noncommutative residue

Noncommutative residue or Wodzicki residue is (up to a constant multiple) the only nontrivial trace functional ResRes on the algebra of pseudodifferential operators (of arbitrary order) on a compact smooth manifold MM. For a ψDO\psi DO PP it can be computed as the coefficient in front of logtlog t in the asymptotic expansion of Tr(Pe tΔ)Tr (P e^{-t \Delta}), where Δ\Delta is the Laplacian, or equivalently, in terms of the usual residue, res s=0Tr(PΔ s)res_{s=0} Tr(P \Delta^{-s}).

If MM is Riemannian, PP an elliptic operator of the negative integer order equal dimM-dim M then, by a result of Connes, the Wodzicki residue coincides with the Dixmier trace?.

  • Mariusz Wodzicki?, Spectral asymmetry and noncommutative residue, PhD thesis, Steklov Institute 1984; Noncommutative residue. I. Fundamentals, in: K-theory, arithmetic and geometry (Moscow, 1984–1986), Lecture Notes in Math. 1289, pp. 320–399, Springer MR923140 doi
  • eom: Wodzicki residue
  • wikipedia noncommutative residue
  • Nigel Kalton, Steven Lord, Denis Potapov, Fedor Sukochev, Traces of compact operators and the noncommutative residue, Adv. Math. 235 (2013) 1-55, arxiv/1210.3423 doi
  • Alain Connes, ; Noncommutative geometry, Academic Press 1984.

Created on January 10, 2014 at 06:43:35. See the history of this page for a list of all contributions to it.