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August 26, 2016

## Minimal Faces and Schur’s Lemma for Embeddings into R^U

In this talk, we will see a generalization of N. Brown’s characterization of extreme points in Hom(N,R^U). In particular, given a separably acting II_1-factor N and an ultrapower of the separably acting hyperfinite II_1-factor R^U, we will show that given \pi: N \rightarrow R^U, the dimension of the minimal face containing [\pi] is one less than the dimension of the center of the relative commutant of \pi. A consequence of this is the “convex independence” of extreme points: the convex hull of n extreme points is an n-vertex simplex. Along the way, we will establish a version of Schur’s Lemma in this context.