Connes's trace theorem for curved noncommutative tori: Application to scalar curvature

In this paper, we prove a version of Connes's trace theorem for noncommutative tori of any dimension n > 2. This allows us to recover and improve earlier versions of this result in dimensions n = 2 and n = 4 by Fathizadeh and Khalkhali. We also recover Connes's integration formula for flat noncommutative tori of McDonald, Sukochev, and Zanin. As a further application, we prove a curved version of this integration formula in terms of the Laplace-Beltrami operator defined by an arbitrary Riemannian metric. For the class of the so-called self-compatible Riemannian metrics (including the conformally flat metrics of Connes and Tretkoff), this shows that Connes's noncommutative integral allows us to recover the Riemannian density. This exhibits a neat link between this notion of noncommutative integral and noncommutative measure theory in the sense of operator algebras. As an application of these results, we set up a natural notion of scalar curvature for curved noncommutative tori.