The theory of pseudodifferential operators on the noncommutative ntorus
Abstract
The methods of spectral geometry are useful for investigating the metric aspects of noncommutative geometry and in these contexts require extensive use of pseudodifferential operators. In a foundational paper, Connes showed that, by direct analogy with the theory of pseudodifferential operators on finitedimensional real vector spaces, one may derive a similar pseudodifferential calculus on noncommutative ntori, 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 GaussBonnet theorem on the noncommutative twotorus 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 pseudodifferential operator and the formula for the symbol of a product of two pseudodifferential operators, we extend these results to finitely generated projective right modules over the noncommutative ntorus. Then we define the corresponding analog of Sobolev spaces and prove equivalents of the Sobolev and Rellich lemmas.
 Publication:

Journal of Physics Conference Series
 Pub Date:
 February 2018
 DOI:
 10.1088/17426596/965/1/012042
 arXiv:
 arXiv:1704.02507
 Bibcode:
 2018JPhCS.965a2042T
 Keywords:

 Mathematics  Operator Algebras;
 Mathematics  Quantum Algebra;
 81R60 (Primary) 81R15 (Secondary)
 EPrint:
 20 pages