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

${\displaystyle \|T\|_{1,\infty }=\sup _{N}{\frac {\sum _{i=1}^{N}\mu _{i}(T)}{\log(N)}}}$

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

${\displaystyle a_{N}={\frac {\sum _{i=1}^{N}\mu _{i}(T)}{\log(N)}}}$.

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

${\displaystyle \operatorname {Tr} _{\omega }(T)=\lim _{\omega }a_{N}}$

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 ${\displaystyle L^{1,\infty }(H)}$ 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

${\displaystyle \zeta _{T}(s)=\operatorname {Tr} (T^{s})=\sum {\mu _{i}^{s}}}$

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.