2-Local uniform isometries between complex Lipschitz algebras

Abstract

Let (X, d) be a metric space and let $$\mathrm{Lip}(X,d) $$ denote the complex algebra of all complex-valued bounded functions f on X for which f is a Lipschitz function on $$\mathrm{(X,d)}$$. In this paper we give a complete description of all 2-local real and complex uniform isometries between $$\mathrm{Lip}(X,d) $$ and $$\mathrm{Lip(Y},\rho \mathrm{)}$$, where (X, d) and $$(Y,\rho )$$ are compact metric spaces. In particular, we show that every 2-local real (complex, respectively) uniform isometry from $$\mathrm{Lip}(X,d) $$ to $$\mathrm{Lip(Y,}\rho \mathrm{)}$$ is a surjective real (complex, respectively) linear uniform isometry.