Commutation theorem for traces: Difference between revisions

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In [[mathematics]], a '''commutation theorem''' 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 [[F.J. 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]]. 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.

==Commutation theorem for finite traces==

Let ''H'' be a [[Hilbert space]] and ''M'' a [[von Neumann algebra]] on ''H'' with a unit vector Ω such that

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The '''commutation theorem''' of Murray and von Neumann states that

:{| border="1" cellspacing="0" cellpadding="5"

|<math>JMJ=M^\prime</math>

|}

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::<math>H_1=H\otimes \ell^2(\Gamma),</math>

:a [[tensor product]] of Hilbert spaces.<ref> ''H''₁ can be identified with the space of square integrable functions on ''X'' x Γ with respect to the [[product measure]].</ref> The '''''group–measure space construction''''' or [[crossed product]] von Neumann algebra

::<math> M = A \rtimes \Gamma</math>

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One of the most important cases of the group–measure space construction is when Γ is the group of integers '''Z''', i.e. the case of a single invertible

measurable transformation ''T''. Here ''T'' must preserve the probability measure μ. Semifinite traces are required to handle the case when ''T'' (or more generally Γ) only preserves an infinite [[equivalence of measures|equivalent]] measure; and the full force of the [[Tomita–Takesaki theory]] is required when there is no invariant measure in the equivalence class, even though the equivalence class of the measure is preserved by ''T'' (or Γ).<ref>{{harvnb|Connes|1979}}</ref><ref name="Takesaki 2002">{{harvnb|Takesaki|2002}}</ref>

==Commutation theorem for semifinite traces==

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for ''a'' in ''M''₀. The operator ''J'' is again called the ''modular conjugation operator'' and extends to a conjugate-linear isometry of ''H'' satisfying ''J''² = I. The commutation theorem of Murray and von Neumann

:{| border="1" cellspacing="0" cellpadding="5"

|<math>JMJ=M^\prime</math>

|}

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==Hilbert algebras==

{{seealsosee also|Tomita–Takesaki theory}}

The theory of Hilbert algebras was introduced by Godement (under the name "unitary algebras"), Segal and Dixmier to formalize the classical method of defining the trace for [[trace class operator]]s starting from [[Hilbert-Schmidt operator]]s.<ref>{{harvnb|Simon|1979}}</ref> Applications in the [[Unitary representation|representation theory of groups]] naturally lead to examples of Hilbert algebras. Every von Neumann algebra endowed with a semifinite trace has a canonical "completed"<ref>Dixmier uses the adjectives ''achevée'' or ''maximale''.</ref> or "full" Hilbert algebra associated with it; and conversely a completed Hilbert algebra of exactly this form can be canonically associated with every Hilbert algebra. The theory of Hilbert algebras can be used to deduce the commutation theorems of Murray and von Neumann; equally well the main results on Hilbert algebras can also be deduced directly from the commutation theorems for traces. The theory of Hilbert algebras was generalised by Takesaki<ref>{{harvnb| name="Takesaki| 2002}}<"/ref> as a tool for proving commutation theorems for semifinite weights in [[Tomita–Takesaki theory]]; they can be dispensed with when dealing with states.<ref name ="rieffel" /><ref>{{harvnb|Pedersen|1979}}</ref><ref>{{harvnb|Bratteli|Robinson|1987}}</ref><ref name ="rieffel" />

====Definition====

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* If (''X'', μ) is an infinite measure space, the algebra ''L''^∞ (''X'') <math>\cap</math> ''L''²(''X'') is a Hilbert algebra with the usual inner product from ''L''²(''X'').

* If ''M'' is a von Neumann algebra with faithful semifinite trace τ, then the *-subalgebra ''M''₀ defined above is a Hilbert algebra with inner product (''a'', '' b'') = τ(''b''*''a'').

* If ''G'' is a [[Haar measure|unimodular]] [[locally compact group]], the convolution algebra ''L''¹(''G'')<math>\cap</math>''L''²(''G'') is a Hilbert algebra with the usual inner product from ''L''²(''G'').

* If (''G'', ''K'') is a [[Gelfand pair]], the convolution algebra ''L''¹(''K''\''G''/''K'')<math>\cap</math>''L''²(''K''\''G''/''K'') is a Hilbert algebra with the usual inner product from ''L''²(''G''); here ''L''^''p''(''K''\''G''/''K'') denotes the closed subspace of ''K''-biinvariant functions in ''L''^''p''(''G'').

* Any dense *-subalgebra of a Hilbert algebra is also a Hilbert algebra.

