Abstract
We study norm convergence and summability of Fourier series in the setting of reduced twisted group C *-algebras of discrete groups. For amenable groups, Følner nets give the key to Fejér summation. We show that Abel-Poisson summation holds for a large class of groups, including e.g. all Coxeter groups and all Gromov hyperbolic groups. As a tool in our presentation, we introduce notions of polynomial and subexponential H-growth for countable groups w.r.t. proper scale functions, usually chosen as length functions. These coincide with the classical notions of growth in the case of amenable groups.
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Akemann, C.A., Ostrand, P.A.: Computing norms in group C *-algebras. Am. J. Math. 98, 1015–1047 (1976)
Backhouse, N.B.: Projective representations of space groups, II: Factor systems. J. Math. Oxford 21, 223–242 (1970)
Backhouse, N.B., Bradley, C.J.: Projective representations of space groups, I: Translation groups. J. Math. Oxford 21, 203–222 (1970)
Bédos, E., Conti, R.: On infinite tensor products of projective unitary representations. Rocky Mt. J. Math. 34, 467–493 (2004)
Bédos, E., Conti, R.: Fourier series and twisted C*-crossed products. In preparation
Bellissard, J.: Gap labelling theorems for Schrödinger operators. In: From number theory to physics, Les Houches, 1989, pp. 538–630. Springer, Berlin (1992)
Bellissard, J.: The noncommutative geometry of aperiodic solids. In: Geometric and topological methods for quantum field theory, Villa de Leya, 2001, pp. 86–156. World Scientific, Singapore (2003)
Berg, C., Reus Christensen, J.P., Ressel, P.: Harmonic Analysis on Semigroups. GTM, vol. 100. Springer, Berlin (1984)
Bożejko, M.: Positive definite bounded matrices and a characterisation of amenable groups. Proc. Am. Math. Soc. 95, 357–360 (1985)
Bożejko, M.: Positive-definite kernels, length functions on groups and a noncommutative von Neumann inequality. Stud. Math. XCV, 107–118 (1989)
Bożejko, M., Fendler, G.: Herz-Schur multipliers and completely bounded multipliers of the Fourier algebra of a locally compact group. Boll. Unione Mat. Ital. A 3(6), 297–302 (1984)
Bożejko, M., Januszkiewicz, T., Spatzier, R.J.: Infinite Coxeter groups do not have Kazhdan’s property. J. Oper. Theory 19, 63–67 (1988)
Bożejko, M., Picardello, M.A.: Weakly amenable groups and amalgameted products. Proc. Am. Math. Soc. 117, 1039–1046 (1993)
Brodzki, J., Niblo, G.: Approximation properties for discrete groups. In: C *-algebras and elliptic theory. Trends in Mathematics, pp. 23–35. Birkhäuser, Basel (2006)
Carey, A.L., Hannabuss, K.A., Mathai, V., McCann, P.: Quantum Hall effect on the hyperbolic plane. Commun. Math. Phys. 190, 629–673 (1998)
Chatterji, I.: Twisted rapid decay. Appendix to [65]
Chatterji, I., Ruane, K.: Some geometric groups with rapid decay. Geom. Funct. Anal. 15, 311–339 (2005)
Chen, X., Wei, S.: Spectral invariant subalgebras of reduced crossed product C *-algebras. J. Funct. Anal. 197, 228–246 (2003)
Cherix, P.-A., Cowling, M., Jolissaint, P., Julg, P., Valette, A.: Groups with the Haagerup property. In: Gromov’s a-T-menability. Progress in Mathematics, vol. 197. Birkhäuser, Basel (2001)
Cohen, J.M.: Operator norms in free groups. Boll. Unione Mat. Ital., B 1, 1055–1065 (2003)
Connes, A.: Compact metric spaces, Fredholm modules, and hyperfiniteness. Ergod. Theory Dyn. Syst. 9, 207–220 (1989)
Connes, A.: Noncommutative Geometry. Academic Press, New York (1994)
Cowling, M.: Sur les coefficients des représentations des groupes Lie simples. Lect. Not. Math. 739, 132–178 (1979)
Cowling, M.: Harmonic analysis on some nilpotent Lie groups (with applications to the representation theory of some semisimple Lie groups). In: Topics in modern harmonic analysis, Turin/Milan, 1982. Ist. Naz.Alta Mat., vols. I, II, pp. 81–123. Francesco Severi, Rome (1983)
Cowling, M., Haagerup, U.: Completely bounded multipliers of the Fourier algebra of a simple Lie group of real rank one. Invent. Math. 96, 507–549 (1989)
Davidson, K.R.: C *-Algebras by Examples. Fields Institute Monographs, vol. 6. Am. Math. Soc., Providence (1996)
