2-Local uniform isometries between complex Lipschitz algebras

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