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Building on the recent determination of the simplicial cohomology groups of the convolution algebra ℓ 1 (ℤ k  + ) [F. Gourdeau, Z. A. Lykova and M. C. White, A Künneth formula in topological homology and its applications to the simplicial cohomology of ℓ 1 (ℤ k  + ),  Studia Math.  166  (2005), 29–54], we investigate what can be said for the cohomology of this algebra with more general symmetric coefficients. Our approach leads us to a discussion of the 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.

HOCHSCHILD HOMOLOGY AND COHOMOLOGY OF ℓ 1 (ℤ k + )