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 $C_{\mathrm{B}\mathrm{S}\mathrm{E}}(\mathrm{\Delta }(A))$ consisting of all BSE-functions on $\mathrm{\Delta }(A)$ where $\mathrm{\Delta }(A)$ denotes the character space of $A$. Indeed, in the case that $A=C_0(X)$ where $X$ is a non-empty locally compact Hausdroff space, we give a complete characterization of $\mathrm{\Delta }(C_{\mathrm{B}\mathrm{S}\mathrm{E}}(\mathrm{\Delta }(A)))$ and in the general case we give a partial answer. Also, using the Fourier algebra, we show that $C_{\mathrm{B}\mathrm{S}\mathrm{E}}(\mathrm{\Delta }(A))$ is not a $C^{\ast }$-algebra in general. Finally for some subsets $E$ of $A^{\ast }$, we define the subspace of BSE-like functions on $\mathrm{\Delta }(A)\cup E$ and give a nice application of this space related to Goldstine’s theorem.