A noncommutative view on topology and order

Understand

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).