Continuity of the Spectrum of Dirac Operators of Spectral Triples for the Spectral Propinquity

The spectral propinquity is a distance, up to unitary equivalence, on the class of metric spectral triples. We prove in this paper that if a sequence of metric spectral triples converges for the propinquity, then the spectra of the Dirac operators for these triples do converge to the spectrum of the Dirac operator at the limit. We obtain this result by first proving that, in an appropriate sense induced by some natural metric, the bounded, continuous functional calculi defined by the Dirac operators also converge. As an application of our work, we see, in particular, that action functionals of a wide class of metric spectral triples are continuous for the spectral propinquity, which clearly connects convergence for the spectral propinquity with the applications of noncommutative geometry to mathematical physics. This fact is a consequence of results on the continuity of the multiplicity of eigenvalues of Dirac operators. In particular, we formalize convergence of adjoinable operators of different C*-correspondences, endowed with appropriate quantum metric data.