# Affiliated operator

> Mediated Wiki article. Canonical URL: https://mediated.wiki/source/Affiliated_operator
> Markdown URL: https://mediated.wiki/source/Affiliated_operator.md
> Source: https://en.wikipedia.org/wiki/Affiliated_operator
> Source revision: 1354564310
> License: Creative Commons Attribution-ShareAlike 4.0 International (https://creativecommons.org/licenses/by-sa/4.0/)

In [mathematics](/source/Mathematics), **affiliated operators** were introduced by [Murray](/source/Francis_Joseph_Murray) and [von Neumann](/source/John_von_Neumann) in the theory of [von Neumann algebras](/source/Von_Neumann_algebras) as a technique for using [unbounded operators](/source/Unbounded_operator) to study modules generated by a single vector. Later [Atiyah](/source/Michael_Francis_Atiyah) and [Singer](/source/Isadore_Singer) showed that [index theorems](/source/Atiyah-Singer_index_theorem) for [elliptic operators](/source/Elliptic_operator) on [closed manifolds](/source/Closed_manifold) with infinite [fundamental group](/source/Fundamental_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 in [L2 cohomology](/source/L2_cohomology), an area between [analysis](/source/Analysis) and [geometry](/source/Geometry) that evolved from the study of such index theorems.

## Definition

Let *M* be a [von Neumann algebra](/source/Von_Neumann_algebra) acting on a [Hilbert space](/source/Hilbert_space) *H*. A [closed](/source/Closed_linear_operator) and densely defined operator *A* is said to be **affiliated** with *M* if *A* commutes with every [unitary operator](/source/Unitary_operator) *U* in the [commutant](/source/Commutant) of *M*. Equivalent conditions are that:

- each unitary *U* in *M'* should leave invariant the graph of *A* defined by G ( A ) = { ( x , A x ) : x ∈ D ( A ) } ⊆ H ⊕ H {\displaystyle G(A)=\{(x,Ax):x\in D(A)\}\subseteq H\oplus H} .

- the projection onto *G*(*A*) should lie in *M*2(*M*).

- each unitary *U* in *M'* should carry *D*(*A*), the [domain](/source/Domain_of_a_function) of *A*, onto itself and satisfy *UAU* = A* there.

- each unitary *U* in *M'* should commute with both operators in the [polar decomposition](/source/Polar_decomposition) of *A*.

The last condition follows by uniqueness of the polar decomposition. If *A* has a polar decomposition

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

it says that the [partial isometry](/source/Partial_isometry) *V* should lie in *M* and that the positive [self-adjoint](/source/Self-adjoint) operator *|A|* should be affiliated with *M*. However, by the [spectral theorem](/source/Spectral_theorem), a positive self-adjoint operator commutes with a unitary operator if and only if each of its spectral projections E ( [ 0 , N ] ) {\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*.

## Measurable operators

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](/source/Gelfand%E2%80%93Naimark%E2%80%93Segal) action of *M* on *H* = *L*2(*M*, τ), [Edward Nelson](/source/Edward_Nelson) proved that the **measurable** affiliated operators do form a [*-algebra](/source/*-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](/source/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 *L*2(*H*) of [Hilbert–Schmidt operators](/source/Hilbert%E2%80%93Schmidt_operator), it gives the well-known theory of non-commutative *L**p* spaces *L**p* (*H*) due to [Schatten](/source/Robert_Schatten) and [von Neumann](/source/John_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](/source/*-algebra), as originally observed in the first paper of [Murray](/source/Francis_Joseph_Murray) and von Neumann. In this case *M* is a [von Neumann regular ring](/source/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](/source/Tomita%E2%80%93Takesaki_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](/source/Alain_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.

## References

- A. Connes, *Non-commutative geometry*, [ISBN](/source/ISBN_(identifier)) [0-12-185860-X](https://en.wikipedia.org/wiki/Special:BookSources/0-12-185860-X)

- [J. Dixmier](/source/J._Dixmier), *Von Neumann algebras*, [ISBN](/source/ISBN_(identifier)) [0-444-86308-7](https://en.wikipedia.org/wiki/Special:BookSources/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](/source/ISBN_(identifier)) [3-540-43566-2](https://en.wikipedia.org/wiki/Special:BookSources/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](/source/ISBN_(identifier)) [3-540-42248-X](https://en.wikipedia.org/wiki/Special:BookSources/3-540-42248-X) [ISBN](/source/ISBN_(identifier)) [3-540-42914-X](https://en.wikipedia.org/wiki/Special:BookSources/3-540-42914-X) [ISBN](/source/ISBN_(identifier)) [3-540-42913-1](https://en.wikipedia.org/wiki/Special:BookSources/3-540-42913-1)

---
Adapted from the Wikipedia article [Affiliated operator](https://en.wikipedia.org/wiki/Affiliated_operator) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Affiliated_operator?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.