Commutators, Commutativity and Dimension in the Socle of a Banach Algebra: A generalized Wedderburn-Artin and Shoda's Theorem

Abstract

As a follow-up to work done in [7], some new insights to the structure of the socle of a semisimple Banach algebra is obtained. In particular, it is shown that the socle is isomorphic as an algebra to the direct sum of tensor products of corresponding left and right minimal ideals. Remarkably, the finite-dimensional case here reduces to the classical Wedderburn-Artin Theorem, and this approach does not use any continuous irreducible representations of the algebra in question. Furthermore, the structure of the socles for which the classical Shoda's Theorem for matrices can be extended, is characterized exactly as those socles which are minimal two-sided ideals. It is then shown that the set of commutators in the socle (i.e. $\left\{xy-yx : x, y \in \mathrm{Soc}\:A \right\}$) is a vector subspace. Finally, we characterize those socles which belong to the center of a Banach algebra and obtain results which suggests that the dimension of certain subalgebras of the socle in fact provides a measure, to some extent, of commutativity.