SPECTRUM OF COMMUTATIVE BANACH-ALGEBRAS AND ISOMORPHISM OF C-ASTERISK-ALGEBRAS RELATED TO LOCALLY COMPACT-GROUPS

Let A be a semisimple commutative regular tauberian Banach algebra wit h spectrum Sigma(A). In this paper, we study the norm spectra of eleme nts of span Sigma(A) and present some applications. In particular, we characterize the discreteness of Sigma(A) in terms of norm spectra. Th e 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 locall y compact group G, let M-2(d)((G) over cap) denote the C-algebra gene rated by left translation operators on L-2(G) and Gd denote the discre te group G. We prove that the Fourier algebra A(G) has property (S) if f 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 an swer 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 if f G is amenable and the Figa-Talamanca-Herz algebra A(p)(G) has proper ty (S) for all p.