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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
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| [[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.
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| ==Commutation theorem for finite traces==
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| 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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| * ''M'' Ω is dense in ''H''
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| * ''M'' ' Ω is dense in ''H'', where ''M'' ' denotes the [[commutant]] of ''M''
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| * (''ab''Ω, Ω) = (''ba''Ω, Ω) for all ''a'', ''b'' in ''M''.
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| The vector Ω is called a ''cyclic-separating trace vector''. It is called a trace vector because the last condition means that the [[matrix coefficient]] corresponding to Ω defines a tracial [[state (functional analysis)|state]] on ''M''. It is called cyclic since Ω generates ''H'' as a topological ''M''-module. It is called separating
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| because if ''a''Ω = 0 for ''a'' in ''M'', then ''aM'''Ω= (0), and hence ''a'' = 0.
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| It follows that the map
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| :<math>Ja\Omega=a^*\Omega</math> | |
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| for ''a'' in ''M'' defines a conjugate-linear isometry of ''H'' with square the identity ''J''<sup>2</sup> = ''I''. The operator ''J'' is usually called the '''modular conjugation operator'''.
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| It is immediately verified that ''JMJ'' and ''M'' commute on the subspace ''M'' Ω, so that
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| :<math>JMJ\subseteq M^\prime.</math>
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| The '''commutation theorem''' of Murray and von Neumann states that
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| :{| border="1" cellspacing="0" cellpadding="5"
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| |<math>JMJ=M^\prime</math>
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| |}
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| One of the easiest ways to see this<ref name="rieffel">{{harvnb|Rieffel|van Daele|1977}}</ref> is to introduce ''K'', the closure of the real
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| subspace ''M''<sub>sa</sub> Ω, where ''M''<sub>sa</sub> denotes the self-adjoint elements in ''M''. It follows that
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| :<math> H=K\oplus iK,</math>
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| an orthogonal direct sum for the real part of inner product. This is just the real orthogonal decomposition for the ±1 eigenspaces of ''J''.
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| On the other hand for ''a'' in ''M''<sub>sa</sub> and ''b'' in ''M'''<sub>sa</sub>, 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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| In particular Ω is a trace vector for ''M''' and ''J'' is unaltered if ''M'' is replaced by ''M'' '. So the opposite inclusion
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| :<math>JM^\prime J\subseteq M</math> | |
|
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| follows by reversing the roles of ''M'' and ''M'''.
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| ===Examples===
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| * 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
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| ::<math>(\lambda(g) f)(x)=f(g^{-1}x),\,\,(\rho(g)f)(x)=f(xg)</math>
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| :for ''f'' in <math>\ell^2(\Gamma)</math> and the commutation theorem implies that
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| ::<math>\lambda(\Gamma)^{\prime\prime}=\rho(\Gamma)^\prime, \,\, \rho(\Gamma)^{\prime\prime}=\lambda(\Gamma)^\prime.</math>
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| :The operator ''J'' is given by the formula
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| ::<math> Jf(g)=\overline{f(g^{-1})}.</math>
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| :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. | |
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| * Another important example is provided by a [[probability space]] (''X'', μ). The [[Abelian von Neumann algebra]] ''A'' = ''L''<sup>∞</sup>(''X'', μ) acts by [[multiplication operator]]s on ''H'' = ''L''<sup>2</sup>(''X'', μ) and the constant function 1 is a cyclic-separating trace vector. It follows that
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| ::<math>A^{\prime}=A,</math>
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| :so that ''A'' is a '''''maximal Abelian subalgebra''''' of ''B''(''H''), the von Neumann algebra of all [[bounded operator]]s on ''H''.
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| * 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''<sup>2</sup>(''X'', μ) according to the formula
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| ::<math>U_g f(x) = f(g^{-1}x),</math>
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| :for ''f'' in ''H'' and normalises the Abelian von Neumann algebra ''A'' = ''L''<sup>∞</sup>(''X'', μ). Let
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| ::<math>H_1=H\otimes \ell^2(\Gamma),</math>
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| :a [[tensor product]] of Hilbert spaces.<ref>''H''<sub>1</sub> 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
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| ::<math> M = A \rtimes \Gamma</math>
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| :is defined to be the von Neumann algebra on ''H''<sub>1</sub> 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''<sub>''g''</sub>.</ref>
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| :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.
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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
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| 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>
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| ==Commutation theorem for semifinite traces==
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| Let ''M'' be a von Neumann algebra and ''M''<sub>+</sub> 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''<sub>+</sub> into [0,∞] such that
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| # <math> \tau(\lambda a + \mu b) = \lambda \tau(a) + \mu \tau(b)</math> for ''a'', ''b'' in ''M''<sub>+</sub> and λ, μ ≥ 0 (''{{visible anchor|semilinearity}}'');
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| # <math> \tau(uau^*)=\tau(a)</math> for ''a'' in ''M''<sub>+</sub> and ''u'' a [[unitary operator]] in ''M'' (''unitary invariance'');
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| # τ is completely additive on orthogonal families of projections in ''M'' (''normality'');
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| # each projection in ''M'' is as orthogonal direct sum of projections with finite trace (''semifiniteness'').
