On Character Space of the Algebra of BSE-functions

Suppose that A is a semi-simple and commutative Banach algebra. In this paper we try to characterize the character space of the Banach algebra CBSE(Δ(A)) consisting of all BSE-functions on Δ(A) where Δ(A) denotes the character space of A. Indeed, in the case that A=C0(X) where X is a non-empty locally compact Hausdroff space, we give a complete characterization of Δ(CBSE(Δ(A))) and in the general case we give a partial answer. Also, using the Fourier algebra, we show that CBSE(Δ(A)) is not a C∗-algebra in general. Finally for some subsets E of A∗, we define the subspace of BSE-like functions on Δ(A)∪E and give a nice application of this space related to Goldstine's theorem.