A description of characters on the infinite wreath product
Abstract
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 {\it wreath} product
Γ≀
S
∞
of
Γ
by
S
∞
. We obtain a full description of unitary
I
I
1
−
factor-representations of
Γ≀
S
∞
in terms of finite characters of
Γ
. Our approach is based on extending Okounkov's classification method for admissible representations of
S
∞
×
S
∞
. Also, we discuss certain examples of representations of type
I
I
1
, where the {\it modular operator} of Tomita-Takesaki expresses naturally by the asymptotic operators, which are important in the characters-theory of infinite symmetric group.