Let G be a locally compact group. The question of whether H¹ L¹(G),M(G), the first Hochschild cohomology group of L¹(G) with coefficients in M(G), is zero was first studied by B. E. Johnson and initiated his development of the theory of amenable Banach algebras. He was able to show that H¹(L¹( G), M( G) = 0 whenever G is amenable, a [SIN]-group, or a matrix group satisfying certain conditions. No group such that H¹(L¹(G),M(G) ≠ 0 is known. In this paper, we approach the problem of whether H¹(L¹(G),M(G) = 0 from several angles. Using weakly almost periodic functions, we show that H¹(L¹(G),L¹(G) is always Hausdorff for unimodular G. We also show that for [IN]-groups, every derivation D : L¹(Gto L¹(G is implemented, not necessarily by an element of M(G), but at least by an element of VN(G), the group von Neumann algebra of G. This applies, in particular, to the group G : = T² ⋊ SL(2,Z}, for which it is unknown whether H}(L¹(G),M(G) = 0. Finally, we analyse the structure of derivations on L¹(G); an important role is played by the closed normal subgroup N of G generated by the elements of G with relatively compact conjugacy classes. We can write an arbitrary derivation D : L¹(G) to L¹(G) as a sum D = DN DN⊥$, where DN and DN⊥ can be tackled with different techniques. Under suitable conditions, all satisfied by T² ⋊ SL(2,Z}, we can show that DN is implemented by an element of VN(G) and that DN⊥ is implemented by a measure.
Proceedings of the London Mathematical Society Vol. 80, Issue 2, p. 360-390