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 operator]]s on a [[compact space|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 system]]s.

== See also ==

* [[Dixmier trace]]

== 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 | author1-link=Victor Guillemin | 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=https://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. | author1-link=Yuri Manin | 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.|author-link=Mariusz Wodzicki|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}}

[[Category:Noncommutative geometry]]