Commutation theorem

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 trace. The first such result was proved by 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
measurable transformation preserving a probability measure. Another important application is in the theory of unitary representations of unimodular locally compact groups, 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.
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
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 "a"Ω = 0 for "a" in "M", then "aM"'Ω= (0), and hence "a" = 0.
It follows that the map
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
The commutation theorem of Murray and von Neumann states that
One of the easiest ways to see this 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
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 ("ab"Ω, 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
follows by reversing the roles of "M" and "M"'.
Examples.
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).
Commutation theorem for semifinite traces.
Let "M" be a von Neumann algebra and "M"+ the set of positive operators in "M". By definition, a semifinite trace (or sometimes just trace) on "M" is a functional τ from "M"+ into [0, ∞] such that
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" = "L"2("M", τ) be the Hilbert space completion of the inner product space
with respect to the inner product
The von Neumann algebra "M" acts by left multiplication on "H" and can be identified with its image. Let
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
is again valid in this case. This result can be proved directly by a variety of methods, but follows immediately from the result for finite traces, by repeated use of the following elementary fact:
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. 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" 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 as a tool for proving commutation theorems for semifinite weights in Tomita–Takesaki theory; they can be dispensed with when dealing with states.
Definition.
A Hilbert algebra is an algebra formula_22 with involution "x"→"x"* and an inner product (,) such that
Properties.
Let "H" be the Hilbert space completion of formula_22 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 
formula_22 on itself by left and right multiplication:
These actions extend continuously to actions on "H". In this case the commutation theorem for Hilbert algebras states that
Moreover if
the von Neumann algebra generated by the operators λ("a"), then
These results were proved independently by and .
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 formula_22) if the map "a" → "xa" of formula_22 into "H" extends to a 
bounded operator on "H", denoted by λ("x"). In this case it is straightforward to prove that:
The commutation theorem follows immediately from the last assertion. In particular
The space of all bounded elements formula_35 forms a Hilbert algebra containing formula_22 as a dense *-subalgebra. It is said to be completed or full because any element in "H" bounded relative to formula_35must actually already lie in formula_35. The functional τ on "M"+ defined by
if "x" = λ(a)*λ(a) and ∞ otherwise, yields a faithful semifinite trace on "M" with
Thus: