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Applications of model theory to C*-dynamics

Gardella, Eusebio and Lupini, Martino (2018) Applications of model theory to C*-dynamics. Journal of Functional Analysis, 275 (7). pp. 1889-1942. ISSN 0022-1236. doi:10.1016/j.jfa.2018.03.020.

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We initiate the study of compact group actions on C*-algebras from the perspective of model theory, and present several applications to C*-dynamics. Firstly, we prove that the continuous part of the central sequence algebra of a strongly self-absorbing action is indistinguishable from the continuous part of the sequence algebra, and in fact equivariantly isomorphic under the Continuum Hypothesis. As another application, we present a unified approach to several dimensional inequalities in C*-algebras, which is done through the notion of order zero dimension for an (equivariant) *-homomorphism. Finiteness of the order zero dimension implies that the dimension of the target algebra can be bounded by the dimension of the domain. The dimension can be, among others, decomposition rank, nuclear dimension, or Rokhlin dimension. As a consequence, we obtain new inequalities for these quantities. As a third application we obtain the following result: if a C*-algebra A absorbs a strongly self-absorbing C*-algebra D, and α is an action of a compact group G on A with finite Rokhlin dimension with commuting towers, then α absorbs any strongly self-absorbing action of G on D. This has a number of interesting consequences, already in the case of the trivial action on D. For example, we deduce that D-stability passes from A to the crossed product. Additionally, in many cases of interest, our result restricts the possible values of the Rokhlin dimension to and ∞, showing a striking parallel to the behavior of the nuclear dimension for simple C*-algebras. We also show that an action of a finite group with finite Rokhlin dimension with commuting towers automatically has the Rokhlin property if the algebra is UHF-absorbing.

Item Type:Article
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URLURL TypeDescription Paper
Lupini, Martino0000-0003-1588-7057
Additional Information:© 2018 Elsevier. Received 24 October 2017, Accepted 29 March 2018, Available online 5 April 2018. The first-named author was partially funded by SFB 878 Groups, Geometry and Actions, and by a postdoctoral fellowship from the Humboldt Foundation. The second-named author was partially supported by the NSF Grant DMS-1600186. Part of this work was carried out by the authors while visiting the Institut Mittag-Leffler in occasion of the program “Classification of operator algebras: complexity, rigidity, and dynamics”. The authors gratefully acknowledge the hospitality and the financial support of the Institute.
Funding AgencyGrant Number
Deutsche Forschungsgemeinschaft (DFG)SFB 878
Alexander von Humboldt FoundationUNSPECIFIED
Institut Mittag-LefflerUNSPECIFIED
Subject Keywords:Model theory for metric structures; Existential theory; Group action; Locally compact second countable group; C*-algebra; Rokhlin dimension; Nuclear dimension; Decomposition rank; Strongly self-absorbing action
Issue or Number:7
Classification Code:MSC: primary: 03C98; 46L55; secondary: 28D05; 46L40; 46M07
Record Number:CaltechAUTHORS:20180406-092947913
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Official Citation:Eusebio Gardella, Martino Lupini, Applications of model theory to C*-dynamics, Journal of Functional Analysis, Volume 275, Issue 7, 2018, Pages 1889-1942, ISSN 0022-1236, (
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:85682
Deposited By: George Porter
Deposited On:09 Apr 2018 15:33
Last Modified:15 Nov 2021 20:31

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