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Simplicial homology of strong semilattices of Banach algebras. (English) Zbl 1217.46031
Summary: Certain semigroups are known to admit a ‘strong semilattice decomposition’ into simpler pieces. We introduce a class of Banach algebras that generalise the \(\ell^{1}\)-convolution algebras of such semigroups, and obtain a disintegration theorem for their simplicial homology. Using this, we show that, for any Clifford semigroup \(S\) of amenable groups, \(\ell^{1}(S)\) is simplicially trivial: this generalises our previous results [the author, Glasg. Math. J. 48, No. 2, 231–245 (2006; Zbl 1112.46056)]. Some other applications are presented.

MSC:
46J05 General theory of commutative topological algebras
46M20 Methods of algebraic topology in functional analysis (cohomology, sheaf and bundle theory, etc.)
16E40 (Co)homology of rings and associative algebras (e.g., Hochschild, cyclic, dihedral, etc.)
43A20 \(L^1\)-algebras on groups, semigroups, etc.
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