A Remark on Contractible Banach Algebras of Operators

For a Banach algebra A, we say that an element M in A ⊗γ A is a hypercommutator if (a ⊗ 1)M = M(1 ⊗ a) for every a ∈ A. A diagonal for a Banach algebra is a hyper-commutator whose image under diagonal mapping is 1. It is well known that a Banach algebra is contractible iff it has a diagonal. The main aim of this note is to show that for any Banach subalgebra A ⊆ L(X) of bounded linear operators on infinite-dimensional Banach space X, which contains the ideal of finite-rank operators, the image of any hyper-commutator of A under the canonical algebra-morphism L(X) ⊗γ L(X) → L(X ⊗γ X), vanishes.