  We establish simplicial triviality of the convolution algebra $\ell^1(S)$,
where $S$ is a band semigroup. This generalizes results of the first author
[Glasgow Math. J. 2005, Houston J. Math. 2010]. 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.
