The F. and M. Riesz theorem and some function spaces (Q1088107)

Let G be a LCA group with dual \(\hat G.\) Let M(G) and \(L^ 1(G)\) be the usual spaces. For a subset E of \(\hat G\), \(M_ E(G)\) denotes the space of all measures in M(G) whose Fourier-Stieltjes transforms vanish off E. Let \(C_ u(G)\) be the space of bounded uniformly continuous functions on G. Let P be a semigroup in \(\hat G\) such that \(P\cup (-P)=\hat G\). Suppose P is not dense in \(\hat G.\) Let \(H^ 1_ P(G)=M_ P(G)\cap L^ 1(G)\), and let \(H_ P^{\infty}(G)=\{g\in L(G):\int_{G}f(x) g(x) dm_ G(x)=0\}\), where \(m_ G\) denotes the Haar measure of G. Then \(H_ P^{\infty}(G)\) is a weak*-closed translation invariant subspace of \(L^{\infty}(G)\). Hence \(H_ P^{\infty}(G)+C_ u(G)\) is a closed subspace of \(L^{\infty}(G)\). On the other hand, Sarason showed that \(H^{\infty}({\mathbb{T}})+C({\mathbb{T}})\) and \(H^{\infty}({\mathbb{R}})+C_ u({\mathbb{R}})\) are closed subalgebras of \(L^{\infty}({\mathbb{T}})\) and \(L^{\infty}({\mathbb{R}})\), respectively. The author obtaines the following theorem. Theorem. The following statements are equivalent: (i) \(H_ P^{\infty}(G)+C_ u(G)\) is an algebra; (ii) \(\mu^*(\gamma \nu)\in L^ 1(G)\) for \(\mu \in M_{P^ c}(G)\), \(\nu \in M_{(-P)^ c}(G)\) and \(\gamma\in \hat G;\) (iii) \(\mu^*(\gamma \nu)\in L^ 1(G)\) for \(\mu,\nu \in M_{P^ c}(G)\) and \(\gamma\in \hat G;\) (iv) \(M_{P^ c}(G)\subset L^ 1(G);\) (v) \(M_ P(G)\subset L^ 1(G);\) (vi) G is isomorphic to \({\mathbb{R}}\times \Delta\) or \({\mathbb{T}}\times \Delta\), where \(\Delta\) is discrete.