With Ilijas Farah. Transactions of the American mathematical society, 2014. (arχiv | journal)

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.

Further notes: This article gives a generalization of a (relatively) recent result concerning the Calkin algebra. It is just one of a number of recent work that mixes set theory and the study of non-separable C*-algebras. I’m really grateful to Ilijas for pulling me into this project, which belongs essentially entirely to him. Thanks to his enthusiasm, I have much new understanding and respect for this remarkable subject.

First, let me define the Calkin algebra. Let $H$ be a separable Hilbert space, and $\mathcal K(H)$ denote the algebra of compact operators on $H$: the norm-closure of the algebra of operators with finite-dimensional range. Then $\mathcal K(H)$ is a two-sided ideal in the algebra $B(H)$ of all bounded linear operators on $H$, and the Calkin algebra is just the quotient $B(H)/\mathcal K(H)$.

In 2007, Phillips and Weaver proved that CH implies that the Calkin algebra has an outer automorphism (i.e., one that isn’t just conjugation by a single element). In this paper, we establish the same result for a much more general class of quotient structures called corona algebra.

Here, if $A\subset B(H)$ then the multiplier algebra of $A$ is the set $M(A)={m\in B(H):mA\subset A\text{ and }Am\subset A}$. The corona of $A$ is the quotient $M(A)/A$. It is easy to see that the corona of $\mathcal K$ is exactly the Calkin algebra, and that the corona of $C(X)$ is exactly $C(\beta X\smallsetminus X)$, the algebra of continuous functions on the corona of $X$. This is where the corona derives its name. We prove:

Theorem. Assume CH holds. If $A$ is separable and is either simple or stable, then the corona of $A$ has $2^{2^{\aleph_0}}$ many automorphisms.

This hypotheses on $A$ are not the weakest possible, but some hypothesis is certainly necessary. There are several algebras $A$ for which it is known that all automorphisms are inner, and there are several cases left open.

Our proof is based on Ilijas’s proof of the result of Phillips and Weaver in the case of the Calkin Algebra. Loosely speaking, we stratify the corona into $\aleph_1$ many layers and then build a complete binary tree of height $\aleph_1$ consisting of partial automorphisms. Here the $\alpha$th level of the tree corresponds to the $\alpha$th stage in the stratification. We take care to ensure that compatible elements of the tree correspond to compatible partial automorphisms, and, by small perturbations at each step, that incompatible elements correspond to distinct partial automorphisms. It follows that the set of branches through the tree give $2^{\aleph_1}$ distinct automorphisms, and assuming CH, this is the desired number.

In the process of the proof, we uncovered a small connection with category theory. I will not elaborate on it here, but rather just note that our proof can be viewed as a computation of the cardinality of a particular derived inverse limit. astly, I should mention that we do not address the (much more important) complementary question. Namely, Ilijas has also shown that Todorčević’s Axiom (also called OCA) implies that the Calkin algebra has no outer automorphisms. It would be very interesting to generalize this much more involved argument to the case of corona algebras as well.