The flip is often discontinuous

TL;DR

This paper investigates the discontinuity of the flip operation on elementary operators in certain Banach algebras, specifically when the algebra is of compact operators on a reflexive Banach space with the approximation property.

Contribution

It demonstrates that the flip operation on elementary operators is discontinuous for algebras of compact operators on reflexive Banach spaces with the approximation property.

Findings

Flip is discontinuous on elementary operators for certain Banach algebras.

Discontinuity occurs when the algebra is of compact operators on a reflexive Banach space.

The result links algebraic properties with topological discontinuities.

Abstract

Let $A$ be a Banach algebra. The flip on $A \otimes A^\op$ is defined through $A \otimes A^\op \ni a \tensor b \mapsto b \tensor a$. If $A$ is ultraprime, $\El(A)$, the algebra of all elementary operators on $A$, can be algebraically identified with $A \otimes A^\op$, so that the flip is well defined on $\El(\A)$. We show that the flip on $\El(A)$ is discontinuous if $A = K(E)$ for a reflexive Banach space $E$ with the approximation property.