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

Lau, Anthony To Ming

doi:10.1090/S0002-9947-1977-0477604-5

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

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