Commutation theorem for traces: Difference between revisions

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

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

{{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

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.

[[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 [[W. Forrest Stinespring|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==

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===Examples===

* One of the simplest cases of the commutation theorem, where it can easily be seen directly, is that of a [[finite group]] Γ acting on the finite-dimensional [[inner product space]] <math>\ell^2(\Gamma)</math> by the left and right [[regular representation]]s λ and ρ. These [[unitary representation]]s are given by the formulas <math display="block">(\lambda(g) f)(x)=f(g^{-1}x),\,\,(\rho(g)f)(x)=f(xg)</math> for ''f'' in <math>\ell^2(\Gamma)</math> and the commutation theorem implies that <math display="block">\lambda(\Gamma)^{\prime\prime} = \rho(\Gamma)^\prime, \,\, \rho(\Gamma)^{\prime\prime}=\lambda(\Gamma)^\prime.</math> The operator ''J'' is given by the formula <math display="block"> Jf(g) = \overline{f(g^{-1})}.</math> Exactly the same results remain true if Γ is allowed to be any [[countable]] [[discrete group]].<ref name="dixmier57">{{harvnb|Dixmier|1957}}</ref> The von Neumann algebra λ(Γ)' ' is usually called the '''''group von Neumann algebra''''' of Γ.

* Another important example is provided by a [[probability space]] (''X'', μ). The [[Abelian von Neumann algebra]] ''A'' = ''L''^∞(''X'', μ) acts by [[multiplication operator]]s on ''H'' = ''L''²(''X'', μ) and the constant function 1 is a cyclic-separating trace vector. It follows that <math display="block">A' = A,</math> so that ''A'' is a '''''maximal Abelian subalgebra''''' of ''B''(''H''), the von Neumann algebra of all [[bounded operator]]s on ''H''.

* The third class of examples combines the above two. Coming from [[ergodic theory]], it was one of von Neumann's original motivations for studying von Neumann algebras. Let (''X'', μ) be a probability space and let Γ be a countable discrete group of measure-preserving transformations of (''X'', μ). The group therefore acts unitarily on the Hilbert space ''H'' = ''L''²(''X'', μ) according to the formula <math display="block">U_g f(x) = f(g^{-1}x),</math> for ''f'' in ''H'' and normalises the Abelian von Neumann algebra ''A'' = ''L''^∞(''X'', μ). Let <math display="block">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 display="block"> M = A \rtimes \Gamma</math> is defined to be the von Neumann algebra on ''H''₁ generated by the algebra <math>A\otimes I</math> and the normalising operators <math>U_g\otimes \lambda(g)</math>.<ref>It should not be confused with the von Neumann algebra on ''H'' generated by ''A'' and the operators ''U''_''g''.</ref>{{pb}} The vector <math>\Omega=1\otimes \delta_1</math> is a cyclic-separating trace vector. Moreover the modular conjugation operator ''J'' and commutant ''M'' ' can be explicitly identified.

::<math>(\lambda(g) f)(x)=f(g^{-1}x),\,\,(\rho(g)f)(x)=f(xg)</math>

:for ''f'' in <math>\ell^2(\Gamma)</math> and the commutation theorem implies that

::<math>\lambda(\Gamma)^{\prime\prime}=\rho(\Gamma)^\prime, \,\, \rho(\Gamma)^{\prime\prime}=\lambda(\Gamma)^\prime.</math>

:The operator ''J'' is given by the formula

::<math> Jf(g)=\overline{f(g^{-1})}.</math>

:Exactly the same results remain true if Γ is allowed to be any [[countable]] [[discrete group]].<ref name="dixmier57">{{harvnb|Dixmier|1957}}</ref> The von Neumann algebra λ(Γ)' ' is usually called the '''''group von Neumann algebra''''' of Γ.

* Another important example is provided by a [[probability space]] (''X'', μ). The [[Abelian von Neumann algebra]] ''A'' = ''L''^∞(''X'', μ) acts by [[multiplication operator]]s on ''H'' = ''L''²(''X'', μ) and the constant function 1 is a cyclic-separating trace vector. It follows that

::<math>A^{\prime}=A,</math>

:so that ''A'' is a '''''maximal Abelian subalgebra''''' of ''B''(''H''), the von Neumann algebra of all [[bounded operator]]s on ''H''.

