Geodesic
An analogue of Schur–Weyl duality for the unitary group of a $\mathrm{II}_1$-factor
Sbornik. Mathematics, Tome 210 (2019) no. 3, pp. 447-472
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An analogue of the classical Schur–Weyl duality is found for the unitary group of an arbitrary $\mathrm{II}_1$-factor. Bibliography: 20 titles.
Keywords: unitary group of a von Neumann factor, Schur–Weyl duality, factor representation, quasi-equivalent representations. 
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N. I. Nessonov. An analogue of Schur–Weyl duality for the unitary group of a $\mathrm{II}_1$-factor. Sbornik. Mathematics, Tome 210 (2019) no. 3, pp. 447-472. http://geodesic.mathdoc.fr/item/SM_2019_210_3_a4/

[1] G. Veil, Klassicheskie gruppy. Ikh invarianty i predstavleniya, IL, M., 1947, 408 pp.; H. Weyl, The classical groups. Their invariants and representations, Princeton Univ. Press, Princeton, NJ, 1939, xii+302 pp. | MR | Zbl

[2] A. A. Kirillov, Elementy teorii predstavlenii, 2-e izd., Nauka, M., 1978, 343 pp. ; A. A. Kirillov, Elements of the theory of representations, Grundlehren Math. Wiss., 220, Springer-Verlag, Berlin–New York, 1976, xi+315 pp. | MR | Zbl | MR | Zbl

[3] D. Beltiţă, K.-H. Neeb, “Polynomial representations of $C^*$-algebras and their applications”, Integral Equations Operator Theory, 86:4 (2016), 545–578 ; arXiv: 1608.02445v1 | DOI | MR | Zbl

[4] M. Takesaki, Theory of operator algebras, v. I, Encyclopaedia Math. Sci., 124, Oper. Alg. Non-commut. Geom., 5, Reprint of the 1st ed., Springer-Verlag, Berlin, 2002, xx+415 pp. | MR | Zbl

[5] N. I. Nessonov, Schur–Weyl duality for the unitary groups of $II_1$-factors, arXiv: 1312.0824

[6] M. Takesaki, Theory of operator algebras, v. II, Encyclopaedia Math. Sci., 125, Oper. Alg. Non-commut. Geom., 6, Springer-Verlag, Berlin, 2003, xxii+518 pp. | DOI | MR | Zbl

[7] A. M. Vershik, N. I. Nessonov, “Stabilnye predstavleniya beskonechnoi simmetricheskoi gruppy”, Izv. RAN. Ser. matem., 79:6 (2015), 93–124 | DOI | MR | Zbl

[8] Yu. A. Neretin, “Kategorii bistokhasticheskikh mer i predstavleniya nekotorykh beskonechnomernykh grupp”, Matem. sb., 183:2 (1992), 52–76 ; Yu. A. Neretin, “Categories of bistochastic measures, and representations of some infinite-dimensional groups”, Russian Acad. Sci. Sb. Math., 75:1 (1993), 197–219 | MR | Zbl | DOI

[9] Yu. A. Neretin, Kategorii simmetrii i beskonechnomernye gruppy, Editorial URSS, M., 1998, 431 pp.; Yu. A. Neretin, Categories of symmetries and infinite-dimensional groups, London Math. Soc. Monogr. (N.S.), 16, The Clarendon Press, Oxford Univ. Press, New York, 1996, xiv+417 pp. | MR | Zbl

[10] G. I. Olshanskii, “Unitarnye predstavleniya $(G,K)$-par, svyazannykh s beskonechnoi simmetricheskoi gruppoi $S(\infty)$”, Algebra i analiz, 1:4 (1989), 178–209 ; G. I. Ol'shanskij, “Unitary representations of $(G,K)$-pairs connected with the infinite symmetric group $S(\infty)$”, Leningrad Math. J., 1:4 (1990), 983–1014 | MR | Zbl

[11] E. Thoma, “Die unzerlegbaren, positiv-definiten Klassenfunktionen der abzählbar unendlichen symmetrischen Gruppe”, Math. Z., 85:1 (1964), 40–61 | DOI | MR | Zbl

[12] N. V. Tsilevich, A. M. Vershik, “Infnite-dimensional Schur–Weyl duality and the Coxeter–Laplace operator”, Comm. Math. Phys., 327:3 (2014), 873–885 | DOI | MR | Zbl

[13] A. A. Kirillov, “Predstavleniya beskonechnomernoi unitarnoi gruppy”, Dokl. AN SSSR, 212:2 (1973), 288–290 | MR | Zbl

[14] I. Penkov, K. Styrkas, “Tensor representations of classical locally finite Lie algebras”, Developments and trends in infinite-dimensional Lie theory, Progr. Math., 288, Birkhäuser Boston, Inc., Boston, MA, 2011, 127–150 | DOI | MR | Zbl

[15] S. Popa, M. Takesaki, “The topological structure of the unitary and automorphism groups of a factor”, Comm. Math. Phys., 155:1 (1993), 93–101 | DOI | MR | Zbl

[16] R. V. Kadison, “Infinite unitary groups”, Trans. Amer. Math. Soc., 72:3 (1952), 386–399 | DOI | MR | Zbl

[17] M. Takesaki, Theory of operator algebras, v. III, Encyclopaedia Math. Sci., 127, Oper. Alg. Non-commut. Geom., 8, Springer-Verlag, Berlin, 2003, xxii+548 pp. | DOI | MR | Zbl

[18] A. Connes, “Periodic automorphisms of the hyperfinite factors of type $\mathrm{II}_1$”, Acta Sci. Math. (Szeged), 39:1-2 (1977), 39–66 | MR | Zbl

[19] A. Daletskii, A. Kalyuzhnyi, “Permutations in tensor products of factors, and $L^2$ cohomology of configuration spaces”, Methods Funct. Anal. Topology, 12:4 (2006), 341–352 | MR | Zbl

[20] T. Enomoto, M. Izumi, “Indecomposable characters on infinite dimensional groups associated with operator algebras”, J. Math. Soc. Japan, 68:3 (2016), 1231–1270 ; arXiv: 1308.6329v1 | DOI | MR | Zbl