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

Let (x, d) be a compact metric space and let k be a nonempty compact subset of x. let α ∈ (0, 1] and let lip(x, k, d^α) denote the banach algebra of all continuous complex-valued functions f on x p(k,d^α)(f) = sup{frac |f(x) − f(y)|{dα(x, y)} : x, y ∈ k, x ≠y} < ∞.when it is equipped with the algebra norm ||f||lip(x,k,d^α) = ||f||x + p(k,d^α)(f), where ||f||x =sup{|f(x)| : x ∈ x}. in this paper we first study the structure of certain ideals of lip(x, k, d^α ).next we show that if k is infinite and int(k) contains a limit point of k then lip(x, k, d^α ) has at least a nonzero continuous point derivation and applying this fact we prove that lip(x, k, d^α ) is not weakly amenable and amenable