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Transactions of the American Mathematical Society

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

 

 

Discontinuous translation invariant functionals


Author: Sadahiro Saeki
Journal: Trans. Amer. Math. Soc. 282 (1984), 403-414
MSC: Primary 43A15; Secondary 43A05
DOI: https://doi.org/10.1090/S0002-9947-1984-0728720-5
MathSciNet review: 728720
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Abstract: Let $ G$ be an infinite $ \sigma$-compact locally compact group. We shall study the existence of many discontinuous translation invariant linear functionals on a variety of translation invariant Fréchet spaces of Radon measures on $ G$. These spaces include the convolution measure algebra $ M(G)$, the Lebesgue spaces $ {L^p}(G)$, where $ 1 \leq p \leq \infty $, and certain combinations thereof. Among other things, it will be shown that $ C(G)$ has many discontinuous translation invariant functionals, provided that $ G$ is amenable. This solves a problem of G. H. Meisters.


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DOI: https://doi.org/10.1090/S0002-9947-1984-0728720-5
Keywords: Translation invariant Fréchet space, discontinuous invariant functional, Radon measure
Article copyright: © Copyright 1984 American Mathematical Society