Invariant states on the wreath product

Abstract

Let ${\mathfrak{S}}_{\infty }$ be the infinity permutation group and $\Gamma$ be a separable topological group. The wreath product $\Gamma \wr {\mathfrak{S}}_{\infty }$ is the semidirect product ${\Gamma }_e^{\infty }⋊{\mathfrak{S}}_{\infty }$ for the usual permutation action of ${\mathfrak{S}}_{\infty }$ on ${\Gamma }_e^{\infty }=\{[{\gamma }_i{]}_{i=1}^{\infty }:{\gamma }_i\in \Gamma ,\text{only finitely many}{\gamma }_i\ne e\}$. In this paper we obtain the full description of indecomposable states $\phi$ on the group $\Gamma \wr {\mathfrak{S}}_{\infty },$ satisfying the condition: φ(sgs^{-1})= φ(g)\text{for each}g\in Γ\wr \mathfrak{S}_\infty,s\in\mathfrak{S}_\infty.

37 pages