Commutation theorem for traces: Difference between revisions

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{{Short description|Identifies the commutant of a specific von Neumann algebra}}
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 [[F.J.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 [[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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:<math>Ja\Omega=a^*\Omega</math>

for ''a'' in ''M'' defines a conjugate-linear isometry of ''H'' with square the identity, ''J''² = ''I''. The operator ''J'' is usually called the '''modular conjugation operator'''.

It is immediately verified that ''JMJ'' and ''M'' commute on the subspace ''M'' Ω, so that<ref>{{harvnb|Bratteli|Robinson|1987|pages=81–82}}</ref>

:<math>JMJ\subseteq M^\prime.</math>

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:<math> H=K\oplus iK,</math>

an orthogonal direct sum for the real part of the inner product. This is just the real orthogonal decomposition for the ±1 eigenspaces of ''J''.

On the other hand for ''a'' in ''M''_sa and ''b'' in ''M'''_sa, the inner product (''ab''Ω, Ω) is real, because ''ab'' is self-adjoint. Hence ''K'' is unaltered if ''M'' is replaced by ''M'' '.

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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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==Commutation theorem for semifinite traces==

Let ''M'' be a von Neumann algebra and ''M''₊ the set of [[positive operator]]s in ''M''. By definition,<ref name="dixmier57" /> a '''semifinite trace''' (or sometimes just '''trace''') on ''M'' is a functional τ from ''M''₊ into [0, ∞] such that

# <math> \tau(\lambda a + \mu b) = \lambda \tau(a) + \mu \tau(b)</math> for ''a'', ''b'' in ''M''₊ and λ, μ ≥ 0 (''{{visible anchor|semilinearity}}'');

# <math> \tau\left(uau^*\right) = \tau(a)</math> for ''a'' in ''M''₊ and ''u'' a [[unitary operator]] in ''M'' (''unitary invariance'');

# τ is completely additive on orthogonal families of projections in ''M'' (''normality'');

# each projection in ''M'' is as orthogonal direct sum of projections with finite trace (''semifiniteness'').

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If τ is a faithful trace on ''M'', let ''H'' = ''L''²(''M'', τ) be the Hilbert space completion of the inner product space

:<math>M_0 = \left\{a \in M| \mid \tau\left(a^*a\right) < \infty\right\}</math>

with respect to the inner product

:<math>(a, b) = \tau\left(b^*a\right).</math>

The von Neumann algebra ''M'' acts by left multiplication on ''H'' and can be identified with its image. Let

:<math>Ja = a^*</math>

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>

|}

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 <math> \supseteq</math>⊇ ''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.

==Hilbert algebras==

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A '''Hilbert algebra'''<ref name="dixmier57" /><ref>{{harvnb|Dixmier|1977}}, Appendix A54–A61.</ref><ref>{{harvnb|Dieudonné|1976}}</ref> is an algebra <math>\mathfrak{A}</math> with involution ''x''→''x''* and an inner product (,) such that

# (''a'', ''b'') = (''b''*, ''a''*) for ''a'', ''b'' in <math>\mathfrak{A}</math>;

# left multiplication by a fixed ''a'' in <math>\mathfrak{A}</math> is a bounded operator;

# * is the adjoint, in other words (''xy'', ''z'') = (''y'', ''x''*''z'');

# the linear span of all products ''xy'' is dense in <math>\mathfrak{A}</math>.

===Examples===

* The Hilbert–Schmidt operators on an infinite-dimensional Hilbert space form a Hilbert algebra with inner product (''a'', ''b'') = Tr (''b''*''a'').

* 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'').

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<math>\mathfrak{A}</math> on itself by left and right multiplication:

:<math> \lambda(a)x = ax,\, \, \rho(a)x = xa.</math>

These actions extend continuously to actions on ''H''. In this case the commutation theorem for Hilbert algebras states that

states that

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

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

|}

Moreover if

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

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

:<math display="block">M = \lambda(\mathfrak{AB})^{\prime\prime},''.</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}}

*{{citation|first=J.|last= Dixmier|authorlink=Jacques Dixmier|title=Von Neumann algebras|publisher=North Holland| isbn=0-444-86308-7 |year=1981}} (English translation)

*{{citation|first=J.|last= Dixmier|authorlink=Jacques Dixmier|title=Les C*-algèbres et leurs représentations|publisher= Gauthier-Villars|year=1969|isbn= 0-7204-0762-1|url-access=registration|url=https://archive.org/details/calgebras0000dixm}}

*{{citation|first=J.|last=Dixmier|authorlink=Jacques Dixmier|title=C* algebras|publisher=North Holland|year=1977|isbn=0-7204-0762-1|url-access=registration|url=https://archive.org/details/calgebras0000dixm}} (English translation)

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*{{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}}

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*{{citation|firstfirst1=F.J.|lastlast1= Murray|authorlink1=F.J.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|

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*{{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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{{Functional analysis}}

{{DEFAULTSORT:Commutation Theorem}}