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.
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 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 immediately verified that JMJ and M commute on the subspace M Ω, so that[1] The commutation theorem of Murray and von Neumann states that One of the easiest ways to see this[2] is to introduce K, the closure of the real subspace Msa Ω, where Msa denotes the self-adjoint elements in M. It follows that an orthogonal direct sum for the real part of the inner product.
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.
If τ is a faithful trace on M, let H = L2(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.
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,[3][8] but follows immediately from the result for finite traces, by repeated use of the following elementary fact: 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.
[9] Applications in the representation theory of groups naturally lead to examples of Hilbert algebras.
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.
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 Godement (1954) and Segal (1953).
In this case it is straightforward to prove that:[15] The commutation theorem follows immediately from the last assertion.