Characters and $II_1$-Factor Representations of Full Groups of Cantor Minimal Systems

TL;DR

This paper characterizes indecomposable characters and $ extrm{II}_1$-factor representations of full groups from Cantor minimal systems, revealing their structure via invariant measures and establishing automatic continuity of certain representations.

Contribution

It provides a detailed description of indecomposable characters and $ extrm{II}_1$-factor representations for full groups of Cantor minimal systems, including new automatic continuity results.

Findings

Indecomposable characters are expressed via invariant measures.

Characters of the full group are products of measure-based characters and homomorphisms.

Finite-type unitary representations without regular subrepresentations are automatically continuous.

Abstract

Let $(X,T)$ be a Cantor minimal system, and let $\Gamma$ denote either its associated topological full group or the full group of a Bratteli diagram associated with $(X,T)$. In this paper we describe the structure of indecomposable (extreme) characters and the associated $\textrm{II}_1$-factor representations for the group $\Gamma$ and its commutator subgroup $\Gamma'$. In particular, we prove that: (1) for every nontrivial indecomposable character $\chi$ of $\Gamma'$, there exists a finite collection (with repetitions allowed) $\{\mu_i\}_{i\in I}$ of $T$-invariant ergodic measures on $X$ such that $\chi(\gamma) = \prod_{i\in I} \mu_i(Fix(\gamma))$, for every $\gamma \in \Gamma'$, where $Fix(\gamma) = \{x\in X : \gamma x = x\}$; and (2) each indecomposable character of $\Gamma$ is the product of an indecomposable character of the form $\prod_{i\in I} \mu_i(Fix(\gamma))$ and a homomorphism from $\Gamma$ into the unit circle. As a consequence, we show that any finite-type unitary representation of $\Gamma'$ that does not contain a regular subrepresentation is automatically continuous with respect to the uniform topology on $\Gamma'$. We also establish a general result on automatic continuity of finite-type unitary representations of infinite groups, which we use in our proofs.