Some remarks about bounded derivations on the Hilbert space of square summable matrices

Abstract

It is known that not every Banach algebra has non-trivialbounded derivations. For instance, consider large families of weightedsemisimple Banach algebras. In particular, we will be concerned withderivations within the concrete frame of the non-abelian,non-unitary, involutive Banach algebra $l^{2}(N^{2})$. Thetheoretical interest in this algebra is based on the well-known fact that itis isomorphic to the class of Hilbert-Schmidt operators acting between twogiven separable Hilbert spaces. In this article, wecharacterize and determine the explicit structure of all boundedderivations on $l^{2}(N^{2})$.