Commutation theorem for traces

<h2 id="commutation-theorem-for-finite-traces">Commutation theorem for finite traces</h2>
Let H be a <a href="/facts/Hilbert_space/aDFMh4lV">Hilbert space</a> and M a <a href="/facts/Von_Neumann_algebra/8NyWUIRH">von Neumann algebra</a> on H with a unit vector Ω such that

<ul><li>M Ω is dense in H</li>
<li>M ' Ω is dense in H, where M ' denotes the <a href="/facts/Commutant/AKhrEUGR">commutant</a> of M</li>
<li>(abΩ, Ω) = (baΩ, Ω) for all a, b in M.</li></ul>
The vector Ω is called a cyclic-separating trace vector. It is called a trace vector because the last condition means that the <a href="/facts/Matrix_coefficient/d3hXkGt3">matrix coefficient</a> corresponding to Ω defines a tracial <a href="/facts/State_(functional_analysis)/HR8Vvjvg">state</a> on M. It is called cyclic since Ω generates H as a topological M-module. It is called separating
because if aΩ = 0 for a in M, then aM'Ω= (0), and hence a = 0.
It follows that the map

J
        a
        Ω
        =
        
          a
          
            ∗
          
        
        Ω
      
    
    {\displaystyle Ja\Omega =a^{*}\Omega }

for a in M defines a conjugate-linear isometry of H with square the identity, J2 = 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<a class="footnote-ref" id="fnref:1" href="#fn:1">1</a>

J
        M
        J
        ⊆
        
          M
          
            ′
          
        
        .
      
    
    {\displaystyle JMJ\subseteq M^{\prime }.}

The commutation theorem of Murray and von Neumann states that

<table><tbody><tr><td> J M J = M ′ {\displaystyle JMJ=M^{\prime }} </td></tr></tbody></table>
One of the easiest ways to see this<a class="footnote-ref" id="fnref:2" href="#fn:2">2</a> is to introduce K, the closure of the real
subspace Msa Ω, where Msa denotes the self-adjoint elements in M. It follows that

H
        =
        K
        ⊕
        i
        K
        ,
      
    
    {\displaystyle H=K\oplus iK,}

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 Msa and b in M'sa, the inner product (abΩ, Ω) is real, because ab is self-adjoint. Hence K is unaltered if M is replaced by M '.
In particular Ω is a trace vector for M' and J is unaltered if M is replaced by M '. So the opposite inclusion

J
        
          M
          
            ′
          
        
        J
        ⊆
        M
      
    
    {\displaystyle JM^{\prime }J\subseteq M}

follows by reversing the roles of M and M'.

<h3>Examples</h3>
<ul><li>One of the simplest cases of the commutation theorem, where it can easily be seen directly, is that of a <a href="/facts/Finite_group/vv2tvogB">finite group</a> Γ acting on the finite-dimensional <a href="/facts/Inner_product_space/HoyjElEy">inner product space</a> 
 
 
 
 
 ℓ
 
 2
 
 
 (
 Γ
 )
 
 
 {\displaystyle \ell ^{2}(\Gamma )}
 
 by the left and right <a href="/facts/Regular_representation/dBND2lRj">regular representations</a> λ and ρ. These <a href="/facts/Unitary_representation/W1XRatG8">unitary representations</a> are given by the formulas 
 
 
 
 (
 λ
 (
 g
 )
 f
 )
 (
 x
 )
 =
 f
 (
 
 g
 
 −
 1
 
 
 x
 )
 ,
 
 
 (
 ρ
 (
 g
 )
 f
 )
 (
 x
 )
 =
 f
 (
 x
 g
 )
 
 
 {\displaystyle (\lambda (g)f)(x)=f(g^{-1}x),\,\,(\rho (g)f)(x)=f(xg)}
 
 for f in 
 
 
 
 
 ℓ
 
 2
 
 
 (
 Γ
 )
 
 
 {\displaystyle \ell ^{2}(\Gamma )}
 
 and the commutation theorem implies that 
 
 
 
 λ
 (
 Γ
 
 )
 
 ′
 ′
 
 
 =
 ρ
 (
 Γ
 
 )
 
 ′
 
 
 ,
 
 
 ρ
 (
 Γ
 
 )
 
 ′
 ′
 
 
 =
 λ
 (
 Γ
 
 )
 
 ′
 
 
 .
 
