Closed convex invariant subsets of $L_{p}(G)$
HTML articles powered by AMS MathViewer
- by Anthony To Ming Lau PDF
- Trans. Amer. Math. Soc. 232 (1977), 131-142 Request permission
Abstract:
Let G be a locally compact group. We characterize in this paper closed convex subsets K of ${L_p}(G),1 \leqslant p < \infty$, that are invariant under all left or all right translations. We prove, among other things, that $K = \{ 0\}$ is the only nonempty compact (weakly compact) convex invariant subset of ${L_p}(G)\;({L_1}(G))$. We also characterize affine continuous mappings from ${P_1}(G)$ into a bounded closed invariant subset of ${L_p}(G)$ which commute with translations, where ${P_1}(G)$ denotes the set of nonnegative functions in ${L_1}(G)$ of norm one. Our results have a number of applications to multipliers from ${L_q}(G)$ into ${L_p}(G)$.References
- Charles A. Akemann, Some mapping properties of the group algebras of a compact group, Pacific J. Math. 22 (1967), 1–8. MR 212587, DOI 10.2140/pjm.1967.22.1
- B. Brainerd and R. E. Edwards, Linear operators which commute with translations. I. Representation theorems, J. Austral. Math. Soc. 6 (1966), 289–327. MR 0206725, DOI 10.1017/S1446788700004286
- K. de Leeuw and I. Glicksberg, Applications of almost periodic compactifications, Acta Math. 105 (1961), 63–97. MR 131784, DOI 10.1007/BF02559535 G.I. Gaudry, Quasimeasures and multiplier problems, Doctoral Dissertation, Australian National Univ., Canberra, Australia, 1966.
- Frederick P. Greenleaf, Invariant means on topological groups and their applications, Van Nostrand Mathematical Studies, No. 16, Van Nostrand Reinhold Co., New York-Toronto, Ont.-London, 1969. MR 0251549
- Edwin Hewitt and Kenneth A. Ross, Abstract harmonic analysis. Vol. I, 2nd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 115, Springer-Verlag, Berlin-New York, 1979. Structure of topological groups, integration theory, group representations. MR 551496, DOI 10.1007/978-1-4419-8638-2
- 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
- Lars Hörmander, Estimates for translation invariant operators in $L^{p}$ spaces, Acta Math. 104 (1960), 93–140. MR 121655, DOI 10.1007/BF02547187
- A. Hulanicki, Means and Følner condition on locally compact groups, Studia Math. 27 (1966), 87–104. MR 195982, DOI 10.4064/sm-27-2-87-104
- J. W. Kitchen Jr., The almost periodic measures on a compact abelian group, Monatsh. Math. 72 (1968), 217–219. MR 230049, DOI 10.1007/BF01362546
- Ronald Larsen, An introduction to the theory of multipliers, Die Grundlehren der mathematischen Wissenschaften, Band 175, Springer-Verlag, New York-Heidelberg, 1971. MR 0435738, DOI 10.1007/978-3-642-65030-7
- Lynn H. Loomis, An introduction to abstract harmonic analysis, D. Van Nostrand Co., Inc., Toronto-New York-London, 1953. MR 0054173
- Shôichirô Sakai, Weakly compact operators on operator algebras, Pacific J. Math. 14 (1964), 659–664. MR 163185, DOI 10.2140/pjm.1964.14.659
- J. G. Wendel, Left centralizers and isomorphisms of group algebras, Pacific J. Math. 2 (1952), 251–261. MR 49911, DOI 10.2140/pjm.1952.2.251 James C.S. Wong, Topological invariant means on locally compact groups, Doctoral Dissertation, Univ. of British Columbia, Vancouver, B.C., 1969.
- James C. S. Wong, Topologically stationary locally compact groups and amenability, Trans. Amer. Math. Soc. 144 (1969), 351–363. MR 249536, DOI 10.1090/S0002-9947-1969-0249536-4
Additional Information
- © Copyright 1977 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 232 (1977), 131-142
- MSC: Primary 43A15
- DOI: https://doi.org/10.1090/S0002-9947-1977-0477604-5
- MathSciNet review: 0477604