The type ${\rm III_1}$ factor generated by regular representations of the infinite dimensional nilpotent group $B_0^\mathbb Z$

TL;DR

This paper proves that the von Neumann algebra generated by regular representations of an infinite-dimensional nilpotent group is a hyperfinite type ${ m III}_1$ factor, extending previous results with a new technique involving crossed products.

Contribution

It establishes the type ${ m III}_1$ hyperfinite nature of the von Neumann algebra for the group $B_0^{ ext{Z}}$, generalizing earlier work on $B_0^{ ext{N}}$ using crossed product techniques.

Findings

The von Neumann algebra is the hyperfinite type ${ m III}_1$ factor.

The crossed product technique removes previous technical measure conditions.

Extension of results from $B_0^{ ext{N}}$ to $B_0^{ ext{Z}}$.

Abstract

We study the von Neumann algebra, generated by the regular representations of the infinite-dimensional nilpotent group $B_0^{\mathbb Z}$. In [14] a condition have been found on the measure for the right von Neumann algebra to be the commutant of the left one. In the present article, we prove that, in this case, the von Neumann algebra generated by the regular representations of group $B_0^{\mathbb Z}$ is the type ${\rm III}_1$ hyperfinite factor. We use a technique, developed in [20] where a similar result was proved for the group $B_0^{\mathbb N}$. The crossed product allows us to remove some technical condition on the measure used in [20]. [14] A.V. Kosyak, Inversion-quasi-invariant Gaussian measures on the group of infinite-order upper-triangular matrices, Funct. Anal. i Priloz. 34, issue 1 (2000) 86--90. [20] A.V. Kosyak, Type ${\rm III_1}$ factors generated by regular representations of infinite dimensional nilpotent group $B_0^{\mathbb N}$, arXiv:0803.3340v1.