  We review and introduce several approaches to the study of centralizer
algebras of the infinite symmetric group $S_\infty$. Our study is led by the
double commutant relationships between finite symmetric groups and partition
algebras; each approach produces a centralizer algebra that is contained in a
partition algebra. Our goal is to incorporate invariants of $S_\infty$, which
ties our work to the study of symmetric functions in non-commuting variables.
We resultantly explore sequence spaces as permutation modules, which yields
families of non-unitary representations of $S_\infty$.
