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

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.