Dixmier Trace Research Articles

The Dixmier trace and asymptotics of zeta functions

We obtain general theorems which enable the calculation of the Dixmier trace in terms of the asymptotics of the zeta function and of the trace of the heat semigroup. We prove our results in a general semi-finite von Neumann algebra. We find for p > 1 that the asymptotics of the zeta function determines an ideal strictly larger than L p , ∞ on which the Dixmier trace may be defined. We also establish stronger versions of other results on Dixmier traces and zeta functions.

Sums of two-dimensional spectral triples

We study countable sums of two-dimensional modules for the continuous complex functions on a compact metric space and show that it is possible to construct a spectral triple which gives the original metric back. This spectral triple will be finitely summable for any positive parameter. We also construct a sum of two-dimensional modules which reflects some aspects of the topological dimensions of the compact metric space, but this will only give the metric back approximately. At the end we make an explicit computation of the last module for the unit interval in $\mathsf R$. The metric is recovered exactly, the Dixmier trace induces a multiple of the Lebesgue integral and the growth of the number of eigenvalues $N(\Lambda)$ bounded by $\Lambda$ behaves, such that $N(\Lambda)/\Lambda$ is bounded, but without limit for $\Lambda\to \infty$.

Gauge Theory and a Dirac Operator on a Noncommutative Space

As a tool to carry out the quantization of gauge theory on a noncommutative space, we present a Dirac operator that behaves as a line element of the canonical noncommutative space. Utilizing this operator, we construct the Dixmier trace, which is the regularized trace for infinite-dimensional matrices. We propose the possibility of solving the cosmological constant problem by applying our gauge theory on the noncommutative space.

Spectral geometry of κ-Minkowski space

After recalling Snyder’s idea [Phys. Rev. 71, 38 (1947)] of using vector fields over a smooth manifold as “coordinates on a noncommutative space,” we discuss a two-dimensional toy-model whose “dual” noncommutative coordinates form a Lie algebra: this is the well-known κ-Minkowski space [Phys. Lett. B 334, 348 (1994)]. We show how to improve Snyder’s idea using the tools of quantum groups and noncommutative geometry. We find a natural representation of the coordinate algebra of κ-Minkowski as linear operators on an Hilbert space (a major problem in the construction of a physical theory), study its “spectral properties,” and discuss how to obtain a Dirac operator for this space. We describe two Dirac operators. The first is associated with a spectral triple. We prove that the cyclic integral of Dimitrijevic et al. [Eur. Phys. J. C 31, 129 (2003)] can be obtained as Dixmier trace associated to this triple. The second Dirac operator is equivariant for the action of the quantum Euclidean group, but it has unbounded commutators with the algebra.

Dixmier traces as singular symmetric functionals and applications to measurable operators

We unify various constructions and contribute to the theory of singular symmetric functionals on Marcinkiewicz function/operator spaces. This affords a new approach to the non-normal Dixmier and Connes–Dixmier traces (introduced by Dixmier and adapted to non-commutative geometry by Connes) living on a general Marcinkiewicz space associated with an arbitrary semifinite von Neumann algebra. The corollaries to our approach, stated in terms of the operator ideal L ( 1 , ∞ ) (which is a special example of an operator Marcinkiewicz space), are: (i) a new characterization of the set of all positive measurable operators from L ( 1 , ∞ ) , i.e. those on which an arbitrary Connes–Dixmier trace yields the same value. In the special case, when the operator ideal L ( 1 , ∞ ) is considered on a type I infinite factor, a bounded operator x belongs to L ( 1 , ∞ ) if and only if the sequence of singular numbers { s n ( x ) } n ⩾ 1 (in the descending order and counting the multiplicities) satisfies ∥ x ∥ ( 1 , ∞ ) ≔ sup N ⩾ 1 1 Log ( 1 + N ) ∑ n = 1 N s n ( x ) < ∞ . In this case, our characterization amounts to saying that a positive element x ∈ L ( 1 , ∞ ) is measurable if and only if lim N → ∞ 1 Log N ∑ n = 1 N s n ( x ) exists; (ii) the set of Dixmier traces and the set of Connes–Dixmier traces are norming sets (up to equivalence) for the space L ( 1 , ∞ ) / L 0 ( 1 ∞ ) , where the space L 0 ( 1 , ∞ ) is the closure of all finite rank operators in L ( 1 , ∞ ) in the norm ∥ . ∥ ( 1 , ∞ ) .

