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Automorphisms of corona algebras, and group cohomology


Authors: Samuel Coskey and Ilijas Farah
Journal: Trans. Amer. Math. Soc. 366 (2014), 3611-3630
MSC (2010): Primary 46L40; Secondary 46L05, 03E50
Published electronically: March 19, 2014
MathSciNet review: 3192609
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Abstract: In 2007 Phillips and Weaver showed that, assuming the Continuum Hypothesis, there exists an outer automorphism of the Calkin algebra. (The Calkin algebra is the algebra of bounded operators on a separable complex Hilbert space, modulo the compact operators.) In this paper we establish that the analogous conclusion holds for a broad family of quotient algebras. Specifically, we will show that assuming the Continuum Hypothesis, if $ A$ is a separable algebra which is either simple or stable, then the corona of $ A$ has nontrivial automorphisms. We also discuss a connection with cohomology theory, namely, that our proof can be viewed as a computation of the cardinality of a particular derived inverse limit.


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Additional Information

Samuel Coskey
Affiliation: The Fields Institute, 222 College Street, Toronto, Ontario, Canada M5T 3J1 — and — Department of Mathematics and Statistics, York University, 4700 Keele Street, North York, Ontario, Canada M3J 1P3
Address at time of publication: Department of Mathematics, Boise State University, 1910 University Drive, Boise, Idaho 83725
Email: scoskey@nylogic.org

Ilijas Farah
Affiliation: Department of Mathematics and Statistics, York University, 4700 Keele Street, North York, Ontario, Canada M3J 1P3 — and — Matematicki Institut, Kneza Mihaila 34, Belgrade, Serbia
Email: ifarah@yorku.ca

DOI: https://doi.org/10.1090/S0002-9947-2014-06146-1
Received by editor(s): August 13, 2012
Published electronically: March 19, 2014
Additional Notes: The second author was partially supported by NSERC
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.