The purpose of this work is to
study the structure of bounded derivations and crossed homomorphisms of
the Banach crossed product A= L1(G,A) of a Banach-∗-algebra A acted
upon by a locally compact group G. As bounded derivations and crossed
homomorphisms are related to 1-cocycles, we first define and study cohomology over
A, generalizing cohomology over group algebras. Then, if G is amenable and A is a
C∗-algebra, or the dual of a Banach space, we show that a bounded derivation
(resp. a crossed homomorphism) on A is equivalent to some couple of a
bounded derivation (resp. a crossed homomorphism) from A to M1(G,A) and a
bounded measure on A with value in the centralizers of A (resp. an element of
A).
