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Affiliated operator

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<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="en" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr">In <a href="Mathematics" title="Mathematics">mathematics</a>, affiliated operators were introduced by <a href="Francis_Joseph_Murray" title="Francis Joseph Murray">Murray</a> and <a href="John_von_Neumann" title="John von Neumann">von Neumann</a> in the theory of <a href="Von_Neumann_algebras" class="mw-redirect" title="Von Neumann algebras">von Neumann algebras</a> as a technique for using <a href="Unbounded_operator" title="Unbounded operator">unbounded operators</a> to study modules generated by a single vector. Later <a href="Michael_Francis_Atiyah" class="mw-redirect" title="Michael Francis Atiyah">Atiyah</a> and <a href="Isadore_Singer" title="Isadore Singer">Singer</a> showed that <a href="Atiyah-Singer_index_theorem" class="mw-redirect" title="Atiyah-Singer index theorem">index theorems</a> for <a href="Elliptic_operator" title="Elliptic operator">elliptic operators</a> on <a href="Closed_manifold" title="Closed manifold">closed manifolds</a> with infinite <a href="Fundamental_group" title="Fundamental group">fundamental group</a> 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 <a href="L2_cohomology" class="mw-redirect" title="L2 cohomology">L2 cohomology</a>, an area between <a href="Analysis" title="Analysis">analysis</a> and <a href="Geometry" title="Geometry">geometry</a> that evolved from the study of such index theorems.

<div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2></div>

Let M be a <a href="Von_Neumann_algebra" title="Von Neumann algebra">von Neumann algebra</a> acting on a <a href="Hilbert_space" title="Hilbert space">Hilbert space</a> H. A <a href="Closed_linear_operator" title="Closed linear operator">closed</a> and densely defined operator A is said to be affiliated with M if A commutes with every <a href="Unitary_operator" title="Unitary operator">unitary operator</a> U in the <a href="Commutant" class="mw-redirect" title="Commutant">commutant</a> of M. Equivalent conditions

are that:

<ul><li>each unitary U in M' should leave invariant the graph of A defined by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G(A)=\{(x,Ax):x\in D(A)\}\subseteq H\oplus H}">

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<annotation encoding="application/x-tex">{\displaystyle G(A)=\{(x,Ax):x\in D(A)\}\subseteq H\oplus H}</annotation>

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</math><img src="./b33c66b1814d4da4a8e705510b01400fb3d53fc9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:39.698ex; height:2.843ex;" alt="{\displaystyle G(A)=\{(x,Ax):x\in D(A)\}\subseteq H\oplus H}" loading="lazy">.</li>

<li>the projection onto G(A) should lie in M2(M).</li>

<li>each unitary U in M' should carry D(A), the <a href="Domain_of_a_function" title="Domain of a function">domain</a> of A, onto itself and satisfy UAU* = A there.</li>

<li>each unitary U in M' should commute with both operators in the <a href="Polar_decomposition" title="Polar decomposition">polar decomposition</a> of A.</li></ul>

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

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<annotation encoding="application/x-tex">{\displaystyle A=V|A|,\,}</annotation>

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</math><img src="./72bc2ba95a0c788f28b913f85002d203aea5c92d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.7ex; height:2.843ex;" alt="{\displaystyle A=V|A|,\,}" loading="lazy"></dd></dl>

it says that the <a href="Partial_isometry" title="Partial isometry">partial isometry</a> V should lie in M and that the positive <a href="Self-adjoint" title="Self-adjoint">self-adjoint</a> operator |A| should be affiliated with M. However, by the <a href="Spectral_theorem" title="Spectral theorem">spectral theorem</a>, a positive self-adjoint operator commutes with a unitary operator if and only if each of its spectral projections <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E([0,N])}">

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<annotation encoding="application/x-tex">{\displaystyle E([0,N])}</annotation>

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</math><img src="./649b3b2625d1615f57fe0dcef82938e94d02034e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.139ex; height:2.843ex;" alt="{\displaystyle E([0,N])}" loading="lazy">

does. This gives another equivalent condition:

<ul><li>each spectral projection of |A| and the partial isometry in the polar decomposition of A lies in M.</li></ul>

<div class="mw-heading mw-heading2"><h2 id="Measurable_operators">Measurable operators</h2></div>

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 <a href="Gelfand%E2%80%93Naimark%E2%80%93Segal" class="mw-redirect" title="Gelfand–Naimark–Segal">Gelfand–Naimark–Segal</a> action of M on H = L2(M, τ), <a href="Edward_Nelson" title="Edward Nelson">Edward Nelson</a> proved that the measurable affiliated operators do form a <a href="*-algebra" title="*-algebra">*-algebra</a> 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 <a href="Convergence_in_measure" title="Convergence in measure">convergence in measure</a>.

It contains all the non-commutative Lp 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 L2(H) of <a href="Hilbert%E2%80%93Schmidt_operator" title="Hilbert–Schmidt operator">Hilbert–Schmidt operators</a>, it gives the well-known theory of non-commutative Lp spaces Lp (H) due to <a href="Robert_Schatten" title="Robert Schatten">Schatten</a> and <a href="John_von_Neumann" title="John von Neumann">von Neumann</a>.

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 <a href="*-algebra" title="*-algebra">*-algebra</a>, as originally observed in the first paper of <a href="Francis_Joseph_Murray" title="Francis Joseph Murray">Murray</a> and von Neumann. In this case M is a <a href="Von_Neumann_regular_ring" title="Von Neumann regular ring">von Neumann regular ring</a>: 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 <a href="Tomita%E2%80%93Takesaki_theory" title="Tomita–Takesaki theory">Tomita–Takesaki theory</a>, it is known that the non-commutative Lp spaces are no longer realised by operators affiliated with the von Neumann algebra. As <a href="Alain_Connes" title="Alain Connes">Connes</a> 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.

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

<ul><li>A. Connes, Non-commutative geometry, <style data-mw-deduplicate="TemplateStyles:r1238218222">

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</style><a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-12-185860-X</bdi></li>

<li>J. Dixmier, Von Neumann algebras, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-444-86308-7</bdi> [Les algèbres d'opérateurs dans l'espace hilbertien: algèbres de von Neumann, Gauthier-Villars (1957 & 1969)]</li>

<li>W. Lück, L2-Invariants: Theory and Applications to Geometry and K-Theory, (Chapter 8: the algebra of affiliated operators) <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>3-540-43566-2</bdi></li>

<li>F. J. Murray and J. von Neumann, Rings of Operators, Annals of Mathematics 37 (1936), 116–229 (Chapter XVI).</li>

<li>E. Nelson, Notes on non-commutative integration, J. Funct. Anal. 15 (1974), 103–116.</li>

<li>M. Takesaki, Theory of Operator Algebras I, II, III, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>3-540-42248-X</bdi> <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>3-540-42914-X</bdi> <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>3-540-42913-1</bdi></li></ul></div><div><div class="zim-footer">

This article is issued from <a class="external text" title="Last edited on 2019-11-03" href="https://en.wikipedia.org/wiki/?title=Affiliated_operator&oldid=924390180">Wikipedia</a>. The text is available under <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Creative Commons Attribution-Share Alike 4.0</a> unless otherwise noted. Additional terms may apply for the media files.

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