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,∞(H) is the space of compact linear operators T on H such that the norm is finite, where the numbers μi(T) are the eigenvalues of |T| arranged in decreasing order. Let The Dixmier trace Trω(T) of T is defined for positive operators T of L1,∞(H) 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: There are many such extensions (such as a Banach limit of α1, α2, α4, α8,...) so there are many different Dixmier traces. As the Dixmier trace is linear, it extends by linearity to all operators of L1,∞(H). If the Dixmier trace of an operator is independent of the choice of limω then the operator is called measurable.

Properties

A trace φ is called normal if φ(sup xα) = sup φ( xα) 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(s)>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 (and in particular is independent of the choice of ω). showed that Wodzicki's noncommutative residue of a pseudodifferential operator on a manifold M of order -dim(M) is equal to its Dixmier trace.