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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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