Semigroup approach to admissible representations of the infinite symmetric group

Let $S(\infty)$ denote the group of finitary permutations of the set $\mathbb N:=\{1,2,3,\dots\}$. It is a countable group admitting a lot of different topologies compatible with the group structure. In particular, such topologies arise from partitions of the set $\mathbb N$ into blocks of infinite size. The corresponding categories of continuous unitary representations of $S(\infty)$ were studied by Nessonov (Sbornik: Mathematics, 2012). We propose a different approach to his classification results based on the so-called semigroup method. Some additional information is also obtained.