Vershik, A. M.; Nessonov, N. I. Stable states and representations of the infinite symmetric group. (English. Russian original) Zbl 1271.20009 Dokl. Math. 86, No. 1, 450-453 (2012); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 445, No. 1, 9-12 (2012). From the introduction: We introduce the class of stable representations, which is a natural extension of the class of representations of finite type. It turns out that stable representations of \(\mathfrak S_{\mathbb N}\) are of type I or II. We also give a complete classification of stable factor representations up to quasiequivalence. At the same time, we obtain an answer to the question posed by the first author [The work of Andrei Okounkov, Notices Am. Math. Soc. 54, No. 3, 391-393 (2007; Zbl 1142.01354)] in connection with G. I. Ol’shanskij’s [Leningr. Math. J. 1, No. 4, 983-1014 (1990); translation from Algebra Anal. 1, No. 4, 178-209 (1989; Zbl 0731.20009)] theory of admissible representations of the group \(\mathfrak S_{\mathbb N}\times\mathfrak S_{\mathbb N}\), that of identifying the components of an admissible representation. Namely, we prove that the set of stable factor representations coincides with the class of representations that can be obtained as the restrictions of admissible irreducible representations of \(\mathfrak S_{\mathbb N}\times\mathfrak S_{\mathbb N}\) to the left and right components (\(\mathfrak S_{\mathbb N}\times e\) and \(e\times\mathfrak S_{\mathbb N}\), respectively). 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 Keywords:infinite symmetric group; stable representations; representations of finite type; stable factor representations; admissible representations; irreducible representations Citations:Zbl 1142.01354; Zbl 0731.20009 PDFBibTeX XMLCite \textit{A. M. Vershik} and \textit{N. I. Nessonov}, Dokl. Math. 86, No. 1, 450--453 (2012; Zbl 1271.20009); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 445, No. 1, 9--12 (2012) Full Text: DOI arXiv References: [1] E. Thoma, Math. Z. 85(1), 40–61 (1964). · Zbl 0192.12402 [2] A. M. Vershik and S. V. Kerov, Funct. Anal. Appl. 15, 246–255 (1981). · Zbl 0507.20006 [3] A. M. Vershik, Notices Am. Math. Soc. 54, 391–393 (2007). [4] G. I. Ol’shanskii, Leningr. Math. J. 1, 983–1014 (1990). [5] M. Takesaki, Theory of Operator Algebras (Springer-Verlag, Berlin, 2005), Vol. 2. · Zbl 0990.46034 [6] A. Yu. Okun’kov, Funct. Anal. Appl. 28, 100–107 (1994). This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.