Homogenous Banach Spaces on the Unit Circle

We prove that a homogeneous Banach space B on the unit circle T can be embedded as a closed subspace of a dual space ΞB contained in the space of bounded Borel measures on T in such a way that the map B → ΞB defines a bijective correspondence between the class of homogeneous Banach spaces on T and the class of prehomogeneous Banach spaces on T. We apply our results to show that the algebra of all continuous functions on T is the only homogeneous Banach algebra on T in which every closed ideal has a bounded approximate identity with a common bound, and that the space of multipliers between two homogeneous Banach spaces is a dual space. Finally, we describe the space ΞB for some examples of homogeneous Banach spaces B on T.