Spectrum of commutative Banach algebras and isomorphism of C*-algebras related to locally compact groups
Volume 129 / 1998
Abstract
Let A be a semisimple commutative regular tauberian Banach algebra with spectrum $Σ_A$. In this paper, we study the norm spectra of elements of $\overline{span} Σ_A$ and present some applications. In particular, we characterize the discreteness of $Σ_A$ in terms of norm spectra. The algebra A is said to have property (S) if, for all $φ ∈ \overline{\span} Σ_A \ {0}$, φ has a nonempty norm spectrum. For a locally compact group G, let $ℳ_2^{d}(Ĝ)$ denote the C*-algebra generated by left translation operators on $L^2(G)$ and $G_{d}$ denote the discrete group G. We prove that the Fourier algebra $A(G)$ has property (S) iff the canonical trace on $ℳ_2^{d}(Ĝ)$ is faithful iff $ℳ_2^{d} (Ĝ)≅ ℳ_2^{d} (Ĝ_{d})$. This provides an 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 $G_{d}$ is amenable iff G is amenable and the Figà-Talamanca-Herz algebra $A_p(G)$ has property (S) for all p.
