Let A be a semisimple commutative regular tauberian Banach algebra with spectrum ${\Sigma }_A$. In this paper, we study the norm spectra of elements of $\overline{span}{\Sigma }_A$ and present some applications. In particular, we characterize the discreteness of ${\Sigma }_A$ in terms of norm spectra. The algebra A is said to have property (S) if, for all $\phi \in \overline{}{\Sigma }_{A}0$, φ has a nonempty norm spectrum. For a locally compact group G, let ${ℳ}_2^d\left(Ĝ\right)$ denote the C*-algebra generated by left translation operators on $L^2\left(G\right)$ and $G_d$ denote the discrete group G. We prove that the Fourier...