de Cannière, J., Haagerup, U.: Multipliers of the Fourier algebras of some simple Lie groups and their discrete subgroups. Am. J. Math. 107, 455–500 (1985)
de la Harpe, P.: Groupes de Coxeter infinis non affines. Exp. Math. 5, 91–96 (1987)
de la Harpe, P.: Groupes hyperboliques, algèbres d’ opérateurs et un théorème de Jolissaint. C. R. Acad. Sci. Paris, Sér. I 307, 771–774 (1988)
de la Harpe, P.: Topics in Geometric Group Theory. Chicago Lectures in Mathematics Series. University of Chicago Press, Chicago (2000)
Dixmier, J.: Les C *-Algèbres et Leurs Représentations. Gauthiers-Villars, Paris (1969)
Dixmier, J.: Les Algèbres d’Opérateurs dans l’Espace Hilbertien (Algèbres de von Neumann). Gauthiers-Villars, Paris (1969)
Dixmier, J.: Topologie Générale. PUF, Paris (1981)
Dorofaeff, B.: The Fourier algebra of SL(2,ℝ)⋊ℝn,n≥2, has no multiplier unit. Math. Ann. 297, 707–724 (1993)
Effros, E.G., Ruan, Z.-J.: Multivariable multiplier for groups and their operator algebras. In: Operator Theory: Operator Algebras and Their Applications, Part I, Durham, NH, 1988. Proc. Symp. Pure Math., vol. 51, pp. 197–218. Am. Math. Soc., Providence (1990)
Exel, R.: Hankel matrices over right ordered amenable groups. Can. Math. Bull. 33, 404–415 (1990)
Eymard, P.: L’algèbre de Fourier d’un groupe localement compact. Bull. Soc. Math. France 92, 181–236 (1964)
Fendler, G.: Simplicity of the reduced C *-algebras of certain Coxeter groups. Illinois J. Math. 47, 883–897 (2003)
Fendler, G., Gröchenig, K., Leinert, M.: Symmetry of weighted L 1-algebras and the GRS-condition. Bull. Lond. Math. Soc. 38, 625–635 (2006)
Figà-Talamanca, A., Picardello, M.A.: Harmonic Analysis on Free Groups. Lectures Notes in Pure and Appl. Math., vol. 87. Dekker, New York (1983)
Flory, V.: Estimating norms in C *-algebras of discrete groups. Math. Ann. 224, 41–52 (1976)
Ghys, E., de la Harpe, P.: Sur les Groupes Hyperboliques d’Après Mikhael Gromov. Progress in Math., vol. 83. Birkhaüser, Basel (1990)
Gröchenig, K., Leinert, M.: Wiener’s lemma for twisted convolution and Gabor frames. J. Am. Math. Soc. 17, 1–18 (2004)
Gromov, M.: Hyperbolic groups. Math. Sci. Res. Inst. Publ. 8, 75–263 (1987)
Haagerup, U.: An example of a nonnuclear C *-algebra, which has the metric approximation property. Invent. Math. 50, 279–293 (1978/79)
Haagerup, U.: Group C *-algebras without the completely bounded approximation property. Unpublished manuscript (1986)
Haagerup, U., Kraus, J.: Approximation properties for group C *-algebras and group von Neumann algebras. Trans. Am. Math. Soc. 344, 667–699 (1994)
Higson, N., Guentner, E.: Group C *-algebras and K-theory. In: Noncommutative geometry. Lecture Notes in Math., vol. 1831, pp. 137–251. Springer, Berlin (2004)
Higson, N., Guentner, E.: Weak amenability of CAT(0)-cubical groups. Preprint (2007)
Howlett, R.B.: On the Schur multipliers of Coxeter groups. J. Lond. Math. Soc. 38, 263–276 (1988)
Humphreys, J.E.: Reflection Groups and Coxeter Groups. Cambridge University Press, Cambridge (1990)
Januszkiewicz, T.: For Coxeter groups z |g| is a coefficient of a uniformly bounded representation. Fund. Math. 174, 79–86 (2002)
Ji, R., Schweitzer, L.: Spectral invariance of smooth crossed products, and rapid decay locally compact groups. K-Theory 10, 283–305 (1996)
Jolissaint, P.: Rapidly decreasing functions in reduced C *-algebras of groups. Trans. Am. Math. Soc. 317, 167–196 (1990)
Jolissaint, P.: K-theory of reduced C *-algebras and rapidly decreasing functions on groups. K-Theory 2, 167–196 (1990)
Jolissaint, P., Valette, A.: Normes de Sobolev et convoluteurs bornés sur L 2(G). Ann. Inst. Fourier (Grenoble) 41, 797–822 (1991)
Kleppner, A.: The structure of some induced representations. Duke Math. J. 29, 555–572 (1962)
Kleppner, A.: Multipliers on Abelian groups. Math. Ann. 158, 11–34 (1965)
Leinert, M.: Faltungsoperatoren auf gewissen diskreten Gruppen. Stud. Math. LII, 149–158 (1974)
Luef, F.: Gabor analysis, noncommutative tori and Feichtinger’s algebra. In: Gabor and Wavelet Frames. IMS Lecture Notes Series, vol. 10. World Scientific, Singapore (2005). arXiv:math/0504146v1
