Affiliated operator

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 ${\displaystyle G(A)=\{(x,Ax):x\in D(A)\}\subseteq H\oplus H}$.
• 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

${\displaystyle A=V|A|,\,}$

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 ${\displaystyle E([0,N])}$ 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 L^p 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 L^p spaces L^p (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 L^p 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.

• A. Connes, Non-commutative geometry, ISBN 0-12-185860-X
• J. Dixmier, Von Neumann algebras, ISBN 0-444-86308-7 [Les algèbres d'opérateurs dans l'espace hilbertien: algèbres de von Neumann, Gauthier-Villars (1957 & 1969)]
• W. Lück, L²-Invariants: Theory and Applications to Geometry and K-Theory, (Chapter 8: the algebra of affiliated operators) ISBN 3-540-43566-2
• F. J. Murray and J. von Neumann, Rings of Operators, Annals of Mathematics 37 (1936), 116–229 (Chapter XVI).
• E. Nelson, Notes on non-commutative integration, J. Funct. Anal. 15 (1974), 103–116.
• M. Takesaki, Theory of Operator Algebras I, II, III, ISBN 3-540-42248-X ISBN 3-540-42914-X ISBN 3-540-42913-1