Type II non-commutative geometry. I. Dixmier trace in von Neumann algebras

We define the notion of Connes–von Neumann spectral triple and consider the associated index problem. We compute the analytic Chern–Connes character of such a generalized spectral triple and prove the corresponding local formula for its Hochschild class. This formula involves the Dixmier trace for II ∞ von Neumann algebras. In the case of foliations, we identify this Dixmier trace with the corresponding measured Wodzicki residue.

Sums of commutators in non-commutative Banach function spaces

Let M be a II ∞-factor and denote by τ its normal faithful semi-finite trace. For any rearrangement invariant Köthe function space X on [0,+∞[, let X( M, τ) be the associated non-commutative Banach function space. This paper is concerned with ideals in M of the form I X ( M, τ)= M∩ X( M, τ) that are contained in L p ( M, τ) for some p>0. It is proved that an element T in I X ( M, τ) is a finite sum of commutators of the form [ A, B] with A∈ I X ( M, τ) and B∈ M if and only if the function t→ 1 t ∫ |λ|>λ t(T) λ dν T(λ) belongs to X, where ν T is the Brown spectral measure of T and t→ λ t ( T) is the non-increasing rearrangement of the function λ→| λ| with respect to ν T . This extends to general Banach function spaces a result obtained by Kalton for quasi-Banach ideals of compact operators and implies that the Dixmier's trace of a quasi-nilpotent element in L 1,∞( M, τ) is always zero.

Dimensions and singular traces for spectral triples, with applications to fractals

Given a spectral triple ( A, H,D) , the functionals on A of the form a↦ τ ω ( a| D| − α ) are studied, where τ ω is a singular trace, and ω is a generalised limit. When τ ω is the Dixmier trace, the unique exponent d giving rise possibly to a non-trivial functional is called Hausdorff dimension, and the corresponding functional the ( d-dimensional) Hausdorff functional. It is shown that the Hausdorff dimension d coincides with the abscissa of convergence of the zeta function of | D| −1, and that the set of α's for which there exists a singular trace τ ω giving rise to a non trivial functional is an interval containing d. Moreover, the endpoints of such traceability interval have a dimensional interpretation. The functionals corresponding to points in the traceability interval are called Hausdorff–Besicovitch functionals. These definitions are tested on fractals in R , by computing the mentioned quantities and showing in many cases their correspondence with classical objects. In particular, for self-similar fractals the traceability interval consists only of the Hausdorff dimension, and the corresponding Hausdorff–Besicovitch functional gives rise to the Hausdorff measure. More generally, for any limit fractal, the described functionals do not depend on the generalized limit ω.

Non-commutative residues for anisotropic pseudo-differential operators in [formula omitted

In this paper we define a non-commutative residue for the algebra of classical anisotropic pseudo-differential operators on R n modulo regularizing operators. Furthermore, by means of the Weyl formula, we show that this trace coincides with the Dixmier trace on operators of order −| M|.

Spectral flow and Dixmier traces

We obtain general theorems which enable the calculation of the Dixmier trace in terms of the asymptotics of the zeta function and of the heat operator in a general semi-finite von Neumann algebra. Our results have several applications. We deduce a formula for the Chern character of an odd L (1,∞) -summable Breuer–Fredholm module in terms of a Hochschild 1-cycle. We explain how to derive a Wodzicki residue for pseudo-differential operators along the orbits of an ergodic R n action on a compact space X. Finally, we give a short proof of an index theorem of Lesch for generalised Toeplitz operators.

Spectral triples and differential calculi related to the Kronecker foliation

Following the ideas of Connes and Moscovici, we describe two spectral triples related to the Kronecker foliation, whose generalized Dirac operators are related to first and second order signature operators. We also consider the corresponding differential calculi Ω D , which are drastically different in the two cases. For the second order signature operator we calculate the Chern character of the spectral triple and the Dixmier trace of certain powers of its Dirac operator. As a side-remark, we give a description of a known calculus on the two-dimensional noncommutative torus in terms of generators and relations.

FREDHOLM THEORY FOR DEGENERATE PSEUDODIFFERENTIAL OPERATORS ON MANIFOLDS WITH FIBERED BOUNDARIES

We consider the calculus Ψ*,* de(X, deΩ½) of double-edge pseudodifferential operators naturally associated to a compact manifold X whose boundary is the total space of a fibration. This fits into the setting of boundary fibration structures, and we discuss the corresponding geometric objects. We construct a scale of weighted double-edge Sobolev spaces on which double-edge pseudodifferential operators act as bounded operators, characterize the Fredholm elements in Ψ*,* de(X) by means of the invertibility of an appropriate symbol map, and describe a K-theoretical formula for the Fredholm index extending the Atiyah–Singer formula for closed manifolds. The algebra of operators of order (0, 0) is shown to be a Ψ*-algebra, hence its K-theory coincides with that of its C *-closure, and we give a description of the corresponding cyclic 6-term exact sequence. We define a Wodzicki-type residue trace on an ideal in Ψ*,* de(X, deΩ½), and we show that it coincides with Dixmier's trace for operators of order –dim X in this ideal. This extends a result of Connes for the closed case.

