mathematics and Statistics, University of Glasgow, University Place, Glasgow G12 8QQ, UK
123
S. Barlak, X. Li
Mathematics Subject Classification Primary 46L05 · 46L40; Secondary 46L80 · 19K35
1 Introduction This paper studies the connection between the UCT problem—one of the remaining major open questions in the structure and classification theory of simple nuclear C*algebras—and Cartan subalgebras of C*-algebras—a concept of recent interest which builds bridges between C*-algebras, topological dynamics, and geometric group theory. The universal coefficient theorem (UCT) for C*-algebras was introduced by Rosenberg and Schochet in [40]. The idea was to formulate a noncommutative analogue of the classical universal coefficient theorem in algebraic topology, very much in line with the general philosophy of viewing C*-algebra theory as noncommutative topology. It facilitates the computation of Kasparov’s bivariant K -theory [18] in terms of topological K -theory for C*-algebras. More precisely, a se
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