This talk will focus on the Calkin algebra of bounded operators modulo compact operators, and the aspects of set theory that have so far proved important in the study of its automorphism group, namely CH, Todorcevic's Axiom (aka the Open Coloring Axiom), and the Proper Forcing Axiom. CH was used by Phillips and Weaver to construct an outer automorphism of the Calkin algebra on a separable Hilbert space, whereas TA was used by Farah to show that every such automorphism is inner. Recently this latter result was extended by myself, Farah, and Schimmerling to automorphisms of the Calkin algebra on any Hilbert space, under the stronger assumption of PFA; whereas the consistency of the existence of outer automorphisms, at least on non-separable Hilbert spaces, is still an open problem.

I hope to sketch the proofs of some of these results.

Notes on this series of talks