Sam Harris, Department of Pure Mathematics, University of Waterloo
"Connes' Embedding Problem & Kirchberg's Conjecture"
We will see that a group $G$ is hyperlinear if and only if its group von Neumann algebra $L(G)$ (coming from the left regular representation of $G$) satisfies the statement of Connes' embedding problem. As time allows, we will also look at a topological structure on the set $vN(H)$ of von Neumann subalgebras of $B(H)$, for a fixed, separable infinite-dimensional Hilbert space $H$. This topology is known as the Effros-Marechal topology. This will allow us to work towards a proof that Connes' embedding problem is equivalent to Kirchberg's conjecture (i.e. that $C^*(F_{\infty}) \odot C^*(F_{\infty})$ has only one $C^*$-norm).
MC 5403