Type III von Neumann algebras in the Theory of Infinite-dimensional Groups

Abstract

<p>We study two examples of von Neumann algebras that are generated by the right and left regular representations of two infinite dimensional groups $\BON$ and $\BOZ$. These regular representations were studied by Alexander Kosyak and depend on a Gaussian measure on a space in which the group is dense. Under a certain general condition on the measure the left von Neumann algebra is the commutant of the right one. In this case we prove that all the algebras are factors and that they are of type III$_1$, according to the classification of Alain Connes.</p>