Compact and quasicompact homomorphisms between differentiable Lipschitz algebras

Abstract

In this note we consider homomorphisms between differentiable Lipschitz algebras $Lip^n(X,\alpha)$ ($0<\alpha \leq 1$) and $lip^n(X,\alpha)$ ($0<\alpha <1$), where $X$ is a perfect compact plane set. We give sufficient conditions implying the compactness and power compactness of these homomorphisms. Moreover, we investigate under what conditions a quasicompact homomorphism between these algebras is power compact. We also give a necessary condition for a homomorphism between these algebras to be quasicompact and in certain cases to be power compact. Finally, using these results, by giving an example we show that there exists a quasicompact homomorphism between these algebras which is not power compact.