A noncommutative view on topology and order

TL;DR

This paper introduces isocones in unital C*-algebras as a noncommutative generalization of order structures, establishing a duality similar to Gelfand-Naimark and classifying isocones in 2x2 matrices.

Contribution

It defines isocones as a new structure on C*-algebras, extending order and topology concepts into the noncommutative setting with a duality framework.

Findings

Isocones generalize continuous non-decreasing functions to noncommutative algebras.

A duality between isocones and certain ordered spaces is established.

Classification of isocones in M_2(C) is provided.

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).