Cartan subalgebras and the UCT problem, II

Barlak, S. and Li, X. (2020) Cartan subalgebras and the UCT problem, II. Mathematische Annalen, 378(1-2), pp. 255-287. (doi: 10.1007/s00208-020-02022-4)

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.

• Item Type: Articles
• Additional Information: The second named author is supported by EPSRC grant EP/M009718/1.
• Status: Published
• Refereed: Yes
• Glasgow Author(s) Enlighten ID: Li, Professor Xin
• Authors: Barlak, S., and Li, X.
• College/School: College of Science and Engineering > School of Mathematics and Statistics > Mathematics
• Journal Name: Mathematische Annalen
• Publisher: Springer
• ISSN: 0025-5831
• ISSN (Online): 1432-1807
• Published Online: 06 July 2020
• Copyright Holders: Copyright © 2020 The Authors
• First Published: First published in Mathematische Annalen 378(1-2): 255-287
• Publisher Policy: Reproduced under a Creative Commons License

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