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Hochschild homology and cohomology of $$\ell ^1 (\mathbb Z_+^k)$$. (English) Zbl 1194.46110
This paper deals with the Hochschild homology and cohomology of $$\ell^1(\mathbb Z^k_+)$$. Let us recall that the simplicial cohomology groups of the convolution algebra $$\ell^1(\mathbb Z^k_+)$$ have been determined by F. Gourdeau, Z. A. Lykova and M. C. White [Stud. Math. 166, No. 1, 29–54 (2005; Zbl 1065.46030)]. It was proved that this cohomology vanishes in degrees $$2$$ and above, which tells us the importance of the coefficients. In this work, the author investigates the computation for more general symmetric coefficients. This leads to a study of the Harrison homology and cohomology. As a byproduct, new results concerning second-degree cohomology are found.

##### MSC:
 46M20 Methods of algebraic topology in functional analysis (cohomology, sheaf and bundle theory, etc.) 46H05 General theory of topological algebras 43A20 $$L^1$$-algebras on groups, semigroups, etc.
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