 
 {\displaystyle \lambda (\Gamma )^{\prime \prime }=\rho (\Gamma )^{\prime },\,\,\rho (\Gamma )^{\prime \prime }=\lambda (\Gamma )^{\prime }.}
 
 The operator J is given by the formula 
 
 
 
 J
 f
 (
 g
 )
 =
 
 
 
 f
 (
 
 g
 
 −
 1
 
 
 )
 
 ¯
 
 
 .
 
 
 {\displaystyle Jf(g)={\overline {f(g^{-1})}}.}
 
 Exactly the same results remain true if Γ is allowed to be any <a href="/facts/Countable/Wj4DmWPv">countable</a> <a href="/facts/Discrete_group/HIjkpJhS">discrete group</a>.<a class="footnote-ref" id="fnref:3" href="#fn:3">3</a> The von Neumann algebra λ(Γ)' ' is usually called the group von Neumann algebra of Γ.</li>
<li>Another important example is provided by a <a href="/facts/Probability_space/dEcv2VrU">probability space</a> (X, μ). The <a href="/facts/Abelian_von_Neumann_algebra/Mljk27ba">Abelian von Neumann algebra</a> A = L∞(X, μ) acts by <a href="/facts/Multiplication_operator/YrvjuoVQ">multiplication operators</a> on H = L2(X, μ) and the constant function 1 is a cyclic-separating trace vector. It follows that 
 
 
 
 
 A
 ′
 
 =
 A
 ,
 
 
 {\displaystyle A'=A,}
 
 so that A is a maximal Abelian subalgebra of B(H), the von Neumann algebra of all <a href="/facts/Bounded_operator/LYmDTIqR">bounded operators</a> on H.</li>
<li>The third class of examples combines the above two. Coming from <a href="/facts/Ergodic_theory/rkqCbIy0">ergodic theory</a>, 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 = L2(X, μ) according to the formula 
 
 
 
 
 U
 
 g
 
 
 f
 (
 x
 )
 =
 f
 (
 
 g
 
 −
 1
 
 
 x
 )
 ,
 
 
 {\displaystyle U_{g}f(x)=f(g^{-1}x),}
 
 for f in H and normalises the Abelian von Neumann algebra A = L∞(X, μ). Let 
 
 
 
 
 H
 
 1
 
 
 =
 H
 ⊗
 
 ℓ
 
 2
 
 
 (
 Γ
 )
 ,
 
 
 {\displaystyle H_{1}=H\otimes \ell ^{2}(\Gamma ),}
 
 a <a href="/facts/Tensor_product/7K3pR1ap">tensor product</a> of Hilbert spaces.<a class="footnote-ref" id="fnref:4" href="#fn:4">4</a> The group–measure space construction or <a href="/facts/Crossed_product/AVtwntCg">crossed product</a> von Neumann algebra 
 
 
 
 M
 =
 A
 ⋊
 Γ
 
 
 {\displaystyle M=A\rtimes \Gamma }
 
 is defined to be the von Neumann algebra on H1 generated by the algebra 
 
 
 
 A
 ⊗
 I
 
 
 {\displaystyle A\otimes I}
 
 and the normalising operators 
 
 
 
 
 U
 
 g
 
 
 ⊗
 λ
 (
 g
 )
 
 
 {\displaystyle U_{g}\otimes \lambda (g)}
 
.<a class="footnote-ref" id="fnref:5" href="#fn:5">5</a> The vector 
 
 
 
 Ω
 =
 1
 ⊗
 
 δ
 
 1
 
 
 
 
 {\displaystyle \Omega =1\otimes \delta _{1}}
 
 is a cyclic-separating trace vector. Moreover the modular conjugation operator J and commutant M ' can be explicitly identified.</li></ul>
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 <a href="/facts/Equivalence_of_measures/S8DCmucU">equivalent</a> measure; and the full force of the <a href="/facts/Tomita%25E2%2580%2593Takesaki_theory/FkYYEB29">Tomita–Takesaki theory</a> 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).<a class="footnote-ref" id="fnref:6" href="#fn:6">6</a><a class="footnote-ref" id="fnref:7" href="#fn:7">7</a>

