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.
First published in Transactions of the American Mathematical Society in volume 366, issue 7 in 2014. Published by the American Mathematical Society. This work is provided under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 license. Details regarding the use of this work can be found at: http://creativecommons.org/licenses/by-nc-nd/4.0/. doi: 10.1090/S0002-9947-2014-06146-1
Coskey, Samuel and Farah, Ilijas. (2014). "Automorphisms of Cornoa Algebras, and Group Cohomology". Transactions of the American Mathematical Society, 366(7), 3611-3630. http://dx.doi.org/10.1090/S0002-9947-2014-06146-1