${\rm II}_1$-factor representations of the infinite symmetric inverse semigroup

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 $\left[ g_{ij}\right]_{i,j\in \mathbb{N}}$ 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 non-negatively definite. A function $f$ said to be indecomposable if the corresponding $\star$-representation $\pi_f$ is a factor-representation. A class of the $R_\infty$-central functions (characters) is defined by the condition $f(rs)=f(sr)$ for all $r,s\in R_\infty$. In this paper we classify all factor-representations of $R_\infty$ that correspond to the $R_\infty$-central positive definite functions.