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

Granirer, Edmond E.

doi:10.1090/S0002-9947-1974-0336241-0

Weakly almost periodic and uniformly continuous functionals on the Fourier algebra of any locally compact group
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Trans. Amer. Math. Soc. 189 (1974), 371-382 Request permission

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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