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Point derivations and cohomologies of Lipschitz algebras

Part of: Topological algebras, normed rings and algebras, Banach algebras General theory of differentiable manifolds Homology and cohomology theories Rings and algebras with additional structure

Published online by Cambridge University Press:  18 July 2019

Kazuhiro Kawamura
Kazuhiro Kawamura*
Affiliation:
Institute of Mathematics, University of Tsukuba, Tsukuba, Ibaraki 305-8571, Japan ([email protected])
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Abstract

For a compact metric space (K, d), LipK denotes the Banach algebra of all complex-valued Lipschitz functions on (K, d). We show that the continuous Hochschild cohomology Hn(LipK, (LipK)*) and Hn(LipK, ℂe) are both infinite-dimensional vector spaces for each n ≥ 1 if the space K contains a certain infinite sequence which converges to a point e ∈ K. Here (LipK)* is the dual module of LipK and ℂe denotes the complex numbers with a LipK-bimodule structure defined by evaluations of LipK-functions at e. Examples of such metric spaces include all compact Riemannian manifolds, compact geodesic metric spaces and infinite compact subsets of ℝ. In particular, the (small) global homological dimension of LipK is infinite for every such space. Our proof uses the description of point derivations by Sherbert [‘The structure of ideals and point derivations in Banach algebras of Lipschitz functions’, Trans. Amer. Math. Soc.111 (1964), 240–272] and directly constructs non-trivial cocycles with the help of alternating cocycles of Johnson [‘Higher-dimensional weak amenability’, Studia Math.123 (1997), 117–134]. An alternating construction of cocycles on the basis of the idea of Kleshchev [‘Homological dimension of Banach algebras of smooth functions is equal to infinity’, Vest. Math. Mosk. Univ. Ser. 1. Mat. Mech.6 (1988), 57–60] is also discussed.

Keywords

Lipschitz algebraHochschild cohomologyglobal homological dimensionde Leeuw mapBanach limit

MSC classification

Primary: 46H99: None of the above, but in this section
Secondary: 55N35: Other homology theories 58A99: None of the above, but in this section 16W99: None of the above, but in this section
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 62 , Issue 4 , November 2019 , pp. 1173 - 1187
Copyright
Copyright © Edinburgh Mathematical Society 2019 

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