nLab noncommutative residue

Contents

Contents

Idea

The 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?.

References

Last revised on October 25, 2023 at 10:08:05. See the history of this page for a list of all contributions to it.