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Right submodules of finite rank for von Neumann dynamical systems

Paul Jolissaint

Abstract

Let $(M,τ,σ,Γ)$ be a (finite) von Neumann dynamical system and let $N$ be a $Γ$-invariant unital von Neumann subalgebra of $M$. If $V\subset L^2(M)$ is a right $N$-submodule whose projection $p_V$ has finite trace in $< M,e_N>$ and is $Γ$-invariant, then we prove that, for every $ε>0$, one can find a $Γ$-invariant submodule $W\subset V$ which has finite rank and such that $Tr(p_V-p_W)<ε$. Furthermore, we also construct a $σ$-cocycle that gives the action of $Γ$ on a basis of $W$. In particular, this answers a question of T. Austin, T. Eisner and T. Tao.

Right submodules of finite rank for von Neumann dynamical systems

Abstract

Let be a (finite) von Neumann dynamical system and let be a -invariant unital von Neumann subalgebra of . If is a right -submodule whose projection has finite trace in and is -invariant, then we prove that, for every , one can find a -invariant submodule which has finite rank and such that . Furthermore, we also construct a -cocycle that gives the action of on a basis of . In particular, this answers a question of T. Austin, T. Eisner and T. Tao.

Paper Structure

This paper contains 4 sections, 4 theorems, 25 equations.

Table of Contents

  1. Introduction
  2. The case of subfactors
  3. The general case
  4. An application to relative weak mixing

Key Result

Proposition 1

Suppose that $(M,\tau,\sigma,\Gamma)$ is a von Neumann dynamical system and that $N$ is a $\Gamma$-invariant type $\mathrm{II}_1$ subfactor of $M$. Let $p$ be a projection in $\langle M,e_N\rangle$ such that $0<t:=\operatorname{Tr}(p)<\infty$ and which is $\Gamma$-invariant. Then $V:=pL^2(M)$ has fi

Theorems & Definitions (5)

  • Proposition 1
  • Proposition 2
  • Definition 3
  • Proposition 4
  • Proposition 5