Automorphisms of corona algebras, and group cohomology

Coskey, Samuel; Farah, Ilijas

doi:10.1090/S0002-9947-2014-06146-1

Automorphisms of corona algebras, and group cohomology
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by Samuel Coskey and Ilijas Farah PDF

Trans. Amer. Math. Soc. 366 (2014), 3611-3630 Request permission

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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