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

TL;DR

This paper explores the geometric structure of embeddings into R^U, revealing relationships between minimal faces, centers of relative commutants, and convex independence, along with a version of Schur's Lemma for II_1-factors.

Contribution

It introduces new geometric and algebraic insights into embeddings into R^U, including minimal face dimensions and a Schur's Lemma variant for II_1-factors.

Findings

Dimension of minimal face is one less than the center of the relative commutant.

Convex hull of n extreme points forms an n-vertex simplex.

Established a version of Schur's Lemma for embeddings of II_1-factors.

Abstract

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.