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

Artem Dudko, Constantine Medynets

Abstract

Let $(X,T)$ be a Cantor minimal system, and let $Γ$ 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 $Γ$ and its commutator subgroup $Γ'$. In particular, we prove that: (1) for every nontrivial indecomposable character $χ$ of $Γ'$, there exists a finite collection (with repetitions allowed) $\{μ_i\}_{i\in I}$ of $T$-invariant ergodic measures on $X$ such that $χ(γ) = \prod_{i\in I} μ_i(Fix(γ))$, for every $γ\in Γ'$, where $Fix(γ) = \{x\in X : γx = x\}$; and (2) each indecomposable character of $Γ$ is the product of an indecomposable character of the form $\prod_{i\in I} μ_i(Fix(γ))$ and a homomorphism from $Γ$ into the unit circle. As a consequence, we show that any finite-type unitary representation of $Γ'$ that does not contain a regular subrepresentation is automatically continuous with respect to the uniform topology on $Γ'$. We also establish a general result on automatic continuity of finite-type unitary representations of infinite groups, which we use in our proofs.

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

Abstract

Let be a Cantor minimal system, and let denote either its associated topological full group or the full group of a Bratteli diagram associated with . In this paper we describe the structure of indecomposable (extreme) characters and the associated -factor representations for the group and its commutator subgroup . In particular, we prove that: (1) for every nontrivial indecomposable character of , there exists a finite collection (with repetitions allowed) of -invariant ergodic measures on such that , for every , where ; and (2) each indecomposable character of is the product of an indecomposable character of the form and a homomorphism from into the unit circle. As a consequence, we show that any finite-type unitary representation of that does not contain a regular subrepresentation is automatically continuous with respect to the uniform topology on . We also establish a general result on automatic continuity of finite-type unitary representations of infinite groups, which we use in our proofs.
Paper Structure (13 sections, 57 theorems, 174 equations, 3 figures)

This paper contains 13 sections, 57 theorems, 174 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Statements of the Main Results
  3. Structure of the Paper
  4. Preliminaries
  5. Group Characters and Finite von Neumann Algebras
  6. Full Groups and Their Algebraic Properties
  7. The Permutation Character on Full Groups
  8. Automatic Continuity of Finite-Type Unitary Representations
  9. Approximating Properties of Full Groups of Bratteli Diagrams
  10. Projections onto Subspaces Invariant under Local Subgroups
  11. The Space of Functions on Invariant Measures
  12. Indecomposable Characters of the Commutator Subgroups
  13. Indecomposable Characters of Full Groups

Key Result

Theorem 1.1

Let $(X, T)$ be a Cantor minimal system and $\Gamma$ be either the full group $\mathcal{F}(\langle T \rangle)$, the AF full group $\mathcal{F}(\langle T \rangle)_{x_0}$, or the commutator subgroup of either. If $\chi$ is an indecomposable character of $\Gamma$, then one of the following holds: If $\Gamma$ is the commutator subgroup of $\mathcal{F}(\langle T \rangle)$ or $\mathcal{F}(\langle T \ra

Figures (3)

  • Figure 1: A Bratteli diagram with a root node $v_0$, $V_1=\{a_1, a_2, a_3\}$, and $V_2 = \{ b_1 , b_2 \}$.
  • Figure 2: A Bratteli diagram with one vertex per level and the exactly $n$ edges between levels $V_{n-1}$ and $V_n$.
  • Figure 3: The Bratteli diagram corresponding to the 2-odometer and the infinite symmetric group $S_{2^\infty}$.

Theorems & Definitions (114)

  • Theorem 1.1
  • Corollary 1.2
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Proposition 2.4
  • Proposition 2.5
  • Definition 2.6
  • Proposition 2.7
  • Remark 2.8
  • ...and 104 more