The relative completion of an 𝐴-Segal algebra is closed

Burnham, James T.

doi:10.1090/S0002-9939-1975-0361786-3

The relative completion of an $A$-Segal algebra is closed
HTML articles powered by AMS MathViewer

by James T. Burnham PDF

Proc. Amer. Math. Soc. 49 (1975), 116-122 Request permission

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

J. T. Burnham, Closed ideals in subalgebras of Banach algebras. I, Proc. Amer. Math. Soc. 32 (1972), 551–555. MR 295078, DOI 10.1090/S0002-9939-1972-0295078-5
J. T. Burnham, Closed ideals in subalgebras of Banach algebras. II. Ditkin’s condition, Monatsh. Math. 78 (1974), 1–3. MR 341095, DOI 10.1007/BF01298190
James T. Burnham, Segal algebras and dense ideals in Banach algebras, Functional analysis and its applications (Internat. Conf., Eleventh Anniversary of Matscience, Madras, 1973; dedicated to Alladi Ramakrishnan), Lecture Notes in Math., Vol. 399, Springer, Berlin, 1974, pp. 33–58. MR 0415204

Relative completions of $A$-Segal algebras

Multipliers of ${L^1}(G)$ into Segal algebras

J. T. Burnham and Richard R. Goldberg, Basic properties of Segal algebras, J. Math. Anal. Appl. 42 (1973), 323–329. Collection of articles dedicated to Salomon Bochner. MR 326304, DOI 10.1016/0022-247X(73)90141-8
J. Cigler, Normed ideals in $L^{1}\,(G)$, Nederl. Akad. Wetensch. Proc. Ser. A 72 = Indag. Math. 31 (1969), 273–282. MR 0250086, DOI 10.1016/1385-7258(69)90015-8
D. H. Dunford, Segal algebras and left normed ideals, J. London Math. Soc. (2) 8 (1974), 514–516. MR 346423, DOI 10.1112/jlms/s2-8.3.514
Hans G. Feichtinger, Zur Idealtheorie von Segal-Algebren, Manuscripta Math. 10 (1973), 307–312 (German, with English summary). MR 324417, DOI 10.1007/BF01332772
N. Aronszajn and E. Gagliardo, Interpolation spaces and interpolation methods, Ann. Mat. Pura Appl. (4) 68 (1965), 51–117. MR 226361, DOI 10.1007/BF02411022

The algebraic radical of a normed ideal in

Edwin Hewitt and Kenneth A. Ross, Abstract harmonic analysis. Vol. II: Structure and analysis for compact groups. Analysis on locally compact Abelian groups, Die Grundlehren der mathematischen Wissenschaften, Band 152, Springer-Verlag, New York-Berlin, 1970. MR 0262773
Michael Leinert, A contribution to Segal algebras, Manuscripta Math. 10 (1973), 297–306. MR 324416, DOI 10.1007/BF01332771
Marc A. Rieffel, Induced Banach representations of Banach algebras and locally compact groups, J. Functional Analysis 1 (1967), 443–491. MR 0223496, DOI 10.1016/0022-1236(67)90012-2
Hans Reiter, $L^{1}$-algebras and Segal algebras, Lecture Notes in Mathematics, Vol. 231, Springer-Verlag, Berlin-New York, 1971. MR 0440280, DOI 10.1007/BFb0060759
N. Th. Varopoulos, The embedding of $A(E)$ into $\overline A(E)$, Thin sets in harmonic analysis (Sem. Inst. Mittag-Leffler, Univ. Stockholm, Stockholm, 1969–1970) Lecture Notes in Pure and Appl. Math., Vol. 2, Dekker, New York, 1971, pp. 131–138, 177–180. MR 0440293