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.