  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
$C_{\rm{BSE}}(\Delta(A))$ consisting of all BSE-functions on $\Delta(A)$ where
$\Delta(A)$ 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 $\Delta(C_{\rm{BSE}}(\Delta(A)))$ and in the
general case we give a partial answer. Also, using the Fourier algebra, we show
that $C_{\rm{BSE}}(\Delta(A))$ is not a $C^*$-algebra in general. Finally for
some subsets $E$ of $A^*$, we define the subspace of BSE-like functions on
$\Delta(A)\cup E$ and give a nice application of this space related to
Goldstine's theorem.
