The theory of pseudo-differential operators on the noncommutative n-torus
- Creators
- Tao, J.
Abstract
The methods of spectral geometry are useful for investigating the metric aspects of noncommutative geometry and in these contexts require extensive use of pseudo-differential operators. In a foundational paper, Connes showed that, by direct analogy with the theory of pseudo-differential operators on finite-dimensional real vector spaces, one may derive a similar pseudo-differential calculus on noncommutative n-tori, and with the development of this calculus came many results concerning the local differential geometry of noncommutative tori for n=2,4, as shown in the groundbreaking paper in which the Gauss–Bonnet theorem on the noncommutative two-torus is proved and later papers. Certain details of the proofs in the original derivation of the calculus were omitted, such as the evaluation of oscillatory integrals, so we make it the objective of this paper to fill in all the details. After reproving in more detail the formula for the symbol of the adjoint of a pseudo-differential operator and the formula for the symbol of a product of two pseudo-differential operators, we extend these results to finitely generated projective right modules over the noncommutative n-torus. Then we define the corresponding analog of Sobolev spaces and prove equivalents of the Sobolev and Rellich lemmas.
Additional Information
© 2018 IOP. Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. I would like to thank Farzad Fathizadeh for suggesting the problem of filling in the details in the pseudo-differential calculus on the noncommutative n-torus. I would also like thank Matilde Marcolli for helping me make plans to take my candidacy exam. I would like to thank Vlad Markovic and Eric Rains for agreeing to be on my candidacy exam committee.Attached Files
Published - Tao_2018_J._Phys._3A_Conf._Ser._965_012042.pdf
Submitted - 1704.02507.pdf
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Additional details
- Eprint ID
- 84834
- Resolver ID
- CaltechAUTHORS:20180214-135938076
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2018-02-14Created from EPrint's datestamp field
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2022-07-12Created from EPrint's last_modified field