Automorphisms of corona algebras, and group cohomology

TL;DR

Under the Continuum Hypothesis, the paper proves that a wide class of corona algebras, including those of simple or stable separable algebras, have nontrivial automorphisms, extending previous results on the Calkin algebra.

Contribution

The paper generalizes the existence of outer automorphisms from the Calkin algebra to a broad family of corona algebras under CH, linking it to group cohomology.

Findings

Corollary: corona of certain separable algebras has nontrivial automorphisms

Connection established between automorphisms and derived inverse limits in cohomology

Extension of Phillips and Weaver's result beyond the Calkin algebra

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.