Point derivations and cohomologies of Lipschitz algebras
Point derivations and cohomologies of Lipschitz algebras
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Lipschitz 代数的点导数和上同调
DOI：
10.1017/s0013091519000142
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发表时间：
2019
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影响因子：
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通讯作者：
Kazuhiro Kawamura
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文献类型：
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作者：
Kazuhiro Kawamura
对于紧致度量空间（K，d），LipK表示（K，d）上所有复值Lipschitz函数的Banach代数。我们证明，如果空间K包含某个收敛于点e的无限序列，则对于每个n ≥ 1，连续Hochschild上同调Hn（LipK，（LipK）*）和Hn（LipK，e）都是无限维向量空间。这里，（LipK）* 是LipK的对偶模，而ε e表示具有LipK-双模结构的复数，该结构由LipK-函数在e处的求值定义。这样的度量空间的例子包括所有紧黎曼流形，紧测地线度量空间和无穷紧子集。特别地，对于每个这样的空间，LipK的（小）全局同调维数是无限的。我们的证明使用了Sherbert的“The structure of ideals and point derivations in Banach algebras of Lipschitz functions”，Trans.Amer.Math.Soc.111（1964），240-272]中对点导子的描述，并借助约翰逊的“Higher-dimensional weak amenability”，Studia Math.123（1997）中的交替上循环直接构造了非平凡上循环，117-134]。在Kleshchev的思想基础上交替构造上循环：“光滑函数的Banach代数的同调维数等于无穷大”，Vest。数学。莫斯克。大学系列1. Mat. Mech.6（1988），57-60]中进行了讨论。
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.