Noncommutative residue invariants for CR and contact manifolds

Noncommutative Residue for Heisenberg Manifolds. I.

In this paper we construct a noncommutative residue for the Heisenberg calculus, that is, for the hypoelliptic calculus on Heisenberg man-ifolds, including on CR and contact manifolds. This noncommutative residue as the residual induced on operators of integer orders by the analytic extension of the usual trace to operators of non-integer orders and it agrees with the integral of the density de...

Trace Expansions and the Noncommutative Residue for Manifolds with Boundary

For a pseudodifferential boundary operator A of order ν ∈ Z and class 0 (in the Boutet de Monvel calculus) on a compact n-dimensional manifold with boundary, we consider the function Tr(AB−s), where B is an auxiliary system formed of the Dirichlet realization of a second order strongly elliptic differential operator and an elliptic operator on the boundary. We prove that Tr(AB−s) has a meromorp...

Dynamical Invariants for Noncommutative Flows

We show that semigroups of endomorphisms of B(H) can often be associated with a dynamical principle; that is, an infinitesimal structure that completely determines the dynamics. These dynamical invariants are similar to the second order differential equations of classical mechanics, in that one can locate a space of momentum operators, a “Riemannian metric”, and a potential. In the simplest cas...

S ep 2 00 6 Gravity and the Noncommutative Residue for Manifolds with Boundary ∗

We prove a Kastler-Kalau-Walze type theorem for the Dirac operator and the signature operator for 3, 4-dimensional manifolds with boundary. As a corollary, we give two kinds of operator theoretic explanations of the gravitational action in the case of 4-dimensional manifolds with flat boundary. Subj. Class.: Noncommutative global analysis; Noncommutative differential geometry. MSC: 58G20; 53A30...

Cr-invariants and the Scattering Operator for Complex Manifolds with Boundary