The Shoda-completion of a Banach algebra

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 L(A). But this extension is so large that it may not preserve the socle of A in the extended algebra 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.