Simplicial cohomology of band semigroup algebras

We establish the simplicial triviality of the convolution algebra $\ell^1(S)$, where $S$ is a band semigroup. This generalizes some results of Choi (Glasgow Math. J. 48 (2006), 231–245; Houston J. Math. 36 (2010), 237–260). To do so, we show that the cyclic cohomology of this algebra vanishes in all odd degrees, and is isomorphic in even degrees to the space of continuous traces on $\ell^1(S)$. Crucial to our approach is the use of the structure semilattice of $S$, and the associated grading of $S$, together with an inductive normalization procedure in cyclic cohomology. The latter technique appears to be new, and its underlying strategy may be applicable to other convolution algebras of interest.