Sufficient Conditions for Density in 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, dα ) denote the Banach algebra of all continuous complex-valued functions f on X for which pα,K(f) = sup{ |f(x)−f(y)| dα(x,y) : x, y ∈ K, x 6= y} < ∞ when equipped the algebra norm ||f||Lip(X,K,dα) = ||f||X + pα,K(f), where ||f||X = sup{|f(x)| : x ∈ X}. We denote by lip(X, K, dα ) the closed subalgebra of Lip(X, K, dα ) consisting of all f ∈ Lip(X, K, dα ) for which |f(x)−f(y)| dα(x,y) → 0 as d(x, y) → 0 with x, y ∈ K. In this paper we obtain a sufficient condition for density of a linear subspace or a subalgebra of Lip(X, K, dα ) in (Lip(X, K, dα ), || · ||Lip(X,K,dα)) (lip(X, K, dα ) in (lip(X, K, dα ), || · ||Lip(X,K,dα)), respectively). In particular, we show that the Lipschitz algebra Lip(X, dα ) is dense in (Lip(X, K, dα ), k · kLip(X,K,dα)) for α ∈ (0, 1] and Lip(X, d) and the little Lipschitz algebra lip(X, dα ) are dense in (lip(X, K, dα ), k · kLip(X,K,dα)) for α ∈ (0, 1).