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

This article is a summary of our talk in QBIC2010. We give an affirmative answer to the question whether there exist Lie algebras for suitable closed subgroups of the unitary group U( {H}) in a Hilbert space {H} with U( {H}) equipped with the strong operator topology. More precisely, for any strongly closed subgroup G of the unitary group U( {M}) in a finite von Neumann algebra {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 /line {M} of all densely defined closed operators affiliated with {M} from the viewpoint of a tensor category.