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A description of characters on the infinite wreath product. (English) Zbl 1143.20007

Let \(\mathfrak S_\infty\) be the group of bijections \(\mathbb{N}\to\mathbb{N}\) having finite sets of non-fixed points. For an arbitrary group \(\Gamma\), \(\mathfrak S_\infty\) admits a natural action on \(\Gamma^\infty\) by automorphisms. Therefore one can form a semidirect product \(\Gamma^\infty\rtimes\mathfrak S_\infty\) called the wreath product \(\Gamma\wr\mathfrak S_\infty\). The authors obtain a description of unitary \(\text{II}_1\)-factor-representations of \(\Gamma\wr\mathfrak S_\infty\).
The approach is based on the semigroup method by G. I. Ol’shanskiĭ [Leningr. Math. J. 1, No. 4, 983-1014 (1990); translation from Algebra Anal. 1, No. 4, 178-209 (1989; Zbl 0731.20009)] and A. Okounkov’s classification method used for admissible representations of \(\mathfrak S_\infty\times\mathfrak S_\infty\) [Funct. Anal. Appl. 28, No. 2, 100-107 (1994); translation from Funkts. Anal. Prilozh. 28, No. 2, 31-40 (1994; Zbl 0830.20029)]. The authors also give examples of type III representations, for which the Tomita-Takesaki modular operator is expressed in terms of the asymptotic operators appearing in the theory of characters of \(\mathfrak S_\infty\).

MSC:

20C32 Representations of infinite symmetric groups
22D10 Unitary representations of locally compact groups
22D25 \(C^*\)-algebras and \(W^*\)-algebras in relation to group representations
46L10 General theory of von Neumann algebras
20E22 Extensions, wreath products, and other compositions of groups
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