The bidual of a locally multiplicatively-convex algebra

Gulick, S. L.

doi:10.2140/pjm.1966.17.71

Cited by 25 publications

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“…**, and o makes i? ** into a Banach algebra [2], However, we can generalize the notion to a wide class of topological algebras, including those with jointly continuous multiplication [11], Any topological algebra whose bidual has this Arens multiplication we call a topological algebra with bidual.…”

Section: S L Gulickmentioning

confidence: 99%

“…Let V be a neighborhood of 0 in E, and let p v be the Minkowski functional on F [13, p. 15]. Let E v be the normed algebra E/PΫ^O), clothed in the quotient topology arising from the semi-normed topology induced by p v on E. If there is a subbasis (^λ)λe^ of neighborhoods of 0 in E such that each E Vκ is a normed algebra, then E is called a locally multiplicatively-convex algebra, and such algebras are topological algebras with biduals [11]. THEOREM 1 )* compact set.…”

Section: Let E and F Be Locally Convex Spaces And Let T: E-+f Be A Cmentioning

confidence: 99%

“…The original discussions on the definition of the Arens multiplication occur in [1,2], are extended in [11], In [2] Arens gives a condition, involving weakly compact sets of the base algebra, which guarantees that the bidual of a commutative Banach algebra be commutative. He asks if there is a necessary condition of the same type.…”

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confidence: 99%

“…For it guarantees that M E correspond to a complex-valued homomorphism on E, so that if y is the identity for E modulo M Ef then the linear space spanned by M E and y is dense in E. We next recall that the second adjoint of a continuous homomorphism is a weak-(i? *) continuous homomorphism [11,Remark 3.11], We can then use verbatim the argument of Theorem 5.3 of [4] to conclude the proof. THEOREM …”

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confidence: 99%

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Commutativity and ideals in the biduals of topological algebras

Gulick¹

1966

Pacific J. Math.

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Section: S L Gulickmentioning

confidence: 99%

Section: Let E and F Be Locally Convex Spaces And Let T: E-+f Be A Cmentioning

confidence: 99%

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confidence: 99%

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Commutativity and ideals in the biduals of topological algebras

Gulick¹

1966

Pacific J. Math.

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“…We say that multiplication is hypocontinuous if it is both right and left hypocontinuous. Gulick [5] has, however, called right hypocontinuity by hypocontinuity.…”

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Topological algebras with a given dual

Chilana¹

1974

Proc. Amer. Math. Soc.

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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.

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The Locally Boundedly Multiplicatively Convex Algebras

Chilana

Sharma

1977

Mathematische Nachrichten

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The purpose of this paper is to introduce the class of locally boundedly niultiplicatively convex algebras. This new class has been compared with other generalizations of BANACH algebras, for instance, with locally multiplicatively convex algebras introduced and studied by ARENS and MICHAEL and A-convex algebras defined and studied by COCHRAN, KEOWN & WILLIAMS. Necessary and sufficient conditions on the dual pair (E, E') are given for E to have a locally boundedly multiplicatively convex topology compatible with duality. Relations between the conditions of local bounded niultiplicative convexity of E with the strong topology and local multiplicative convexity of the bidunl with ARENS product and that of the associated bornological structure have been studied.A good number of examples have been given from Function spaces, in particular, weighted function spaces and algebras with the mixed topology.

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The bidual of a locally multiplicatively-convex algebra

Cited by 25 publications

References 17 publications

Commutativity and ideals in the biduals of topological algebras

Commutativity and ideals in the biduals of topological algebras

Topological algebras with a given dual

The Locally Boundedly Multiplicatively Convex Algebras

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