Type \rm III₁ factors generated by regular representations of infinite dimensional nilpotent group B₀^{\mathbb N}

2008/03/23 by Kosyak, Alexandre · 1 citation
#17B65 #22E65 #28C20 #28D25 #FOS: Mathematics #Operator Algebras (math.OA) #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.0803.3340

Abstract

We study the von Neumann algebra, generated by the unitary representations of infinite-dimensional groups nilpotent group B₀^{\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_∞ 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 B₀^{\mathbb N} is type \rm III₁ hyperfinite factor. The case of the nilpotent group B₀^{\mathbb Z} of infinite in both directions matrices will be studied in [6].

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• The type \rm III₁ factor generated by regular representations of the infinite dimensional nilpotent group B₀^\mathbb Z

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