In mathematics, a singular trace is a trace on a space of linear operators of a separable Hilbert space that vanishes on operators of finite rank. Singular traces are a feature of infinite-dimensional Hilbert spaces such as the space of square-summable sequences and spaces of square-integrable functions. Linear operators on a finite-dimensional Hilbert space have only the zero functional as a singular trace since all operators have finite rank. For example, matrix algebras have no non-trivial singular traces and the matrix trace is the unique trace up to scaling. American mathematician Gary Weiss and, later, British mathematician Nigel Kalton observed in the infinite-dimensional case that there are non-trivial singular traces on the ideal of trace class operators. Therefore, in distinction to the finite-dimensional case, in infinite dimensions the canonical operator trace is not the unique trace up to scaling. The operator trace is the continuous extension of the matrix trace from finite rank operators to all trace class operators, and the term singular derives from the fact that a singular trace vanishes where the matrix trace is supported, analogous to a singular measure vanishing where Lebesgue measure is supported. Singular traces measure the asymptotic spectral behaviour of operators and have found applications in the noncommutative geometry of French mathematician Alain Connes. In heuristic terms, a singular trace corresponds to a way of summing numbers a1, a2, a3, ... that is completely orthogonal or 'singular' with respect to the usual sum a1 + a2 + a3 + ... . This allows mathematicians to sum sequences like the harmonic sequence (and operators with similar spectral behaviour) that are divergent for the usual sum. In similar terms a (noncommutative) measure theory or probability theory can be built for distributions like the Cauchy distribution (and operators with similar spectral behaviour) that do not have finite expectation in the usual sense.
Origin By 1950 French mathematician Jacques Dixmier, a founder of the semifinite theory of von Neumann algebras, thought that a trace on the bounded operators of a separable Hilbert space would automatically be normal up to some trivial counterexamples. 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 trace on weak trace-class operators, disproving his earlier view. Singular traces based on Dixmier's construction are called Dixmier traces. Independently and by different methods, German mathematician Albrecht Pietsch (de) investigated traces on ideals of operators on Banach spaces. In 1987 Nigel Kalton answered a question of Pietsch by showing that the operator trace is not the unique trace on quasi-normed proper subideals of the trace-class operators on a Hilbert space. József Varga independently studied a similar question. 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. A consequence of the results of Weiss and the spectral condition of Kalton was the existence of non-trivial singular traces on trace class operators. 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.
Definition A trace φ on a two-sided ideal J of the bounded linear operators B(H) on a separable Hilbert space H is a linear functional φ:J → C {\displaystyle \mathbb {C} } such that φ(AB) = φ(BA) for all operators A from J and B from B(H). That is, a trace is a linear functional on J that vanishes on the commutator subspace Com(J) of J. A trace φ is singular if φ(A) = 0 for every A from the subideal of finite rank operators F(H) within J.
Existence and characterisation 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, 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. Here μ: J+ → j+ is the map from a positive operator to its singular values. A singular trace φ corresponds to a symmetric functional f on the sequence space j that vanishes on c00, the sequences with a finite number of non-zero terms. The characterisation parallels the construction of the usual operator trace where
T r ( A ) = ∑ n = 0 ∞ μ ( n , A ) = ∑ μ ( A ) {\displaystyle {\rm {Tr}}(A)=\sum _{n=0}^{\infty }\mu (n,A)=\sum \mu (A)}
for A a positive trace class operator. The trace class operators and the sequence space of summable sequences are in Calkin correspondence. (The sum Σ is a symmetric functional on the space of summable sequences.)
Existence 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. There are ideals that admit infinitely many linearly independent non-zero singular traces. For example, the commutator subspace of the ideal of weak trace-class operators contains the ideal of trace class operators and every positive operator in the commutator subspace of the weak trace class is trace class. Consequently, every trace on the weak trace class ideal is singular and the co-dimension of the weak trace class ideal commutator subspace is infinite. Not all of the singular traces on the weak trace class ideal are Dixmier traces.
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