The Gelfand transform, positive linear functionals, and positive-definite functions

In this note, unless we say otherwise every vector space or algebra we speak about is over C. If A is a Banach algebra and e ∈ A satisfies xe = x and ex = x for all x ∈ A, and also ‖e‖ = 1, we say that e is unity and that A is unital. If A is a unital Banach algebra and x ∈ A, the spectrum of x is the set σ(x) of those λ ∈ C for which λe−x is not invertible. It is a fact that if A is a unital Banach algebra and x ∈ A, then σ(x) 6= ∅. If A and B are Banach algebras and T : A → B is a map, we say that T is an isomorphism of Banach algebras if T is an algebra isomorphism and an isometry.