Dudko, A. V.; Nessonov, N. I. A description of characters on the infinite wreath product. (English) Zbl 1143.20007 Methods Funct. Anal. Topol. 13, No. 4, 301-317 (2007). 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 \(\mathfrak S_\infty\). Reviewer: Anatoly N. Kochubei (Kyïv) Cited in 3 Documents 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 Keywords:countable symmetric group; permutation groups; wreath products; factor-representations; modular operators Citations:Zbl 0731.20009; Zbl 0830.20029 PDFBibTeX XMLCite \textit{A. V. Dudko} and \textit{N. I. Nessonov}, Methods Funct. Anal. Topol. 13, No. 4, 301--317 (2007; Zbl 1143.20007) Full Text: arXiv