On character space of the algebra of BSE-functions

TL;DR

This paper characterizes the character space of BSE-functions on the spectrum of a semi-simple commutative Banach algebra, providing complete results for certain cases and partial insights generally, with applications to Goldstine's theorem.

Contribution

It offers a complete characterization of the character space for BSE-functions on C_0(X) and partial results in the general case, also showing that this algebra is not a C*-algebra.

Findings

Complete characterization for C_0(X) case

Partial characterization in the general case

C_BSE(Δ(A)) is not a C*-algebra in general

Abstract

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.