A domain-theoretic Banach–Alaoglu theorem

Plotkin, Gordon

doi:10.1017/s0960129506005172

Cited by 13 publications

(14 citation statements)

References 16 publications

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“…Plotkin's Banach-Alaoglu Theorem [16] is also subsumed by the previous example. In fact, for d-cones scalar multiplication always preserves the way-below relation and scott-continuous additive maps are automatically homogeneous.…”

Section: Conesmentioning

confidence: 86%

“…Instead of linear functionals we consider continuous algebra homomorphisms into a test algebra R replacing the reals. Our main result, Theorem 5.8, concerns the compactness of the space C * of continuous homomorphisms from X to R. It extends Plotkin's Theorem 2 in [16]. The methods rely on an idea due to A. Jung [2].…”

Section: Introductionmentioning

confidence: 90%

“…We call it the lower-open topology following Plotkin [16] in the case where R = R + . By the preceding proposition it has a subbasis for the closed sets of the form…”

Section: Function Spacesmentioning

confidence: 99%

“…A binary operation + preserves the way-below relation if x 1 y 1 , x 2 y 2 =⇒ x 1 + x 2 y 1 + y 2 ; in this case, the way-below relation has been called additive in [16].…”

Section: A General Bourbaki-alaoglu Theoremmentioning

confidence: 99%

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Weak topologies and compactness in asymmetric functional analysis

Keimel

2015

Topology and its Applications

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The classical Bourbaki-Alaoglu theorem asserts that the polar U • of a neighborhood of 0 in a locally convex topological vector space V is a weak * compact subset of the dual space V * . G. Plotkin has proved an analogous theorem in the framework of continuous d-cones, a kind of asymmetric version of topological vector spaces in Domain Theory. In this paper we extend Plotkin's results. We consider topological spaces X with a finitary continuous algebraic structure in the sense of universal algebra instead of a cone structure. Linear functionals are replaced by continuous algebra homomorphisms into a test algebra R replacing the reals. Our main result, Theorem 5.8, concerns the compactness of the space C * of continuous homomorphisms from X to R under appropriate hypotheses. We exhibit conditions under which C * is not only a space but also an algebra, as in the classical situation. This leads us to the notion of entropicity in the sense of universal algebra. The background for our investigation is Domain Theory and its use in denotational semantics (see [8]). Thus our spaces are strongly non-Hausdorff. This paper can be seen as a contribution to asymmetric topology and analysis.

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Section: Conesmentioning

confidence: 86%

Section: Introductionmentioning

confidence: 90%

“…We call it the lower-open topology following Plotkin [16] in the case where R = R + . By the preceding proposition it has a subbasis for the closed sets of the form…”

Section: Function Spacesmentioning

confidence: 99%

“…A binary operation + preserves the way-below relation if x 1 y 1 , x 2 y 2 =⇒ x 1 + x 2 y 1 + y 2 ; in this case, the way-below relation has been called additive in [16].…”

Section: A General Bourbaki-alaoglu Theoremmentioning

confidence: 99%

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Weak topologies and compactness in asymmetric functional analysis

Keimel

2015

Topology and its Applications

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“…Taking the lower semicontinuous envelope yields a continuous retraction on the the lower semicontinuous monoid homomorphisms. This technique has first been applied by Jung [2] and is heavily used in [28,21]. In [8] it is mentioned that in the proof of Theorem 3.7 on the compactness of the space of traces the same idea has been communicated to the authors by E. Kirchberg.…”

Section: The Dual M * Of a Precuntz Semigroupmentioning

confidence: 99%

The Cuntz semigroup and domain theory

Keimel

2017

Soft Comput

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Domain theory has its origins in Mathematics and Theoretical Computer Science. Mathematically it combines order and topology. Its central concepts have their origin in the idea of approximating ideal objects by their relatively finite or, more generally, relatively compact parts.The development of domain theory in recent years was mainly motivated by question in denotational semantics and the theory of computation. But since 2008, domain theoretical notions and methods are used in the theory of C * -algebras in connection with the Cuntz semigroup. This paper is largely expository. It presents those notions of domain theory that seem to be relevant for the theory of Cuntz semigroups and have sometimes been developed independently in both communities. It also contains a new aspect in presenting results of Elliott, Ivanescu and Santiago on the cone of traces of a C * -algebra as a particular case of the dual of a Cuntz semigroup.

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