Simplicial homology of strong semilattices of Banach algebras

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, $\lp{1}(S)$ is simplicially trivial: this generalises results in \cite{YC_GMJ}. Some other applications are presented.