Homogeneous Banach spaces on the unit circle

We prove that a homogeneous Banach space ${\mathcal B}$ on the unit circle ${\mathbb T}$ can be embedded as a closed subspace of a dual space $\Xi_{{\mathcal B}}^{\ast}$ contained in the space of bounded Borel measures on ${\mathbb T}$ in such a way that the map ${\mathcal B}\mapsto\Xi_{{\mathcal B}}^{\ast}$ defines a bijective correspondence between the class of homogeneous Banach spaces on ${\mathbb T}$ and the class of prehomogeneous Banach spaces on ${\mathbb T}$.

We apply our results to show that the algebra of all continuous functions on ${\mathbb T}$ is the only homogeneous Banach algebra on ${\mathbb T}$ 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 $\Xi_{{\mathcal B}}^{\ast}$ for some examples of homogeneous Banach spaces ${\mathcal B}$ on ${\mathbb T}$.