Quasicompact and Riesz unital endomorphisms of real Lipschitz algebras of complex-valued functions

We first show that a bounded linear operator T on a real Banach space E is quasicompact (Riesz, respectively) if and only if T′: EC → EC is quasicompact (Riesz, respectively), where the complex Banach space EC is a suitable complexification of E and T′ is the complex linear operator on EC associated with T. Next, we prove that every unital endomorphism of real Lipschitz algebras of complex-valued functions on compact metric spaces with Lipschitz involutions is a composition operator. Finally, we study some properties of quasicompact and Riesz unital endomorphisms of these algebras.