Singular traces measure the asymptotic spectral behaviour of operators and have found applications in the noncommutative geometry of French mathematician Alain Connes.
By 1950 French mathematician Jacques Dixmier, a founder of the semifinite theory of von Neumann algebras,[5] thought that a trace on the bounded operators of a separable Hilbert space would automatically be normal[clarification needed] up to some trivial counterexamples.
[6]: 217 Over the course of 15 years Dixmier, aided by a suggestion of Nachman Aronszajn and inequalities proved by Joseph Hersch, developed an example of a non-trivial yet non-normal[clarification needed] trace on weak trace-class operators,[7] disproving his earlier view.
Independently and by different methods, German mathematician Albrecht Pietsch (de) investigated traces on ideals of operators on Banach spaces.
[10] To solve the question of uniqueness of the trace on the full ideal of trace-class operators, Kalton developed a spectral condition for the commutator subspace of trace class operators following on from results of Gary Weiss.
[2][6]: 185 Also independently, and from a different direction, Mariusz Wodzicki investigated the noncommutative residue, a trace on classical pseudo-differential operators on a compact manifold that vanishes on trace class pseudo-differential operators of order less than the negative of the dimension of the manifold.
A trace φ is singular if φ(A) = 0 for every A from the subideal of finite rank operators F(H) within J. Singular traces are characterised by the spectral Calkin correspondence between two-sided ideals of bounded operators on Hilbert space and rearrangement invariant sequence spaces.
Using the spectral characterisation of the commutator subspace due to Ken Dykema, Tadeusz Figiel, Gary Weiss and Mariusz Wodzicki,[12] to every trace φ on a two-sided ideal J there is a unique symmetric functional f on the corresponding Calkin sequence space j such that for every positive operator A belonging to J.
A non-zero trace φ exists on a two-sided ideal J of operators on a separable Hilbert space if the co-dimension of its commutator subspace is not zero.
Equation (3) is the precise statement that singular traces measure asymptotic spectral behaviour of operators.
For example, in quantum statistical mechanics the expectation of an observable S is calculated against a fixed trace-class energy density operator T by the formula where vT belongs to (l∞)* ≅ l1.
Here Pn is the projection operator onto the one-dimensional subspace spanned by the energy eigenstate en.
[15] For a product ST where S is bounded and T is selfadjoint and belongs to a two-sided ideal J then for any trace φ on J.