Operator algebras and Logic seminar, II: Ilijas Farah

Speaker: Ilijas Farah (York University)

Title: Ulam-stability of C*-algebras

Abstract: This talk is loosely connected to the first talk. The proof that, under suitable set-theoretic assumptions, every automorphism of the Calkin algebra is inner involves several diverse techniques, one of which will be the subject of this talk.

If G and H are groups and H is equipped with a metric $d$, then for $\varepsilon>0$, an $\varepsilon$-homomorphism $\psi:  G \to  H$ is a map such that $d(\psi(xy),\psi(x)\psi(y))<\varepsilon$ for all $x,y$ in $G$. S. Ulam asked under what conditions on $G$ and $H$ can every $\varepsilon$-homomorphism be uniformly approximated with a true homomorphism.

Analogous `Ulam-stabiity' questions have been studied in diverse contexts, and a part of the proof that all automorphisms of the Calkin algebra are inner consists of proving Ulam-stability for finite-dimensional C*-algebras. Most questions about Ulam-stability of C*-algebras are open. Part of this is work in progress with V. Alekseev and A. Thom.