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

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

2008-03-23

arxiv.org/abs/0803.3340v2

Mathematics

Operator Algebras

25 pages

Scientific paper

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].

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