A Remark on Contractible Banach Algebras of Operators

TL;DR

This paper investigates the properties of hyper-commutators in Banach algebras of operators, showing that their images vanish under certain conditions, which relates to the concept of contractibility.

Contribution

It demonstrates that in specific Banach subalgebras of bounded operators, hyper-commutators' images always vanish, extending understanding of contractibility in operator algebras.

Findings

Hyper-commutators' images vanish in certain subalgebras

Contractibility relates to the existence of diagonals in Banach algebras

Results apply to subalgebras containing finite-rank operators

Abstract

For a Banach algebra $A$, we say that an element $M$ in $A\otimes^\gamma A$ is a hyper-commutator if $(a\otimes 1)M=M(1\otimes a)$ for every $a\in A$. 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 $A\subseteq\mathcal{L}(X)$ 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 $\mathcal{L}(X)\otimes^\gamma\mathcal{L}(X)\to\mathcal{L}(X\otimes^\gamma X)$, vanishes.