Positivity and Noncommutative Analysis
Positivity and Noncommutative Analysis pp 471-510 | Cite as
A Residue Formula for Locally Compact Noncommutative Manifolds
Denis Potapov
Fedor Sukochev
Dominic Vella
Dmitriy Zanin
First Online: 10 August 2019
Part of the Trends in Mathematics book series (TM)
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 Open image in new window 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.
Dixmier trace Connes integration formula Noncommutative plane
Dedicated to Ben de Pagter on the occasion of his 65th birthday
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
$$\displaystyle \begin{aligned}\|g_p\|{}_{W_{n,2}(0,1)}=\mathcal{O}(1),\quad p\downarrow1.\end{aligned}$$
Let (δuf)(t) = f(ut). We write
$$\displaystyle \begin{aligned}g_p=h\cdot \big(f-(p-1)(\delta_{p-1}f)\big),\end{aligned}$$
$$\displaystyle \begin{aligned}h(t)=\frac{t}{2}\coth\Big(\frac{t}{2}\Big),\quad f(t)=\frac 1t\tanh\Big(\frac{t}{2}\Big),\quad t\in\mathbb{R}.\end{aligned}$$
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
$$\displaystyle \begin{aligned}\big(f-(p-1)(\delta_{p-1}f)\big)^{(k)}=f^{(k)}-(p-1)^{k+1}\delta_{p-1}(f^{(k)}).\end{aligned}$$
$$\displaystyle \begin{aligned}\Big\|\big(f-(p-1)(\delta_{p-1}f)\big)^{(k)}\Big\|{}_{\infty}\leq \big(1+(p-1)^{k+1}\big)\|f^{(k)}\|{}_{\infty}.\end{aligned} $$
$$\displaystyle \begin{aligned}\|g_p\|{}_{W_{n,2}(1,\infty)}=\mathcal{O}\big((p-1)^{-\frac 12}\big),\quad p\downarrow1.\end{aligned}$$
$$\displaystyle \begin{aligned}g_p=h\cdot (f-\delta_{p-1}f),\end{aligned}$$
$$\displaystyle \begin{aligned}h(t)=\frac 12\coth\Big(\frac{t}{2}\Big),\quad f(t)=\tanh\Big(\frac{t}{2}\Big),\quad t\in\mathbb{R}.\end{aligned}$$
$$\displaystyle \begin{aligned}\|g_p\|{}_{W_{n,2}(1,\infty)}\leq\|h\|{}_{W_{n,\infty}(1,\infty)}\|f-\delta_{p-1}f\|{}_{W_{n,2}(1,\infty)}.\end{aligned}$$
$$\displaystyle \begin{aligned}(f-\delta_{p-1}f)^{(k)}=f^{(k)}-(p-1)^k\delta_{p-1}(f^{(k)}).\end{aligned}$$
$$\displaystyle \begin{aligned} \big\|(f-\delta_{p-1}f)^{(k)}\big\|{}_{L_2(1,\infty)} &\leq\|f^{(k)}\|{}_{L_2(1,\infty)}+(p-1)^k\big\|\delta_{p-1}(f^{(k)})\big\|{}_{L_2(1,\infty)} \\&\leq\big(1+(p-1)^{k-\frac 12}\big)\cdot\|f^{(k)}\|{}_{L_2(0,\infty)}. \end{aligned} $$
For gp defined as above, \(\|g_p\|{ }_{{W_{n,2}}}=\mathcal {O}\big ((p-1)^{-\frac 12}\big )\) as p ↓ 1.
As gp 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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© Springer Nature Switzerland AG 2019
1.School of Mathematics and StatisticsUniversity of New South WalesSydneyAustralia
Potapov D., Sukochev F., Vella D., Zanin D. (2019) A Residue Formula for Locally Compact Noncommutative Manifolds. In: Buskes G. et al. (eds) Positivity and Noncommutative Analysis. Trends in Mathematics. Birkhäuser, Cham
First Online 10 August 2019
DOI https://doi.org/10.1007/978-3-030-10850-2_24
Publisher Name Birkhäuser, Cham