On the Algebras VN(H) and VN(H)(*) of an Ultraspherical Hypergroup H

Abstract

Let H be an ultraspherical hypergroup and let A(H) be the Fourier algebra associated with H. In this paper, we study the dual and the double dual of A(H). We prove among other things that the subspace of all uniformly continuous functionals on A(H) forms a C∗ -algebra. We also prove that the double dual A(H)∗∗ is neither commutative nor semisimple with respect to the Arens product, unless the underlying hypergroup H is finite. Finally, we study the unit elements of A(H)∗∗.