Topological Algebras with a Given Dual

Abstract

Abstract.Given an algebra E and a total subspace E' of its algebraic dual, we obtain necessary and sufficient conditions in terms of E' for the existence of an /4-convex or a locally /n-convex topology on E compatible with duality (E, E'). It has also been proved that if E with the weak topology w(E, £') is the closed linear hull of a bounded set and has hypocontinuous multiplication then it is locally /«-convex.1. Introduction. Let £ be a complex (or real) algebra and £' be a total subspace of the algebraic dual £*. To avoid repetitions we use the notation, terminology and results in [3] and [4] without specifications. An algebra with a locally convex linear topology for which multiplication is separately continuous will be called a locally convex algebra. An absolutely convex set B in £ is called right (left) A-convex if it absorbs Bx (xB) for each x e E, it will be called A-convex if it is both right and left Aconvex. A locally convex algebra is called (right, left) A-convex if there exists a basis of (right, left) ^-convex neighbourhoods of zero. Multiplication in a locally convex algebra will be said to be right (left) hypocontinuous if given a neighbourhood U of o and a bounded set B there exists a neighbourhood V of o satisfying VB<^ U (BV<= U). We say that multiplication is hypocontinuous if it is both right and left hypocontinuous. Key words and phrases, (right, left) /í-convex algebra, locally /«-convex algebra, topology compatible with duality, (right, left) multiplicative translate of a functional, collectionwise multiplicative set of functionals, collectionwise (right, left) multiplicativetranslation invariant set of functionals, hypocontinuous and jointly continuous multiplication.