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

Document Type: Research Paper


1 Department of Mathematics, Payame Noor University, Tehran, 19359-3697, Iran

2 Department of Mathematics, Faculty of Science, Arak University, Arak, Iran


Let $(X,d)$ be a compact
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
$X$ for which
$$p_{(K,d^\alpha)}(f)=\sup\{\frac{|f(x)-f(y)|}{d^\alpha(x,y)} : x,y\in K , x\neq y\}<\infty$$
when it is equipped with the algebra norm $||f||_{{\rm Lip}(X, K, d^ {\alpha})}= ||f||_X+ p_{(K,d^{\alpha})}(f)$, where $||f||_X=\sup\{|f(x)|:~x\in X \}$.  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.