On the Krull property in topological algebras. (English) Zbl 1180.46035

The paper deals with Krull and \(Q'\)-algebras. The author gives sufficient conditions under which a topological algebra is an annihilator algebra if and only if it is a Krull algebra. It is also shown that, in the case of a semisimple regular annihilator topological algebra \(E\), \(E\) is a Krull algebra if and only if every proper closed left (or right) ideal is contained in a proper left (respectively, right) annihilator ideal. Let \(E\) be a topological algebra and \(I\) a two-sided ideal. Some properties, like being a Krull algebra or a \(Q'\)-algebra, of the quotient algebra \(E/I\) are induced from similar properties of \(E\). In the last section, some properties of orthocompleted algebras and subalgebras of Krull algebras are stated. The question when the Krull property of a tensor product algebra is inherited by its tensor factors is studied. Four open questions are formulated.
Reviewer: Mart Abel (Tartu)


46H05 General theory of topological algebras
46H10 Ideals and subalgebras
46K05 General theory of topological algebras with involution
46M05 Tensor products in functional analysis
46H20 Structure, classification of topological algebras