* The third class of examples combines the above two. Coming from [[ergodic theory]], it was one of von Neumann's original motivations for studying von Neumann algebras. Let (''X'', μ) be a probability space and let Γ be a countable discrete group of measure-preserving transformations of (''X'', μ). The group therefore acts unitarily on the Hilbert space ''H'' = ''L''²(''X'', μ) according to the formula

::<math>U_g f(x) = f(g^{-1}x),</math>

:for ''f'' in ''H'' and normalises the Abelian von Neumann algebra ''A'' = ''L''^∞(''X'', μ). Let

::<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>

:is defined to be the von Neumann algebra on ''H''₁ generated by the algebra <math>A\otimes I</math> and the normalising operators <math>U_g\otimes \lambda(g)</math>.<ref>It should not be confused with the von Neumann algebra on ''H'' generated by ''A'' and the operators ''U''_''g''.</ref>

:The vector <math>\Omega=1\otimes \delta_1</math> is a cyclic-separating trace vector. Moreover the modular conjugation operator ''J'' and commutant ''M'' ' can be explicitly identified.

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

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|}

is again valid in this case. This result can be proved directly by a variety of methods,<ref name="dixmier57" /><ref>{{harvnb|Takesaki|1979|pages=324–325}}</ref> but follows immediately from the result for finite traces, by repeated use of the following elementary fact:

:''If'' ''M''₁ ⊇ ''M''₂ ''are two von Neumann algebras such that'' ''p''_''n'' ''M''₁ = ''p''_''n'' ''M''₂ ''for a family of projections'' ''p''_''n'' ''in the commutant of'' ''M''₁ ''increasing to'' ''I'' ''in the [[strong operator topology]], then'' ''M''₁ = ''M''₂.

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Moreover if

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

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The commutation theorem follows immediately from the last assertion. In particular

:<math display="block">M_0M = \lambda(\mathfrak{B})''.</math>

The space of all bounded elements <math>\mathfrak{B}</math> forms a Hilbert algebra containing <math>\mathfrak{A}</math> as a dense *-subalgebra. It is said to be '''completed''' or '''full''' because any element in ''H'' bounded relative to <math>\mathfrak{B}</math> must actually already lie in <math>\mathfrak{B}</math>. The functional τ on ''M''₊ defined by

* ''M'' = λ(<math>\mathfrak{B}</math>)".

:<math display="block"> \tau(x) = (a,a)</math>

if ''x'' = ''λ''(''a'')*''λ''(''a'') and ∞ otherwise, yields a faithful semifinite trace on ''M'' with

The space of all bounded elements <math>\mathfrak{B}</math> forms a Hilbert algebra containing <math>\mathfrak{A}</math> as a dense *-subalgebra. It is said to be '''completed''' or '''full''' because any element in ''H'' bounded relative to <math>\mathfrak{B}</math>must actually already lie in <math>\mathfrak{B}</math>. The functional τ on ''M''₊ defined by

<math display="block">M_0 = \mathfrak{B}.</math>

:<math> \tau(x) = (a,a)</math>

if ''x'' = λ(a)*λ(a) and ∞ otherwise, yields a faithful semifinite trace on ''M'' with

:<math>M_0 = \mathfrak{B}.</math>

Thus:

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==References==

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*{{citation|first=A.|last=Connes|authorlink=Alain Connes|title=Sur la théorie non commutative de l’intégrationl'intégration|series=Lecture Notes in Mathematics|volume=(Algèbres d'Opérateurs)|publisher=Springer-Verlag|year=1979|pages=19–143|isbn=978-3-540-09512-5}}

*{{citation|first=J.|last = Dieudonné|authorlink=Jean Dieudonné|title=Treatise on Analysis, Vol. II |year=1976|publisher=Academic Press|isbn=0-12-215502-5}}

*{{citation|first=J.|last= Dixmier|authorlink=Jacques Dixmier|title=Les algèbres d'opérateurs dans l'espace hilbertien: algèbres de von Neumann|publisher= Gauthier-Villars |year=1957}}

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*{{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|issue=1|publisher=Annals of Mathematics|jstor=1969832}}

*{{citation|firstfirst1=F.J.|lastlast1= Murray|authorlink1=Francis Joseph Murray|first2= J. |last2=von Neumann |authorlink2=John von Neumann

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

publisher=Academic Press|isbn=0-12-549450-5}}

*{{citation|lastlast1=Rieffel|firstfirst1= 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|doi=10.2140/pjm.1977.69.187|doi-access=free}}

*{{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|issue=3|publisher=Annals of Mathematics|jstor=1969729}} (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|isbn = 0-521-22286-9}}

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*{{citation|first=M. |last=Takesaki |title=Theory of Operator Algebras II|publisher=Springer-Verlag|isbn=3-540-42248-X |year=2002}}

{{Functional analysis}}