Laplace–Beltrami operators on noncommutative tori

Abstract

In this paper, we construct Laplace-Beltrami operators associated with arbitrary Riemannian metrics on noncommutative tori of any dimension. These operators enjoy the main properties of the Laplace-Beltrami operators on ordinary Riemannian manifolds. The construction takes into account the non-triviality of the group of modular automorphisms. On the way we introduce notions of Riemannian density and Riemannian volumes for noncommutative tori.

Pseudodifferential calculus on noncommutative tori, I. Oscillating integrals

“…In [69] a Riemannian metric on A θ is given by a positive invertible matrix g " pg ij q P M n pA θ q whose entries are selfadjoint elements of A θ . Its determinant is defined by detpgq :" exp`Trrlogpgqs˘, where logpgq P M n pA θ q is defined by holomorphic functional calculus and Tr is the matrix trace (see [45]). The determinant detpgq is a positive invertible element of A θ , and so νpgq :" a detpgq is a positive invertible element of A θ .…”

“…The determinant detpgq is a positive invertible element of A θ , and so νpgq :" a detpgq is a positive invertible element of A θ . Let g´1 " pg ij q be the inverse matrix of g. In [45] the Laplace-Beltrami operator associated with g is the 2nd order differential operator ∆ g : A θ Ñ A θ given by…”

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

“…We seek for a curved version of the flat integration formula of [43], i.e., an extension involving the Laplace-Beltrami operator ∆ g associated with an arbitrary Riemannian metric g on T n θ which was recently constructed in [33]. We refer to Section 9 for the precise definition of Riemannian metrics on T n θ in the sense of [33,48]. They are given by symmetric positive invertible matrices g " pg ij q with entries in A θ .…”

“…Thus, if we let Hg be the Hilbert space of the GNS representation of Aθ, then have a˚-representation a Ñ a˝of A θ in Hθ , where a˝" J g a˚J g " νpgq´1 2 aνpgq 1 2 acts by left multiplication. The Laplace-Beltrami operator ∆ g : A θ Ñ A θ of [33] is an elliptic 2nd order differential operator with principal symbol νpgq´1 2 |ξ| 2 g νpgq 1 2 , where |ξ| g :" p ř g ij ξ i ξ j q 1 2 and g´1 " pg ij q is the inverse matrix of g. This is also an essentially selfadjoint operator on Hg with non-negative spectrum. Our curved integration formula (Theorem 10.6) then states that, for every a P A θ , the operator a˝∆´n 2 g is strongly measurable, and we have (1.4)ż a˝∆´n 2 g "ĉ n τ " aνpgq ‰ ,ĉ n :" 1 n |S n´1 |, whereνpgq :" |S n´1 |´1 ş S n´1 |ξ|´n g d n´1 ξ is the so-called spectral Riemannian density (see Section 10).…”

“…[25,33]). The determinant of any h P GLǹ pA θ q is defined bydetphq " exp " Trplog hq ‰ , h P GLǹ pA θ q P A`θ ,…”