Topological dynamical systems associated to II_1 factors

TL;DR

This paper explores the rich topological and dynamical structures of the space of *-homomorphisms from II_1 factors to the hyperfinite II_1 factor, revealing new insights into their automorphism groups and associated dynamical systems.

Contribution

It introduces a detailed topological framework for the space of homomorphisms from II_1 factors to R, analyzing automorphism actions and dynamical systems, especially for free product factors.

Findings

Hom(N,R) is infinite-dimensional and convex-like for non-hyperfinite N.

Outer automorphism group acts by affine homeomorphisms on Hom(N,R).

Dynamical systems for free group factors are shown to be isomorphic.

Abstract

If $N \subset \R$ is a separable II$_1$-factor, the space $\Hom(N,\R)$ of unitary equivalence classes of unital *-homomorphisms $N \to \R$ is shown to have a surprisingly rich structure. If $N$ is not hyperfinite, $\Hom(N,\R)$ is an infinite-dimensional, complete, metrizeable topological space with convex-like structure, and the outer automorphism group $\Out(N)$ acts on it by "affine" homeomorphisms. (If $N \cong R$, then $\Hom(N,\R)$ is just a point.) Property (T) is reflected in the extreme points -- they're discrete in this case. For certain free products $N = \Sigma \ast R$, every countable group acts nontrivially on $\Hom(N, \R)$, and we show the extreme points are not discrete for these examples. Finally, we prove that the dynamical systems associated to free group factors are isomorphic.