Invariant states on the wreath product

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