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Simplicial homology and Hochschild cohomology of Banach semilattice algebras. (English) Zbl 1112.46056
Let $$S$$ be a semilattice, i.e., $$S$$ is a commutative semigroup with the property that every element $$s\in S$$ satisfies $$s^2=s$$. Let $$\ell^1(S)$$ be the corresponding Banach convolution algebra. H. G. Dales and J. Duncan [in: Proceedings of the 13th international conference on Banach algebras (1998; Zbl 0938.46046)] showed that the Hochschild cohomology groups $$H^1(\ell^1(S),M)$$ and $$H^2(\ell^1(S),M)$$ are trivial for all symmetric $$\ell^1(S)$$-bimodules $$M$$. This result has been extended to the third cohomology groups by F. Gourdeau, A. Pourabbas and M. C. White [Can. Math. Bull. 50, No. 1, 56–70 (2007; Zbl 1136.43001)]. In the paper under review, this result is now extended to all higher cohomology groups.

##### MSC:
 46M20 Methods of algebraic topology in functional analysis (cohomology, sheaf and bundle theory, etc.) 46J40 Structure and classification of commutative topological algebras 43A20 $$L^1$$-algebras on groups, semigroups, etc.
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