In this thesis, we give an expanded account of simplicial triviality of the convolution algebra ?? 1 (S) of a band semigroup S stablished by Choi (2012). Computing the Hochschild cohomology of Banach algebras has remained a difficult task, even when restricted to the stroked="f" filled="f" path="m@4@5l@4@11@9@11@9@5xe" o:preferrelative="t" o:spt="75" coordsize="21600,21600" 1 -convolution algebras of semigroups. Choi has shown that the simplicial cohomology of the semigroup algebra ?? 1 (S) vanishes when S is a normal band; that is a band S for which for all . However, the techniques were unable to handle the case of general band semigroups. We note that bands comprise a rich and interesting justify; LINE-HEIGHT: normal; MARGIN: 0cm 0cm 0pt; unicode-bidi: embed; DIRECTION: ltr" - The cyclic cohomology of ?? 1 (S) is isomorphic in even degrees to the space of continuous traces on ?? 1 (S), and vanishes in odd degrees; - The simplicial cohomology of ?? 1 (S) vanishes in all strictly positive degrees. 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 covolution algebras of interest. The techniques used in establishing these results resemble those in the earlier work of Gourdeau, Johnson and White (2005), in that one performs explicit calculations with cyclic cochains, and then used the Connes-Tzygan long exact sequence to calculate the simplicial cohomology. As in that work, the decition to work with cyclic cohomology is forced upon us by the nature of our construction, and is not merely incidental. Some of the results appear to generalize to the setting of Banach algebras which are ?? 1 -graded over a semilattice. In particular, it seems that similar calculations would provide an alternative approach to some of Choi’s existing results for Clifford semigroups. However, we shall focus throughout on the case of band semigroup algebras in order to keep the exposition reasonably self-contained. One approach which one might be tempted to adopt, in order to prove that band semigroup algebras have trivial cyclic cohomology, is to exhaust the band by finitely generated bands and cobound the cocycle on increasingly large sets. This is even more tempting when one recalls that finitely generated bands are finite. However, one encounters problems with this approach. It is difficult to obtain uniform control of the norms of the coboundaries as we take increasingly large generating sets for these bands. This is true even in the commutative case. Another feathere is that finite band algebras are, in general, neither semisimple nor amenable, which makes their trivial simplicial cohomology surprising.