uniform norm

Let (X‚ d) and (Y‚ ρ) be compact metric spaces‚ τ be a Lipschitz involution on (X‚ d) and η be Lipschitz involution on (Y‚ ρ). Suppose that  for all x∈X‚ for all y∈Y‚ , A is a real subalgebra of C(X‚ τ) which contains Lip(X‚ d‚ τ) and B is a real subalgebra of C(Y‚ η) which contains Lip(Y‚ ρ‚ η). We prove that if T:A→B is a surjetive -homogenous norm-additive in modulus map then there exists a unique bijection  such that  for all f∈A ‚ y∈Y and . Applying this fact‚ we show that if  (X‚d) and (Y‚ρ) are compact metric spaces and  is a surjective -homogenous norm-additive in modulus map then there exists a Lipschitz homeomorphism  from  to such that  for all  and y∈Ỵ