פעילויות השבוע

Title: Existence of outer automorphisms of the Calkin algebra is undecidable in ZFC

Speaker: N. Christopher Phillips, University of Oregon and Ben Gurion University of the Negev

Abstract:

The Calkin algebra $Q$ is the quotient of the algebra $L(H)$ of bounded operators on a separable infinite dimensional Hilbert space $H$ by the ideal of compact operators (the closure of the ideal of finite rank operators). It is an explicit simple $C^*$-algebra, first studied by Calkin in 1941. It takes a few lines to prove that every automorphism of $L(H)$ is inner, that is, of the form $a\mapsto ua u^{-1}$ for some unitary $u$ in $L(H)$. Are all automorphisms of $Q$ inner? Despite the concrete description of $Q$, this is undecidable in ZFC. Assuming the Continuum Hypothesis (CH), there are outer (that is, not inner) automorphisms (joint with Weaver, 2007). Assuming the Open Coloring Axiom (OCA; also called Todorcevic‘s Axiom), all automorphisms of $Q$ are inner (Farah, 2011).

In these talks, we will outline proofs of both results. The talks are intended to be accessible to people in both operator algebras and set theory. We will follow Farah‘s reproof of the existence of outer automorphisms under CH, which uses much less $C^*$-algebra machinery than the original proof, and uses some of the same ingredients as the proof of nonexistence under OCA.

We will very briefly say something about later results which have been proved, as well as problems which remain open, involving generalizations of the Calkin algebra, such as outer multiplier algebras of $C^*$-algebras and $l^p$ Calkin algebras. It remains open whether the existence of orientation reversing automorphisms of the original Calkin algebra is consistent with ZFC.