Bounded operator: Difference between revisions
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In [[algebra]], the '''commutant''' of a [[subset]] ''S'' of a [[semigroup]] (such as an [[algebra over a field|algebra]] or a [[group (mathematics)|group]]) ''A'' is the subset ''S''′ of elements of ''A'' [[commutative operation|commuting]] with every element of ''S''.<ref name=Accardi>{{cite book|last=Luigi Accardi, Franco Fagnola|title=Quantum Interacting Particle Systems: Lecture Notes of the Volterra-CIRM International School, Trento, Italy, 23-29 September 2000|year=2002|publisher=World Scientific|isbn=9789812381040|pages=29-30|url=http://books.google.com/books?id=SVxuwaHuUyUC&pg=PA29&dq=semigroup+%22commutant%22+definition&hl=en&sa=X&ei=oqqGUpv3KseoiAKd4IDYBg&ved=0CEkQ6AEwBQ#v=onepage&q=semigroup%20%22commutant%22%20definition&f=false}}</ref> In other words, | |||
:<math>S'=\{x\in A: sx=xs\ \mbox{for}\ \mbox{every}\ s\in S\}.</math> | |||
''S''′ forms a [[subsemigroup]]. This generalizes the concept of [[centralizer]] in [[group theory]]. | |||
==Properties== | |||
*<math>S' = S''' = S'''''</math> - A commutant is its own [[bicommutant]]. | |||
*<math>S'' = S'''' = S''''''</math> - A [[bicommutant]] is its own bicommutant. | |||
==See also== | |||
*[[Bicommutant]] | |||
*[[von Neumann bicommutant theorem]] | |||
==References== | |||
{{reflist}} | |||
[[Category:Group theory]] | |||
Revision as of 05:22, 10 January 2014
In algebra, the commutant of a subset S of a semigroup (such as an algebra or a group) A is the subset S′ of elements of A commuting with every element of S.[1] In other words,
S′ forms a subsemigroup. This generalizes the concept of centralizer in group theory.
Properties
- - A commutant is its own bicommutant.
- - A bicommutant is its own bicommutant.
See also
References
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