Type ${\rm III_1}$ factors generated by regular representations of infinite dimensional nilpotent group $B_0^{\mathbb N}$

TL;DR

This paper characterizes the von Neumann algebra generated by regular representations of an infinite-dimensional nilpotent group, establishing it as a type ${ m III}_1$ hyperfinite factor, advancing understanding of infinite-dimensional group representations.

Contribution

It determines the type of von Neumann algebra generated by regular representations of the infinite-dimensional nilpotent group $B_0^{ ext N}$ as a type ${ m III}_1$ hyperfinite factor, which was previously unknown.

Findings

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

Conditions for irreducibility of representations are discussed.

The work extends classification of factors generated by infinite-dimensional groups.

Abstract

We study the von Neumann algebra, generated by the unitary representations of infinite-dimensional groups nilpotent group $B_0^{\mathbb N}$. The conditions of the irreducibility of the regular and quasiregular representations of infinite-dimensional groups (associated with some quasi-invariant measures) are given by the so-called Ismagilov conjecture (see [1,2,9-11]). In this case the corresponding von Neumann algebra is type ${\rm I}_\infty$ factor. When the regular representation is reducible we find the sufficient conditions on the measure for the von Neumann algebra to be factor (see [13,14]). In the present article we determine the type of corresponding factors. Namely we prove that the von Neumann algebra generated by the regular representations of infinite-dimensional nilpotent group $B_0^{\mathbb N}$ is type ${\rm III}_1$ hyperfinite factor. The case of the nilpotent group $B_0^{\mathbb Z}$ of infinite in both directions matrices will be studied in [6].