Translation-invariant linear forms on 𝐿_{𝑝}(𝐺)

Rosenblatt, Joseph

doi:10.1090/S0002-9939-1985-0784168-5

Translation-invariant linear forms on $L_ p(G)$
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by Joseph Rosenblatt

Proc. Amer. Math. Soc. 94 (1985), 226-228

DOI: https://doi.org/10.1090/S0002-9939-1985-0784168-5

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Abstract:

Let $G$ be a compact group such that the identity representation of $G$ is not contained in the regular representation on $L_2^0(G,{\lambda _G})$ of $G$ with the discrete topology. Then any left translation invariant linear form on ${L_p}(G),1 < p < \infty$, is continuous and must be a constant times the Haar integral. This shows that many classical matrix groups $G$ admit only continuous left translation invariant linear forms on ${L_p}(G),1 < p < \infty$.

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Ergodic group actions