noncommutative residue
{{Short description|Mathematical concept}}
In mathematics, noncommutative residue, defined independently by M. {{harvtxt|Wodzicki|1984}} and {{harvtxt|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. {{harvtxt|Adler|1978}} and Y. {{harvtxt|Manin|1978}} in the context of one-dimensional integrable systems.
See also
References
- {{Citation | last1=Adler | first1=M. | title=On a trace functional for formal pseudo differential operators and the symplectic structure of the Korteweg-de Vries type equations | doi=10.1007/BF01410079 | doi-access=free | mr=520927 | date=1978 | journal=Inventiones Mathematicae | issn=0020-9910 | volume=50 | issue=3 | pages=219–248}}
- {{Citation | last1=Guillemin | first1=Victor | title=A new proof of Weyl's formula on the asymptotic distribution of eigenvalues | doi=10.1016/0001-8708(85)90018-0 | doi-access=free | mr=772612 | year=1985 | journal=Advances in Mathematics | issn=0001-8708 | volume=55 | issue=2 | pages=131–160}}
- {{Citation | last1=Kassel | first1=Christian | title=Le résidu non commutatif (d'après M. Wodzicki) | url=http://www.numdam.org/item?id=SB_1988-1989__31__199_0 | mr=1040574 | year=1989 | journal=Astérisque | issn=0303-1179 | issue=177 | pages=199–229}}
- {{Citation | last1=Manin | first1=Ju. I. | title=Current problems in mathematics, Vol. 11 (Russian) | publisher=Akad. Nauk SSSR Vsesojuz. Inst. Naučn. i Tehn. Informacii, Moscow | mr=0501136 | year=1978 | chapter=Algebraic aspects of nonlinear differential equations | pages=5–152}}
- {{citation|last=Wodzicki|first=M.|title=Spectral asymmetry and noncommutative residue|series=PhD thesis|publisher=Steklov institute of mathematics|year=1984|place=Moscow}}
- {{Citation | last1=Wodzicki | first1=Mariusz | title=K-theory, arithmetic and geometry (Moscow, 1984--1986) | publisher=Springer-Verlag | location=Berlin, New York | series=Lecture Notes in Math. | doi= 10.1007/BFb0078372 | mr=923140 | year=1987 | volume=1289 | chapter=Noncommutative residue. I. Fundamentals | pages=320–399}}