Semnan UniversityInternational Journal of Nonlinear Analysis and Applications2008-68228120170401The structure of ideals, point derivations, amenability and weak amenability of extended Lipschitz algebras38940449310.22075/ijnaa.2016.493ENMalihehMayghaniDepartment of Mathematics, Payame Noor University, Tehran, 19359-3697, IranDavoodAlimohammadiDepartment of Mathematics, Faculty of Science, Arak University,
Arak, IranJournal Article20160317Let $(X,d)$ be a compact<br />metric space and let $K$ be a nonempty compact subset of $X$. Let $alpha in (0, 1]$ and let ${rm Lip}(X,K,d^ alpha)$ denote the Banach algebra of all continuous complex-valued functions $f$ on<br />$X$ for which<br />$$p_{(K,d^alpha)}(f)=sup{frac{|f(x)-f(y)|}{d^alpha(x,y)} : x,yin K , xneq y}<infty$$<br />when it is equipped with the algebra norm $||f||_{{rm Lip}(X, K, d^ {alpha})}= ||f||_X+ p_{(K,d^{alpha})}(f)$, where $||f||_X=sup{|f(x)|:~xin X }$. In this paper we first study the structure of certain ideals of ${rm Lip}(X,K,d^alpha)$. Next we show that if $K$ is infinite and ${rm int}(K)$ contains a limit point of $K$ then ${rm Lip}(X,K,d^alpha)$ has at least a nonzero continuous point derivation and applying this fact we prove that ${rm Lip}(X,K,d^alpha)$ is not weakly amenable and amenable.https://ijnaa.semnan.ac.ir/article_493_c33ba9a36bbd03e49c6d1e1671e3f46e.pdf