Topological algebras and Mackey topologies

Abstract

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-

Algèbres $A$-convexes et problème de Michael

“…Encore une fois, étant métrisables, t et M(x) coïncideraient si elles avaient les mêmes bornés. Cette proposition permet de donner une réponse partielle à la question (3.6) de [2]. Avec les notations de [2] on a Proposition 5.6.…”

“…Cette proposition permet de donner une réponse partielle à la question (3.6) de [2]. Avec les notations de [2] on a Proposition 5.6. Si x(E, E1) est tonnelée, alors "L(E, E') et x(E, E1) coincident.…”

Representation of 𝐴-convex algebras

“…The proof of the first part is given in [2]; the second part follows from the fact that if/is an algebraic homomorphism then [f(A*)]<= [f(A)]* and similar straightforward calculations.…”

“…Basic results on A-convex algebras are found in [1], [2] and [3]. In order to make this note relatively self-contained we briefly repeat pertinent definitions.…”

“…A basic reference to locally «i-convex algebras is [5]. Examples of ^-convex algebras (see [2]) include: (1) all locally «i-convex algebras; (2) the space (Cb(x), ß) of bounded C-valued functions on a locally compact Hausdorff space with the strict topology; and (3) multiplier algebras.…”