<h2 id="commutation-theorem-for-semifinite-traces">Commutation theorem for semifinite traces</h2>
Let M be a von Neumann algebra and M+ the set of <a href="/facts/Positive_operator/kvTXwFHj">positive operators</a> in M. By definition,<a class="footnote-ref" id="fnref:8" href="#fn:8">8</a> a semifinite trace (or sometimes just trace) on M is a functional τ from M+ into [0, ∞] such that

<ol><li>
 
 
 
 τ
 (
 λ
 a
 +
 μ
 b
 )
 =
 λ
 τ
 (
 a
 )
 +
 μ
 τ
 (
 b
 )
 
 
 {\displaystyle \tau (\lambda a+\mu b)=\lambda \tau (a)+\mu \tau (b)}
 
 for a, b in M+ and λ, μ ≥ 0 (semilinearity);</li>
<li>
 
 
 
 τ
 
 (
 
 u
 a
 
 u
 
 ∗
 
 
 
 )
 
 =
 τ
 (
 a
 )
 
 
 {\displaystyle \tau \left(uau^{*}\right)=\tau (a)}
 
 for a in M+ and u a <a href="/facts/Unitary_operator/rkypfQQ9">unitary operator</a> in M (unitary invariance);</li>
<li>τ is completely additive on orthogonal families of projections in M (normality);</li>
<li>each projection in M is as orthogonal direct sum of projections with finite trace (semifiniteness).</li></ol>
If in addition τ is non-zero on every non-zero projection, then τ is called a faithful trace.
If τ is a faithful trace on M, let H = L2(M, τ) be the Hilbert space completion of the inner product space

M
 
 0
 
 
 =
 
 {
 
 a
 ∈
 M
 ∣
 τ
 
 (
 
 
 a
 
 ∗
 
 
 a
 
 )
 
 <
 ∞
 
 }
 
 
 
 {\displaystyle M_{0}=\left\{a\in M\mid \tau \left(a^{*}a\right)<\infty \right\}}

with respect to the inner product

(
        a
        ,
        b
        )
        =
        τ
        
          (
          
            
              b
              
                ∗
              
            
            a
          
          )
        
        .
      
    
    {\displaystyle (a,b)=\tau \left(b^{*}a\right).}

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

J
        a
        =
        
          a
          
            ∗
          
        
      
    
    {\displaystyle Ja=a^{*}}

for a in M0. The operator J is again called the modular conjugation operator and extends to a conjugate-linear isometry of H satisfying J2 = I. The commutation theorem of Murray and von Neumann

<table><tbody><tr><td> J M J = M ′ {\displaystyle JMJ=M^{\prime }} </td></tr></tbody></table>
is again valid in this case. This result can be proved directly by a variety of methods,<a class="footnote-ref" id="fnref:9" href="#fn:9">9</a><a class="footnote-ref" id="fnref:10" href="#fn:10">10</a> but follows immediately from the result for finite traces, by repeated use of the following elementary fact:

If M1 ⊇ M2 are two von Neumann algebras such that pn M1 = pn M2 for a family of projections pn in the commutant of M1 increasing to I in the <a href="/facts/Strong_operator_topology/awhSW5op">strong operator topology</a>, then M1 = M2.
<h2 id="hilbert-algebras">Hilbert algebras</h2>
See also: <a href="/facts/Tomita%25E2%2580%2593Takesaki_theory/FkYYEB29">Tomita–Takesaki theory</a>
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 <a href="/facts/Trace_class_operator/2AmWENA3">trace class operators</a> starting from <a href="/facts/Hilbert%25E2%2580%2593Schmidt_operator/Xhy17yn6">Hilbert–Schmidt operators</a>.<a class="footnote-ref" id="fnref:11" href="#fn:11">11</a> Applications in the <a href="/facts/Unitary_representation/W1XRatG8">representation theory of groups</a> naturally lead to examples of Hilbert algebras. Every von Neumann algebra endowed with a semifinite trace has a canonical "completed"<a class="footnote-ref" id="fnref:12" href="#fn:12">12</a> 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<a class="footnote-ref" id="fnref:13" href="#fn:13">13</a> as a tool for proving commutation theorems for semifinite weights in <a href="/facts/Tomita%25E2%2580%2593Takesaki_theory/FkYYEB29">Tomita–Takesaki theory</a>; they can be dispensed with when dealing with states.<a class="footnote-ref" id="fnref:14" href="#fn:14">14</a><a class="footnote-ref" id="fnref:15" href="#fn:15">15</a><a class="footnote-ref" id="fnref:16" href="#fn:16">16</a>

