nLab noncommutative residue

Idea

The noncommutative residue or Wodzicki residue is (up to a constant multiple) the only nontrivial trace functional $Res$ on the algebra of pseudodifferential operators (of arbitrary order) on a compact smooth manifold $M$. For a $\psi DO$ $P$ it can be computed as the coefficient in front of $log t$ in the asymptotic expansion of $Tr (P e^{-t \Delta})$, where $\Delta$ is the Laplacian, or equivalently, in terms of the usual residue, $res_{s=0} Tr(P \Delta^{-s})$.

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

References

• Mariusz Wodzicki, Spectral asymmetry and noncommutative residue, PhD thesis, Steklov Institute (1984)

• Mariusz Wodzicki, Noncommutative residue. I. Fundamentals, in: K-theory, arithmetic and geometry, Lecture Notes in Math. 1289, Springer (1987) 320-399 [doi:10.1007%2FBFb0078372, MR923140]

• 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.

category: analysis, noncommutative geometry