A description of characters on the infinite wreath product

Abstract

Let ${\mathfrak{S}}_{\infty }$ be the infinity permutation group and $\Gamma$ an arbitrary group. Then ${\mathfrak{S}}_{\infty }$ admits a natural action on ${\Gamma }^{\infty }$ by automorphisms, so one can form a semidirect product ${\Gamma }^{\infty }⋊{\mathfrak{S}}_{\infty }$, known as the {\it wreath} product $\Gamma \wr {\mathfrak{S}}_{\infty }$ of $\Gamma$ by ${\mathfrak{S}}_{\infty }$. We obtain a full description of unitary $I{I}_1-$factor-representations of $\Gamma \wr {\mathfrak{S}}_{\infty }$ in terms of finite characters of $\Gamma$. Our approach is based on extending Okounkov's classification method for admissible representations of ${\mathfrak{S}}_{\infty }\times {\mathfrak{S}}_{\infty }$. Also, we discuss certain examples of representations of type $I{I}_1$, where the {\it modular operator} of Tomita-Takesaki expresses naturally by the asymptotic operators, which are important in the characters-theory of infinite symmetric group.

33 pages. We receive the full description of the finite characters on the infinite wreath product