Semigroup approach to admissible representations of the infinite symmetric group

Let $S(\mathrm{\infty })$ 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 $S(\mathrm{\infty })$ 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.