${\mathrm{II}}_1$-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 $r_1,r_2,\dots ,r_n\in R_{\infty }$ the matrix $\left[f\left(r_i{r}_j^{\star }\right)\right]$ 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.

11 pages