Extensions of Topological Algebras

We prove that, in the class of commutative topological algebras with separately continuous multiplication, an element is permanently singular if and only if it is a topological divisor of zero. This extends the result given by R. Arens [1] for the Banach algebra case. We also give sufficient conditions for non-removability of ideals in commutative topological algebras with jointly continuous multiplication. AMS Subject Classification (1980): 46J10 Introduction. By a topological algebra we mean a topological vector space with a jointly continuous multiplication making of it a complex algebra. The topology of a topological algebra A can be given by a system U of zero-neighbourhoods satisfying the following properties: (i) For every V ∈ U , there exists W ∈ U such that W + W ⊂ V. (ii) For every V ∈ U and α ∈ C with |α| ≤ 1, αV ⊂ V. (iii) Every V ∈ U is absorbent. (iv) For every V ∈ U , there exists W ∈ U such that W ·W ⊂ V. Every algebra in this paper will be a commutative complex algebra with unit element denoted usually by e. A locally convex algebra is a topological algebra with a system of convex zeroneighbourhoods. The topology of a locally convex algebra A can be given by a directed system of seminorms {| · |α : α ∈ D} (in this case, (iv) above can be written as follows: for every α ∈ D there exists β ∈ D such that |xy|α ≤ |x|β |y|β for all x, y ∈ A). Let A and B be topological algebras with units eA and eB , respectively. We say that B is an extension of A if there exists a unit preserving, injective algebra homomorphism f : A → B such that A is topologically isomorphic to its image f(A). In this case, we identify A with f(A) and simply write A ⊂ B. Let A be a topological algebra and I ⊂ A an ideal. We say that I is removable if there exists an extension B ⊃ A such that I is not contained in any proper ideal of B. It is easy to see that this condition is equivalent to the existence of a finite number of elements x1, . . . , xk ∈ I and y1, . . . , yk ∈ B such that x1y1 + · · · + xkyk = e. An ideal which is not removable will be called non-removable. The notion of non-removable ideal was introduced by R. Arens [2]. Non-removable ideals in commutative Banach algebras have been studied, e.g., in [2], [6], [4] and [5], and in topological algebras in [8], [9] and [10]. *The second and fourth named authors have been partially supported by a research project from La Consejeŕıa de Educación y Ciencia de La Junta de Andalućıa. The third named author has been supported by a research grant from El Ministerio de Educación y Ciencia. 1