A Result on the diagonal of contractible operator Banach algebras

A Banach algebra A is called contractible if for any Banach A-bimodule E, every continuous derivation from A into E is inner. One of the oldest unconfirmed conjectures in amenability  says that every contractible Banach algebra is finite dimensional. It is well-known that a Banach algebra A is contractible if and only if its unital and has a diagonal, that is a member M in the Banach algebra A⊗πA such that satisfies in ∆(M)=1 and a⊗1M=M(1⊗a) for every a in A. In this note we show that any diagonal of a contractible Banach algebra of operators on an infinite dimensional Banach space has a specific null property.