Weak locally multiplicatively-convex algebra

Warner, Seth

doi:10.2140/pjm.1955.5.1025

Cited by 19 publications

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References 2 publications

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“…A sequentially complete, normed algebra is complete. An advertibly complete, normed algebra is a (?-algebra by Theorem 7 of [25]. Theorem 2.…”

Section: Any Normed Algebra £ Is I-bornologicalmentioning

confidence: 98%

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Inductive limits of normed algebras

Warner¹

1956

Trans. Amer. Math. Soc.

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For a Hausdorff locally convex space E with topological dual E', the following properties are equivalent:(B 1) Every bound linear transformation from P into any locally convex space is continuous; (B 2) No strictly stronger locally convex topology on E has the same bound sets; (B 3) Every bornivore set is a neighborhood of zero (a bornivore set is a convex, equilibrated set absorbing every bound set); (B 4) E is the (linear) inductive limit [7, pp. 61-66] of normed spaces {Ea} with respect to linear maps ga:Ea-*P such that U" ga(Ea) =P; (B 5) The topology of P is r(E, E') (the strongest locally convex topology on P yielding E' as topological dual), and every bound linear form on P is continuous. ( . Our purpose here is to formulate the analogous notion of the algebraic inductive limit of locally m-convex algebras and to study those locally mconvex algebras which are algebraic inductive limits of normed algebras.In §1 we summarize briefly the basic definitions and some of the elementary results in the theory of locally w-convex algebras. In §2, after defining the notion of the algebraic inductive limit of locally m-convex algebras in an obvious way, we consider the following problem. If E is an algebra, \Ea\ a family of locally w-convex algebras, ga:P"->P a homomorphism for all a, there exist on P both the algebraic inductive limit topology with respect to locally w-convex algebras Ea and homomorphisms ga, and the linear inductive limit topology with respect to locally convex spaces Ea and linear maps ga; when do they coincide ? This question is important, for the topologies of certain important locally convex spaces (occurring, for example, in the theory of distributions and the theory of integration) are defined as linear inductive limits of locally convex spaces which are also locally wz-convex algebras, with respect to linear maps which are also homomorphisms of the algebras considered. To establish that such topologies are actually locally w-convex, and thus fall within the purview of topological algebra, it is desirable to have general criteria insuring that the two inductive limit topologies coincide.

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“…A sequentially complete, normed algebra is complete. An advertibly complete, normed algebra is a (?-algebra by Theorem 7 of [25]. Theorem 2.…”

Section: Any Normed Algebra £ Is I-bornologicalmentioning

confidence: 98%

“…Lemma E.4 of [16, p. 77] [25] shows that the sequence { -E"=i ^'In-i converges to the adverse of x for 3^ and hence also for the weaker topology induced on EB by 3. Thus 5C [x\ -E"-i x" exists and is the adverse of x].…”

Section: Any Normed Algebra £ Is I-bornologicalmentioning

confidence: 99%

Inductive limits of normed algebras

Warner¹

1956

Trans. Amer. Math. Soc.

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“…From our Theorem 3.9 any cartesian product of normed algebras is a bi-lmc algebra, since normed algebras are evidently bi-lmc algebras. We mention too that the problem of telling when Imc algebras remain Imc when endowed with the strong topology is related to questions posed by Warner in [21] and [22]. In [22] he finds necessary and sufficient conditions for (E, t s ) to be Imc.…”

Section: Proof Let Abcsementioning

confidence: 99%

“…In [22] he finds necessary and sufficient conditions for (E, t s ) to be Imc. In [21] he leaves as an open question one which is equivalent to the following: is an Imc algebra still Imc if it is endowed with the Mackey topology?…”

Section: Proof Let Abcsementioning

confidence: 99%

The bidual of a locally multiplicatively-convex algebra

Gulick¹

1966

Pacific J. Math.

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“…For information about locally m-convex algebras see [ó], [l], [8] and [9 ]; for A -convex algebras see [3] and [4]. An equivalent definition of A -convex algebra is the following: an A -convex algebra is an algebra E with a topology defined via a family P of seminorms such that for p in P and x in E, there are constants M(p, x) and N(p, x)…”

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confidence: 99%

Topological algebras and Mackey topologies

Cochran¹

1971

Proc. Amer. Math. Soc.

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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-

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Weak locally multiplicatively-convex algebra

Cited by 19 publications

References 2 publications

Inductive limits of normed algebras

Inductive limits of normed algebras

The bidual of a locally multiplicatively-convex algebra

Topological algebras and Mackey topologies

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