Commutation theorem

• 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 ${\displaystyle \ell ^{2}(\Gamma )}$ by the left and right regular representations λ and ρ. These unitary representations are given by the formulas

${\displaystyle (\lambda (g)f)(x)=f(g^{-1}x),\,\,(\rho (g)f)(x)=f(xg)}$

for f in ${\displaystyle \ell ^{2}(\Gamma )}$ and the commutation theorem implies that

${\displaystyle \lambda (\Gamma )^{\prime \prime }=\rho (\Gamma )^{\prime },\,\,\rho (\Gamma )^{\prime \prime }=\lambda (\Gamma )^{\prime }.}$

The operator J is given by the formula

${\displaystyle Jf(g)={\overline {f(g^{-1})}}.}$

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

${\displaystyle A^{\prime }=A,}$

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

${\displaystyle U_{g}f(x)=f(g^{-1}x),}$

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

${\displaystyle H_{1}=H\otimes \ell ^{2}(\Gamma ),}$

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

${\displaystyle M=A\rtimes \Gamma }$

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

The vector ${\displaystyle \Omega =1\otimes \delta _{1}}$ is a cyclic-separating trace vector. Moreover the modular conjugation operator J and commutant M ' can be explicitly identified.

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]

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

1. (a, b) = (b*, a*) for a, b in ${\displaystyle {\mathfrak {A}}}$;
2. left multiplication by a fixed a in ${\displaystyle {\mathfrak {A}}}$ 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 ${\displaystyle {\mathfrak {A}}}$.

• 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) ${\displaystyle \cap }$ 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)${\displaystyle \cap }$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)${\displaystyle \cap }$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.

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

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

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

${\displaystyle \lambda ({\mathfrak {A}})^{\prime \prime }=\rho ({\mathfrak {A}})^{\prime }}$

Moreover if

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

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

${\displaystyle JMJ=M^{\prime }}$

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

• 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(${\displaystyle {\mathfrak {B}}}$)".

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

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

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

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

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.