The Shoda-completion of a Banach algebra

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.