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

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