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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.

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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