Let M be a von Neumann algebra acting on a Hilbert space H. A closed and densely defined operator A is said to be affiliated with M if A commutes with every unitary operator U in the commutant of M. Equivalent conditions
are that:
- each unitary U in M' should leave invariant the graph of A defined by .
- the projection onto G(A) should lie in M2(M).
- each unitary U in M' should carry D(A), the domain of A, onto itself and satisfy UAU* = A there.
- each unitary U in M' should commute with both operators in the polar decomposition of A.
The last condition follows by uniqueness of the polar decomposition. If A has a polar decomposition
it says that the partial isometry V should lie in M and that the positive self-adjoint operator |A| should be affiliated with M. However, by the spectral theorem, a positive self-adjoint operator commutes with a unitary operator if and only if each of its spectral projections
does. This gives another equivalent condition:
- each spectral projection of |A| and the partial isometry in the polar decomposition of A lies in M.
In general the operators affiliated with a von Neumann algebra M need not necessarily be well-behaved under either addition or composition. However in the presence of a faithful semi-finite normal trace τ and the standard Gelfand–Naimark–Segal action of M on H = L2(M, τ), Edward Nelson proved that the measurable affiliated operators do form a *-algebra with nice properties: these are operators such that τ(I − E([0,N])) < ∞
for N sufficiently large. This algebra of unbounded operators is complete for a natural topology, generalising the notion of convergence in measure.
It contains all the non-commutative Lp spaces defined by the trace and was introduced to facilitate their study.
This theory can be applied when the von Neumann algebra M is type I or type II. When M = B(H) acting on the Hilbert space L2(H) of Hilbert–Schmidt operators, it gives the well-known theory of non-commutative Lp spaces Lp (H) due to Schatten and von Neumann.
When M is in addition a finite von Neumann algebra, for example a type II1 factor, then every affiliated operator is automatically measurable, so the affiliated operators form a *-algebra, as originally observed in the first paper of Murray and von Neumann. In this case M is a von Neumann regular ring: for on the closure of its image |A| has a measurable inverse B and then T = BV* defines a measurable operator with ATA = A. Of course in the classical case when X is a probability space and M = L∞ (X), we simply recover the *-algebra of measurable functions on X.
If however M is type III, the theory takes a quite different form. Indeed in this case, thanks to the Tomita–Takesaki theory, it is known that the non-commutative Lp spaces are no longer realised by operators affiliated with the von Neumann algebra. As Connes showed, these spaces can be realised as unbounded operators only by using a certain positive power of the reference modular operator. Instead of being characterised by the simple affiliation relation UAU* = A, there is a more complicated bimodule relation involving the analytic continuation of the modular automorphism group.