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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)

##### MSC:
 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