Dixmier trace


In mathematics, the Dixmier trace, introduced by, is a non-normal trace on a space of linear operators on a Hilbert space larger than the space of trace class operators. Dixmier traces are examples of singular traces.
Some applications of Dixmier traces to noncommutative geometry are described in.

Definition

If H is a Hilbert space, then L1,∞ is the space of compact linear operators T on H such that the norm
is finite, where the numbers μi are the eigenvalues of |T| arranged in decreasing order. Let
The Dixmier trace Trω of T is defined for positive operators T of L1,∞ to be
where limω is a scale-invariant positive "extension" of the usual limit, to all bounded sequences. In other words, it has the following properties:
  • limω ≥ 0 if all αn ≥ 0
  • limω = lim whenever the ordinary limit exists
  • limω = limω
There are many such extensions so there are many different Dixmier traces.
As the Dixmier trace is linear, it extends by linearity to all operators of L1,∞.
If the Dixmier trace of an operator is independent of the choice of limω then the operator is called measurable.

Properties

  • Trω is linear in T.
  • If T ≥ 0 then Trω ≥ 0
  • If S is bounded then Trω = Trω
  • Trω does not depend on the choice of inner product on H.
  • Trω = 0 for all trace class operators T, but there are compact operators for which it is equal to 1.
A trace φ is called normal if φ = sup φ for every bounded increasing directed family of positive operators. Any normal trace on is equal to the usual trace, so the Dixmier trace is an example of a non-normal trace.

Examples

A compact self-adjoint operator with eigenvalues 1, 1/2, 1/3,... has Dixmier trace equal to 1.
If the eigenvalues μi of the positive operator T have the property that
converges for Re>1 and extends to a meromorphic function near s=1 with at most a simple pole at s=1, then the Dixmier trace
of T is the residue at s=1.
showed that Wodzicki's noncommutative residue of a pseudodifferential operator on a manifold M of order -dim is equal to its Dixmier trace.