Sam Harris, Department of Pure Mathematics, University of Waterloo
"Connes' Embedding Problem & Kirchberg's Conjecture (continued)"
We will wrap up the proof that the embedding problem is equivalent to Kirchberg's conjecture. We will then look at an invariant related to von Neumann algebras that satisfy Connes' embedding problem, known as Brown's invariant. This invariant is the set of normal, injective $*$-homomorphisms from a tracial von Neumann algebra $N$ into an ultrapower of the hyperfinite $II_1$-factor that preserve the trace (modulo unitary equivalence). Surprisingly, this invariant can be endowed with a convex structure. This yields the fact that the image of such a homomorphism is a factor (i.e. has trivial center in the ultrapower of the hyperfinite $II_1$-factor) if and only if the associated point in the convex set is an extreme point.
MC 5403