  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).
