Some algebraic and homological properties of Lipschitz algebras and their second duals

Abstract

Let (X, d) be a metric space and α > 0. We study homological properties and different types of amenability of Lipschitz algebras LipαX and their second duals. Precisely, we first provide some basic properties of Lipschitz algebras, which are important for metric geometry to know how metric properties are reflected in simple properties of Lipschitz functions. Then we show that all of these properties are equivalent to either uniform discreteness or finiteness of X. Finally, some results concerning the character space and Arens regularity of Lipschitz algebras are provided.