Simplicial (Co)-homology of $\ell ^{1}(\mathbb{Z}_{+})$

Farhat, Yasser ; Gourdeau, Frédéric

Canadian mathematical bulletin, Tome 62 (2019) no. 4, pp. 756-766

Résumé

We consider the unital Banach algebra $\ell ^{1}(\mathbb{Z}_{+})$ and prove directly, without using cyclic cohomology, that the simplicial cohomology groups ${\mathcal{H}}^{n}(\ell ^{1}(\mathbb{Z}_{+}),\ell ^{1}(\mathbb{Z}_{+})^{\ast })$ vanish for all $n\geqslant 2$. This proceeds via the introduction of an explicit bounded linear operator which produces a contracting homotopy for $n\geqslant 2$. This construction is generalised to unital Banach algebras $\ell ^{1}({\mathcal{S}})$, where ${\mathcal{S}}={\mathcal{G}}\cap \mathbb{R}_{+}$ and ${\mathcal{G}}$ is a subgroup of $\mathbb{R}_{+}$.

Détail

DOI : 10.4153/S0008439518000644

Mots-clés : simplicial homology, Banach algebra, simplicial cohomology