Extended Abstract. A Banach algebra A is called contractible if for any Banach A-bimodule E, every continuous derivation from A into E is inner. Contractibility is among of the various notions of amenability which have been defined after a very bold paper of Johnson on cohomology of Banach algebras. It is well-known that full matrix algebras and their finite direct sum are contractible. Also, every finite-dimensional contractible Banach algebra is of this form.  One of the oldest unconfirmed conjectures in amenability says that every contractible Banach algebra is finite dimensional. The special case of this conjecture which is still unconfirmed says that for any Banach space X, if the Banach algebra B(X) of all bounded linear operators on X, is contractible then X 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⊗_π A such that satisfies in ∆(M)=1 and (a⊗1)M=M(1⊗a) for every a in A. Here,  ⊗_π denotes the completed projective tensor product of Banach spaces, and ∆: A⊗_π A→A is the unique bounded linear operator defined by (a⊗b)⟼ab.
For a Banach space X of finite dimension m, B(X) is isomorphic to the matrix algebra M_m. It is well-known that M_m has a unique diagonal of the form m^-1 Σ_ij δ_ij⊗δ_ji where δ_ij’s denote the standard basis of M_m.
The aim of this short note is to prove a property (Theorem 6) for the diagonal of any contractible B(X) when X is infinite-dimensional. For the proof we use the famous estimate of Kadec and Snobar on the norms of projection operators on finite-dimensional subspaces. We hope that this property of diagonal and the technics we have used here, helps to solve the mentioned conjecture on finite-dimensionality of X.
Consider the following operator:
Ψ: B(X)⊗_π B(X)⟶B(X⊗_π X),       Ψ(T⊗S)(x⊗y)=T(x)⊗T(y).
Then Ψ is a Banach algebra homomorphism with norm equals to 1. We denote the image of M ∈ B(X)⊗_π B(X) under Ψ by M^op. The main result of this note is Theorem 6:
Theorem 5. Suppose that there is an infinite-dimensional Banach space X such that B(X) is contractible. Then for every diagonal M of B(X) we have M^op=0.