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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==
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| 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
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| are that:
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| *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>.
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| *the projection onto ''G''(''A'') should lie in ''M''<sub>2</sub>(''M'').
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| *each unitary ''U'' in ''M''' should carry ''D''(''A''), the [[Domain of a function|domain]] of ''A'', onto itself and satisfy ''UAU* = A'' there.
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| *each unitary ''U'' in ''M''' should commute with both operators in the [[polar decomposition]] of ''A''.
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| The last condition follows by uniqueness of the polar decomposition. If ''A'' has a polar decomposition
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| :<math>A=V|A|, \, </math>
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| 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>
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| does. This gives another equivalent condition:
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| *each spectral projection of |''A''| and the partial isometry in the polar decomposition of ''A'' should lie in ''M''.
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| == Measurable operators ==
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| 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'' = ''L''<sup>2</sup>(''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''])) < ∞
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| for ''N'' sufficiently large. This algebra of unbounded operators is complete for a natural topology, generalising the notion of [[convergence in measure]].
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| It contains all the non-commutative ''L''<sup>''p''</sup> spaces defined by the trace and was introduced to facilitate their study.
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| 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''<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]].
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| 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'' = ''BV''<sup>*</sup> defines a measurable operator with ''ATA'' = ''A''. Of course in the classical case when ''X'' is a probability space and ''M'' = ''L''<sup>∞</sup> (''X''), we simply recover the *-algebra of measurable functions on ''X''.
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| 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> = ''A'', there is a more complicated bimodule relation involving the analytic continuation of the modular automorphism group.
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| == References ==
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| * A. Connes, ''Non-commutative geometry'', ISBN 0-12-185860-X
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| * 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)]
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| * 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
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| * F. J. Murray and J. von Neumann, ''Rings of Operators'', Annals of Math. '''37''' (1936), 116–229 (Chapter XVI).
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| * E. Nelson, ''Notes on non-commutative integration'', J. Funct. Anal. '''15''' (1974), 103–116.
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| * 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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| [[Category:Operator theory]]
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| [[Category:Von Neumann algebras]]
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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)