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

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.