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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Closed convex invariant subsets of $L_{p}(G)$

Author: Anthony To Ming Lau
Journal: Trans. Amer. Math. Soc. 232 (1977), 131-142
MSC: Primary 43A15
MathSciNet review: 0477604
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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 ${L_p}(G)$.

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Keywords: Locally compact group, left translations, right translations, convex sets, multipliers, operators commuting with translations, <IMG WIDTH="29" HEIGHT="38" ALIGN="MIDDLE" BORDER="0" SRC="images/img2.gif" ALT="${L_p}$">-spaces, measure algebra, convolution, multipliers, affine mappings
Article copyright: © Copyright 1977 American Mathematical Society