# Dixmier trace

In mathematics, the Dixmier trace, introduced by Jacques Dixmier (1966), 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 (Connes 1994).

## Definition

If H is a Hilbert space, then L1,∞(H) is the space of compact linear operators T on H such that the norm

$\|T\|_{1,\infty} = \sup_N\frac{\sum_{i=1}^N \mu_i(T)}{\log(N)}$

is finite, where the numbers μi(T) are the eigenvalues of |T| arranged in decreasing order. Let

$a_N = \frac{\sum_{i=1}^N \mu_i(T)}{\log(N)}$.

The Dixmier trace Trω(T) of T is defined for positive operators T of L1,∞(H) to be

$\operatorname{Tr}_\omega(T)= \lim_\omega a_N$

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(αn) ≥ 0 if all αn ≥ 0 (positivity)
• limω(αn) = lim(αn) whenever the ordinary limit exists
• limω(α1, α1, α2, α2, α3, ...) = limω(αn) (scale invariance)

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

• Trω(T) is linear in T.
• If T ≥ 0 then Trω(T) ≥ 0
• If S is bounded then Trω(ST) = Trω(TS)
• Trω(T) does not depend on the choice of inner product on H.
• Trω(T) = 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 xα) = sup φ( xα) for every bounded increasing directed family of positive operators. Any normal trace on $L^{1,\infty}(H)$ 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

$\zeta_T(s)= \operatorname{Tr}(T^s)= \sum{\mu_i^s}$

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).

Connes (1988) showed that Wodzicki's noncommutative residue (Wodzicki 1984) of a pseudodifferential operator on a manifold is equal to its Dixmier trace.