Commutation theorem for traces: Difference between revisions

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{{short description|Identifies the commutant of a specific von Neumann algebra acting on a Hilbert space}}

{{short description|none}}

In [[mathematics]], a '''commutation theorem for traces''' explicitly identifies the [[commutant]] of a specific [[von Neumann algebra]] acting on a [[Hilbert space]] in the presence of a [[Von Neumann algebra#Weights, states, and traces|trace]].

The first such result was proved by [[Francis Joseph Murray]] and [[John von Neumann]] in the 1930s and applies to the von Neumann algebra generated by a [[discrete group]] or by the [[dynamical system]] associated with a [[ergodic theory|measurable transformation]] preserving a [[probability measure]].

[[ergodic theory|measurable transformation]] preserving a [[probability measure]]. Another important application is in the theory of [[unitary representation]]s of [[Haar measure|unimodular]] [[locally compact group]]s, where the theory has been applied to the [[regular representation]] and other closely related representations. In particular this framework led to an abstract version of the [[Plancherel theorem]] for unimodular locally compact groups due to [[Irving Segal]] and Forrest Stinespring and an abstract [[Plancherel theorem for spherical functions]] associated with a [[Gelfand pair]] due to [[Roger Godement]]. Their work was put in final form in the 1950s by [[Jacques Dixmier]] as part of the theory of '''Hilbert algebras'''. It was not until the late 1960s, prompted partly by results in [[algebraic quantum field theory]] and [[quantum statistical mechanics]] due to the school of [[Rudolf Haag]], that the more general non-tracial [[Tomita–Takesaki theory]] was developed, heralding a new era in the theory of von Neumann algebras.

It was not until the late 1960s, prompted partly by results in [[algebraic quantum field theory]] and [[quantum statistical mechanics]] due to the school of [[Rudolf Haag]], that the more general non-tracial [[Tomita–Takesaki theory]] was developed, heralding a new era in the theory of von Neumann algebras.

==Commutation theorem for finite traces==