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Quantum Differentiability on Noncommutative Euclidean Spaces

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Abstract

We study the topic of quantum differentiability on quantum Euclidean d-dimensional spaces (otherwise known as Moyal d-spaces), and we find conditions that are necessary and sufficient for the singular values of the quantised differential to have decay \(O(n^{-\alpha })\) for \(0 < \alpha \le \frac{1}{d}\). This result is substantially more difficult than the analogous problems for Euclidean space and for quantum d-tori.

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Notes

  1. It is meaningful to write \(\mathrm {rank}(\theta )/2\), since the rank of an antisymmetric matrix is always even.

  2. See [63, pp. 38] for the definition of this function space.

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Acknowledgements

The authors would like to thank to anonymous referees for numerous helpful comments and corrections to Lemma 3.12. We also extend our gratitude to Galina Levitina for noticing a gap in our original proof of Theorem 1.5. This work was done when the third author was visiting the University of New South Wales; he wishes to express his gratitude to the first two authors for their kind hospitality. The authors are supported by Australian Research Council (Grant No. FL170100052); X. Xiong is also partially supported by the National Natural Science Foundation of China (Grant No. 11301401).

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Authors and Affiliations

  1. School of Mathematics and Statistics, UNSW, Kensington, NSW, 2052, Australia

    Edward McDonald, Fedor Sukochev & Xiao Xiong

  2. Institute for Advanced Study in Mathematics, Harbin Institute of Technology, Harbin, 150001, China

    Xiao Xiong

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  1. Edward McDonald
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Correspondence to Edward McDonald.

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McDonald, E., Sukochev, F. & Xiong, X. Quantum Differentiability on Noncommutative Euclidean Spaces. Commun. Math. Phys. 379, 491–542 (2020). https://doi.org/10.1007/s00220-019-03605-2

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