Algebras of Fourier transforms with closed restrictions

Abstract:

Let G denote a compact abelian group and let B denote a Banach subalgebra of A, the algebra of complex-valued functions on G whose Fourier series is absolutely convergent. If B contains the constant functions, separates the points of G, and if the restriction algebra, $B(E)$, is closed in $A(E)$ for every closed subset E of G, then $B = A$.