Classification of indecomposable states on the infinite symmetric inverse semigroup invariant under the infinite symmetric group. Semifinite case

TL;DR

This paper classifies all semifinite factor-representations of the infinite symmetric inverse semigroup that are invariant under the infinite symmetric group, expanding understanding of their structure and invariant functions.

Contribution

It provides a complete classification of semifinite factor-representations of the infinite symmetric inverse semigroup invariant under the infinite symmetric group.

Findings

Classification of all semifinite factor-representations.

Identification of invariant positive definite functions.

Extension of representation theory for infinite semigroups.

Abstract

Let $\mathbb{N}$ be a set of the natural numbers. Symmetric inverse semigroup $R_\infty$ is the semigroup of all infinite 0-1 matrices $[g_{ij}]$ with at most one 1 in each row and each column such that $g_{ii}=1$ on the complement of a finite set. The binary operation in $R_\infty$ is the ordinary matrix multiplication. It is clear that infinite symmetric group $\mathfrak{S}_\infty$ is a subgroup of $R_\infty$. The map $\star:\left[ g_{ij}\right]\mapsto\left[ g_{ji}\right]$ is an involution on $R_\infty$. We call a function $f$ on $R_\infty$ positive definite if for all $r_1, r_2, \ldots, r_n\in R_\infty$ the matrix $\left[ f\left( r_ir_j^\star\right)\right]$ is Hermitian and positive semi-definite. A function $f$ said to be indecomposable if the corresponding $\ast$-representation $\pi_f$ is a factor-representation. A class of the $\mathfrak{S}_\infty$-invariant functions is defined by the condition $f(rs)=f(sr)$ for all $r\in R_\infty$ and $s\in\mathfrak{S}_\infty$. In this paper we classify all semifinite factor-representations of $R_\infty$ that correspond to the $\mathfrak{S}_\infty$-invariant positive definite functions.