Closed Ideals, Point Derivations and Weak Amenability of Extended Little Lipschitz Algebras

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 Lip(X,K,d^\alpha) denote the Banach algebra of all continuous complexvalued functions f on for which p_{(K,d^\alpha)(f)=\sup\{\frac{f(x)f(y)}{d^\alpha(x,y)}:\ x,y\in K,\ x\neq y\} when equipped with the algebra norm \|f\|_{Lip(X,K,d^\alpha}=\|f\|_X+p_{(K,d^\alpha)(f), where \|f\|_X=\sup\{|f(x)|\ |\ x\in X\}. We denote by lip(X,K,d^\alpha) the closed subalgebra of Lip(X,K,d^\alpha) consisting of all f\in Lip(X,K,d^\alpha) for which \frac{f(x)f(y)}{d^\alpha(x,y)}\longrightarrow 0 as d(x,y)\longrightarrow x,y\in K. In this paper we show that every proper closed ideal of (lip(X,K,d^\alpha),\|.\|_{Lip(X,K,d^\alpha)})is the intersection of all maximal ideals containing it. We also prove that every continuous point derivation of lip(X,K,d^\alpha) is zero. Next we show that lip(X,K,d^\alpha) is weakly amenable if $\alpha\in(0,1/2). We also prove that Lip(\mathbb{T},K,d^{1/2}) is weakly amenable, where \mathbb{T}=\{x\in\mathbb{C}|\ |z|=1\}, d is the Euclidean metric on T and K is a nonempty compact set in (\mathbb{T},d).