  Building on the recent determination of the simplicial cohomology groups of
the convolution algebra ${\ell}^1({\mathbb Z}_+^k)$ [Gourdeau, Lykova, White,
2005] we investigate what can be said for cohomology of this algebra with more
general symmetric coefficients. Our approach leads us to a discussion of
Harrison homology and cohomology in the context of Banach algebras, and a
development of some of its basic features. As an application of our techniques
we reprove some known results on second-degree cohomology.
