University of MaraghehSahand Communications in Mathematical Analysis2322-580709120180101Quasicompact and Riesz unital endomorphisms of real Lipschitz algebras of complex-valued functions1142424010.22130/scma.2018.24240ENMaliheh MayghaniDepartment of Mathematics, Payame Noor University, P. O. Box: 19359-3697, Tehran, Iran.Davood AlimohammadiDepartment of Mathematics, Faculty of Science, Arak University, Arak 38156-8-8349, Iran.Journal Article20161127We first show that a bounded linear operator $ T $ on a real Banach space $ E $ is quasicompact (Riesz, respectively) if and only if $T': E_{mathbb{C}}longrightarrow E_{mathbb{C}}$ is quasicompactÂ (Riesz, respectively), where the complex Banach space $E_{mathbb{C}}$ is a suitable complexification of $E$ and $T'$ is the complex linear operator on $E_{mathbb{C}}$ 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.http://scma.maragheh.ac.ir/article_24240_91e55951d6b21d67e1abf159e8c6f90f.pdf