Free group representations: Duplicity and oddity on the boundary
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- by Waldemar Hebisch, Gabriella Kuhn and Tim Steger PDF
- Trans. Amer. Math. Soc. 375 (2022), 1825-1860 Request permission
Abstract:
We present a powerful theorem for proving the irreducibility of tempered unitary representations of the free group.References
- S. Adams, Boundary amenability for word hyperbolic groups and an application to smooth dynamics of simple groups, Topology 33 (1994), no. 4, 765–783. MR 1293309, DOI 10.1016/0040-9383(94)90007-8
- Adrien Boyer and Łukasz Garncarek, Asymptotic Schur orthogonality in hyperbolic groups with application to monotony, Trans. Amer. Math. Soc. 371 (2019), no. 10, 6815–6841. MR 3939562, DOI 10.1090/tran/7653
- Kenneth R. Davidson, $C^*$-algebras by example, Fields Institute Monographs, vol. 6, American Mathematical Society, Providence, RI, 1996. MR 1402012, DOI 10.1090/fim/006
- Jacques Dixmier, $C^*$-algebras, North-Holland Mathematical Library, Vol. 15, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977. Translated from the French by Francis Jellett. MR 0458185
- Jacques Dixmier, von Neumann algebras, North-Holland Mathematical Library, vol. 27, North-Holland Publishing Co., Amsterdam-New York, 1981. With a preface by E. C. Lance; Translated from the second French edition by F. Jellett. MR 641217
- Alessandro Figà-Talamanca and Massimo A. Picardello, Spherical functions and harmonic analysis on free groups, J. Functional Analysis 47 (1982), no. 3, 281–304. MR 665019, DOI 10.1016/0022-1236(82)90108-2
- Alessandro Figà-Talamanca and Tim Steger, Harmonic analysis for anisotropic random walks on homogeneous trees, Mem. Amer. Math. Soc. 110 (1994), no. 531, xii+68. MR 1219707, DOI 10.1090/memo/0531
- James Glimm, Type I $C^{\ast }$-algebras, Ann. of Math. (2) 73 (1961), 572–612. MR 124756, DOI 10.2307/1970319
- Greg Hjorth, When is an equivalence relation classifiable?, Proceedings of the International Congress of Mathematicians, Vol. II (Berlin, 1998), 1998, pp. 23–32. MR 1648053
- Gabriella Kuhn and Tim Steger, More irreducible boundary representations of free groups, Duke Math. J. 82 (1996), no. 2, 381–436. MR 1387235, DOI 10.1215/S0012-7094-96-08218-6
- M. Gabriella Kuhn and Tim Steger, Monotony of certain free group representations, J. Funct. Anal. 179 (2001), no. 1, 1–17. MR 1807250, DOI 10.1006/jfan.2000.3663
- M. Gabriella Kuhn and Tim Steger, Free group representations from vector-valued multiplicative functions. I, Israel J. Math. 144 (2004), 317–341. MR 2121544, DOI 10.1007/BF02916716
- M. Gabriella Kuhn, Sandra Saliani, and Tim Steger, Free group representations from vector-valued multiplicative functions, II, Math. Z. 284 (2016), no. 3-4, 1137–1162. MR 3563271, DOI 10.1007/s00209-016-1692-z
- M. Gabriella Kuhn, Sandra Saliani, and Tim Steger, Free group representations from vector-valued multiplicative functions, iii, 2020.
- M. Gabriella Kuhn, Amenable actions and weak containment of certain representations of discrete groups, Proc. Amer. Math. Soc. 122 (1994), no. 3, 751–757. MR 1209424, DOI 10.1090/S0002-9939-1994-1209424-3
- William L. Paschke, Pure eigenstates for the sum of generators of the free group, Pacific J. Math. 197 (2001), no. 1, 151–171. MR 1810213, DOI 10.2140/pjm.2001.197.151
- T. Pytlik and R. Szwarc, An analytic family of uniformly bounded representations of free groups, Acta Math. 157 (1986), no. 3-4, 287–309. MR 857676, DOI 10.1007/BF02392596
- Carlo Pensavalle and Tim Steger, Tensor products with anisotropic principal series representations of free groups, Pacific J. Math. 173 (1996), no. 1, 181–202. MR 1387798, DOI 10.2140/pjm.1996.173.181
- Iain Raeburn, Allan M. Sinclair, and Dana P. Williams, Equivariant completely bounded operators, Pacific J. Math. 139 (1989), no. 1, 155–194. MR 1010790, DOI 10.2140/pjm.1989.139.155
- Walter Rudin, Functional analysis, McGraw-Hill Series in Higher Mathematics, McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1973. MR 0365062
- W. Forrest Stinespring, Positive functions on $C^*$-algebras, Proc. Amer. Math. Soc. 6 (1955), 211–216. MR 69403, DOI 10.1090/S0002-9939-1955-0069403-4
- M. Takesaki, Theory of operator algebras. II, Encyclopaedia of Mathematical Sciences, vol. 125, Springer-Verlag, Berlin, 2003. Operator Algebras and Non-commutative Geometry, 6. MR 1943006, DOI 10.1007/978-3-662-10451-4
- Hisaaki Yoshizawa, Some remarks on unitary representations of the free group, Osaka Math. J. 3 (1951), 55–63. MR 41854
Additional Information
- Waldemar Hebisch
- Affiliation: Mathematical Institute, University of Wrocław, 50-384 Wrocław, pl. Grunwaldzki 2/4, Poland
- MR Author ID: 263309
- ORCID: 0000-0002-0930-4460
- Email: hebisch@mail.math.uni.wroc.pl
- Gabriella Kuhn
- Affiliation: Dipartimento di Matematica e Applicazioni, Università di Milano Bicocca, via Cozzi 53, 20125 Milano, Italy
- MR Author ID: 210437
- Email: mariagabriella.kuhn@unimib.it
- Tim Steger
- Affiliation: Matematica, Università degli Studi di Sassari, Via Piandanna 4, 07100 Sassari, Italy
- MR Author ID: 248120
- Email: steger@uniss.it
- Received by editor(s): October 12, 2020
- Received by editor(s) in revised form: July 31, 2021
- Published electronically: December 3, 2021
- © Copyright 2021 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 375 (2022), 1825-1860
- MSC (2020): Primary 22D10, 43A65; Secondary 22E45, 22E40
- DOI: https://doi.org/10.1090/tran/8546
- MathSciNet review: 4378081