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Weak topologies and compactness in asymmetric functional analysis. (English) Zbl 1323.46049

The Banach-Alaoglu theorem asserts that the polar of any neighborhood of the origin of a locally convex space is compact for the weak\(^*\)-topology. G. Plotkin [Math. Struct. Comput. Sci. 16, No. 2, 299–311 (2006; Zbl 1107.06006)] extended this result to \(d\)-cones – continuous domains endowed with a compatible cone structure. In this paper, the author works with universal algebras with a signature \(\Omega\) of operations, each of preassigned arity. He supposes the universal algebra \(A\) endowed with an order compatible to each operation in \(\Omega\) and with a topology for which the mapping \(\omega:A^n\to A\) is continuous for each \(\omega\in\Omega\). The author extends to this framework notions and results from semantic domain theory and proves an Alaoglu-Bourbaki type theorem.
Let us quote from the abstract: “We consider topological spaces \(X\) with a finitary continuous algebraic structure in the sense of a 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. Thus our spaces are strongly non-Hausdorff. This paper can be seen as a contribution to asymmetric topology and analysis.”

MSC:

46S99 Other (nonclassical) types of functional analysis
06B35 Continuous lattices and posets, applications
08A70 Applications of universal algebra in computer science
22A99 Topological and differentiable algebraic systems
46A50 Compactness in topological linear spaces; angelic spaces, etc.
46A40 Ordered topological linear spaces, vector lattices
68Q55 Semantics in the theory of computing

Citations:

Zbl 1107.06006
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References:

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