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.