Minimal faces and Schur's Lemma for embeddings into R^U

In the context of N. Brown's Hom(N,R^U), we establish 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. We also show the "convex independence" of extreme points in the sense that the convex hull of n extreme points is an n-vertex simplex. Along the way, we establish a version of Schur's Lemma for embeddings of II$_1$-factors.