# Automorphisms of the Calkin Algebra

Let H be a separable infinite dimensional Hilbert space, for example, the space l 2 of all square summable sequences.

Let L(H) be the algebra of all continuous linear maps from H to H, and let K(H) be the closure in L(H) of the set of continuous linear maps which have finite rank. Then K(H) is an ideal in L(H), and we can form the quotient algebra Q = L(H)/K(H). It is called the Calkin algebra, and is an example of a C*-algebra.

Question: Does the Calkin algebra have outer automorphisms, that is, automorphisms not of the form a 7→ uau−1 for suitable u ∈ Q?

It turns out that this question is undecidable in ZFC. In this talk, we will outline a proof (joint work with Nik Weaver) that, assuming the Continuum Hypothesis, outer automorphisms exist. In fact, there are more automorphisms than there are possible choices for u. (However, Ilijas Farah proved that it is consistent with ZFC that Q has no

outer automorphisms.) Along the way, we will see a bad aspect of the model theory of Q. The failure of the appropriate statement makes the proof more difficult. This talk is intended to be accessible to set theorists with little knowledge of C*-algebras.