1. 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

• M Ω is dense in H
• M ' Ω is dense in H, where M ' denotes the commutant of M
• (abΩ, Ω) = (baΩ, Ω) for all a, b in M.

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 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

[math]\displaystyle{ Ja\Omega=a^*\Omega }[/math]

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

[math]\displaystyle{ JMJ\subseteq M^\prime. }[/math]

The commutation theorem of Murray and von Neumann states that

[math]\displaystyle{ JMJ=M^\prime }[/math]

One of the easiest ways to see this^[1] is to introduce K, the closure of the real subspace M_sa Ω, where M_sa denotes the self-adjoint elements in M. It follows that

[math]\displaystyle{ H=K\oplus iK, }[/math]

an orthogonal direct sum for the real part of 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 (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

[math]\displaystyle{ JM^\prime J\subseteq M }[/math]

follows by reversing the roles of M and M'.

1.1. 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]\displaystyle{ \ell^2(\Gamma) }[/math] by the left and right regular representations λ and ρ. These unitary representations are given by the formulas

[math]\displaystyle{ (\lambda(g) f)(x)=f(g^{-1}x),\,\,(\rho(g)f)(x)=f(xg) }[/math]

for f in [math]\displaystyle{ \ell^2(\Gamma) }[/math] and the commutation theorem implies that

[math]\displaystyle{ \lambda(\Gamma)^{\prime\prime}=\rho(\Gamma)^\prime, \,\, \rho(\Gamma)^{\prime\prime}=\lambda(\Gamma)^\prime. }[/math]

The operator J is given by the formula

[math]\displaystyle{ Jf(g)=\overline{f(g^{-1})}. }[/math]

Exactly the same results remain true if Γ is allowed to be any countable discrete group.^[2] 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 operators on H = L²(X, μ) and the constant function 1 is a cyclic-separating trace vector. It follows that

[math]\displaystyle{ A^{\prime}=A, }[/math]

so that A is a maximal Abelian subalgebra of B(H), the von Neumann algebra of all bounded operators 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 = L²(X, μ) according to the formula

[math]\displaystyle{ 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]\displaystyle{ H_1=H\otimes \ell^2(\Gamma), }[/math]

a tensor product of Hilbert spaces.^[3] The group–measure space construction or crossed product von Neumann algebra

[math]\displaystyle{ M = A \rtimes \Gamma }[/math]

is defined to be the von Neumann algebra on H₁ generated by the algebra [math]\displaystyle{ A\otimes I }[/math] and the normalising operators [math]\displaystyle{ U_g\otimes \lambda(g) }[/math].^[4]

The vector [math]\displaystyle{ \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 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 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).^[5][6]

2. Commutation Theorem for Semifinite Traces

Let M be a von Neumann algebra and M₊ the set of positive operators in M. By definition,^[2] a semifinite trace (or sometimes just trace) on M is a functional τ from M₊ into [0, ∞] such that

1. [math]\displaystyle{ \tau(\lambda a + \mu b) = \lambda \tau(a) + \mu \tau(b) }[/math] for a, b in M₊ and λ, μ ≥ 0 (semilinearity);
2. [math]\displaystyle{ \tau\left(uau^*\right) = \tau(a) }[/math] for a in M₊ and u a unitary operator in M (unitary invariance);
3. τ is completely additive on orthogonal families of projections in M (normality);
4. each projection in M is as orthogonal direct sum of projections with finite trace (semifiniteness).

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

[math]\displaystyle{ M_0 = \left\{a \in M \mid \tau\left(a^*a\right) \lt \infty\right\} }[/math]

with respect to the inner product

[math]\displaystyle{ (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]\displaystyle{ Ja = a^* }[/math]

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

[math]\displaystyle{ JMJ = M^\prime }[/math]

is again valid in this case. This result can be proved directly by a variety of methods,^[2] 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 strong operator topology, then M₁ = M₂.

3. Hilbert Algebras

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 operators starting from Hilbert–Schmidt operators.^[7] Applications in the representation theory of groups naturally lead to examples of Hilbert algebras. Every von Neumann algebra endowed with a semifinite trace has a canonical "completed"^[8] 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^[6] as a tool for proving commutation theorems for semifinite weights in Tomita–Takesaki theory; they can be dispensed with when dealing with states.^[1][9][10]

3.1. Definition

A Hilbert algebra^[2][11][12] is an algebra [math]\displaystyle{ \mathfrak{A} }[/math] with involution x→x* and an inner product (,) such that

1. (a, b) = (b*, a*) for a, b in [math]\displaystyle{ \mathfrak{A} }[/math];
2. left multiplication by a fixed a in [math]\displaystyle{ \mathfrak{A} }[/math] is a bounded operator;
3. * is the adjoint, in other words (xy, z) = (y, x*z);
4. the linear span of all products xy is dense in [math]\displaystyle{ \mathfrak{A} }[/math].

3.2. 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]\displaystyle{ \cap }[/math] L²(X) is a Hilbert algebra with the usual inner product from L²(X).
• 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).
• If G is a unimodular locally compact group, the convolution algebra L¹(G)[math]\displaystyle{ \cap }[/math]L²(G) is a Hilbert algebra with the usual inner product from L²(G).
• If (G, K) is a Gelfand pair, the convolution algebra L¹(K\G/K)[math]\displaystyle{ \cap }[/math]L²(K\G/K) is a Hilbert algebra with the usual inner product from L²(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.

3.3. Properties

Let H be the Hilbert space completion of [math]\displaystyle{ \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 [math]\displaystyle{ \mathfrak{A} }[/math] on itself by left and right multiplication:

[math]\displaystyle{ \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

[math]\displaystyle{ \lambda(\mathfrak{A})^{\prime\prime} = \rho(\mathfrak{A})^\prime }[/math]

Moreover if

[math]\displaystyle{ M = \lambda(\mathfrak{A})^{\prime\prime}, }[/math]

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

[math]\displaystyle{ JMJ = M^\prime }[/math]

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 [math]\displaystyle{ \mathfrak{A} }[/math]) if the map a → xa of [math]\displaystyle{ \mathfrak{A} }[/math] into H extends to a bounded operator on H, denoted by λ(x). In this case it is straightforward to prove that:^[13]

Thus:

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.

• Jx is also a bounded element, denoted x*, and λ(x*) = λ(x)*;
• a → ax is given by the bounded operator ρ(x) = Jλ(x*)J on H;
• M ' is generated by the ρ(x)'s with x bounded;
• λ(x) and ρ(y) commute for x, y bounded.

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

• M = λ([math]\displaystyle{ \mathfrak{B} }[/math])".

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

[math]\displaystyle{ \tau(x) = (a,a) }[/math]

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

[math]\displaystyle{ M_0 = \mathfrak{B}. }[/math]

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.