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 generalizes 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).

In this two-part paper we propose an extension of Connes' notion of even spectral triple to the Lorentzian setting. This extension, which we call a spectral spacetime, is discussed in part II where several natural examples are given which are not covered by the previous approaches to the problem. Part I only deals with the commutative and continuous case of a manifold. It contains all the necessary material for the generalization to come in part II, namely the characterization of the signature of the metric in terms of a time-orientation 1-form and a natural Krein product on spinor fields. It turns out that all the data available in Noncommutative Geometry (the algebra of functions, the Krein space of spinor fields, the representation of the algebra on it, the Dirac operator, charge conjugation and chirality), but nothing more, play a role in this characterization. Thus, only space and time oriented spin manifolds of even dimension are considered for a noncommutative generalization in this approach. We observe that these are precisely the kind of manifolds on which the modern theories of spacetime and matter are defined.

We investigate the representation of diffeomorphisms in Connes' Spectral Triples formalism. By encoding the metric and spin structure in a moving frame, it is shown on the paradigmatic example of spin semi-Riemannian manifolds that the bimodule of noncommutative 1-forms Ω 1 is an invariant structure in addition to the chirality, real structure and Krein product. Adding Ω 1 and removing the Dirac operator from an indefinite Spectral Triple we obtain a structure which we call an algebraic background. All the Dirac operators compatible with this structure then form the configuration space of a noncommutative Kaluza-Klein theory. In the case of the Standard Model, this configuration space is stricty larger than the one obtained from the fluctuations of the metric, and contains in addition to the usual gauge fields the Z ′ B−L -boson, a complex scalar field σ, which is known to be required in order to obtain the correct Higgs mass in the Spectral Standard Model, and flavour changing fields. The latter are invariant under automorphisms and can be removed without breaking the symmetries. It is remarkable that, starting from the conventional Standard Model algebra C ⊕ H ⊕ M3(C), the "accidental" B − L symmetry is necessarily gauged in this framework.

We define almost-commutative algebraic backgrounds and give conditions on them allowing us to compute their configuration space in terms of those of the continuous and finite parts. We apply these results to a background with finite algebra C⊕H⊕M3(C) and find that the configuration space is larger than the one obtained from the fluctuations of the metric and contains in addition to the Standard Model (SM) gauge fields, the ZB-L′-boson, a complex scalar field σ, and flavor changing fields. The latter can be removed similarly to centralizing fields in the gravity model studied in the first part. The remaining fields belong to a U(1)B-L-extension of the SM.

An analogy with real Clifford algebras on even-dimensional vector spaces suggests to assign a couple of space and time dimensions modulo 8 to any algebra (represented over a complex Hilbert space) containing two self-adjoint involutions and an anti-unitary operator with specific commutation relations.It is shown that this assignment is compatible with the tensor product: the space and time dimensions of the tensor product are the sums of the space and time dimensions of its factors. This could provide an interpretation of the presence of such algebras in P T -symmetric Hamiltonians or the description of topological matter.This construction is used to build an indefinite (i.e. pseudo-Riemannian) version of the spectral triples of noncommutative geometry, defined over Krein spaces instead of Hilbert spaces. Within this framework, we can express the Lagrangian (both bosonic and fermionic) of a Lorentzian almost-commutative spectral triple. We exhibit a space of physical states that solves the fermion-doubling problem. The example of quantum electrodynamics is described.