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A Residue Formula for Locally Compact Noncommutative Manifolds

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Positivity and Noncommutative Analysis

Part of the book series: Trends in Mathematics ((TM))

Abstract

The most significant result in the integration theory for locally compact noncommutative spaces was proven in Carey et al. (J Funct Anal 263(2):383–414, 2012). We review and improve on this result. For bounded, positive operators A, B on a separable Hilbert space \(\mathcal {H}\), if \(AB\in \mathcal {M}_{1,\infty }\), \([A^{\frac 12},B]\in \mathcal {L}_1\) and the limit exists, then AB is Dixmier measurable and, for any dilation-invariant extended limit ω, we have

$$\displaystyle \operatorname {{\mathrm {Tr}}}_\omega (AB)=\lim _{p\downarrow 1}(p-1) \operatorname {{\mathrm {Tr}}}(B^pA^p). $$

The proof of this result is facilitated by the efficient use of recent advances in the theory of singular traces and double operator integrals. The Dixmier traces of pseudo-differential operators of the form \(M_f(1-\Delta )^{-\frac {d}2}\) on \(\mathbb {R}^d\), where f is a Schwartz function on \(\mathbb {R}^d\) and Δ denotes the Laplacian on \(L_2(\mathbb {R}^d)\), are swiftly recovered using this result. An analogous result for the noncommutative plane is also obtained.

Dedicated to Ben de Pagter on the occasion of his 65th birthday

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Acknowledgements

The authors express their gratitude to Steven Lord for the extraordinary help he contributed related to the presentation of this text and the verification of its results. Denis Potapov, Fedor Sukochev and Dmitriy Zanin gratefully acknowledge the financial support from the Australian Research Council. Dominic Vella gratefully acknowledges the support received through the Australian Government Research Training Program Scholarship.

Appendix: Proof of Lemma 3

Let \(n\in \mathbb {N}\) and p ≥ 1. In the following, for −∞≤ a < b ≤∞, the Sobolev space of p-integrable functions on the interval (a, b) may be defined by

$$\displaystyle \begin{aligned}W_{n,p}(a,b)\mathrel{\mathop:}=\Big\{ f\in L_p(a,b)\;:\; \|\partial^\alpha f\|{}_{L_p(a,b)}<\infty,\,\text{ for multi-indices s.t. }|\alpha|\leq n \Big\},\end{aligned}$$

with corresponding norm given by

$$\displaystyle \begin{aligned}\|f\|{}_{W_{n,p}(a,b)}\mathrel{\mathop:}= \sum_{\alpha:|\alpha|\leq m}\Big\|\frac{\partial^{\alpha_1}}{\partial t_1^{\alpha_1}}\cdots\frac{\partial^{\alpha_d}}{\partial t_d^{\alpha_d}}(f)\Big\|{}_{L_p(a,b)},\quad f\in W_{n,p}(a,b).\end{aligned}$$

Lemma 1

For \(n\in \mathbb {N},\) we have

$$\displaystyle \begin{aligned}\|g_p\|{}_{W_{n,2}(0,1)}=\mathcal{O}(1),\quad p\downarrow1.\end{aligned}$$

Proof

Let (δ u f)(t) = f(ut). We write

$$\displaystyle \begin{aligned}g_p=h\cdot \big(f-(p-1)(\delta_{p-1}f)\big),\end{aligned}$$

where

$$\displaystyle \begin{aligned}h(t)=\frac{t}{2}\coth\Big(\frac{t}{2}\Big),\quad f(t)=\frac 1t\tanh\Big(\frac{t}{2}\Big),\quad t\in\mathbb{R}.\end{aligned}$$

By Leibniz rule, we have

$$\displaystyle \begin{aligned} \|g_p\|{}_{W_{n,2}(0,1)} &\leq\Big\|h\cdot \big(f-(p-1)(\delta_{p-1}f)\big)\Big\|{}_{W_{n,\infty}(0,1)} \\&\leq\|h\|{}_{W_{n,\infty}(0,1)}\big\|f-(p-1)(\delta_{p-1}f)\big\|{}_{W_{n,\infty}(0,1)}. \end{aligned} $$

