  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.
