  Let $G$ be a discrete group. We give a decomposition theorem for the
Hochschild cohomology of $\ell^1(G)$ with coefficients in certain $G$-modules.
Using this we show that if $G$ is commutative-transitive, the canonical
inclusion of bounded cohomology of $G$ into simplicial cohomology of
$\ell^1(G)$ is an isomorphism.
