8 - Algebras with Minimal Ideals

Summary

Introduction

In this chapter we investigate the role which minimal left ideals play in those algebras which possess them. (Minimal here simply means minimal under inclusion among all non-zero left ideals.) Significant results usually depend on assuming that the algebra is semiprime. However, the stronger assumption of semisimplicity, which is common in other areas of our theory, is comparatively seldom needed here. Furthermore, norms behave particularly well on minimal ideals so that a number of results can be proved for normed algebras without assuming completeness. In particular, any norm is spectral on many of the algebras considered in this chapter.

We will naturally be concerned mainly with algebras which not only have some minimal left ideals, but “enough” minimal left ideals in some sense. Such algebras axe quite special, but they have a correspondingly rich theory. Many classes of algebras with “enough” minimal left ideals have been defined, and towards the end of the section we will prove all the inclusions which hold between these various classes. (These results are summarized in a diagram of implications in Theorem 8.8.11.) However, we will concentrate attention on the class of modular annihilator algebras which was introduced by Bertram Yood [1958] and has been intensively studied by Yood [1964] and by Bruce A. Barnes [1964], [1966], [1968a], [1968b], and [1971a]. This class seems to give a particularly good compromise between axioms strong enough to give a significant theory and axioms weak enough to be satisfied by most important examples. Furthermore, modular annihilator algebras are defined purely algebraically while many of the other classes axe defined in terms of mixed algebraic and topological criteria. This provides a particularly elegant and straightforward theory.