Herz-Schur multipliers and non-uniformly bounded representations of locally compact groups

• Autorzy: Steenstrup T.
• Języki publikacji: EN

Abstrakty

EN

Let G be a second countable, locally compact group and let φ be a continuous Herz-Schur multiplier on G. Our main result gives the existence of a (not necessarily uniformly bounded) strongly continuous representation π of G on a Hilbert space ℋ , together with vectors ξ, η ϵ ℋ, such that φ(y⁻¹x) = ⟨π(x)ξ, π(y⁻¹) * η⟩ for x, y ϵ G and sup_{x ϵ G} ∥π(x)ξ sup_{y ϵ G} ∥π(y⁻¹) * η∥ = ∥φ∥ _M0A(G). Moreover, we obtain control over the growth of the representation in the sense that ∥π(g)∥≤ exp ( c/2 d(g, e)) for g ϵ G, where e ϵ G is the identity element, c is a constant, and d is a metric on G.

Słowa kluczowe

EN

Herz-Schur multipliers; non-uniformly bounded representations; free groups

• Wydawca: Wrocław University of Science and Technology
• Czasopismo: Probability and Mathematical Statistics
• Rocznik: 2013
• Tom: Vol. 33, Fasc. 2
• Strony: 213--223
• Opis fizyczny: Bibliogr. 10 poz.

Twórcy

autor

Steenstrup T.

• Department of Mathematics and Computer Science, University of Southern Denmark, Campusvej 55, DK-5230 Odense M, Denmark

Bibliografia

• [1] M. Bożejko and G. Fendler, Herz-Schur multipliers and completely bounded multipliers of the Fourier algebra of a locally compact group, Boll. Unione Mat. Ital. (6) 3-A (1984), pp. 297-302.
• [2] M. Bożejko and G. Fendler, Herz-Schur multipliers and uniformly bounded representations of discrete groups, Arch. Math. (Basel) 57 (1991), pp. 290-298.
• [3] J. De Cannière and U. Haagerup, Multipliers of the Fourier algebras of some simple Lie groups and their discrete subgroups, Amer. J. Math. 107 (1985), pp. 455-500.
• [4] P. Eymard, L’algèbre de Fourier d’un groupe localement compact, Bull. Soc. Math. France 92 (1964), pp. 181-236.
• [5] U. Haagerup and A. Przybyszewska, Proper metrics on locally compact groups, and proper affine isometric actions on Banach spaces, arXiv:math/0606794 (2006).
• [6] C. Herz, Une généralisation de la notion de transformée de Fourier-Stieltjes, Ann. Inst. Fourier (Grenoble) 24 (1974), pp. 145-157.
• [7] P. Jolissaint, A characterization of completely bounded multipliers of Fourier algebras, Colloq. Math. 63 (1992), pp. 311-313.
• [8] G. Pisier, Are unitarizable groups amenable?, Progr. Math. 248 (2005), pp. 323-362.
• [9] T. Steenstrup, Fourier multiplier norms of spherical functions on the generalized Lorentz groups, arXiv:0911.4977 (2009).
• [10] R. A. Struble, Metrics in locally compact groups, Compos. Math. 28 (1974), pp. 217-222.

Uwagi

Opracowanie rekordu w ramach umowy 509/P-DUN/2018 ze środków MNiSW przeznaczonych na działalność upowszechniającą naukę (2019).