Vanishing of the third simplicial cohomology group of 𝑙¹(𝐙₊)

Gourdeau, Frédéric; White, Michael

doi:10.1090/S0002-9947-01-02738-6

Vanishing of the third simplicial cohomology group of $l^1(\mathbf {Z}_+)$
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by Frédéric Gourdeau and Michael C. White PDF

Trans. Amer. Math. Soc. 353 (2001), 2003-2017 Request permission

Abstract:

We show that $\mathcal {H}^3(l^1(\mathbf {Z}_+),l^1(\mathbf {Z}_+)’)=0$. We first use the Connes-Tzygan exact sequence to prove that this is equivalent to the vanishing of the third cyclic cohomology group $\mathcal {H}C^3(\mathcal {I},\mathcal {I}’)$, where $\mathcal {I}$ is the non-unital Banach algebra $l^1(\mathbf {N})$, and then prove that $\mathcal {H}C^3(\mathcal {I},\mathcal {I}’)=0$.

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