For 0 ≤ k ≤ n, we have

$$\displaystyle \begin{aligned}\big(f-(p-1)(\delta_{p-1}f)\big)^{(k)}=f^{(k)}-(p-1)^{k+1}\delta_{p-1}(f^{(k)}).\end{aligned}$$

Hence,

$$\displaystyle \begin{aligned}\Big\|\big(f-(p-1)(\delta_{p-1}f)\big)^{(k)}\Big\|{}_{\infty}\leq \big(1+(p-1)^{k+1}\big)\|f^{(k)}\|{}_{\infty}.\end{aligned} $$

□

Lemma 2

For \(n\in \mathbb {N},\) we have

$$\displaystyle \begin{aligned}\|g_p\|{}_{W_{n,2}(1,\infty)}=\mathcal{O}\big((p-1)^{-\frac 12}\big),\quad p\downarrow1.\end{aligned}$$

Proof

Let (δ u f)(t) = f(ut). We write

$$\displaystyle \begin{aligned}g_p=h\cdot (f-\delta_{p-1}f),\end{aligned}$$

where

$$\displaystyle \begin{aligned}h(t)=\frac 12\coth\Big(\frac{t}{2}\Big),\quad f(t)=\tanh\Big(\frac{t}{2}\Big),\quad t\in\mathbb{R}.\end{aligned}$$

By Leibniz rule, we have

$$\displaystyle \begin{aligned}\|g_p\|{}_{W_{n,2}(1,\infty)}\leq\|h\|{}_{W_{n,\infty}(1,\infty)}\|f-\delta_{p-1}f\|{}_{W_{n,2}(1,\infty)}.\end{aligned}$$

For 0 ≤ k ≤ n, we have

$$\displaystyle \begin{aligned}(f-\delta_{p-1}f)^{(k)}=f^{(k)}-(p-1)^k\delta_{p-1}(f^{(k)}).\end{aligned}$$

Hence,

$$\displaystyle \begin{aligned} \big\|(f-\delta_{p-1}f)^{(k)}\big\|{}_{L_2(1,\infty)} &\leq\|f^{(k)}\|{}_{L_2(1,\infty)}+(p-1)^k\big\|\delta_{p-1}(f^{(k)})\big\|{}_{L_2(1,\infty)} \\&\leq\big(1+(p-1)^{k-\frac 12}\big)\cdot\|f^{(k)}\|{}_{L_2(0,\infty)}. \end{aligned} $$

□

Lemma 3

For g p defined as above, \(\|g_p\|{ }_{{W_{n,2}}}=\mathcal {O}\big ((p-1)^{-\frac 12}\big )\) as p ↓ 1.

Proof

As g p is even, we have

$$\displaystyle \begin{aligned}\|g_p\|{}_{{W_{n,2}}}\leq 2\big(\|g_p\|{}_{W_{n,2}(0,1)}+\|g_p\|{}_{W_{n,2}(1,\infty)}\big).\end{aligned}$$

The assertion follows from the preceding lemmas. □

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Authors and Affiliations

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

    Denis Potapov, Fedor Sukochev, Dominic Vella & Dmitriy Zanin

Authors
  1. Denis Potapov
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  2. Fedor Sukochev
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  3. Dominic Vella
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  4. Dmitriy Zanin
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Corresponding author

Correspondence to Fedor Sukochev .

Editor information

Editors and Affiliations

  1. Department of Mathematics, University of Mississippi, Oxford, MS, USA

    Gerard Buskes

  2. Mathematical Institute, Leiden University, Leiden, The Netherlands; Department of Mathematics and Applied Mathematics, University of Pretoria, Pretoria, South Africa

    Marcel de Jeu

  3. College of Science and Engineering, Flinders University, Adelaide, Australia

    Peter Dodds

  4. Department of Mathematics, University of South Carolina, Columbia, SC, USA

    Anton Schep

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

    Fedor Sukochev

  6. Delft Institute of Applied Mathematics, Delft University of Technology, Delft, The Netherlands

    Jan van Neerven

  7. Mathematical Sciences Research Centre, Queen’s University Belfast, Belfast, UK

    Anthony Wickstead

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Potapov, D., Sukochev, F., Vella, D., Zanin, D. (2019). A Residue Formula for Locally Compact Noncommutative Manifolds. In: Buskes, G., et al. Positivity and Noncommutative Analysis. Trends in Mathematics. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-10850-2_24

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