×

zbMATH — the first resource for mathematics

Harmonic analysis of weighted \(L^p\)-algebras. (English) Zbl 1247.43004
Throughout \(G\) will always denote a compactly generated locally compact group. It is well known that \(L^1(G)\) is a Banach algebra, where the multiplication is ordinary convolution of functions. For \(1 < p < \infty \) it is also well known that \(L^p(G)\) is not a convolution algebra. However, for certain weight functions \(\omega\) on \(G, L^p(G, \omega)\) is a Banach \(\ast\)-algebra. The aim of the paper under review is to extend the theory of convolution algebras to \(L^p(G, \omega)\) when it is a Banach \(\ast\)-algebra. It is also assumed in this paper that \(G\) has polynomial growth, in addition to the properties mentioned above. The authors give conditions on the weight \(\omega\) that guarantees \(L^p(G, \omega)\) is a Banach \(\ast\)-algebra. Furthermore, they say \((G, \omega)\) satisfies (LPAlg) if \(\omega\) meets these particular conditions. Several examples of weights \(\omega\) are given for which \((G, \omega)\) satisfies (LPAlg). An example of a weight \(\omega\) is also given for which \(L^p(G, \omega)\) is not an algebra. We shall say \(\omega\) satisfies (BDna) if \(\omega\) satisfies a Beurling-Domar type condition.
The authors proceed to give results involving symmetry, regularity, the weak Wiener property, and the Wiener property, when \((G, \omega)\) satisfies (LPAlg). An example of such a result is the following:
Let \((G, \omega)\) be (LPAlg). Let us assume that the weight \(\omega\) also satisfies (BDna). Then the algebra \(L^p(G, \omega)\) has the weak Wiener property.
Let \(p \geq 1\) be a real number. In the last section of the paper a weight \(\omega\) is constructed, depending on \(p\), on \(\mathbb{R}\) for which the commutative algebra \(L^p(G, \omega)\) is symmetric. The authors then construct an infinite-dimensional topologically irreducible representation of \(L^p(G, \omega)\). This is interesting since all topologically irreducible \(\ast\)-representations of \(L^p(G, \omega)\) are one-dimensional.

