A Remark on Contractible Banach Algebras of Operators

2022/11/07 by Maysam Maysami Sadr, Sadr, Maysam Maysami
Mathematics · #Advanced Topics in Algebra #Advanced Operator Algebra Research #Advanced Banach Space Theory

paper · pdf · doi:10.48550/arxiv.2211.03493

Abstract

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∈ A. A diagonal for a Banach algebra is a hyper-commutator which its 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\subseteqL(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)⊗^γL(X)\toL(X⊗^γX), vanishes.

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