The structure of ideals, point derivations, amenability and weak amenability of extended Lipschitz algebras

Let (X,d) be a compactmetric space and let K be a nonempty compact subset of X. Let \alpha \in (۰, ۱] and let {\rm Lip}(X,K,d^ \alpha) denote the Banach algebra of all  continuous complex-valued functions f onX for whichp_{(K,d^\alpha)}(f)=\sup\{\frac{|f(x)-f(y)|}{d^\alpha(x,y)} : x,y\in K , x\neq y\}<\inftywhen 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)|:~x\in 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.

Keywords:

Amenability , Banach function algebra , extended Lipschitz algebra , point derivation , weak amenability