# Hokkaido Mathematical Journal

## Hokkaido Mathematical Journal, 41 (2012) pp.31-99

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### 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 $¥overline{¥mathfrak{M}}$ of all densely defined closed operators affiliated with $¥mathfrak{M}$ from the viewpoint of a tensor category.

MSC(Primary) 22E65 46L51 finite von Neumann algebra, unitary group, affiliated operator, measurable operator, strong resolvent topology, tensor category, infinite dimensional Lie group, infinite dimensional Lie algebra