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| If in addition τ is non-zero on every non-zero projection, then τ is called a '''faithful trace'''.
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| If τ is a faithul trace on ''M'', let ''H'' = ''L''<sup>2</sup>(''M'', τ) be the Hilbert space completion of the inner product space
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| :<math>M_0=\{a\in M| \tau(a^*a) <\infty\}</math>
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| with respect to the inner product
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| :<math>(a,b)=\tau(b^*a).</math>
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| The von Neumann algebra ''M'' acts by left multiplication on ''H'' and can be identified with its image. Let
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| :<math>Ja=a^*</math>
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| for ''a'' in ''M''<sub>0</sub>. The operator ''J'' is again called the ''modular conjugation operator'' and extends to a conjugate-linear isometry of ''H'' satisfying ''J''<sup>2</sup> = I. The commutation theorem of Murray and von Neumann
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| :{| border="1" cellspacing="0" cellpadding="5"
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| |<math>JMJ=M^\prime</math>
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| |}
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| is again valid in this case. This result can be proved directly by a variety of methods,<ref name="dixmier57" /> but follows immediately from the result for finite traces, by repeated use of the following elementary fact:
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| :''If'' ''M''<sub>1</sub> <math> \supseteq</math> ''M''<sub>2</sub> ''are two von Neumann algebras such that'' ''p''<sub>''n''</sub> ''M''<sub>1</sub> = ''p''<sub>''n''</sub> ''M''<sub>2</sub> ''for a family of projections'' ''p''<sub>''n''</sub> ''in the commutant of'' ''M''<sub>1</sub> ''increasing to'' ''I'' ''in the [[strong operator topology]], then'' ''M''<sub>1</sub> = ''M''<sub>2</sub>.
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| ==Hilbert algebras==
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| {{see also|Tomita–Takesaki theory}}
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| 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 name="Takesaki 2002"/> 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>
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| ===Definition===
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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
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| # (''a'',''b'')=(''b''*,''a''*) for ''a'', ''b'' in <math>\mathfrak{A}</math>;
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| # left multiplication by a fixed ''a'' in <math>\mathfrak{A}</math> is a bounded operator;
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| # * is the adjoint, in other words (''xy'',''z'') = (''y'', ''x''*''z'');
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| # the linear span of all products ''xy'' is dense in <math>\mathfrak{A}</math>.
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| ===Examples===
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| * The Hilbert-Schmidt operators on an infinite-dimensional Hilbert space form a Hilbert algebra with inner product (''a'',''b'') = Tr (''b''*''a'').
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| * If (''X'', μ) is an infinite measure space, the algebra ''L''<sup>∞</sup> (''X'') <math>\cap</math> ''L''<sup>2</sup>(''X'') is a Hilbert algebra with the usual inner product from ''L''<sup>2</sup>(''X'').
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| * If ''M'' is a von Neumann algebra with faithful semifinite trace τ, then the *-subalgebra ''M''<sub>0</sub> defined above is a Hilbert algebra with inner product (''a'', '' b'') = τ(''b''*''a'').
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| * If ''G'' is a [[Haar measure|unimodular]] [[locally compact group]], the convolution algebra ''L''<sup>1</sup>(''G'')<math>\cap</math>''L''<sup>2</sup>(''G'') is a Hilbert algebra with the usual inner product from ''L''<sup>2</sup>(''G'').
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| * If (''G'', ''K'') is a [[Gelfand pair]], the convolution algebra ''L''<sup>1</sup>(''K''\''G''/''K'')<math>\cap</math>''L''<sup>2</sup>(''K''\''G''/''K'') is a Hilbert algebra with the usual inner product from ''L''<sup>2</sup>(''G''); here ''L''<sup>''p''</sup>(''K''\''G''/''K'') denotes the closed subspace of ''K''-biinvariant functions in ''L''<sup>''p''</sup>(''G'').
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| * Any dense *-subalgebra of a Hilbert algebra is also a Hilbert algebra.
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| ===Properties===
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| Let ''H'' be the Hilbert space completion of <math>\mathfrak{A}</math> with respect to the inner product and let ''J'' denote the extension of the involution to a conjugate-linear involution of ''H''. Define a representation λ and an anti-representation ρ of
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| <math>\mathfrak{A}</math> on itself by left and right multiplication:
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| :<math> \lambda(a)x=ax,\, \, \rho(a)x=xa.</math>
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| These actions extend continuously to actions on ''H''. In this case the commutation theorem for Hilbert algebras
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| states that
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| :{| border="1" cellspacing="0" cellpadding="5"
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| |<math>\lambda(\mathfrak{A})^{\prime\prime}=\rho(\mathfrak{A})^\prime</math>
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| |}
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| Moreover if
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| :<math>M=\lambda(\mathfrak{A})^{\prime\prime},</math>
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| the von Neumann algebra generated by the operators λ(''a''), then
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| :{| border="1" cellspacing="0" cellpadding="5"
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| |<math>JMJ=M^\prime</math>
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| |}
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| These results were proved independently by {{harvtxt|Godement|1954}} and {{harvtxt|Segal|1953}}.
