2-Local isometries between Banach algebras of continuous functions with involution

Abstract

We give a description of 2-local real isometries between $$C(X,\tau )$$ and $$C(Y,\eta )$$ where X and Y are compact Hausdorff spaces, X is also first countable and $$\tau $$ and $$\eta $$ are topological involutions on X and Y, respectively. In particular, we show that every 2-local real isometry T from $$C(X,\tau )$$ to $$ C(Y,\eta )$$ is a surjective real linear isometry whenever X is also separable.