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

Part of: Homological methods Abstract harmonic analysis Topological algebras, normed rings and algebras, Banach algebras

Published online by Cambridge University Press:  17 December 2018

Yasser Farhat  and
Frédéric Gourdeau
Yasser Farhat
Affiliation:
Academic Support Department, Abu Dhabi Polytechnic, P.O. Box 111499, Abu Dhabi, UAE Email: [email protected]
Frédéric Gourdeau
Affiliation:
Département de mathématiques et de statistique, Université Laval, 1045, avenue de la Médecine, Québec, QC G1V 0A6 Email: [email protected]
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Abstract

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 $\mathbb{R}_{+}$.

Keywords

simplicial homologyBanach algebrasimplicial cohomology

MSC classification

Primary: 46H20: Structure, classification of topological algebras
Secondary: 43A20: $L^1$-algebras on groups, semigroups, etc. 16E40: (Co)homology of rings and algebras (e.g. Hochschild, cyclic, dihedral, etc.)
Type
Article
Information
Canadian Mathematical Bulletin , Volume 62 , Issue 4 , December 2019 , pp. 756 - 766
Copyright
© Canadian Mathematical Society 2018 

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Footnotes

This work was partially supported by the National Sciences and Engineering Research Council of Canada.

References

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