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states that

:{| border="1" cellspacing="0" cellpadding="5"

|<math>\lambda(\mathfrak{A})^{\prime\prime}=\rho(\mathfrak{A})^\prime</math>

|}

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Moreover if

:<math>M=\lambda(\mathfrak{A})^{\prime\prime},</math>

the von Neumann algebra generated by the operators λ(''a''), then

:{| border="1" cellspacing="0" cellpadding="5"

|<math>JMJ=M^\prime</math>

|}

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==Notes==

{{reflist|2}}

==References==

*{{citation|first=O.|last=Bratteli|first2=D.W.|last2=Robinson|title=Operator Algebras and Quantum Statistical Mechanics 1, Second Edition|publisher=Springer-Verlag|year=1987|id=ISBN 3-540-17093-6}}

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*{{citation|first=J.|last=Dixmier|authorlink=Jacques Dixmier|title=C* algebras|publisher=North Holland|year=1977|id=ISBN 0720407621}} (English translation)

*{{citation|first=R.|last=Godement|authorlink=Roger Godement|title=Mémoire sur la théorie des caractères dans les groupes localement compacts unimodulaires|journal=J. Math. Pures Appl.|volume= 30|year=1951|pages=1–110}}

*{{citation|first=R.|last=Godement|authorlink=Roger Godement|title=Théorie des caractères. I. Algèbres unitaires|journal=Ann. Of Math.|volume= 59|year=1954|pages=47–62|doi=10.2307/1969832|url=http://jstor.org/stable/1969832|issue=1|publisher=Annals of Mathematics}}

*{{citation|first=F.J.|last= Murray|authorlink1=F.J. Murray|first2= J. |last2=von Neumann |authorlink2=John von Neumann|url=http://links.jstor.org/sici?sici=0003-486X%28193601%292%3A37%3A1%3C116%3AOROO%3E2.0.CO%3B2-Y

|title=On rings of operators| journal= Ann. Of Math. (2) |volume= 37 |year=1936|pages=116–229|doi=10.2307/1968693|jstor=10.2307/1968693|issue=1|publisher=Annals of Mathematics}}

*{{citation|first=F.J.|last= Murray|authorlink1=F.J. Murray|first2= J. |last2=von Neumann|authorlink2=John von Neumann |url=http://links.jstor.org/sici?sici=0002-9947%28193703%2941%3A2%3C208%3AOROOI%3E2.0.CO%3B2-9

|title=On rings of operators II|journal= Trans. Amer. Math. Soc. |volume= 41 |year=1937|pages= 208–248|doi=10.2307/1989620|issue=2|jstor=10.2307/1989620|publisher=American Mathematical Society}}

*{{citation|first=F.J.|last= Murray|authorlink1=F.J. Murray|first2= J. |last2=von Neumann |authorlink2=John von Neumann|url=http://links.jstor.org/sici?sici=0003-486X%28194310%292%3A44%3A4%3C716%3AOROOI%3E2.0.CO%3B2-O

|title=On rings of operators IV|journal= Ann. Of Math. (2) |volume= 44 |year=1943|pages= 716–808|doi=10.2307/1969107|jstor=10.2307/1969107|issue=4|publisher=Annals of Mathematics}}

*{{citation|last=Pedersen|first=G.K.|title=C* algebras and their automorphism groups|series=London Mathematical Society Monographs|volume=14|year=1979|

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*{{citation|last=Rieffel|first= M.A.|last2= van Daele|first2=A.|title=A bounded operator approach to Tomita–Takesaki theory|journal=Pacific J. Math.|volume= 69 |year=1977|pages= 187–221}}

*{{citation|last=Segal|first=I.E.| authorlink=Irving Segal|title=A non-commutative extension of abstract integration|journal=Ann. Of Math. |volume=57|year=1953|pages= 401–457|doi=10.2307/1969729|url=http://jstor.org/stable/1969729|issue=3|publisher=Annals of Mathematics}} (Section 5)

*{{citation|last=Simon|first= B.|authorlink=Barry Simon|title=Trace ideals and their applications|series=London Mathematical Society Lecture Note Series|volume= 35|publisher= Cambridge University Press|year= 1979|id = ISBN 0-521-22286-9}}

*{{citation|first=M. |last=Takesaki |title=Theory of Operator Algebras II|publisher=Springer-Verlag|id= ISBN 3-540-42914-X|year=2002}}

{{DEFAULTSORT:Commutation Theorem}}

[[Category:Von Neumann algebras]]

[[Category:Representation theory of groups]]