Invariant states on the wreath product

TL;DR

This paper characterizes all indecomposable invariant states on the wreath product of a topological group with the infinite permutation group, providing a complete description of such states.

Contribution

It offers a full classification of invariant states on the wreath product group, extending understanding of its representation theory and invariant measures.

Findings

Complete description of indecomposable invariant states

Characterization of states invariant under conjugation

Advancement in understanding wreath product group representations

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.