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

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

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

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     title = {An analytic series of irreducible representations of the free group},
     journal = {Annales de l'Institut Fourier},
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     year = {1988},
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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. https://aif.centre-mersenne.org/articles/10.5802/aif.1124/

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

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

[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 | Zbl

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

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

[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 | Zbl

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

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

[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 | Zbl

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

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

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