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 $\Gamma^\infty_e \rtimes \mathfrak{S}_\infty$ for the usual permutation action of $\mathfrak{S}_\infty$ on $\Gamma^\infty_e=\{[\gamma_i]_{i=1}^\infty : \gamma_i\in \Gamma,\textit{only finitely many}\gamma_i\neq e\}$. In this paper we obtain the full description of indecomposable states $\varphi$ on the group $\Gamma\wr\mathfrak{S}_\infty,$ satisfying the condition: \varphi(sgs^{-1})= \varphi(g)\text{for each}g\in \Gamma\wr \mathfrak{S}_\infty,s\in\mathfrak{S}_\infty.