A description of characters on the infinite wreath product

الأصل · EN

Let S∞ be the infinity permutation group and Γ an arbitrary group. Then S∞ admits a natural action on Γ∞ by automorphisms, so one can form a semidirect product Γ∞ S∞, known as the wreath product Γ∞ of Γ by S∞. We obtain a full description of unitary II₁-factor-representations of Γ∞ in terms of finite characters of Γ. Our approach is based on extending Okounkov's classification method for admissible representations of S∞∞. Also, we discuss certain examples of representations of type II₁, where the modular operator of Tomita-Takesaki expresses naturally by the asymptotic operators, which are important in the characters-theory of infinite symmetric group.