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| The proof relies on the notion of "bounded elements" in the Hilbert space completion ''H''.
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| An element of ''x'' in ''H'' is said to be '''bounded''' (relative to <math>\mathfrak{A}</math>) if the map ''a'' → ''xa'' of <math>\mathfrak{A}</math> into ''H'' extends to a
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| bounded operator on ''H'', denoted by λ(''x''). In this case it is straightforward to prove that:<ref>{{harvnb|Godement|1954|pp=52–53}}</ref>
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| * ''Jx'' is also a bounded element, denoted ''x''*, and λ(''x''*) = λ(''x'')*;
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| * ''a'' → ''ax'' is given by the bounded operator ρ(''x'') = ''J''λ(''x''*)''J'' on ''H'';
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| * ''M'' ' is generated by the ρ(''x'')'s with ''x'' bounded;
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| * λ(''x'') and ρ(''y'') commute for ''x'', ''y'' bounded.
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| The commutation theorem follows immediately from the last assertion. In particular
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| * ''M'' = λ(<math>\mathfrak{B}</math>)".
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| 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''<sub>+</sub> defined by
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| :<math> \tau(x) = (a,a)</math>
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| if ''x'' =λ(a)*λ(a) and ∞ otherwise, yields a faithful semifinite trace on ''M'' with
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| :<math> M_0=\mathfrak{B}.</math>
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| Thus:
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| :{| border="1" cellspacing="0" cellpadding="5"
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| |'''''There is a one-one correspondence between von Neumann algebras on H with faithful semifinite trace and full Hilbert algebras with Hilbert space completion H.'''''
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| |}
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| ==See also==
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| * [[von Neumann algebra]]
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| * [[Affiliated operator]]
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| * [[Tomita–Takesaki theory]]
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| ==Notes==
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| {{reflist|2}}
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| ==References==
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| *{{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=A.|last=Connes|authorlink=Alain Connes|title=Sur la théorie non commutative de l’intégration|series=Lecture Notes in Mathematics|volume=(Algèbres d'Opérateurs)|publisher=Springer-Verlag|year=1979|pages=19–143|id=ISBN 978-3-540-09512-5}}
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| *{{citation|first=J.|last = Dieudonné|authorlink=Jean Dieudonné|title=Treatise on Analysis, Vol. II |year=1976|publisher=Academic Press|id=ISBN 0-12-215502-2}}
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| *{{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=J.|last= Dixmier|authorlink=Jacques Dixmier|title=Von Neumann algebras|publisher=North Holland| isbn=0-444-86308-7 |year=1981}} (English translation)
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| *{{citation|first=J.|last= Dixmier|authorlink=Jacques Dixmier|title=Les C*-algèbres et leurs représentations|publisher= Gauthier-Villars|year=1969|id= ISBN 0-7204-0762-1}}
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| *{{citation|first=J.|last=Dixmier|authorlink=Jacques Dixmier|title=C* algebras|publisher=North Holland|year=1977|id=ISBN 0-7204-0762-1}} (English translation)
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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}}
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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|first=F.J.|last= Murray|authorlink1=F.J. Murray|first2= J. |last2=von Neumann |authorlink2=John von Neumann
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| |title=On rings of operators| journal= Ann. Of Math. (2) |volume= 37 |year=1936|pages=116–229|doi=10.2307/1968693|jstor=1968693|issue=1|publisher=Annals of Mathematics}}
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| *{{citation|first=F.J.|last= Murray|authorlink1=F.J. Murray|first2= J. |last2=von Neumann|authorlink2=John von Neumann
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| |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=1989620|publisher=American Mathematical Society}}
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| *{{citation|first=F.J.|last= Murray|authorlink1=F.J. Murray|first2= J. |last2=von Neumann |authorlink2=John von Neumann
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| |title=On rings of operators IV|journal= Ann. Of Math. (2) |volume= 44 |year=1943|pages= 716–808|doi=10.2307/1969107|jstor=1969107|issue=4|publisher=Annals of Mathematics}}
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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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| publisher=Academic Press|id=ISBN 0-12-549450-5}}
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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}}
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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)
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| *{{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}}
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| *{{citation|first=M. |last=Takesaki |title=Theory of Operator Algebras II|publisher=Springer-Verlag|id= ISBN 3-540-42914-X|year=2002}}
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| {{DEFAULTSORT:Commutation Theorem}}
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| [[Category:Von Neumann algebras]]
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| [[Category:Representation theory of groups]]
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| [[Category:Ergodic theory]]
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| [[Category:Theorems in functional analysis]]
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| [[Category:Theorems in representation theory]]
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