ASL winter meeting, San Diego, January 2013 (slides)

Abstract: In 2007 Phillips and Weaver showed that assuming CH, there exists an outer automorphism of the Calkin algebra. Here, the Calkin algebra is the algebra of bounded operators on a separable Hilbert space modulo the compact operators; it can be thought of as the non-commutative analog of $\mathcal P(\mathbb N)/\mathrm{Fin}$. The Calkin algebra is just a special case of a quotient construction called the corona construction: for $A\subset B(\mathcal H)$ the corona of $A$ is $M(A)/A$, where $M(A)$ denotes the multiplier algebra of $A$. The corona construction can be thought of as the non-commutative analog of the Stone-Cech remainder.

A series of recent results show that assuming CH, various classes of corona algebras have outer automorphisms. In all cases, the techniques have come from logic: either using a set-theoretic construction or a model-theoretic arguments. In this talk we will very briefly sketch the case when $A$ is separable and either simple or stable. At the heart of the argument is the construction of a tree of partial automorphisms of height $\aleph_1$.

This is joint work with Ilijas Farah.