The cyclic and simplicial cohomology of $l^1(\mathbf {N})$

Abstract:

Let $\mathcal {A}=l^1(\mathbf Z_+)$ be the unital semigroup algebra of $\mathbf N$. We show that the cyclic cohomology groups $\mathcal {H}C^n(\mathcal {A},\mathcal {A}’)$ vanish when $n$ is odd and are one dimensional when $n$ is even ($n\ge 2$). Using Connes’ exact sequence, these results are used to show that the simplicial cohomology groups $\mathcal {H}^n(\mathcal {A},\mathcal {A}’)$ vanish for $n\ge 2$. The results obtained are extended to unital algebras $l^1(S)$ for some other semigroups of $\mathbf {R}$.