%0 Journal Article %T Quasicompact and Riesz unital endomorphisms of real Lipschitz algebras of complex-valued functions %J Sahand Communications in Mathematical Analysis %I University of Maragheh %Z 2322-5807 %A Mayghani, Maliheh %A Alimohammadi, Davood %D 2018 %\ 01/01/2018 %V 09 %N 1 %P 1-14 %! Quasicompact and Riesz unital endomorphisms of real Lipschitz algebras of complex-valued functions %K Complexification %K Lipschitz algebra %K Lipschitz involution %K Quasicompact operator %K Riesz operator %K Unital endomorphism %R 10.22130/scma.2018.24240 %X We 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 quasicompact  (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. %U https://scma.maragheh.ac.ir/article_24240_91e55951d6b21d67e1abf159e8c6f90f.pdf