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).

If H is a Hilbert space, then L^1,∞(H) is the space of compact linear operators T on H such that the norm

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

The Dixmier trace Tr_ω(T) of T is defined for positive operators T of L^1,∞(H) to be

where lim_ω 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 L^1,∞(H). If the Dixmier trace of an operator is independent of the choice of lim_ω then the operator is called measurable.

A trace φ is called normal if φ(sup x_α) = sup φ( x_α) 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.

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