A Remark on Contractible Banach Algebras of Operators · arXivDeskAbstract
For a Banach algebra A, we say that an element M in A⊗γA is a hyper-commutator if (a⊗1)M=M(1⊗a)
for every
. A diagonal for a Banach algebra is a hyper-commutator which its image under diagonal mapping is
. 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
, which contains the ideal of finite-rank operators, the image of any hyper-commutator of
under the canonical algebra-morphism
L(X)⊗γL(X)→L(X⊗γX) , vanishes.