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The descriptive classification of some classes of C∗-algebras

Kechris, Alexander S. (1998) The descriptive classification of some classes of C∗-algebras. In: Proceedings of the Sixth Asian Logic Conference. World Scientific , River Edge, NJ, pp. 121-149. ISBN 9789810234324.

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We introduce here a parametrization of separable C*-algebras by a standard Borel space and study the descriptive complexity of various canonical classes of C*-algebras in this parametrization. This can be viewed as providing an analog of the corresponding classification of classes of von Neumann algebras (acting on a fixed separable Hilbert space) in the Effros Borel space of von Neumann algebras (see for example Nielsen [13]). However, in contrast with the von Neumann case, where most interesting classes (like: factors, type I, II, III, hyperfinite) turn out to be Borel, in the C*-algebra case many important classes turn out to be co-analytic but not Borel. This makes the situation more interesting from the set-theoretic point of view, and leads to further questions, like the construction of canonical co-analytic norms, which we also address here. Finally, we relate the above to work of Sutherland [17] on parametrization of Polish groups and indicate how one might be able to show that various familiar classes of (second countable) locally compact groups are non-Borel as well.

Item Type:Book Section
Additional Information:© 1998 World Scientific. We would like to thank E. Effros for many valuable conversations about this subject. Research and preparation of this paper were partially supported by NSF Grants DMS-9317509 and 9619880.
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MathSciNet ReviewMR1789734
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ID Code:38625
Deposited By: Tony Diaz
Deposited On:23 May 2013 15:46
Last Modified:03 Oct 2019 04:59

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