Groupoid approach to the dynamical system of commutative von Neumann Algebras

TL;DR

This paper explores the dynamical systems of commutative von Neumann algebras using a groupoid approach, analyzing automorphism groups and ergodic actions on measure spaces within a rigorous algebraic framework.

Contribution

It introduces a novel groupoid-based method to study the dynamics of commutative von Neumann algebras and their automorphism groups in the context of ergodic theory.

Findings

Proper action of automorphism group on measure space

Existence of slices at each point in the generalized space

Representation of ergodic action on a commutative von Neumann algebra

Abstract

The automorphism group $Aut(X,\mu)$ of a compact, complete metric space $X$ with a Radon measure $\mu$ is a subgroup of $\mathcal{U}(L^2(X,\mu))$-the unitary group of operators on $L^2(X,\mu)$. The $Aut(X,\mu)$-action on the generalized space $\mathcal{M}(X)$ is a proper action. Hence, there exists a slice at each point of the generalized space $\mathcal{M}(X)$. Measure Groupoid (virtual group) is subsequently employed to analyze the resulting dynamical system as that of the ergodic action of the commutative algebra (a lattice) $C(X)$ on the generalized space $\mathcal{M}(X)$ which is represented on a commutative von Neumann algebra.