Lie Group-Lie Algebra Correspondences of Unitary Groups in Finite von Neumann Algebras

TL;DR

This paper establishes a correspondence between Lie algebras and certain subgroups of the unitary group in finite von Neumann algebras, showing that generators of strongly continuous one-parameter subgroups form a complete topological Lie algebra.

Contribution

It proves the existence of Lie algebras for strongly closed subgroups of unitary groups in finite von Neumann algebras and characterizes affiliated operator algebras via tensor categories.

Findings

Generators form a complete topological Lie algebra

Characterization of affiliated operator algebras

Affirmative answer to the existence of Lie algebras in this setting

Abstract

We give an affirmative answer to the question whether there exist Lie algebras for suitable closed subgroups of the unitary group $U(\mathcal{H})$ in a Hilbert space $\mathcal{H}$ with $U(\mathcal{H})$ equipped with the strong operator topology. More precisely, for any strongly closed subgroup $G$ of the unitary group $U(\mathfrak{M})$ in a finite von Neumann algebra $\mathfrak{M}$, we show that the set of all generators of strongly continuous one-parameter subgroups of $G$ forms a complete topological Lie algebra with respect to the strong resolvent topology. We also characterize the algebra $\mathfrak{M}$ of all densely defined closed operators affiliated with $\mathfrak{M}$ from the viewpoint of a tensor category.