Noncommutative Residues, Dixmier's Trace, and Heat Trace Expansions on Manifolds with Boundary

A Noncommutative Residue on Tori and a Semiclassical Limit

We define a noncommutative residue for classical Euclidean pseudodifferential operators on a torus of arbitrary dimension. We prove that, up to multiplication by a constant, it is the unique trace on the algebra of classical pseudodifferential operators modulo infinitely smoothing operators. In the case of the two torus, we show that the noncommutative residue is the semiclassical limit of a noncommutative residue defined on classical pseudodifferential operators on noncommutative two tori.

The canonical trace and the noncommutative residue on the noncommutative torus

Using a global symbol calculus for pseudodifferential operators on tori, we build a canonical trace on classical pseudodifferential operators on noncommutative tori in terms of a canonical discrete sum on the underlying toroidal symbols. We characterise the canonical trace on operators on the noncommutative torus as well as its underlying canonical discrete sum on symbols of fixed (resp.\ any) non--integer order. On the grounds of this uniqueness result, we prove that in the commutative setup, this canonical trace on the noncommutative torus reduces to Kontsevich and Vishik's canonical trace w…

The noncommutative residue and canonical trace in the light of Stokes' and continuity properties

We show that the noncommutative residue density, resp. the cut-off regularised integral are the only closed linear, resp. continuous closed linear forms on certain classes of symbols. This leads to alternative proofs of the uniqueness of the noncommutative residue, resp. the canonical trace as linear, resp. continuous linear forms on certain classes of classical pseudodifferential operators which vanish on brackets. The uniqueness of the canonical trace actually holds on classes of classical pseudodifferential with vanishing residue density which include non integer order operators in all dime…

关于锥流形上的非交换留数与热迹展开

On the Noncommutative Residue and the Heat Trace Expansion on Conic Manifolds

本文研究了锥流形上锥伪微分算子的热迹渐近展开，显式给出了包含 $\log t$ 和 $(\log t)^2$ 新项的完整展开式，并据此恢复了自然代数上的所有迹泛函。

Trace Expansions and the Noncommutative Residue for Manifolds with Boundary

For a pseudodifferential boundary operator A of integer order νand class zero (in the Boutet de Monvel calculus) on a compact n-dimensional manifold with boundary, we consider the function Trace(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 Trace(AB^{-s}) has a meromorphic extension to the complex plane with poles at the half-integers s = (n+ν-j)/2, j = 0,1,... (possibly double for s<0), and we prove that its residue at zero equals the noncommutative r…

An equivariant noncommutative residue

Let $\gp$ be a finite group acting on a compact manifold $M$ and $\maA(M)$ denote the algebra of classical complete symbols on $M$. We determine all traces on the cross-product algebra $\maA(M) \rtimes Γ$. These traces appear as residues of certain meromorphic 'zeta' functions and can be considered as equivariant generalization of the non-commutative residue trace. The local formula for these traces depends on more than one component of the complete asymptotic expansion. For instance, the local formula for these traces depends also on derivatives in the normal directions to fixed point manifol…