Advances in Dixmier traces and applications

Jacques Dixmier constructed a trace in the 1960s on an ideal larger than the trace class. In 1988 Alain Connes developed Dixmier’s trace and used it centrally in noncommutative geometry, extending classical Yang-Mills actions, the noncommutative residue of Adler, Manin, Wodzicki and Guillemin, and integration of differential forms. Independent of Dixmier’s construction and Connes development, Albrecht Pietsch identified a bijective correspondence between traces on two-sided ideals and shift invariant functionals in the 1980s. At the same time Kalton and Figiel identified the commutator subspace of trace class operators, showing that there exist traces different from ‘the trace’ on the trace class ideal. The commutator approach was subsequently developed in the 1990s for arbitrary ideals by Dykema, Figiel, Weiss and Wodzicki. We survey recent advances in singular traces, of which Dixmier’s trace is an example, based on the approaches of Dixmier, Connes, Pietsch, Kalton, Figiel and the approach of Dykema, Figiel, Weiss and Wodzicki. The results include the bijective association of positive traces with Banach limits, the characterisation of Dixmier traces within this bijection, Lidskii and Fredholm formulations of singular traces as the summation of divergent sums of eigenvalues and expectation values, and their calculation using zeta function residues, heat semigroup asymptotics and symbols of integral operators. There are basic implications of these advances for users in noncommutative geometry such as the redundancy of the requirement for invariance properties of the extended limit used in Dixmier’s trace, the capacity to calculate traces for resolvents of non-smooth partial differential operators and the characterisation of independence from which singular trace is used in terms of the rate of log divergence of the series of energy expectation values—a more physically suitable criteria to impose, or to test the satisfaction of, than series of generally intractable singular values of products of operators. We also survey recent applications in noncommutative geometry such as calculation of traces using noncommutative symbols, that Connes’ Hochschild Character formula holds for any trace, and extensions of Connes’ results for quantum differentiability for Euclidean space and the noncommutative torus.