## Pure Mathematics Seminar: Valerio Capraro

**Valerio Capraro**, University of Southampton

**Wednesday 21 May 2014, 1500-1600
A54, Postgraduate Statistics Centre Lecture Theatre**

#### Groups associated to II_1 factors

Let M,N be II_1-factors. We study the set of morphisms from M to Nbarotimes B(H), where H is a separable Hilbert space. We prove that this set has a natural structure of topological commutative semigroup which always satisfies cancellation and then it embeds into its Grothendieck group. We discuss some examples where this group is trivial and others where it is enormous. If N is an ultrapower of a McDuff II_1-factor, we show that this group carries a natural vector space structure and a natural metric. It is, in fact, a Banach space with natural actions of outer automorphism groups.

The case N=R^omega, an ultrapower of the hyperfinite II_1-factor, is particularly relevant: existence of extreme points of Hom(M,N), seen as a (convex, closed, and bounded) subset of Hom(M,Nbarotimes B(H)), is equivalent to a problem of Sorin Popa concerning existence of embeddings of M into R^omega with factorial commutant.

This talk is mainly based on the joint paper with Nate Brown "Groups associated to II_1-factors", Journal of Functional Analysis 264 (2013) 493-507.