The derivation problem is a familiar one concerning group algebras, particularly L 1 (G) and von Neumann algebras. In this paper, we study the Banach bimodule ℓ p (G), which is generated by the ℓ p norm over a specific class of groups with well-organized conjugacy classes. For this case, we will demonstrate that all ℓ p (G) derivations are inner.

In this paper we give a way of equipping the derivation algebra of a group algebra with the structure of a graded algebra. The derived group is used as the grading group. For the proof, the identification of the derivation with the characters of the adjoint action groupoid is used. These results also allow us to obtain the analogous structure of a graded algebra for outer derivations. A non-trivial graduation is

A description of the algebra of outer derivations of a group algebra of a finitely presented discrete group is given in terms of the Cayley complex of the groupoid of the adjoint action of the group. This task is a smooth version of Johnson's problem concerning the derivations of a group algebra. It is shown that the algebra of outer derivations is isomorphic to the group of the one-dimensional cohomology with compact supports of the Cayley complex over the field of complex numbers.

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