Derivations and cohomologies of Lipschitz algebras

Abstract

For a compact metric space (M, d), \({\mathrm {Lip}}M\) denotes the Banach algebra of all complex-valued Lipschitz functions on (M, d). Motivated by a classical result of de Leeuw, we give a canonical construction of a compact Hausdorff space \({\hat{M}}\) and a continuous surjection \(\pi :{\hat{M}} \rightarrow M\) which may viewed as a metric analogue of the unit sphere bundle over a Riemannian manifold. It is shown that, for each \(n \ge 1\) the continuous Hochschild cohomology \({\mathrm {H}}^{n}({\mathrm {Lip}}M, C({\hat{M}}))\) has the infinite rank as a \({\mathrm {Lip}}M\)-module, if the metric space (M, d) admits a local geodesic structure, for example, if M is a compact Riemannian manifold or a non-positively curved metric space. Here \(C({\hat{M}})\) denotes the algebra of all complex-valued continuous functions on \({\hat{M}}\). On the other hand, if the coefficient \(C({\hat{M}})\) is replaced with C(M), then it is shown that \({\mathrm {H}}^{1}({\mathrm {Lip}}M,C(M)) = 0\) for each compact Lipschitz manifold M.