Spectrum of commutative Banach algebras and isomorphism of C*-algebras related to locally compact groups

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 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 is an element of span Sigma(A) \ {0}, cp has a nonempty norm spectrum. For a locally compact group G, let M-2(d)((G) over cap) denote the C*-algebra generated by left translation operators on L-2(G) and Gd denote the discrete group G. We prove that the Fourier algebra A(G) has property (S) iff the canonical trace on M-2(d)((G) over cap) is faithful iff M-2(d)((G) over cap) congruent to M-2(d)((G(d)) over cap) This provides san answer to the isomorphism problem of the two C*-algebras and generalizes the so-called "uniqueness theorem" on the group algebra L-1(G) of a locally compact abelian group G. We also prove that Gd is amenable iff G is amenable and the Figa-Talamanca-Herz algebra A(p)(G) has property (S) for all p.