Derivations on Group Algebras

Let G be a locally compact group. The question of whether H 1 L 1 ( G ), M ( G ), the first Hochschild cohomology group of L 1 ( 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 H 1 ( L 1 ( G ), M ( G ) = 0 whenever G is amenable, a [SIN] ‐group, or a matrix group satisfying certain conditions. No group such that H 1 ( L 1 ( G ), M ( G ) ≠ 0 is known. In this paper, we approach the problem of whether H 1 ( L 1 ( G ), M ( G ) = 0 from several angles. Using weakly almost periodic functions, we show that H 1 ( L 1 ( G ), L 1 ( G ) is always Hausdorff for unimodular G . We also show that for [IN] ‐groups, every derivation D : L 1 ( G to L 1 ( 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 : = T 2 ⋊ SL(2,Z}, for which it is unknown whether H } 1 ( L 1 ( G ), M ( G ) = 0. Finally, we analyse the structure of derivations on L 1 ( 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 : L 1 ( G ) to L 1 ( G ) as a sum D = D N D N ⊥$, where D N and D N ⊥ can be tackled with different techniques. Under suitable conditions, all satisfied by T 2 ⋊ SL(2,Z}, we can show that D N is implemented by an element of VN( G ) and that D N⊥ is implemented by a measure. 1991 Mathematics Subject Classification : 22D05, 22D25, 43A10, 43A20, 46H25, 46L10, 46M20, 47B47, 47B48.