Luef, F.: On spectral invariance of non-commutative tori. In: Operator Theory, Operator Algebras, and Applications. Contemp. Math., vol. 414, pp. 131–146. Am. Math. Soc., Providence (2006)
Mackey, G.: Unitary representations of group extensions I. Acta Math. 99, 265–311 (1958)
Marcolli, M., Mathai, V.: Twisted index theory on good orbifolds, I: noncommutative Bloch theory. Commun. Contemp. Math. 1, 553–587 (1999)
Marcolli, M., Mathai, V.: Towards the fractional quantum Hall effect: a noncommutative geometry perspective. Preprint (2005)
Mathai, V.: Heat kernels and the range of the trace on completions of twisted group algebras. With an appendix by Indira Chatterji. Contemp. Math. 398, 321–345 (2006)
Mercer, R.: Convergence of Fourier series in discrete crossed products of von Neumann algebras. Proc. Am. Math. Soc. 94, 254–258 (1985)
Nebbia, C.: Multipliers and asymptotic behaviour of the Fourier algebra of nonamenable groups. Proc. Am. Math. Soc. 84, 549–554 (1982)
Ol’shanskiĭ, A.Yu.: Almost every group is hyperbolic. Int. J. Algebra Comput. 2, 1–17 (1992)
Ozawa, N.: Weak amenability of hyperbolic groups. Preprint (2007)
Ozawa, N., Rieffel, M.: Hyperbolic group C *-algebras and free-product C *-algebras as compact quantum metric spaces. Can. J. Math. 57, 1056–1079 (2005)
Packer, J.A.: C *-algebras generated by projective representations of the discrete Heisenberg group. J. Oper. Theory 18, 41–66 (1987)
Packer, J.A.: Twisted group C *-algebras corresponding to nilpotent discrete groups. Math. Scand. 64, 109–122 (1989)
Packer, J.A., Raeburn, I.: Twisted crossed product of C *-algebras. Math. Proc. Camb. Philos. Soc. 106, 293–311 (1989)
Packer, J.A., Raeburn, I.: Twisted crossed product of C *-algebras II. Math. Ann. 287, 595–612 (1990)
Packer, J.A., Raeburn, I.: On the structure of twisted group C *-algebras. Trans. Am. Math. Soc. 334, 685–718 (1992)
Paterson, A.: Amenability. Math. Surveys and Monographs, vol. 29. Am. Math. Soc., Providence (1988)
Paulsen, V.: Completely Bounded Maps and Operator Algebras. Cambridge University Press, Cambridge (2002)
Picardello, M.A.: Positive definite functions and Lp convolutions operators on amalgams. Pac. J. Math. 123, 209–221 (1986)
Pier, J.P.: Amenable Locally Compact Groups. Wiley, New York (1984)
Pisier, G.: Similarity Problems and Completely Bounded Maps, 2nd edn. Lect. Notes in Math., vol. 1618. Springer, Berlin (2001)
Sauvageot, J.-L.: Strong Feller noncommutative kernels, strong Feller semigroups and harmonic analysis. In: Operator Algebras and Quantum Field Theory, Rome, 1996, pp. 105–110. International Press, Cambridge (1997)
Sauvageot, J.-L.: Strong Feller semigroups on C *-algebras. J. Oper. Theory 42, 83–102 (1999)
Schweitzer, L.B.: Dense m-convex Fréchet subalgebras of operator algebra crossed products by Lie groups. Int. J. Math. 4, 601–673 (1993)
Valette, A.: Les représentations uniformement bornées associées à un arbre réel. Bull. Soc. Math. Belg., Ser. A 42, 747–760 (1990)
Valette, A.: Weak amenability of right-angled Coxeter groups. Proc. Am. Math. Soc. 119, 1331–1334 (1993)
Valette, A.: Introduction to the Baum-Connes Conjecture. Lectures in Mathematics ETH Zürich. Birkhäuser, Basel (2002)
Wagon, S.: The Banach-Tarski Paradox. Encyclopedia of Math. and its Appl., vol. 24. Cambridge University Press, Cambridge (1985)
Weaver, N.: Lipschitz algebras and derivations of von Neumann algebras. J. Funct. Anal. 139, 261–300 (1996)
Zeller-Meier, G.: Produits croisés d’une C *-algèbre par un groupe d’automorphismes. J. Math. Pures Appl. 47, 101–239 (1968)
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Communicated by Karlheinz Gröchenig.
E. Bédos was partially supported by the Norwegian Research Council.
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Bédos, E., Conti, R. On Twisted Fourier Analysis and Convergence of Fourier Series on Discrete Groups. J Fourier Anal Appl 15, 336–365 (2009). https://doi.org/10.1007/s00041-009-9067-z
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DOI: https://doi.org/10.1007/s00041-009-9067-z
Keywords
- Twisted group C *-algebra
- Fourier series
- Fejér summation
- Abel-Poisson summation
- Amenable group
- Haagerup property
- Length function
- Polynomial growth
- Subexponential growth