Some results on the Hochschild cohomology of group algebras

In this paper we investigate structure of the second cohomology of a discrete group $G$. First, for a $G$-set $S$ we show that an isomorphism of vector spaces from $\mathcal{H}^n(\ell^1(G),\ell^\infty(S))$ onto $\bigoplus_{i\in I}^\infty \mathcal{H}^n(\ell^1(G),\ell^\infty({\mathcal{C}_i}))$ exists, where $\{\mathcal{C}_{i}\}_{i\in I}$ is the set of orbits of $S$. Next we define the notion of pseudoderivation and apply it for the calculation of $\mathcal{H}_{b,2}^2(\ell^1(G),\ell^\infty(S))$.