On Lie GroupLie Algebra Correspondences of Unitary Groups in Finite Von Neumann Algebras
Abstract
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 oneparameter 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.
 Publication:

Quantum BioInformatics IV
 Pub Date:
 January 2011
 DOI:
 10.1142/9789814343763_0003
 arXiv:
 arXiv:1005.4850
 Bibcode:
 2011qbi4.conf...29A
 Keywords:

 finite von Neumann algebra;
 unitary group;
 affiliated operator;
 measurable operator;
 strong resolvent topology;
 tensor category;
 infinite dimensional Lie group;
 infinite dimensional Lie algebra;
 Mathematics  Operator Algebras;
 Mathematics  Functional Analysis;
 22E65;
 46L51
 EPrint:
 56 pages