<h3>Definition</h3>
A Hilbert algebra<a class="footnote-ref" id="fnref:17" href="#fn:17">17</a><a class="footnote-ref" id="fnref:18" href="#fn:18">18</a><a class="footnote-ref" id="fnref:19" href="#fn:19">19</a> is an algebra 
 
 
 
 
 
 A
 
 
 
 
 {\displaystyle {\mathfrak {A}}}
 
 with involution x→x* and an inner product (,) such that

<ol><li>(a, b) = (b*, a*) for a, b in 
 
 
 
 
 
 A
 
 
 
 
 {\displaystyle {\mathfrak {A}}}
 
;</li>
<li>left multiplication by a fixed a in 
 
 
 
 
 
 A
 
 
 
 
 {\displaystyle {\mathfrak {A}}}
 
 is a bounded operator;</li>
<li>* is the adjoint, in other words (xy, z) = (y, x*z);</li>
<li>the linear span of all products xy is dense in 
 
 
 
 
 
 A
 
 
 
 
 {\displaystyle {\mathfrak {A}}}
 
.</li></ol>
<h3>Examples</h3>
<ul><li>The Hilbert–Schmidt operators on an infinite-dimensional Hilbert space form a Hilbert algebra with inner product (a, b) = Tr (b*a).</li>
<li>If (X, μ) is an infinite measure space, the algebra L∞ (X) 
 
 
 
 ∩
 
 
 {\displaystyle \cap }
 
 L2(X) is a Hilbert algebra with the usual inner product from L2(X).</li>
<li>If M is a von Neumann algebra with faithful semifinite trace τ, then the *-subalgebra M0 defined above is a Hilbert algebra with inner product (a, b) = τ(b*a).</li>
<li>If G is a <a href="/facts/Haar_measure/2IcmTcuU">unimodular</a> <a href="/facts/Locally_compact_group/AGPC8vnY">locally compact group</a>, the convolution algebra L1(G)
 
 
 
 ∩
 
 
 {\displaystyle \cap }
 
L2(G) is a Hilbert algebra with the usual inner product from L2(G).</li>
<li>If (G, K) is a <a href="/facts/Gelfand_pair/1nr6GewY">Gelfand pair</a>, the convolution algebra L1(K\G/K)
 
 
 
 ∩
 
 
 {\displaystyle \cap }
 
L2(K\G/K) is a Hilbert algebra with the usual inner product from L2(G); here Lp(K\G/K) denotes the closed subspace of K-biinvariant functions in Lp(G).</li>
<li>Any dense *-subalgebra of a Hilbert algebra is also a Hilbert algebra.</li></ul>
<h3>Properties</h3>
Let H be the Hilbert space completion of 
 
 
 
 
 
 A
 
 
 
 
 {\displaystyle {\mathfrak {A}}}
 
 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

A
 
 
 
 
 {\displaystyle {\mathfrak {A}}}
 
 on itself by left and right multiplication:

λ
        (
        a
        )
        x
        =
        a
        x
        ,
        
        
        ρ
        (
        a
        )
        x
        =
        x
        a
        .
      
    
    {\displaystyle \lambda (a)x=ax,\,\,\rho (a)x=xa.}

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

<table><tbody><tr><td> λ ( A ) ′ ′ = ρ ( A ) ′ {\displaystyle \lambda ({\mathfrak {A}})^{\prime \prime }=\rho ({\mathfrak {A}})^{\prime }} </td></tr></tbody></table>
Moreover if

M
        =
        λ
        (
        
          
            A
          
        
        
          )
          
            ′
            ′
          
        
        ,
      
    
    {\displaystyle M=\lambda ({\mathfrak {A}})^{\prime \prime },}

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

<table><tbody><tr><td> J M J = M ′ {\displaystyle JMJ=M^{\prime }} </td></tr></tbody></table>
These results were proved independently by Godement (1954) and Segal (1953).
The proof relies on the notion of "bounded elements" in the Hilbert space completion H.
An element of x in H is said to be bounded (relative to 
 
