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 alphain(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,yin k, xneq 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)| | xin x}. we denote by lip(x,k,d^alpha) the closed subalgebra of lip(x,k,d^alpha) consisting of all fin lip(x,k,d^alpha) for which frac{f(x)f(y)}{d^alpha(x,y)}longrightarrow 0 as d(x,y)longrightarrow x,yin 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 $alphain(0,1/2). we also prove that lip(mathbb{t},k,d^{1/2}) is weakly amenable, where mathbb{t}={xinmathbb{c}| |z|=1}, d is the euclidean metric on t and k is a nonempty compact set in (mathbb{t},d).