Cartan subalgebras and the UCT problem, II

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 $$\mathcal O_2$$ O 2 ) 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.

Every classifiable simple C*-algebra has a Cartan subalgebra

“…Generally speaking, Cartan subalgebras allow us to introduce ideas from geometry and dynamical systems to the study of C*-algebras. More concretely, in view of [3,4], we expect that our answer to Question 1.1 will lead to progress on the UCT question.…”

“…Moreover, G. Szabó and S. Vaes independently found groupoid models for W, again using constructions different from ours. Furthermore, independently from [4] and the present paper, similar tools to the ones in [4, § 3] were developed in [2], which give rise to groupoid models for Z and W as well as other examples.…”

“…The key tool for all the results in this paper is an improved version of [4,Theorem 3.6], which allows us to construct Cartan subalgebras in inductive limit C*-algebras. The C*-algebraic formulation reads as follows.…”

“…In this section, we improve the main result in [4, § 3] and give a C*-algebraic interpretation. Let us first recall [4,Theorem 3.6]. We use the same notations and definitions as in [4,36].…”

“…Recently, the notion of Cartan subalgebras in C*-algebras [25,36] has attracted attention, due to connections to topological dynamics [26][27][28] and the UCT question [3,4]. In particular the reformulation of the UCT question in [3,4] raises the following natural question (see [29,Question 5.9], [42,Question 16] and [5, Problems 1 and 2]):…”