 
 
 
 
 A
 
 
 
 
 {\displaystyle {\mathfrak {A}}}
 
) if the map a → xa of 
 
 
 
 
 
 A
 
 
 
 
 {\displaystyle {\mathfrak {A}}}
 
 into H extends to a 
bounded operator on H, denoted by λ(x). In this case it is straightforward to prove that:<a class="footnote-ref" id="fnref:20" href="#fn:20">20</a>

<ul><li>Jx is also a bounded element, denoted x*, and λ(x*) = λ(x)*;</li>
<li>a → ax is given by the bounded operator ρ(x) = Jλ(x*)J on H;</li>
<li>M ' is generated by the ρ(x)'s with x bounded;</li>
<li>λ(x) and ρ(y) commute for x, y bounded.</li></ul>
The commutation theorem follows immediately from the last assertion. In particular

M
        =
        λ
        (
        
          
            B
          
        
        
          )
          ″
        
        .
      
    
    {\displaystyle M=\lambda ({\mathfrak {B}})''.}

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

τ
        (
        x
        )
        =
        (
        a
        ,
        a
        )
      
    
    {\displaystyle \tau (x)=(a,a)}

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

M
          
            0
          
        
        =
        
          
            B
          
        
        .
      
    
    {\displaystyle M_{0}={\mathfrak {B}}.}

Thus:

<table><tbody><tr><td>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.</td></tr></tbody></table>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Von_Neumann_algebra/8NyWUIRH">von Neumann algebra</a></li>
<li><a href="/facts/Affiliated_operator/rHYfzwEu">Affiliated operator</a></li>
<li><a href="/facts/Tomita%25E2%2580%2593Takesaki_theory/FkYYEB29">Tomita–Takesaki theory</a></li></ul>
<h2 id="notes">Notes</h2>

