The Banach-Stone Theorem for Banach algebras

Abstract: 

Thomas Tonev The University of Montana, Missoula, USA The classical Banach-Stone Theorem, which can be found in any standard text on functional analysis, tells us that all linear isometries between the spaces C(X) and C(Y ) are weighted composition operators. There were many attempts in the past to extend and generalize this theorem. We prove the Banach-Stone Theorem for Banach algebras. Let X, Y be locally compact Hausdorff spaces and A, B be algebras of bounded continuous functions on X, Y such that their uniform closures A, B are function algebras, not necessarily with units. If T : A → B is a linear surjection that preserves the sup-norms (i.e. kT fk = kfk for all f ∈ A), then there is a homeomorphism ψ: δB → δA between the corresponding Choquet boundaries and a continuous function α: δB → C with |α| = 1 such that (T f)(y) = α(y) f ψ(y)
for all f, g ∈ A and y ∈ δB, i.e. T is an α-weighted ψ-composition operator. If A (but not necessarily B) has a unit, then the same conclusion (with α = T1 ) holds for any sup-norm-linear operator T (i.e. such that λ (T f) + µ (T g) = kλf + µgk, f, g ∈ A, λ, µ ∈ C) with the property T i = i(T1 )