Spectral distances on RCD spaces (Q6635469)

In this paper the author studies the link between several ``spectral distances''. In particular, in [Geom. Funct. Anal. 4, No. 4, 373--398 (1994; Zbl 0806.53044)] \textit{P. Bérard} et al. have defined a spectral distance \(d^t_{\mathrm{Spec}}\) between two smooth manifolds and the author asks the two questions:\N\N(Q1) Is it possible to generalize the spectral distance \(d^t_{\mathrm{Spec}}\) to a nonsmooth setting?\\\N(Q2) Can we give a necessary and sufficient condition for the validity of \(d^t_{\mathrm{Spec}}\)-convergence in terms of the measured Gromov-Hausdorff (mGH) convergence?\N\NHe gives a positive answer to these questions in the frame of metric measured spaces with Ricci curvature bounded from below, the so-called RCD spaces. Then, considering another notion of spectral convergence of Riemannian manifolds (also based on the heat kernels) given by \textit{A. Kasue} and \textit{H. Kumura} [Tôhoku Math. J. (2) 46, No. 2, 147--179 (1994; Zbl 0814.53035); Tôhoku Math. J. (2) 48, No. 1, 71--120 (1996; Zbl 0853.58100)], the author studies the following question:\N\N(Q3) After generalizing the spectral distance in the sense of Kasue-Kumura to a nonsmooth setting, can we give a necessary and sufficient condition for the validity of this spectral-convergence in terms of the mGH convergence?\N\NThe second main result, Theorem 1.4, provides a positive answer to (Q3).