Derivations from Banach algebras
by
H. G. Dales
Department of Pure Mathematics, University of Leeds

Let A be an algebra, and let E be an A-bimodule. A derivation from A to E is a linear map D:A ® E such that D(ab) = Da ·b + a ·Db for all a, b Î A.

Now let A be a Banach algebra, and let E be a Banach A-bimodule. We consider the space Z ¹ (A, E) of continuous derivations from A to E. For example, for each x Î E, the map a ® a ·x - x ·a is such a map; these maps form the space N ¹ (A, E) of inner derivations. We study the quotient space H ¹ (A, E) = Z¹ (A, E)/N¹(A, E).

A Banach algebra A is amenable if H¹ (A, E¢) = {0} for each E, and weakly amenable if H¹ (A, A¢) = {0}. Here E¢ is the dual of the Banach A-bimodule E. We shall review some classical theorems about amenable and weakly amenable Banach algebras in various classes, and state recent important theorems of C.J. Read and B.E. Johnson.

Let G be a locally compact group, with group algebra L¹ (G) and measure algebra M(G). We shall discuss continuous derivations from these Banach algebras, and prove that M(G) is amenable if and only if G is discrete and amenable as a group.

(T)

Date received: April 14, 2000