Stable representations of the infinite symmetric group

We study the notion of a stable unitary representation of a group (or a $\star$-representation of a $\mathbf C^\star$-algebra) with respect to some group of automorphisms of the group (or algebra). In the case of the group of finitary permutations of a countable set we give a complete description, up to quasi-equivalence, of the representations which are stable with respect to the group of all automorphisms of the group. In particular, we solve an old question concerning factor representations associated with Ol'shansky–Okun'kov admissible representations. It is proved that these representations are induced by factor representations of type ${\rm II}_1$ of two-block Young subgroups. The class of stable representations will be the subject of further research.