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.