- Affiliated operator
In

mathematics ,**affiliated operators**were introduced by Murray and von Neumann in the theory ofvon Neumann algebras as a technique for usingunbounded operator s to study modules generated by a single vector. Later Atiyah and Singer showed that index theorems forelliptic operator s onclosed manifold s 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 in L² cohomology, an area betweenanalysis andgeometry that evolved from the study of such index theorems.**Definition**Let "M" be a

von Neumann algebra acting on aHilbert space "H". A closed and densely defined operator "A" is said to be**affiliated**with "M" if "A" commutes with everyunitary operator "U" in thecommutant of "M". Equivalent conditionsare that:*each unitary "U" in "M"' should leave invariant the graph of "A" defined by $G(A)=\{(x,Ax):xin\; D(A)\}\; subseteq\; Hoplus\; H$.

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

_{2}(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 :$A=V|A|$, it says that the

partial isometry "V" should lie in "M" and that the positiveself-adjoint operator "|A|" should be affiliated with "M". However, by thespectral theorem , a positive self-adjoint operator commutes with a unitary operator if and only if each of its spectral projections $E(\; [0,N]\; )$does. This gives another equivalent condition:*each spectral projection of "|A|" and the partial isometry in the polar decomposition of "A" should lie 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 $au$ and the standard

Gelfand-Naimark-Segal action of "M" on "H"="L"^{2}("M",$au$),Edward Nelson proved that the**measurable**affiliated operators do form a*-algebra with nice properties: these are operators such that $au$("I"-"E"( [0,"N"] ))<∞for "N" sufficiently large. This algebra of unbounded operators is complete for a natural topology, generalising the notion ofconvergence 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") ofHilbert-Schmidt operator s, it gives the well-known theory of non-commutative "L"^{"p"}spaces "L"^{"p"}("H") due to Schatten andvon Neumann .When "M" is in addition a

**finite**von Neumann algebra, for example a type II_{1}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 avon 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 theTomita-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.**References*** J.Brohan, communication privée.

* 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, "L"^{2}-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 Math.**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

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