Factorization in Banach algebras and the general strict topology

Sentilles, F. Dennis; Taylor, Donald C.

doi:10.1090/s0002-9947-1969-0247437-9

Cited by 60 publications

(30 citation statements)

References 12 publications

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“…In the next section we will obtain some further answers to this question with a summation in §9. To close out this section we obtain for these topologies other analogues of the results in [3] and [35]. It is also clear that 4.8 is true with ß replaced by ß0.…”

Section: Definitions and Preliminariessupporting

confidence: 57%

“…The proof will be a good deal clearer if we retrieve the following special result in the locally compact setting; this lemma can also be observed as a Corollary to [35,Theorem 4.1].…”

Section: Definitions and Preliminariesmentioning

confidence: 90%

“…In fact, w*-countably compact subsets are equicontinuous in that context. Proof, (a) If Fe X'ß, then F e X'ßa for some a and hence F is bounded and we can view FeX' according to [35,Theorem 4.1]. By 3.3, F e X'ßa for every a. Conversely, Fe f)a X'ßa implies Fe Xe again by 3.3.…”

Section: Definitions and Preliminariesmentioning

confidence: 99%

“…According to [34], CQ defines a "strict" topology ßQ on C(X), this topology generated by the seminorms p^(f)= \\jX\\, fe C(X), one for each £ e C0. Clearly, if P((f) = 0 for every £ e C0 then/=0, so that ßQ is Hausdorff and [33], [34] and [35] apply.…”

Section: Definitions and Preliminariesmentioning

confidence: 99%

“…For the topology ß0 we have the analogue of the descriptions found in [6] and [35] of the ß-equicontinuous sets(e). Given (a), it is clear that H is bounded in norm since the norm bounded set Bx in C(X) is ßo-bounded.…”

Section: Definitions and Preliminariesmentioning

confidence: 99%

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Bounded Continuous Functions on a Completely Regular Space

Sentilles¹

1972

Transactions of the American Mathematical Society

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138

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Abstract. Three locally convex topologies on C(X) are introduced and developed, and in particular shown to coincide with the strict topology on locally compact A1 and yield dual spaces consisting of tight, -r-additive and x where p. is a bounded regular Borel measure on X. This yields a very satisfactory relationship between the topology on X, the space C(X), a natural class of linear functionals on it, and those measures on Jfthat measure at least the sets determined by the topology on X in the usual way, the Borel sets.This kind of representation was subsequently extended to locally compact spaces : first to functionals on the space C0(X) of continuous functions vanishing at infinity, and then further, to the bounded continuous functions on X. The last result, due to R. C. Buck, demanded the use of a locally convex topology, the strict topology, rather than a norm topology. In both extensions the same satisfactory relationship between measure and topology was obtained.In this paper we begin the development of locally convex topologies for C(X) which extend this kind of representation to its last reasonable setting, completely regular Hausdorff spaces. This setting appears to be ultimate in the sense of Hewitt's example of a regular space upon which the only continuous functions are constants. Definitions and preliminaries.The actual work of integral representation of linear forms has been done by other authors, going back to Aleksandrov [1] and, following his work, by Varadarajan [39], and later Knowles [21] and more recently Kirk [20] and Moran ([24], [25]). Our work relies heavily on theirs and will not extend the representations they have obtained but will relate these works to earlier

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Section: Definitions and Preliminariessupporting

confidence: 57%

“…The proof will be a good deal clearer if we retrieve the following special result in the locally compact setting; this lemma can also be observed as a Corollary to [35,Theorem 4.1].…”

Section: Definitions and Preliminariesmentioning

confidence: 90%

Section: Definitions and Preliminariesmentioning

confidence: 99%

Section: Definitions and Preliminariesmentioning

confidence: 99%

Section: Definitions and Preliminariesmentioning

confidence: 99%

See 3 more Smart Citations

Bounded Continuous Functions on a Completely Regular Space

Sentilles¹

1972

Transactions of the American Mathematical Society

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138

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Locally convex topologies on spaces of continuous vector functions

Katsaras

1976

Mathematische Nachrichten

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The space of vector‐valued integrable functions under certain locally convex topologies

Maghsoudi

2012

Mathematische Nachrichten

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Let E be a Banach space, Ω a locally compact space, and μ a positive Radon measure on Ω. In this paper, through extending to Lebesgue‐Bochner spaces, we show that the topology β1 introduced by Singh is a type of strict topology. We then investigate various properties of this locally convex topology. In particular, we show that the strong dual of L1(μ, E) can be identified with a Banach space. We also show that the topology β1 is a metrizable, barrelled or bornological space if and only if Ω is compact. Finally, we consider the generalized group algebra \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$L^1(G, \mathbf {A})$\end{document} under certain natural locally convex topologies. As an application of our results, we prove that \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$L^1(G,\mathbf {A})$\end{document} under the topology β1 is a complete semi‐topological algebra.

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Factorization in Banach algebras and the general strict topology

Cited by 60 publications

References 12 publications

Bounded Continuous Functions on a Completely Regular Space

Bounded Continuous Functions on a Completely Regular Space

Locally convex topologies on spaces of continuous vector functions

The space of vector‐valued integrable functions under certain locally convex topologies

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