Dixmier Traces and Unitary Representations

As is known, for any two given unitary representations of a topological group in Hilbert spaces, a new unitary representation in the corresponding space of Hilbert-Schmidt mappings can be defined. In the paper, a modification of this construction is suggested that uses the Dixmier trace (instead of the ordinary operator trace). The case of the group of second-order real unimodular matrices is studied in detail.

Trotter–Kato Product Formula and Symmetrically Normed Ideals

It is proven that the Trotter product formula converges in the norm of a symmetrically normed ideal of compact operators away from t0>0 if the Kac operator (the transfer matrix) F(t)=e−tB/2e−tAe−tB/2 belongs to this ideal for t=t0. The result is generalized to the Trotter–Kato product formula. Moreover, if the perturbation B is small relative to A, then error bounds for convergence are obtained. The results apply to the Dixmier trace.

Dixmier's trace for boundary value problems

Let X be a smooth manifold with boundary of dimension n > 1. The operators of order −n and type zero in Boutet de Monvel's calculus form a subset of Dixmier's trace ideal \(\) for the Hilbert space \(\) of L2 sections in vector bundles E over X, F over ∂X.

Symmetric Functionals and Singular Traces

We study the construction and properties of positive linear functionals on symmetric spaces of measurable functions which are monotone with respect to submajorization. The construction of such functionals may be lifted to yield the existence of singular traces on certain non-commutative Marcinkiewicz spaces which generalize the notion of Dixmier trace.

The Connes–Lott Program on the Sphere

We describe a classical Schwinger-type model as a study of the projective modules over the algebra of complex-valued functions on the sphere. On these modules, classified by π2(S2), we construct hermitian connections with values in the universal differential envelope. Instead of describing matter by the usual Dirac spinors yielding the standard Schwinger model on the sphere, we apply the Connes–Lott program to the Hilbert space of complexified inhomogeneous forms with its Atiyah–Kähler structure. This Hilbert space splits in two minimal left ideals of the Clifford algebra preserved by the Dirac–Kähler operator D=i(d-δ). The induced representation of the universal differential envelope, in order to recover its differential structure, is divided by the unwanted differential ideal and the obtained quotient is the usual complexified de Rham exterior algebra with Clifford action on the "spinors" of the Hilbert space. The subsequent steps of the Connes–Lott program allow to define a matter action, and the field action is obtained using the Dixmier trace which reduces to the integral of the curvature squared.

Complete determination of the singularity structure of zeta functions

Series of extended Epstein type provide examples of non-trivial zeta functions with important physical applications. The regular part of their analytic continuation is seen to be a convergent or an asymptotic series. Their singularity structure is completely determined in terms of the Wodzicki residue in its generalized form, which is proven to yield the residua of all the poles of the zeta function, and not just that of the rightmost pole (obtainable from the Dixmier trace). The calculation is a very down-to-earth application of these powerful functional analytical methods in physics.

On the structure of a differential algebra used by Connes and Lott

We investigate the structure of a differential algebra associated with the simplest twopoint K-cycle used by Connes and Lott for physical model building. We show that, in this case, the factorization of the universal differential algebra with respect to the canonical ideal can be performed explicitly. This enables us to present a complete analysis of the structure of the differential algebra under investigation, including explicit multiplication and differentiation rules. Moreover, we get explicit formulae for the scalar product, induced by the Dixmier trace, of elements of the differential algebra of arbitrary degree.

Hamilton formalism in non-cummutative geometry

We study the Hamilton formalism for Connes-Lott models, i.e. for Yang-Mills theory in non-commutative geometry. The starting point is an associative ∗-algebra A which is of the form A = C (I, A s), where A s is itself an associative ∗-algebra. With appropriate choice of a K-cycle over A it is possible to identify the time-like part of the generalized differential algebra constructed out of A. We define the non-commutative analogue of integration on space-like surfaces via the Dixmier trace restricted to the representation of the space-like part A s of the algebra. Due to this restriction it is possible to define the Lagrange function resp. Hamilton function also for Minkowskian space-time. We identify the phase-space and give a definition of the Poisson bracket for Yang-Mills theory in non-commutative geometry. This general formalism is applied to a model on a two-point space and to a model on Minkowski space-time x two-point space.