Abstract.Let £ be a locally »re-convex algebra with dual space E'. In a recent paper S. Warner asked if the finest locally »re-convex topology on E compatible with E' was the mackey topology. It is shown that this is not the case. A similar result is given for this question in the 4-convex algebra case. For any A -convex algebra, a construction is given of an associated locally »reconvex algebra. It is shown that this associated locally »re-convex topology is always the compact-open topology for the space Cb(S) with the strict topology.Seth Warner [9] extended the idea of bornological linear space to the case of locally »z-convex algebras. For a given locally »z-convex algebra E with dual space E', he noted the existence of a finest locally »z-convex topology, %(P, P')> compatible with the given duality. In this note we show that x(E, E') does not necessarily coincide with the mackey topology t(P, E'). This answers a question presented by Warner [9, p. 215, Question 3]. The class of A -convex algebras introduced in [3] and [4] provide a similar situation. There is a finest A -convex topology, S(P, E'), compatible with a given duality and we show that 2 (P, E') is not necessarily a mackey topology.We give a method to construct the finest locally w-convex topology coarser than a given A -convex topology. Let 5 denote a locally compact hausdorff space, Cb(S) the space of bounded continuous complexvalued functions on S, ß the strict topology introduced by Buck [2] and k the compact-open topology. We use the description obtained to show that the finest locally w-convex topology coarser than ß is precisely k. Thus, there are no locally m-convex topologies between ß and k.2. Preliminaries. In this section the basic definitions are given and a description of the strict topology is listed for use in §3. Throughout this note P will denote an algebra over R or C and topology will always mean locally convex linear topology.(2.1) Definition. A convex balanced absorbing subset V of E is called m-convex if V-VE V (i.e. if V is idempotent). A convex bal-
Abstract.Necessary and sufficient conditions are given in terms of E' that a weak topology w(E, E') on an algebra E be biconvex. The main condition is that each element g of E' contain a weakly closed subspace L of finite codimension such that g is bounded on all multiplicative translates of L. For weak topologies, A -convexity (which assumes only separate continuity of multiplication) is equivalent to joint continuity of multiplication.
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