Modular annihilator algebras

Abstract

The elements of minimal left (right) ideals in a semi-prime modular annihilator algebra A completely characterized by the property of being singles not in rad A. An element s of A is called single if whenever asb = 0 for some a, b in A then at least one of as, sb is zero. B(X) of bounded operators on X containing it. In a number of operator algebras the converse is also true (see [7], [lo], [12]) making the property of being equivalent to being an operator of rank one. In an abstract semi-prime modular annihilator Banach, the property of being rank one is meaningless,'but the link between single elements and rank oneness is crucial. To be specific, we show that such Banach algebras can faithfully represented on a Banach space X in such a way that the image of an element s of A is of rank one if and only if the element is single and s 4 rad A. Also we completely characterize the minimal left (right) ideals of A by showing that such ideals are precisely the ideals of the form As(sA) where s is single, s 4 rad A. Finally, we characterize the socle of A as the set of all finite sums of single elements that are not in the radical of A.In general, the terminology and the notation used will be as in F. F. Bonsall and J. Duncan in [3]. All the algebras considered will be over the complex field.Mathematics subject classification numbers, Primary 1980/85.