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Derivations and cohomologies of Lipschitz algebras

  • Kazuhiro KawamuraEmail author
Original Paper

Abstract

For a compact metric space (Md), \({\mathrm {Lip}}M\) denotes the Banach algebra of all complex-valued Lipschitz functions on (Md). 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 (Md) 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.

Keywords

Lipschitz algebra Hochschild cohomology De Leeuw map Tangent bundle Stone–Čech compactifications 

Mathematics Subject Classification

46H99 55N35 58A99 16W99 

Notes

Acknowledgements

The author is grateful to the referees for their comments which were helpful to improve the exposition of the paper. Kazuhiro Kawamura is supported by JSPS KAKENHI Grant number 17K05241.

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Copyright information

© Tusi Mathematical Research Group (TMRG) 2019

Authors and Affiliations

  1. 1.Institute of MathematicsUniversity of TsukubaIbarakiJapan

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