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 algebra of all $f \in C(X)$ for which $f|_{K} \in {\rm Lip(K, d^{\alpha})}$. In this paper we first study the structure of certain ideals of the algebra ${\rm Lip}(X,K,d^\alpha)$. 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.
