Weighted group algebra as an ideal in its second dual space
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- by F. Ghahramani PDF
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Abstract:
For a locally compact group $G$ let ${L^1}(G,\omega \lambda )$ be a weighted group algebra. We characterize compact and weakly compact multipliers on ${L^1}(G,\omega \lambda )$. This characterization is employed to find a necessary and sufficient condition for ${L^1}(G,\omega \lambda )$ to be an ideal in its second dual space, where the second dual is equipped with an Arens product. In the special case where $\omega (t) = 1(t \in G)$, we deduce a result due to K. P. Wong that if $G$ is a compact group, then ${L^1}(G,\lambda )$ is an ideal in its second dual space and its converse due to S. Watanabe.References
- Charles A. Akemann, Some mapping properties of the group algebras of a compact group, Pacific J. Math. 22 (1967), 1–8. MR 212587
- Richard Arens, Operations induced in function classes, Monatsh. Math. 55 (1951), 1–19. MR 44109, DOI 10.1007/BF01300644
- Richard Arens, The adjoint of a bilinear operation, Proc. Amer. Math. Soc. 2 (1951), 839–848. MR 45941, DOI 10.1090/S0002-9939-1951-0045941-1
- W. G. Bade and H. G. Dales, Norms and ideals in radical convolution algebras, J. Functional Analysis 41 (1981), no. 1, 77–109. MR 614227, DOI 10.1016/0022-1236(81)90062-8
- Sterling K. Berberian, Lectures in functional analysis and operator theory, Graduate Texts in Mathematics, No. 15, Springer-Verlag, New York-Heidelberg, 1974. MR 0417727
- J. Duncan and S. A. R. Hosseiniun, The second dual of a Banach algebra, Proc. Roy. Soc. Edinburgh Sect. A 84 (1979), no. 3-4, 309–325. MR 559675, DOI 10.1017/S0308210500017170 N. Dunford and J. Schwartz, Linear operators, Part 1, Interscience, New York, 1958.
- F. Ghahramani, Homomorphisms and derivations on weighted convolution algebras, J. London Math. Soc. (2) 21 (1980), no. 1, 149–161. MR 576191, DOI 10.1112/jlms/s2-21.1.149
- F. Ghahramani, Compact elements of weighted group algebras, Pacific J. Math. 113 (1984), no. 1, 77–84. MR 745596
- Frederick P. Greenleaf, Norm decreasing homomorphisms of group algebras, Pacific J. Math. 15 (1965), 1187–1219. MR 194911
- Michael Grosser, $L^{1}(G)$ as an ideal in its second dual space, Proc. Amer. Math. Soc. 73 (1979), no. 3, 363–364. MR 518521, DOI 10.1090/S0002-9939-1979-0518521-9
- David L. Johnson, A characterization of compact groups, Proc. Amer. Math. Soc. 74 (1979), no. 2, 381–382. MR 524322, DOI 10.1090/S0002-9939-1979-0524322-8
- Hans Reiter, Classical harmonic analysis and locally compact groups, Clarendon Press, Oxford, 1968. MR 0306811
- Shôichirô Sakai, Weakly compact operators on operator algebras, Pacific J. Math. 14 (1964), 659–664. MR 163185
- Seiji Watanabe, A Banach algebra which is an ideal in the second dual space, Sci. Rep. Niigata Univ. Ser. A 11 (1974), 95–101. MR 383079 —, A Banach algebra which is an ideal in the second dual space II, Sci. Rep. Niigata Univ. Ser. A 13 (1976), 43-48.
- Pak-ken Wong, On the Arens product and annihilator algebras, Proc. Amer. Math. Soc. 30 (1971), 79–83. MR 281005, DOI 10.1090/S0002-9939-1971-0281005-2
Additional Information
- © Copyright 1984 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 90 (1984), 71-76
- MSC: Primary 43A22; Secondary 43A15, 46J99, 47B99
- DOI: https://doi.org/10.1090/S0002-9939-1984-0722417-9
- MathSciNet review: 722417