On the Existence of Minimal Ideals in a Banach Algebra

Introduction. In this paper we give a sufficient condition that an algebra have a minimal left (or right) ideal. Specifically, we prove that if A is a complex semisimple Banach algebra with the property that the spectrum of every element in A is at most countable, then A has a minimal left ideal. If A is an ^4*-algebra, we prove that A has a minimal left ideal if the spectrum of every self-adjoint element of A is at most countable. These two basic results are given in §2, and some applications of them are given in §§3 and 4. The main result of §3 is Theorem 3.1 which is a variant of a theorem concerning £*-algebras that was proved through the separate efforts of M. A. Naïmark [5] and A. Rosenberg [7]. In §4 we prove that a complex semisimple Banach algebra with the property that the spectrum of every element in the algebra has no nonzero limit points is a modular annihilator algebra. This together with a previous result of the author shows that modular annihilator Banach algebras are characterized by this property.