Abstract: We consider von Neumann algebras generated by regular representations of the infinite-dimensional group B_0^\mathbb Z of infinite, finite-order upper triangular matrices. These regular representations were defined and studied by Alexander Kosyak. They depend on a Gaussian measure on the group of infinite (arbitrary order) upper triangular matrices. A certain condition on the measure implies that the right regular representation is reducible and that the von Neumann algebra generated by the right regular representation is the commutant of the left one. In this case we prove that these von Neumann algebras are type III_1 factors, according to the classification of Alain Connes.

Abstract: Hamiltonian (or symplectic) diffeomorphisms are well-known to preserve volume. Gromov's work in the eighties implied that the smallest factor in a polydisk also corresponds to an invariant of symplectic embeddings. In dimensions at least 6 we will ask to what extent the areas of other factors place restrictions on symplectic embeddings. Specific constructions will show that any such restrictions must be fairly weak, but they nevertheless exist and remarkably our constructions turn out to be in some sense sharp. We will derive some implications for Hofer's metric on groups of Hamiltonian diffeomorphisms.