Lie group-Lie algebra correspondences of unitary groups in finite von Neumann algebras (Q664874)

The instructive paper under review is devoted to an extensive investigation on some Lie theoretic aspects of the unitary group of a finite von Neumann algebra when that group is endowed with the strong operator topology. One of the main results is contained in Theorem 4.6 and says that if \({\mathfrak M}\) is a finite von Neumann algebra and \(G\) is a closed subgroup of the unitary group \(U({\mathfrak M})\) with respect to the strong operator topology, then the set \({\mathfrak g}\) of infinitesimal generators of strongly continuous one-parameter subgroups of \(G\) has the natural structure of a complete topological real Lie algebra with respect to the strong operator topology. It is also noted that \({\mathfrak g}\) is a Lie subalgebra of the associative algebra \(\overline{\mathfrak M}\) of all operators affiliated to \({\mathfrak M}\). A wealth of valuable information is also obtained in this connection. As a mere sample of such a piece of information, let us mention that Theorem 5.5 provides a categoy theoretic characterization of the algebras of operators affiliated to finite von Neumann algebras. More specifically, one introduces a suitable tensor category consisting of topological algebras of unbounded operators and proves that the correspondence \({\mathfrak M}\mapsto\overline{\mathfrak M}\) essentially provides an equivalence between that category and the category of finite von Neumann algebras with the morphisms given by the \(\sigma\)-weakly continuous unital \(*\)-homomorphisms. We should also mention that the unitary groups \(U({\mathfrak M})\) endowed with the strong operator topology in general fail to be infinite-dimensional Lie groups in the usual sense. Nevertheless it was pointed out by \textit{D. Beltiţă} [Forum Math. 22, No. 2, 241--253 (2010; Zbl 1188.22002)] that their Lie theoretic features can be studied within the framework of the topological groups with Lie algebras, in the sense of \textit{K. H. Hofmann} and \textit{S. A. Morris} [The Lie theory of connected pro-Lie groups. A structure theory for pro-Lie algebras, pro-Lie groups, and connected locally compact groups. Zürich: European Mathematical Society (EMS) (2007; Zbl 1153.22006)].