Inmathematics,affiliated operators were introduced byMurray andvon Neumann in the theory ofvon Neumann algebras as a technique for usingunbounded operators to study modules generated by a single vector. LaterAtiyah andSinger showed thatindex theorems forelliptic operators onclosed manifolds with infinitefundamental group could naturally be phrased in terms of unbounded operators affiliated with the von Neumann algebra of the group. Algebraic properties of affiliated operators have proved important inL² cohomology, an area betweenanalysis andgeometry that evolved from the study of such index theorems.

LetM be avon Neumann algebra acting on aHilbert spaceH. Aclosed and densely defined operatorA is said to beaffiliated withM ifA commutes with everyunitary operatorU in thecommutant ofM. Equivalent conditionsare that:

• each unitaryU inM' should leave invariant the graph ofA defined by${\displaystyle G(A)=\{(x,Ax):x\in D(A)\}\subseteq H\oplus H}$.
• the projection ontoG(A) should lie inM₂(M).
• each unitaryU inM' should carryD(A), thedomain ofA, onto itself and satisfyUAU* = A there.
• each unitaryU inM' should commute with both operators in thepolar decomposition ofA.

The last condition follows by uniqueness of the polar decomposition. IfA has a polar decomposition

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

it says that thepartial isometryV should lie inM and that the positiveself-adjoint operator|A| should be affiliated withM. However, by thespectral 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 ofA lies inM.

In general the operators affiliated with a von Neumann algebraM 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 standardGelfand–Naimark–Segal action ofM onH = L²(M, τ),Edward Nelson proved that themeasurable affiliated operators do form a*-algebra with nice properties: these are operators such that τ(I − E([0,N])) < ∞forN sufficiently large. This algebra of unbounded operators is complete for a natural topology, generalising the notion ofconvergence in measure.It contains all the non-commutativeL^p spaces defined by the trace and was introduced to facilitate their study.

This theory can be applied when the von Neumann algebraM istype I ortype II. WhenM = B(H) acting on the Hilbert spaceL²(H) ofHilbert–Schmidt operators, it gives the well-known theory of non-commutativeL^p spacesL^p (H) due toSchatten andvon Neumann.

WhenM is in addition afinite 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 ofMurray and von Neumann. In this caseM is avon Neumann regular ring: for on the closure of its image|A| has a measurable inverseB and thenT = BV^* defines a measurable operator withATA = A. Of course in the classical case whenX is a probability space andM = L^∞ (X), we simply recover the *-algebra of measurable functions onX.

If howeverM istype III, the theory takes a quite different form. Indeed in this case, thanks to theTomita–Takesaki theory, it is known that the non-commutativeL^p spaces are no longer realised by operators affiliated with the von Neumann algebra. AsConnes 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 relationUAU^* = 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 Mathematics37 (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-XISBN 3-540-42914-XISBN 3-540-42913-1

• Operator theory
• Von Neumann algebras