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

Journal: Sahand Communications in Mathematical Analysis (Vol.9, No. 1)

Publication Date:

Authors : ; ;

Page : 1-14

Keywords : Complexification; Lipschitz algebra; Lipschitz involution; Quasicompact operator; Riesz operator; Unital endomorphism;

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Abstract

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.

Last modified: 2018-02-03 20:15:59