Existence of infinite-dimensional Lie algebra for a unitary group on a Hilbert space and related aspects (Q2905537)

The authors study the following problem: Let \(\mathcal{H}\) be an infinite dimensional Hilbert space, \(U(\mathcal{H})\) be its unitary group equipped with the strong operator topology and \(G\) be a strongly closed subgroup of \(U(\mathcal{H})\). Is there a natural Lie algebra for \(G\)? Applying non-commutative integration theory the authors show that the natural candidate for the answer (the set of all skew-adjoint operators for which the corresponding one-parameter unitary subgroups belong to \(G\)) indeed gives a positive answer to this question in the case when \(G\) is a strongly closed subgroup of the unitary group of a finite von Neumann algebra. Furthermore, they show that the Lie algebraic operations for this Lie algebra are continuous with respect to the strong resolvent topology and that the Lie algebra is complete as a uniform space. The analysis is based on comparison between measure topology induced by the tracial state and the strong resolvent topology.NEWLINENEWLINEFor the entire collection see [Zbl 1248.46002].