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Nonclassifiability of UHF $ L^p$-operator algebras


Authors: Eusebio Gardella and Martino Lupini
Journal: Proc. Amer. Math. Soc. 144 (2016), 2081-2091
MSC (2010): Primary 47L10, 03E15; Secondary 47L30
DOI: https://doi.org/10.1090/proc/12859
Published electronically: October 1, 2015
MathSciNet review: 3460169
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Abstract: For $ p\in [1,\infty )$, we prove that simple, separable, monotracial UHF $ L^{p}$-operator algebras are not classifiable up to (complete) isomorphism using countable structures, such as K-theoretic data, as invariants. The same assertion holds even if one only considers UHF $ L^{p}$-operator algebras of tensor product type obtained from a diagonal system of similarities. For $ p=2$, it follows that separable nonselfadjoint UHF operator algebras are not classifiable by countable structures up to (complete) isomorphism. Our results, which answer a question of N. Christopher Phillips, rely on Borel complexity theory, and particularly Hjorth's theory of turbulence.


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Additional Information

Eusebio Gardella
Affiliation: Department of Mathematics, Deady Hall, University of Oregon, Eugene, Oregon 97403-1222 – and – Fields Institute for Research in Mathematical Sciences, 222 College Street, Toronto, Ontario M5T 3J1, Canada
Address at time of publication: Mathematisches Institut der WWU Munster, Einsteinstraße 62, 48149 Münster, Germany
Email: gardella@uni-muenster.de

Martino Lupini
Affiliation: Department of Mathematics and Statistics, N520 Ross, 4700 Keele Street, Toronto, Ontario M3J 1P3, Canada – and – Fields Institute for Research in Mathematical Sciences, 222 College Street, Toronto, Ontario M5T 3J1, Canada
Address at time of publication: Fakultät für Mathematik, Universität Wien, Oskar-Morgenstern-Platz 1, Room 02.126, 1090 Wien, Austria
Email: martino.lupini@univie.ac.at

DOI: https://doi.org/10.1090/proc/12859
Keywords: $L^p$-operator algebra, nonselfadjoint operator algebra, UHF algebra, Borel complexity, turbulence
Received by editor(s): March 18, 2015
Received by editor(s) in revised form: May 30, 2015, and June 7, 2015
Published electronically: October 1, 2015
Additional Notes: The first author was partially supported by the US National Science Foundation under Grant DMS-1101742
The second author was supported by the York University Susan Mann Dissertation Scholarship
This work was initiated while the authors were at the Banff International Research Station for the occasion of the workshop “Dynamics and C*-algebras: Amenability and Soficity”. The hospitality of the BIRS center is gratefully acknowledged.
Communicated by: Adrian Ioana
Article copyright: © Copyright 2015 American Mathematical Society