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Advances in Dixmier traces and applications

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Abstract

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.

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Notes

  1. 1.

    Symmetric sequence spaces are also called rearrangement invariant sequence spaces. The reader of the literature should be warned that some texts refer to rearrangement invariant spaces solid under Hardy-Littlewood submajorisation as symmetric spaces. The same object in other texts is referred to as fully symmetric spaces.

  2. 2.

    Z(E) is a linear subspace of E and the algebraic dual of E∕Z(E) admits a functional \(\tilde {f}\) such that \(\tilde {f}([x]) \neq 0\). Let f be the extension of \(\tilde {f}\) to E vanishing on Z(E).

  3. 3.

    One of the strongest invariances generally considered are traces Trω where ω = ω ∘ M is an extended limit invariant under M. This set of traces are characterised by what might be considered ‘infinitely’ factorisable Banach limits where θ = θ ∘ C is a Banach limit invariant under C [203].

  4. 4.

    That this is a bijective correspondence follows from using the identification of Dixmier traces with fully symmetric functionals in [122].

  5. 5.

    The function (s − 1)Tr(BA s) is bounded for s ≥ 1 if and only if \(0 \leq A \in \mathcal {M}_{1,\infty }\) [28, Theorem 4.5].

  6. 6.

    Note that this function is not bounded when \(0 \leq A \in \mathcal {M}_{1,\infty }\) belongs to the dual of the Macaev ideal. The logarithmic mean is applied to obtain a bounded function in this case [28, 198].

  7. 7.

    For example, the statement 1. (a) ⇔ (b) in Theorem 4.2, that

    $$\displaystyle \begin{aligned}\mathrm{Tr}_\omega(A) = c, \quad c \text{ const.}, A \in \mathcal{L}_{1,\infty} \end{aligned}$$

    is equivalent to

    $$\displaystyle \begin{aligned}\lim_{n\to \infty} \frac 1{\log(n+1)} \sum_{k=0}^n \lambda(k,A)= c \end{aligned}$$

    for an eigenvalue sequence λ(n, A), n ≥ 0, is false on \(\mathcal {M}_{1,\infty }\). It is true when \(0 \leq A \in \mathcal {M}_{1,\infty }\) [147], but false for arbitrary operators in the ideal [199, Corollary 11]. The maximal ideal on which the statement 1. (a) ⇔ (b) remains true has been identified [199, p. 3058]—it is not \(\mathcal {L}_{1,\infty }\).

  8. 8.

    A spectral triple \((\mathcal {A},D,H)\) consists of a ∗-algebra \(\mathcal {A} \subset \mathcal {L}(H)\) and a self-adjoint operator D : Dom(D) → H such that \([D,a] \in \mathcal {L}(H)\). Connes and Moscovici defined the dimension spectrum [55, II.1]. Let Sd = ∪BSd(B) where B belongs to the algebra generated by a, [D, a], \(a \in \mathcal {A}\) and \(\mathrm {Sd}(B) \subset \mathbb {C}\) is the set such that Tr(B|D|z) is analytic on \(\mathbb {C} \setminus \mathrm {Sd}(B)\). It is usually assumed that Sd is a discrete set, but not that poles with imaginary components should be excluded.

  9. 9.

    For pseudodifferential operators this can be weakened to M ψB = B; this implies \(BM_\psi - B \in \mathcal {L}_1\) [150, Example 10.2.23], which is sufficient. Throughout this section the condition BM ψ = B on \(B \in \mathcal {L}(L_2(\mathbb {R}^d))\) can be replaced by \(BM_\psi - B \in \mathcal {L}_1\).

  10. 10.

    Even though \(\Psi _{\mathrm {cl}}^0\) is a ∗-subalgebra of bounded operators on \(L_2(\mathbb {R}^d)\), the principal symbol map on \(\Psi _{\mathrm {cl}}^0\) cannot be defined directly with π 0. There are operators in \(\Psi _{\mathrm {cl}}^0\) whose commutators are not compact operators on \(L_2(\mathbb {R}^d)\) [57, Lemma 10.5]. Cordes considers several subalgebras of \(\Psi ^0_{\mathrm {cl}}\) with compact commutators, the maximal one being the ∗-subalgebra of operators B whose total symbol \(\sigma (B)(x,\xi ) \in C_b^\infty (\mathbb {R}^d \times \mathbb {R}^d)\) has all derivatives in \(C_0^\infty (\mathbb {R}^d \times \mathbb {R}^d)\) [57, p. 133], and the ∗-subalgebra of operators B whose total symbol satisfy [58] [204, Chap. IV]

    $$\displaystyle \begin{aligned}\partial^\alpha_x \partial^\beta_\xi \sigma(B)(x,\xi) = O((1+|x|{}^2)^{-|\alpha|}) O((1+|\xi|{}^2)^{-|\beta|}), \quad x,\xi \in \mathbb{R}^d . \end{aligned}$$
  11. 11.

    As one of the algebras is commutative, it is in particular nuclear and all C ∗-norms on \(\mathcal {A}_1\otimes _{\mathrm {alg}}\mathcal {A}_2\) coincide.

  12. 12.

    S is the subspace of measurable functions \(f \in L_0(\mathbb {R}^d)\) with a distribution function

    $$\displaystyle \begin{aligned}n_f(\lambda) = m( \{ s \in \mathbb{R}^d : |f(s)| > \lambda \} ) , \quad \lambda > 0 , m\text{ Lebesgue measure,} \end{aligned}$$

    that is finite as λ →∞. Equivalently, the function μ(s, f), s > 0 is finite valued.

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Acknowledgements

The authors gratefully thank Alain Connes, Albrecht Pietsch and Raphael Ponge for historical comments. The authors also thank Albrecht Pietsch and Aleksandr Usachev for discussion on the PhD thesis of Jozsef Varga.

This research was supported by the Australian Research Council.

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Authors and Affiliations

  1. School of Mathematics and Statistics, University of New South Wales, Sydney, NSW, Australia

    Steven Lord, Fedor A. Sukochev & Dmitriy Zanin

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  1. Steven Lord
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  1. Department of Physics Institute for Mathematics, Astrophysics and Particle Physics, Nijmegen, The Netherlands

    Ali Chamseddine

  2. Department of Mathematics, Johns Hopkins University, Baltimore, MD, USA

    Caterina Consani

  3. Department of Mathematics, Pennsylvania State University, University Park, PA, USA

    Nigel Higson

  4. Department of Mathematics, The University of Western Ontario, London, ON, Canada

    Masoud Khalkhali

  5. Department of Mathematics, Ohio State University, Columbus, OH, USA

    Henri Moscovici

  6. Department of Mathematics, Texas A&M University, College Station, TX, USA

    Guoliang Yu

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Lord, S., Sukochev, F.A., Zanin, D. (2019). Advances in Dixmier traces and applications. In: Chamseddine, A., Consani, C., Higson, N., Khalkhali, M., Moscovici, H., Yu, G. (eds) Advances in Noncommutative Geometry. Springer, Cham. https://doi.org/10.1007/978-3-030-29597-4_9

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