<ul><li>Bratteli, O.; Robinson, D.W. (1987), Operator Algebras and Quantum Statistical Mechanics 1, Second Edition, Springer-Verlag, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 3-540-17093-6</li>
<li><a href="/facts/Alain_Connes/tjFJ9Iou">Connes, A.</a> (1979), Sur la théorie non commutative de l'intégration, Lecture Notes in Mathematics, vol. (Algèbres d'Opérateurs), Springer-Verlag, pp. 19–143, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-3-540-09512-5</li>
<li><a href="/facts/Jean_Dieudonn%25C3%25A9/gWagzMdu">Dieudonné, J.</a> (1976), Treatise on Analysis, Vol. II, Academic Press, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-12-215502-5</li>
<li><a href="/facts/Jacques_Dixmier/eIgzkHvX">Dixmier, J.</a> (1957), Les algèbres d'opérateurs dans l'espace hilbertien: algèbres de von Neumann, Gauthier-Villars</li>
<li><a href="/facts/Jacques_Dixmier/eIgzkHvX">Dixmier, J.</a> (1981), Von Neumann algebras, North Holland, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-444-86308-7 (English translation)</li>
<li><a href="/facts/Jacques_Dixmier/eIgzkHvX">Dixmier, J.</a> (1969), <a href="https://archive.org/details/calgebras0000dixm">Les C*-algèbres et leurs représentations</a>, Gauthier-Villars, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-7204-0762-1</li>
<li><a href="/facts/Jacques_Dixmier/eIgzkHvX">Dixmier, J.</a> (1977), <a href="https://archive.org/details/calgebras0000dixm">C* algebras</a>, North Holland, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-7204-0762-1 (English translation)</li>
<li><a href="/facts/Roger_Godement/jdjb03n0">Godement, R.</a> (1951), "Mémoire sur la théorie des caractères dans les groupes localement compacts unimodulaires", J. Math. Pures Appl., 30: 1–110</li>
<li><a href="/facts/Roger_Godement/jdjb03n0">Godement, R.</a> (1954), "Théorie des caractères. I. Algèbres unitaires", Ann. of Math., 59 (1), Annals of Mathematics: 47–62, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.2307%2F1969832">10.2307/1969832</a>, <a href="/facts/JSTOR_(identifier)/YTeVmaJ7">JSTOR</a> <a href="https://www.jstor.org/stable/1969832">1969832</a></li>
<li><a href="/facts/Francis_Joseph_Murray/HsnrRXR1">Murray, F.J.</a>; <a href="/facts/John_von_Neumann/aUk6urof">von Neumann, J.</a> (1936), "On rings of operators", Ann. of Math., 2, 37 (1), Annals of Mathematics: 116–229, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.2307%2F1968693">10.2307/1968693</a>, <a href="/facts/JSTOR_(identifier)/YTeVmaJ7">JSTOR</a> <a href="https://www.jstor.org/stable/1968693">1968693</a></li>
<li><a href="/facts/Francis_Joseph_Murray/HsnrRXR1">Murray, F.J.</a>; <a href="/facts/John_von_Neumann/aUk6urof">von Neumann, J.</a> (1937), "On rings of operators II", Trans. Amer. Math. Soc., 41 (2), American Mathematical Society: 208–248, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.2307%2F1989620">10.2307/1989620</a>, <a href="/facts/JSTOR_(identifier)/YTeVmaJ7">JSTOR</a> <a href="https://www.jstor.org/stable/1989620">1989620</a></li>
<li><a href="/facts/Francis_Joseph_Murray/HsnrRXR1">Murray, F.J.</a>; <a href="/facts/John_von_Neumann/aUk6urof">von Neumann, J.</a> (1943), "On rings of operators IV", Ann. of Math., 2, 44 (4), Annals of Mathematics: 716–808, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.2307%2F1969107">10.2307/1969107</a>, <a href="/facts/JSTOR_(identifier)/YTeVmaJ7">JSTOR</a> <a href="https://www.jstor.org/stable/1969107">1969107</a></li>
<li>Pedersen, G.K. (1979), C* algebras and their automorphism groups, London Mathematical Society Monographs, vol. 14, Academic Press, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-12-549450-5</li>
<li>Rieffel, M.A.; van Daele, A. (1977), "A bounded operator approach to Tomita–Takesaki theory", Pacific J. Math., 69: 187–221, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.2140%2Fpjm.1977.69.187">10.2140/pjm.1977.69.187</a></li>
<li><a href="/facts/Irving_Segal/SasFUqI8">Segal, I.E.</a> (1953), "A non-commutative extension of abstract integration", Ann. of Math., 57 (3), Annals of Mathematics: 401–457, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.2307%2F1969729">10.2307/1969729</a>, <a href="/facts/JSTOR_(identifier)/YTeVmaJ7">JSTOR</a> <a href="https://www.jstor.org/stable/1969729">1969729</a> (Section 5)</li>
<li><a href="/facts/Barry_Simon/BImmcE2I">Simon, B.</a> (1979), Trace ideals and their applications, London Mathematical Society Lecture Note Series, vol. 35, Cambridge University Press, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-521-22286-9</li>
<li>Takesaki, M. (1979), Theory of Operator Algebras I, Springer-Verlag, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 3-540-42914-X</li>
