Bilinear isometries on subspaces of continuous functions

Abstract

Let A and B be strongly separating linear subspaces of C0(X) and C0(Y ), respectively, and assume that ∂A ̸= ∅ (∂A stands for the set of generalized peak points for A) and ∂B ≠ ∅. Let T : A×B −→ C0(Z) be a bilinear isometry. Then there exist a nonempty subset Z0 of Z, a surjective continuous mapping h : Z0 −→ ∂A × ∂B and a norm-one continuous function a : Z0 −→ K such that T(f,g)(z) = a(z)f(πx(h(z))g(πy(h(z)) for all z ∈ Z0 and every pair (f, g) ∈ A × B. These results can be applied, for example, to non-unital function algebras.