Ideals in Topological Rings

Abstract

We present here an investigation of the theory of one-sided ideals in a topological ring R. One of our aims is to discuss the question of "left" properties versus "right" properties. A problem of this sort is to decide if (a) all the modular maximal right ideals of R are closed if and only if all the modular maximal left ideals of R are closed. It is shown that this is the case if R is a quasi-Q-ring, that is, if R is bicontinuously isomorphic to a dense subring of a Q-ring (for the notion of a Q-ring see (6) or §2). All normed algebras are quasi-Q-rings. Also (a) holds if R is a semisimple ring with dense socle.

On the Existence of Minimal Ideals in a Banach Algebra

“…Let B be the closure of A in the operator norm. Then by [2, Lemma 2.6] SB is dense in B, and by [8,Lemma 3.11,p. 41], this implies that £ is a modular annihilator £*-algebra of operators.…”

“…It remains to be shown that an irreducible modular annihilator £*-algebra of operators on a Hubert space 3tf is the algebra of completely continuous operators on df. But by [8,Theorem 4.1,p. 42] such an algebra must be dual and then the result follows by [6,Corollary (4.10.20),p.…”

“…Definitions of most of the concepts mentioned in this paper may be found in C. Rickart's book [6] (although our notations often differ). Information concerning modular annihilator algebras may be found in [1] or [8].…”

The strong radical and the left regular representation

“…We call A modular annihilator if every maximal modular left (right) ideal of A has a nonzero right (left) annihilator. A semisimple Banach algebra with dense socle is modular annihilator [9 We will also be interested in the right multiplication operators R a , where, for each a e A, R a (x) = xa for all x e A.…”

Multipliers on complemented Banach algebras

“…We call A a modular annihilator algebra if every maximal modular left (right) ideal of A has a nonzero right (left) annihilator. A semisimple Banach algebra with dense socle is modular annihilator [15,Lemma 3.11,p. 41].…”