Geodesic
Simplicial (Co)-homology of $\ell ^{1}(\mathbb{Z}_{+})$
Canadian mathematical bulletin, Tome 62 (2019) no. 4, pp. 756-766

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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 $\mathbb{R}_{+}$.
DOI : 10.4153/S0008439518000644
Mots-clés : simplicial homology, Banach algebra, simplicial cohomology 
Farhat, Yasser; Gourdeau, Frédéric. Simplicial (Co)-homology of $\ell ^{1}(\mathbb{Z}_{+})$. Canadian mathematical bulletin, Tome 62 (2019) no. 4, pp. 756-766. doi: 10.4153/S0008439518000644
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[1] Bade, W. G., Curtis, P. C. Jr., and Dales, H. G., Amenability and weak amenability for Beurling and Lipschitz algebras . Proc. Lond. Math. Soc. 55(1987), 359–377. Google Scholar | DOI

[2] Choi, Y., Gourdeau, F., and White, M. C., Simplicial cohomology of band semigroup algebras . Proc. Roy. Soc. Edinburgh Sect. A 142(2012), no. 4, 715–744. Google Scholar | DOI

[3] Dales, H. G. and Duncan, J., Second order cohomology in groups of some semigroup algebras . In: Banach Algebras ’97 . Walter de Gruyter, Berlin, 1998, pp. 101–117. Google Scholar

[4] Farhat, Y., Homologie et cohomologie de quelques algèbres de Banach. Ph.D. thesis, Université Laval, Québec, QC, 2013. Google Scholar

[5] Gourdeau, F., Johnson, B. E., and White, M. C., The cyclic and simplicial cohomology of l 1(ℕ) . Trans. Amer. Math. Soc. 357(2005), no. 12, 5097–5113. Google Scholar

[6] Gourdeau, F. and White, M. C., Vanishing of the third simplicial cohomology group of l 1(ℤ) . Trans. Amer. Math. Soc. 353(2001), no. 5, 2003–2017. Google Scholar

[7] Gourdeau, F. and White, M. C., The cyclic and simplicial cohomology of the bicyclic semigroup algebra . Q. J. Math. 62(2011), 607–624. Google Scholar | DOI

[8] Gourdeau, F. and White, M. C., The cyclic and simplicial cohomology of the Cuntz semigroup algebra . J. Math. Anal. Appl. 386(2012), no. 2, 921–932. Google Scholar | DOI

[9] Helemskii, A. Ya., Banach cyclic (co)homology and the Connes–Tzygan exact sequence . J. Lond. Math. Soc. 46(1992), 449–462. Google Scholar | DOI

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