A noncommutative view on topology and order

Abstract

In this paper we put forward the definition of particular subsets on a unital C*-algebra, that we call isocones, and which reduce in the commutative case to the set of continuous non-decreasing functions with real values for a partial order relation defined on the spectrum of the algebra, which satisfies a compatibility condition with the topology (complete separateness). We prove that this space/algebra correspondence is a dual equivalence of categories, which is in fact only a mild generalization of the Gelfand-Naimark duality. Thus we can expect that general isocones could serve to define a notion of noncommutative ordered spaces. We also explore some basic algebraic constructions involving isocones, and classify those which are defined in M_2(C).

31 pages. To appear in Journal of Geometry and Physics

References in corpus (3)

• Further results on the smoothability of Cauchy hypersurfaces and Cauchy time functions
• Causal Sets: Discrete Gravity (Notes for the Valdivia Summer School)
• Canonical quantization and the spectral action, a nice example

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