Abstract
The paper is devoted to weighted spaces ℒ w p (G) on a locally compact group G. If w is a positive measurable function on G, then the space ℒ w p (G), p ≥ 1, is defined by the relation ℒ w p (G) = {f: fw ∈ ℒ p (G)}. The weights w for which these spaces are algebras with respect to the ordinary convolution are treated. It is shown that, for p > 1, every sigma-compact group admits a weight defining such an algebra. The following criterion is proved (which was known earlier for special cases only): a space ℒ w p (G) is an algebra if and only if the function w is semimultiplicative. It is proved that the invariance of the space ℒ w p (G) with respect to translations is a sufficient condition for the existence of an approximate identity in the algebra ℒ w p (G). It is also shown that, for a nondiscrete group G and for p > 1, no approximate identity of an invariant weighted algebra can be bounded.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
I. M. Gel’fand, D. A. Raikov, and G. E. Shilov, Commutative Normed Rings (Fizmatgiz, Moscow, 1960; Chelsea Publishing Co., New York, 1964).
J. Wermer, “On a class of normed rings,” Ark. Mat. 2(6), 537–551 (1954).
E. Hewitt and K. A. Ross, Abstract Harmonic Analysis, Vols. I, II (Academic Press, Inc., Publishers, New York; Springer-Verlag, Berlin-Göttingen-Heidelberg, 1963; Springer-Verlag, New York-Berlin, 1970; Mir, Moscow, 1975).
Yu. N. Kuznetsova, “Weighted L p-algebras on groups,” Funktsional. Anal. i Prilozhen. 40(3), 82–85 (2006) [Funct. Anal. Appl. 40 (3), 234–236 (2006)].
R. E. Edwards, “The stability of weighted Lebesgue spaces,” Trans. Amer. Math. Soc. 93(3), 369–394 (1959).
H. G. Feichtinger, “Gewichtsfunktionen auf lokalkompakten Gruppen,” Österreich. Akad. Wiss. Math.-Natur. Kl. Sitzungsber. II 188(8–10), 451–471 (1979).
S. Grabiner, “Weighted convolution algebras as analogues of Banach algebras of power series,” in Radical Banach Algebras and Automatic Continuity, Lecture Notes inMath., Proc. Conf. held at California State Univ., Long Beach, Calif., July 17–31, 1981 (Springer, Berlin-New York, 1983), Vol. 975, pp. 282–289
A. Bruckner, “Differentiation of integrals,” Amer. Math. Monthly 78 (9, part II) (1971).
A. Ionescu Tulcea and C. Ionescu Tulcea, Topics in the Theory of Lifting, in Ergeb. Math. Grenzgeb. (Springer-Verlag, Berlin-New York, 1969), Vol. 48.
G. I. Gaudry, “Multipliers of weighted Lebesgue and measure spaces,” Proc. London Math. Soc. (3) 19, 327–340 (1969).
Volker Runde, Lectures on Amenability, in Lecture Notes in Math. (Springer-Verlag, Berlin-Heidelberg, 2002), Vol. 1774.
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © Yu. N. Kuznetsova, 2008, published in Matematicheskie Zametki, 2008, Vol. 84, No. 4, pp. 567–576.
Rights and permissions
About this article
Cite this article
Kuznetsova, Y.N. Invariant Weighted Algebras ℒ w p (G). Math Notes 84, 529–537 (2008). https://doi.org/10.1134/S0001434608090241
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0001434608090241