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Weakly almost periodic and uniformly continuous functionals on the Fourier algebra of any locally compact group


Author: Edmond E. Granirer
Journal: Trans. Amer. Math. Soc. 189 (1974), 371-382
MSC: Primary 43A60
DOI: https://doi.org/10.1090/S0002-9947-1974-0336241-0
MathSciNet review: 0336241
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Abstract: We define for any locally compact group G, the space of bounded uniformly continuous functionals on Ĝ, $ UCB(\hat G)$, in the context of P. Eymard [Bull. Soc. Math. France 92 (1964), 181-236. MR 37 #4208] (for notations see next section). For $ u \in A(G)$ let $ {u^ \bot } = \{ \phi \in VN(G);\phi [A(G)u] = 0\} $.

Theorem. If for some norm separable subspace $ X \subset VN(G)$ and some positive definite $ 0 \ne u \in A(G),UCB(\hat G) \subset $ norm closure $ [W(\hat G) + X + {u^ \bot }]$ then G is discrete. If G is discrete then $ UCB(\hat G) \subset AP(\hat G) \subset W(\hat G)$.


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DOI: https://doi.org/10.1090/S0002-9947-1974-0336241-0
Article copyright: © Copyright 1974 American Mathematical Society

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