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

We first show that every unital endomorphism of real Lipschitz algebras of complex-valued functions on compact metric spaces with Lipschitz involutions is a composition operator. Next, we establish a formula for essential spectral radius of a unital endomorphism T of these algebras under a condition which is equivalent to the quasicompactness of the endomorphism T. We also conclude a necessary and sufficient condition for a unital endomorphism of these algebras to be Riesz.