Some Dense Subspaces and Subalgebras 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, dα) denote the Banach algebra of all continuous complex-valued functions f on X for which pα,K (f ) = sup{ |f (x)−f (y)| : x, y ∈ K, x ̸= 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, d ) the closed subalgebra of Lip(X, K, dα) consisting of all f ∈ Lip(X, K, dα) for which |f (x)−f (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), ∥ · ∥Lip(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), ∥ · ∥Lip(X,K,dα )) for α ∈ (0, 1