Hochschild homology and cohomology of $\ell^1(\mathbb Z_+^k)$

Choi, Yemon (2010) Hochschild homology and cohomology of $\ell^1(\mathbb Z_+^k)$. The Quarterly Journal of Mathematics, 61 (1). pp. 1-28. ISSN 0033-5606

Abstract

Building on the recent determination of the simplicial cohomology groups of the convolution algebra ℓ1(ℤk+) [F. Gourdeau, Z. A. Lykova and M. C. White, A Künneth formula in topological homology and its applications to the simplicial cohomology of ℓ1(ℤk+), 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.