The Shoda-completion of a Banach algebra

TL;DR

This paper introduces the Shoda-completion, a minimal extension of a Banach algebra that preserves the socle and resolves the issue of expressing traceless socle elements as commutators, unlike in the original algebra.

Contribution

It develops the concept of the Shoda-completion, an extension of Banach algebras that maintains the socle structure while fixing the commutator representation problem.

Findings

The Shoda-completion preserves the socle structure.

It enables expressing traceless socle elements as commutators.

The extension is minimal and natural.

Abstract

In stark contrast to the case of finite rank operators on a Banach space, the socle of a general, complex, semisimple, and unital Banach algebra $A$ may exhibit the `pathological' property that not all traceless elements of the socle of $A$ can be expressed as the commutator of two elements belonging to the socle. The aim of this paper is to show how one may develop an extension of $A$ which removes the aforementioned problem. A naive way of achieving this is to simply embed $A$ in the algebra of bounded linear operators on $A$, i.e. the natural embedding of $A$ into $\mathcal L(A)$. But this extension is so large that it may not preserve the socle of $A$ in the extended algebra $\mathcal L(A)$. Our proposed extension, which we shall call the Shoda-completion of $A$, is natural in the sense that it is small enough for the socle of $A$ to retain the status of socle elements in the extension.