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In [[mathematics]], '''affiliated operators''' were introduced by [[Francis Joseph Murray (mathematician)|Murray]] and [[John von Neumann|von Neumann]] in the theory of [[von Neumann algebras]] as a technique for using [[unbounded operator]]s to study modules generated by a single vector. Later [[Michael Francis Atiyah|Atiyah]] and [[Isadore Singer|Singer]] showed that [[Atiyah-Singer index theorem|index theorems]] for [[elliptic operator]]s on [[closed manifold]]s with infinite [[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|L<sup>2</sup> cohomology]], an area between [[analysis]] and [[geometry]] that evolved from the study of such index theorems.
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==Definition==
Let ''M'' be a [[von Neumann algebra]] acting on a [[Hilbert space]] ''H''. A [[closed linear operator|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 <math> G(A)=\{(x,Ax):x\in D(A)\} \subseteq H\oplus H</math>.
 
*the projection onto ''G''(''A'') should lie in ''M''<sub>2</sub>(''M'').
 
*each unitary ''U'' in ''M''' should carry ''D''(''A''), the [[Domain of a function|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
:<math>A=V|A|, \, </math>
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 <math> E([0,N]) </math>
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 &tau; and the standard [[Gelfand&ndash;Naimark&ndash;Segal]] action of ''M'' on ''H''&nbsp;=&nbsp;''L''<sup>2</sup>(''M'',&nbsp;&tau;), [[Edward Nelson]] proved that the '''measurable''' affiliated operators do form a [[*-algebra]] with nice properties: these are operators such that &tau;(''I''&nbsp;&minus;&nbsp;''E''([0,''N'']))&nbsp;<&nbsp;&infin;
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''<sup>''p''</sup> 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''&nbsp;=&nbsp;''B''(''H'') acting on the Hilbert space ''L''<sup>2</sup>(''H'') of [[Hilbert–Schmidt operator]]s, it gives the well-known theory of non-commutative ''L''<sup>''p''</sup> spaces ''L''<sup>''p''</sup> (''H'') due to Schatten and [[von Neumann]].  
 
When ''M'' is in addition a '''finite''' von Neumann algebra, for example a type II<sub>1</sub> factor, then every affiliated operator is automatically measurable, so the affiliated operators form a [[*-algebra]], as originally observed in the first paper of [[Francis Joseph Murray (mathematician)|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''&nbsp;=&nbsp;''BV''<sup>*</sup> defines a measurable operator with ''ATA''&nbsp;=&nbsp;''A''. Of course in the classical case when ''X'' is a probability space and ''M''&nbsp;=&nbsp;''L''<sup>&infin;</sup> (''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''<sup>''p''</sup> spaces are no longer realised by operators affiliated with the von Neumann algebra. As [[Alain Connes|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''<sup>*</sup>&nbsp;=&nbsp;''A'', there is a more complicated bimodule relation involving the analytic continuation of the modular automorphism group.
 
== References ==
* 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<sup>2</sup>-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&ndash;229 (Chapter XVI).
* E. Nelson, ''Notes on non-commutative integration'', J. Funct. Anal. '''15''' (1974), 103&ndash;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
 
[[Category:Operator theory]]
[[Category:Von Neumann algebras]]

Latest revision as of 11:44, 10 January 2015

Hi there, I am Sophia. To climb is something she would by no means give up. She functions as a travel agent but soon she'll be on her own. My spouse and I live in Kentucky.

Feel free to visit my website: real psychics (http://www.chk.woobi.co.kr)