Частично центральные состояния на бесконечной симметрической группе

Let $\mathfrak{S}_\infty$ be the group of all finite bijections $\mathbb{N}\to\mathbb{N}$. Denote by $\widehat{\mathfrak{S}}_{\infty}^2$ the set of all unitary irreducible {\it admissible} representations of $\mathfrak{S}_\infty^2=\mathfrak{S}_\infty\times \mathfrak{S}_\infty$. We study the factor representations of $\mathfrak{S}_\infty$ that are the restrictions of the representations from $\widehat{\mathfrak{S}}_{\infty}^2$ to $\mathfrak{S}_\infty\times\mathbf{e}$, where $\mathbf{e}$ is the unit element of $\mathfrak{S}_\infty$. It turn out that these representations are of type ${\rm I}$, ${\rm II}_1$ or ${\rm II}_\infty$. The full description for the classes of the quasiequivalent representations is given.