An algebraic formulation of causality for noncommutative geometry

We propose an algebraic formulation of the notion of causality for spectral triples corresponding to globally hyperbolic manifolds with a well-defined noncommutative generalization. The causality is given by a specific cone of Hermitian elements respecting an algebraic condition based on the Dirac operator and a fundamental symmetry. We prove that in the commutative case the usual notion of causality is recovered. We show that, when the dimension of the manifold is even, the result can be extended in order to have an algebraic constraint suitable for a Lorentzian distance formula.