Some applications of Dixmier traces to noncommutative geometry are described in (Connes 1994).
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.
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.
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).