An analytic series of irreducible representations of the free group
Annales de l'Institut Fourier, Volume 38 (1988) no. 1, p. 87-110

Let ${\mathbf{F}}_{k}$ be a free group on $k$ generators. We construct the series of uniformly bounded representations ${\prod }_{z}$ of ${\mathbf{F}}_{k}$ acting on the common Hilbert space, depending analytically on the complex parameter z, $1/\left(2k-1\right)<|z|<1$, such that each representation ${\prod }_{z}$ is irreducible. If $z$ is real or $|z|=1/\left(\sqrt{2k-1}\right)$ then ${\prod }_{z}$ is unitary; in other cases ${\prod }_{z}$ cannot be made unitary. For $z\ne {z}^{\prime }$ representations ${\prod }_{z}$ and ${\prod }_{{z}^{\prime }}$ are congruent modulo compact operators.

Soit ${\mathbf{F}}_{k}$ un groupe libre avec $k$ générateurs. On construit une série des représentations uniformément bornées ${\prod }_{z}$ de ${\mathbf{F}}_{k}$ qui opèrent sur un espace de Hilbert commun. Les représentations ${\prod }_{z}$ sont irréductibles et dépendent analytiquement d’un paramètre complexe $z$ tel que $1/\left(2k-1\right)<|z|<1$. Pour $z$ réel ou $|z|=1/\left(\sqrt{2k-1}\right)$ les ${\prod }_{z}$ sont unitaires; autrement ${\prod }_{z}$ ne sont pas unitarisables. Pour $z\ne {z}^{\prime }$ les différences ${\prod }_{z}-{\prod }_{{z}^{\prime }}$ sont des opérateurs compacts.

@article{AIF_1988__38_1_87_0,
author = {Szwarc, Ryszard},
title = {An analytic series of irreducible representations of the free group},
journal = {Annales de l'Institut Fourier},
publisher = {Imprimerie Louis-Jean},
volume = {38},
number = {1},
year = {1988},
pages = {87-110},
doi = {10.5802/aif.1124},
zbl = {0634.22003},
mrnumber = {89j:22023},
language = {en},
url = {http://www.numdam.org/item/AIF_1988__38_1_87_0}
}

Szwarc, Ryszard. An analytic series of irreducible representations of the free group. Annales de l'Institut Fourier, Volume 38 (1988) no. 1, pp. 87-110. doi : 10.5802/aif.1124. http://www.numdam.org/item/AIF_1988__38_1_87_0/

[1] P. Cartier, Harmonic analysis on trees, Proc. Sympos. Pure Math. Amer. Math. Soc., 26 (1972), 419-424. | MR 49 #3038 | Zbl 0309.22009

[2] J. M. Cohen, Operator norms on free groups, Boll. Un. Math. Ital., 1-B (1982), 1055-1065. | MR 85d:22011 | Zbl 0518.46050

[3] A. Connes, The Chern character in K-homology, preprint.

[4] A. Figà-Talamanga, M.A. Picardello, Spherical functions and harmonic analysis on free groups, J. Funct. Anal., 47 (1982), 281-304. | MR 83m:22018 | Zbl 0489.43008

[5] A. Figà-Talamanga, M.A. Picardello, Harmonic analysis on free groups, Lecture Notes in Pure Appl. Math., M. Dekker, New York 1983. | Zbl 0536.43001

[6] U. Haagerup, An example of a non-nuclear C*-algebra which has the metric approximation property, Invent. Math., 50 (1979), 279-293. | MR 80j:46094 | Zbl 0408.46046

[7] R. A. Kunze, E. M. Stein, Uniformly bounded representations and harmonic analysis of the 2 x 2 real unimodular group, Amer. J. Math., 82 (1960), 1-62. | MR 29 #1287 | Zbl 0156.37104

[8] A. M. Mantero, A. Zappa, The Poisson transform on free groups and uniformly bounded representations, J. Funct. Anal., 47 (1983), 372-400. | MR 85b:22010 | Zbl 0532.43006

[9] M. Pimsner, D. Voiculescu, K-groups of reduced crossed products by free groups, J. Oper. Theory, 8 (1982), 131-156. | MR 84d:46092 | Zbl 0533.46045

[10] T. Pytlik, Radial functions on free groups and a decomposition of the regular representation into irreducible components, J. Reine Angew. Math., 326 (1981), 124-135. | MR 84a:22017 | Zbl 0464.22004

[11] T. Pytlik, R. Szwarc, An analytic family of uniformly bounded representations of free groups, Acta Math., 157 (1986), 287-309. | MR 88e:22014 | Zbl 0681.43011

[12] H. Yoshizawa, Some remarks on unitary representations of the free group, Osaka Math. J., 3 (1951), 55-63. | MR 13,10h | Zbl 0045.30103