Right submodules of finite rank for von Neumann dynamical systems

TL;DR

This paper studies the structure of right submodules in von Neumann dynamical systems, showing how to approximate invariant submodules with finite-rank ones and constructing a cocycle for the group action.

Contribution

It introduces a method to approximate invariant submodules with finite-rank submodules and constructs a cocycle describing the group action, answering a question by Austin, Eisner, and Tao.

Findings

Existence of finite-rank invariant submodules approximating given submodules

Construction of a $\sigma$-cocycle for the group action on a basis

Resolution of a question posed by Austin, Eisner, and Tao

Abstract

Let $(M,\tau,\sigma,\Gamma)$ be a (finite) von Neumann dynamical system and let $N$ be a $\Gamma$-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 $\Gamma$-invariant, then we prove that, for every $\epsilon>0$, one can find a $\Gamma$-invariant submodule $W\subset V$ which has finite rank and such that $Tr(p_V-p_W)<\epsilon$. Furthermore, we also construct a $\sigma$-cocycle that gives the action of $\Gamma$ on a basis of $W$. In particular, this answers a question of T. Austin, T. Eisner and T. Tao.