<li>Takesaki, M. (2002), Theory of Operator Algebras II, Springer-Verlag, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 3-540-42248-X</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Bratteli & Robinson 1987, pp. 81–82 - Bratteli, O.; Robinson, D.W. (1987), Operator Algebras and Quantum Statistical Mechanics 1, Second Edition, Springer-Verlag, ISBN 3-540-17093-6 <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Rieffel & van Daele 1977 - Rieffel, M.A.; van Daele, A. (1977), "A bounded operator approach to Tomita–Takesaki theory", Pacific J. Math., 69: 187–221, doi:10.2140/pjm.1977.69.187 <a href="https://doi.org/10.2140%2Fpjm.1977.69.187" target="_blank">https://doi.org/10.2140%2Fpjm.1977.69.187</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Dixmier 1957 - Dixmier, J. (1957), Les algèbres d'opérateurs dans l'espace hilbertien: algèbres de von Neumann, Gauthier-Villars <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">H1 can be identified with the space of square integrable functions on X x Γ with respect to the product measure. <a href="/wiki/Product_measure" target="_blank">/wiki/Product_measure</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">It should not be confused with the von Neumann algebra on H generated by A and the operators Ug. <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">Connes 1979 - Connes, A. (1979), Sur la théorie non commutative de l'intégration, Lecture Notes in Mathematics, vol. (Algèbres d'Opérateurs), Springer-Verlag, pp. 19–143, ISBN 978-3-540-09512-5 <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">Takesaki 2002 - Takesaki, M. (2002), Theory of Operator Algebras II, Springer-Verlag, ISBN 3-540-42248-X <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
<li id="fn:8">Dixmier 1957 - Dixmier, J. (1957), Les algèbres d'opérateurs dans l'espace hilbertien: algèbres de von Neumann, Gauthier-Villars <a href="#fnref:8" class="footnote-back-ref">↩</a></li>
<li id="fn:9">Dixmier 1957 - Dixmier, J. (1957), Les algèbres d'opérateurs dans l'espace hilbertien: algèbres de von Neumann, Gauthier-Villars <a href="#fnref:9" class="footnote-back-ref">↩</a></li>
<li id="fn:10">Takesaki 1979, pp. 324–325 - Takesaki, M. (1979), Theory of Operator Algebras I, Springer-Verlag, ISBN 3-540-42914-X <a href="#fnref:10" class="footnote-back-ref">↩</a></li>
<li id="fn:11">Simon 1979 - Simon, B. (1979), Trace ideals and their applications, London Mathematical Society Lecture Note Series, vol. 35, Cambridge University Press, ISBN 0-521-22286-9 <a href="#fnref:11" class="footnote-back-ref">↩</a></li>
<li id="fn:12">Dixmier uses the adjectives achevée or maximale. <a href="#fnref:12" class="footnote-back-ref">↩</a></li>
<li id="fn:13">Takesaki 2002 - Takesaki, M. (2002), Theory of Operator Algebras II, Springer-Verlag, ISBN 3-540-42248-X <a href="#fnref:13" class="footnote-back-ref">↩</a></li>
<li id="fn:14">Rieffel & van Daele 1977 - Rieffel, M.A.; van Daele, A. (1977), "A bounded operator approach to Tomita–Takesaki theory", Pacific J. Math., 69: 187–221, doi:10.2140/pjm.1977.69.187 <a href="https://doi.org/10.2140%2Fpjm.1977.69.187" target="_blank">https://doi.org/10.2140%2Fpjm.1977.69.187</a> <a href="#fnref:14" class="footnote-back-ref">↩</a></li>
<li id="fn:15">Pedersen 1979 - Pedersen, G.K. (1979), C* algebras and their automorphism groups, London Mathematical Society Monographs, vol. 14, Academic Press, ISBN 0-12-549450-5 <a href="#fnref:15" class="footnote-back-ref">↩</a></li>
<li id="fn:16">Bratteli & Robinson 1987 - Bratteli, O.; Robinson, D.W. (1987), Operator Algebras and Quantum Statistical Mechanics 1, Second Edition, Springer-Verlag, ISBN 3-540-17093-6 <a href="#fnref:16" class="footnote-back-ref">↩</a></li>
<li id="fn:17">Dixmier 1957 - Dixmier, J. (1957), Les algèbres d'opérateurs dans l'espace hilbertien: algèbres de von Neumann, Gauthier-Villars <a href="#fnref:17" class="footnote-back-ref">↩</a></li>
<li id="fn:18">Dixmier 1977, Appendix A54–A61. - Dixmier, J. (1977), C* algebras, North Holland, ISBN 0-7204-0762-1 <a href="https://archive.org/details/calgebras0000dixm" target="_blank">https://archive.org/details/calgebras0000dixm</a> <a href="#fnref:18" class="footnote-back-ref">↩</a></li>
<li id="fn:19">Dieudonné 1976 - Dieudonné, J. (1976), Treatise on Analysis, Vol. II, Academic Press, ISBN 0-12-215502-5 <a href="#fnref:19" class="footnote-back-ref">↩</a></li>
<li id="fn:20">Godement 1954, pp. 52–53 - Godement, R. (1954), "Théorie des caractères. I. Algèbres unitaires", Ann. of Math., 59 (1), Annals of Mathematics: 47–62, doi:10.2307/1969832, JSTOR 1969832 <a href="https://doi.org/10.2307%2F1969832" target="_blank">https://doi.org/10.2307%2F1969832</a> <a href="#fnref:20" class="footnote-back-ref">↩</a></li>
</ol>

Commutation theorem for traces open-in-new

Commutation theorem for traces