Noncommutative residue

In mathematics, noncommutative residue, defined independently by M. Wodzicki (1984) and Guillemin (1985), is a certain trace on the algebra of pseudodifferential operators on a compact differentiable manifold that is expressed via a local density. In the case of the circle, the noncommutative residue had been studied earlier by M. Adler (1979) and Y. Manin (1978) in the context of one-dimensional integrable systems.

See also

• Dixmier trace

References

• Adler, M. (1978), "On a trace functional for formal pseudo differential operators and the symplectic structure of the Korteweg-de Vries type equations", Inventiones Mathematicae, 50 (3): 219–248, doi:10.1007/BF01410079, ISSN 0020-9910, MR 520927 
• Guillemin, Victor (1985), "A new proof of Weyl's formula on the asymptotic distribution of eigenvalues", Advances in Mathematics, 55 (2): 131–160, doi:10.1016/0001-8708(85)90018-0, ISSN 0001-8708, MR 772612 
• Kassel, Christian (1989), "Le résidu non commutatif (d'après M. Wodzicki)", Astérisque (177): 199–229, ISSN 0303-1179, MR 1040574 
• Manin, Ju. I. (1978), "Algebraic aspects of nonlinear differential equations", Current problems in mathematics, Vol. 11 (Russian), Akad. Nauk SSSR Vsesojuz. Inst. Naučn. i Tehn. Informacii, Moscow, pp. 5–152, MR 0501136 
• Wodzicki, M. (1984), Spectral asymmetry and noncommutative residue, PhD thesis, Moscow: Steklov institute of mathematics 
• Wodzicki, Mariusz (1987), "Noncommutative residue. I. Fundamentals", K-theory, arithmetic and geometry (Moscow, 1984--1986), Lecture Notes in Math., 1289, Berlin, New York: Springer-Verlag, pp. 320–399, doi:10.1007/BFb0078372, MR 923140