Stable representations of the infinite symmetric group

摘要

We study the notion of a stable unitary representation of a group (or a star-representation of a 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 II1 of two-block Young subgroups. The class of stable representations will be the subject of further research.