MSC:
43A15 \(L^p\)-spaces and other function spaces on groups, semigroups, etc.
22D15 Group algebras of locally compact groups
22D20 Representations of group algebras
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Atzmon, A., Irreducible representations of abelian groups, Lecture notes in math., 1359, 83-92, (1987)
[2] D. Alexandre, Idéaux minimaux d’algèbres de groupes, PhD thesis, Metz (2000).
[3] Bernstein, S.N., Le problème de l’approximation des fonctions continues sur tout l’axe réel et l’une de ses applications, Bull. math. soc. France, 52, 399-410, (1924) · JFM 50.0195.01
[4] A. Beurling, Sur les intégrales de Fourier absolument convergentes et leur application à une transformation fonctionnelle, 9ème Congrès des Mathématiciens scandinaves, août 1938, Helsinki (1939), pp. 345-366.
[5] A. Beurling, Sur le spectre des fonctions, Analyse Harmonique, Colloques Internationaux du Centre National de la Recherche Scientifique, no. 15, CNRS Paris (1949), pp. 9-29.
[6] Blanco, A.; Kaijser, S.; Ransford, T.J., Real interpolation of Banach algebras and factorization of weakly compact homomorphisms, J. funct. anal., 217, 126-141, (2004) · Zbl 1078.46050
[7] Bonsall, F.F.; Duncan, J., Complete normed algebras, () · Zbl 0271.46039
[8] Dahlke, S.; Fornasier, M.; Gröchenig, K., Optimal adaptive computations in the Jaffard algebra and localized frames, J. approx. theory, 162, 1, 153-185, (2010) · Zbl 1184.42029
[9] Dixmier, J., Opérateurs de rang fini dans LES représentations unitaires, Inst. hautes etudes sci. publ. math., 13-25, (1960) · Zbl 0100.32303
[10] Domar, Y., Harmonic analysis based on certain commutative Banach algebras, Acta math., 96, 1-66, (1956) · Zbl 0071.11302
[11] Dziubanski, J.; Ludwig, J.; Molitor-Braun, C., Functional calculus in weighted group algebras, Rev. mat. complut., 17, Núm. 2, 321-357, (2004) · Zbl 1049.43001
[12] Erdelyi, A., ()
[13] El-Fallah, O.; Nikol’skiĭ, N.K.; Zarrabi, M., Estimates for resolvents in Beurling-Sobolev algebras, Saint |St. Petersburg math. J., 10, 6, 901-964, (1999)
[14] Feichtinger, H.G., Gewichtsfunktionen auf lokalkompakten gruppen, Sitzber. österr. akad. wiss. abt. II, 188, 8-10, 451-471, (1979) · Zbl 0447.43004
[15] Fendler, G.; Gröchenig, K.; Leinert, M., Symmetry of weighted \(L^1\)-algebras and the GRS-condition, Bull. London math. soc., 38, 4, 625-635, (2006) · Zbl 1096.43002
[16] Fendler, G.; Gröchenig, K.; Leinert, M.; Ludwig, J.; Molitor-Braun, C., Weighted group algebras on groups of polynomial growth, Math. Z., 245, 791-821, (2003) · Zbl 1050.43003
[17] Glushko, V.P.; Savchenko, Yu.B., Higher-order degenerate elliptic equations: spaces, operators, boundary-value problems, J. math. sci., 39, 6, 3088-3148, (1987) · Zbl 0657.35064
[18] Godement, R., Théorèmes taubériens et théorie spectrale, Ann. sci. ec. norm. supér., 64, 119-138, (1947) · Zbl 0033.37601
[19] Gröchenig, K.; Leinert, M., Wiener’s lemma for twisted convolution and Gabor frames, J. amer. math. soc., 17, 1, 1-18, (2004) · Zbl 1037.22012
[20] Gröchenig, K.; Leinert, M., Symmetry and inverse closedness of matrix algebras and functional calculus for infinite matrices, Trans. amer. math. soc., 358, 6, 2695-2711, (2006) · Zbl 1105.46032
[21] Hewitt, E.; Ross, K.A., Abstract harmonic analysis I, II., (1997), Springer, 3rd printing
[22] Hulanicki, A., On the spectrum of convolution operators on groups with polynomial growth, Invent. math., 17, 135-142, (1972) · Zbl 0264.43007
[23] Hulanicki, A., A functional calculus for rockland operators on nilpotent Lie groups, Studia math., 78, 253-266, (1984) · Zbl 0595.43007
[24] Kaniuth, E., ()
[25] Kerman, R.; Sawyer, E., Convolution algebras with weighted rearrangement-invariant norm, Studia math., 108, 2, 103-126, (1994) · Zbl 0838.46020
[26] Kudryavtsev, L.D.; Nikol’skiĭ, S.M., Spaces of differentiable functions of several variables and imbedding theorems, (), 1140, Transl. from Russian
[27] Kuznetsova, Yu.N., Weighted \(L^p\)-algebras on groups, Funktsional. anal. i prilozhen., 40, 3, 82-85, (2006), Translation in Funct. Anal. Appl. 40 (3) (2006), 234-236
[28] Kuznetsova, Yu.N., Invariant weighted algebras \(L_p^w(G)\), Mat. zametki, 84, 4, 567-576, (2008)
[29] Kuznetsova, Yu.N., Constructions of regular algebras \(L_p^w(G)\), Mat. sb., 200, 2, 75-88, (2009) · Zbl 1172.46036
[30] Kuznetsova, Yu.N., Example of a weighted algebra \(L_p^w(G)\) on an uncoutable discrete group, J. math. anal. appl., 353, 660-665, (2009) · Zbl 1165.43005
[31] Leptin, H., On group algebras of nilpotent Lie groups, Studia math., 47, 37-49, (1973) · Zbl 0258.22009
[32] Leptin, H., Ideal theory in group algebras of locally compact groups, Invent. math., 31, 259-278, (1976) · Zbl 0328.22012
[33] Leptin, H., Symmetrie in banachschen algebren, Arch. math., 27, 4, 394-400, (1976) · Zbl 0327.46060
[34] Losert, V., On the structure of groups with polynomial growth II, J. London math. soc., 63, 640-654, (2001) · Zbl 1010.22008
[35] Ludwig, J., A class of symmetric and a class of Wiener group algebras, J. funct. anal., 31, 187-194, (1979) · Zbl 0402.22003
[36] Ludwig, J., Irreducible representations of exponential solvable Lie groups and operators with smooth kernels, J. reine angew. math., 339, 1-26, (1983) · Zbl 0492.22007
[37] Naimark, M.A., Normed algebras, (1972), Wolters-Noordhoff Publishing
[38] Mosak, R., ()
[39] Nikol’skiĭ, N.K., Selected problems of weighted approximation and spectral analysis, Trudy math. inst. Steklov, 120, (1974)
[40] Pytlik, T., On the spectral radius of elements in group algebras, Bull. acad. polon. sci. Sér. sci. math. astronom. phys., 21, 899-902, (1973) · Zbl 0275.43003
[41] Pytlik, T., Symbolic calculus on weighted group algebras, Studia math., 73, 169-176, (1982) · Zbl 0504.43005
[42] Read, C.J., Quasinilpotent operators and the invariant subspace problem, J. London math. soc., 2, 56, 595-606, (1997) · Zbl 0892.47005
[43] Saeki, S., The \(L^p\)-conjecture and young’s inequality, Illinois J. math., 34, 3, 614-627, (1990) · Zbl 0701.22003
[44] Segal, I.E., The group algebra of a locally compact group, Trans. amer. math. soc., 61, 69-105, (1947) · Zbl 0032.02901
[45] Šilov, G.E., On regular normed rings, Trav. inst. math. Stekloff, 21, (1947)
[46] Urbanik, K., A proof of a theorem of zelasko on \(L^p\)-algebras, Colloquium math., VIII, 121-123, (1961), fasc. 1 · Zbl 0095.10302
[47] Vretblad, A., Spectral analysis in weighted \(L^1\) spaces on \(\mathbb{R}\), Ark. mat., 11, 109-138, (1973) · Zbl 0258.46047
[48] Wermer, J., On a class of normed rings, Ark. mat., 2, 537-551, (1954), Hf. 6 · Zbl 0055.09902
[49] Wiener, N., The Fourier intergral and certain of its applications, (1933), Cambridge University Press Cambridge, Reprinted by Dover Publ., Inc., New-York (1958 · JFM 59.0416.01
[50] Zelasko, W., On the algebras \(L_p\) of locally compact groups, Colloq. math., VIII, 115-120, (1961), fasc. 1
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.