Cartan subalgebras in C*-algebras. Existence and uniqueness

Li, X. and Renault, J. (2019) Cartan subalgebras in C*-algebras. Existence and uniqueness. Transactions of the American Mathematical Society, 372(3), pp. 1985-2010. (doi: 10.1090/tran/7654)

Abstract

We initiate the study of Cartan subalgebras in C*-algebras, with a particular focus on existence and uniqueness questions. For homogeneous C*-algebras, these questions can be analyzed systematically using the theory of fiber bundles. For group C*-algebras, while we are able to find Cartan subalgebras in C*-algebras of many connected Lie groups, there are classes of (discrete) groups, for instance non-abelian free groups, whose reduced group C*-algebras do not have any Cartan subalgebras. Moreover, we show that uniqueness of Cartan subalgebras usually fails for classifiable C*-algebras. However, distinguished Cartan subalgebras exist in some cases, for instance in nuclear uniform Roe algebras.

• Item Type: Articles
• Additional Information: The first named author was supported by EPSRC grant EP/M009718/1.
• Status: Published
• Refereed: Yes
• Glasgow Author(s) Enlighten ID: Li, Professor Xin
• Authors: Li, X., and Renault, J.
• College/School: College of Science and Engineering > School of Mathematics and Statistics > Mathematics
• Journal Name: Transactions of the American Mathematical Society
• Publisher: American Mathematical Society
• ISSN: 0002-9947
• ISSN (Online): 1088-6850
• Published Online: 12 April 2019
• Copyright Holders: Copyright © 2019 American Mathematical Society
• First Published: First published in Transactions of the American Mathematical Society 372(3): 1985-2010
• Publisher Policy: Reproduced in accordance with the publisher copyright policy