Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

The relative completion of an $ A$-Segal algebra is closed


Author: James T. Burnham
Journal: Proc. Amer. Math. Soc. 49 (1975), 116-122
MSC: Primary 46H10; Secondary 43A20
DOI: https://doi.org/10.1090/S0002-9939-1975-0361786-3
MathSciNet review: 0361786
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The main result is this theorem: If the Banach algebra $ A$ has bounded approximate right units and $ B$ is an $ A$-Segal algebra, then the relative completion of $ B$ with respect to $ A$ is an $ A$-Segal algebra. Furthermore, $ B$ is a closed ideal of its relative completion with respect to $ A$.


References [Enhancements On Off] (What's this?)

  • [1] James T. Burnham, Closed ideals in subalgebras of Banach algebras. I, Proc. Amer. Math. Soc. 32 (1972), 551-555. MR 0295078 (45:4146)
  • [2] -, Closed ideals in subalgebras of Banach algebras. II: Ditkin's condition, Monatsh. Math. 78 (1974), 1-3. MR 0341095 (49:5845)
  • [3] -, Segal algebras and dense ideals in Banach algebras, Proc. Internat. Conf. on Functional Analysis (Madras, India, 1973), Lecture Notes in Math., vol. 399, Springer-Verlag, Berlin and New York, 1974. MR 0415204 (54:3295)
  • [4] -, Relative completions of $ A$-Segal algebras, Bull. Amer. Math. Soc. (to appear).
  • [5] James T. Burnham and Richard R. Goldberg, Multipliers of $ {L^1}(G)$ into Segal algebras, Acta Math. Sinica (to appear).
  • [6] -, Basic properties of Segal algebras, J. Math. Anal. Appl. 42 (1973), 323-329. MR 0326304 (48:4648)
  • [7] J. Cigler, Normed ideals in $ {L^1}(G)$, Nederl. Akad. Wetensch. Proc. Ser. A 72 = Indag. Math. 31 (1969), 273-282. MR 40 #3327. MR 0250086 (40:3327)
  • [8] B. Dunford, Segal algebras and left normed ideals, J. London Math. Soc. (to appear). MR 0346423 (49:11148)
  • [9] H. G. Feichtinger, Zur Idealtheorie von Segal-Algebren, Manuscripta Math. 10 (1973), 307-312. MR 0324417 (48:2769)
  • [10] N. Aronszajn and E. Gagliardo, Interpolation spaces and interpolation methods, Ann. Mat. Pura Appl. (4) 68 (1965), 51-117. MR 37 #1951. MR 0226361 (37:1951)
  • [11] Colin C. Graham, The algebraic radical of a normed ideal in $ {L^1}(G)$, preprint, 1973.
  • [12] E. Hewitt and K. A. Ross, Abstract harmonic analysis. Vol. II: Structure and analysis for compact groups. Analysis on locally compact Abelian groups, Die Grundlehren der math. Wissenschaften, Band 152, Springer-Verlag, Berlin and New York, 1970. MR 41 #7378; erratum, 42, p. 1825. MR 0262773 (41:7378)
  • [13] Michael Leinert, A contribution to Segal algebras, Manuscripta Math. 10 (1973), 297-306. MR 0324416 (48:2768)
  • [14] M. A. Rieffel, Induced Banach representations of Banach algebras and locally compact groups, J. Functional Analysis 1 (1967), 443-491. MR 36 #6544. MR 0223496 (36:6544)
  • [15] Hans Reiter, $ {L^1}$-algebras and Segal algebras, Lecture Notes in Math., vol. 231, Springer-Verlag, Berlin and New York, 1971. MR 0440280 (55:13158)
  • [16] N. Th. Varopoulos, "The embedding of $ A(E)$ into $ \tilde A(E)$", Chapter XI in: Thin sets in harmonic analysis, edited by L. A. Lindahl and F. Paulsen, Lecture Notes in Pure and Appl. Math., Dekker, New York, 1971. MR 0440293 (55:13168)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 46H10, 43A20

Retrieve articles in all journals with MSC: 46H10, 43A20


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1975-0361786-3
Keywords: Banach algebras, Banach modules, closed ideals, approximate identities, Segal algebras
Article copyright: © Copyright 1975 American Mathematical Society

American Mathematical Society