Hochschild homology and cohomology of {\ell}^1({\mathbb Z}_+^k)

Building on the recent determination of the simplicial cohomology groups of the convolution algebra ${\mathrm{\ell }}^1({\mathbb{Z}}_{+}^k)$ [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.