Nonunital operator systems and noncommutative convexity

Kennedy, M., Kim, S.-J. and Manor, N. (2023) Nonunital operator systems and noncommutative convexity. International Mathematics Research Notices, 2023(5), pp. 4408-4455. (doi: 10.1093/imrn/rnab349)

Abstract

Abstract We establish the dual equivalence of the category of generalized (i.e., potentially nonunital) operator systems and the category of pointed compact noncommutative (nc) convex sets, extending a result of Davidson and the 1st author. We then apply this dual equivalence to establish a number of results about generalized operator systems, some of which are new even in the unital setting. For example, we show that the maximal and minimal C*-covers of a generalized operator system can be realized in terms of theC*-algebra of continuous nc functions on its nc quasistate space, clarifying recent results of Connes and van Suijlekom. We also characterize “C*-simple” generalized operator systems, that is, generalized operator systems with a simple minimal C*-cover, in terms of their nc quasistate spaces. We develop a theory of quotients of generalized operator systems that extends the theory of quotients of unital operator systems. In addition, we extend results of the 1st author and Shamovich relating to nc Choquet simplices. We show that a generalized operator system is a C*-algebra if and only if its nc quasistate space is an nc Bauer simplex with zero as an extreme point, and we show that a second countable locally compact group has Kazhdan’s property (T) if and only if for every action of the group on a C*-algebra, the set of invariant quasistates is the quasistate space of a C*-algebra.

• Item Type: Articles
• Additional Information: This work was supported by the Canadian Natural Sciences and Engineering Research Council Discovery [2018-202107 to M.K.]; the Canadian Natural Sciences and Engineering Research Council PGS-D Scholarship [396162013 to S.-J.K., 401226864 to N.M.]; and the European Research Council [817597 to S.-J.K.] under the European Union's Horizon 2020 research and innovation programme.
• Status: Published
• Refereed: Yes
• Glasgow Author(s) Enlighten ID: Kim, Mr Se Jin
• Authors: Kennedy, M., Kim, S.-J., and Manor, N.
• College/School: College of Science and Engineering > School of Mathematics and Statistics > Mathematics
• Journal Name: International Mathematics Research Notices
• Publisher: Oxford University Press
• ISSN: 1073-7928
• ISSN (Online): 1687-0247
• Published Online: 19 January 2022
• Copyright Holders: Copyright © 2022 The Authors
• First Published: First published in International Mathematics Research Notices 2023(5): 4408-4455
• Publisher Policy: Reproduced in accordance with the publisher copyright policy

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