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von Neumann algebra in nLab
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content="application/xhtml+xml;charset=utf-8" /><title></title></head> <body> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <h4 id="algebra">Algebra</h4> <div class="hide"><div> <ul> <li><a class="existingWikiWord" href="/nlab/show/algebra">algebra</a>, <a class="existingWikiWord" href="/nlab/show/higher+algebra">higher algebra</a></li> <li><a class="existingWikiWord" href="/nlab/show/universal+algebra">universal algebra</a></li> <li><a class="existingWikiWord" href="/nlab/show/monoid">monoid</a>, <a class="existingWikiWord" href="/nlab/show/semigroup">semigroup</a>, <a class="existingWikiWord" href="/nlab/show/quasigroup">quasigroup</a></li> <li><a class="existingWikiWord" href="/nlab/show/nonassociative+algebra">nonassociative algebra</a></li> <li><a class="existingWikiWord" href="/nlab/show/associative+unital+algebra">associative unital algebra</a></li> <li><a class="existingWikiWord" href="/nlab/show/commutative+algebra">commutative algebra</a></li> <li><a class="existingWikiWord" href="/nlab/show/Lie+algebra">Lie algebra</a>, <a class="existingWikiWord" href="/nlab/show/Jordan+algebra">Jordan algebra</a></li> <li><a class="existingWikiWord" href="/nlab/show/Leibniz+algebra">Leibniz algebra</a>, <a class="existingWikiWord" href="/nlab/show/pre-Lie+algebra">pre-Lie algebra</a></li> <li><a class="existingWikiWord" href="/nlab/show/Poisson+algebra">Poisson algebra</a>, <a class="existingWikiWord" href="/nlab/show/Frobenius+algebra">Frobenius algebra</a></li> <li><a class="existingWikiWord" href="/nlab/show/lattice">lattice</a>, <a class="existingWikiWord" href="/nlab/show/frame">frame</a>, <a class="existingWikiWord" href="/nlab/show/quantale">quantale</a></li> <li><a class="existingWikiWord" href="/nlab/show/Boolean+ring">Boolean ring</a>, <a class="existingWikiWord" href="/nlab/show/Heyting+algebra">Heyting algebra</a></li> <li><a class="existingWikiWord" href="/nlab/show/commutator">commutator</a>, <a class="existingWikiWord" href="/nlab/show/center">center</a></li> <li><a class="existingWikiWord" href="/nlab/show/monad">monad</a>, <a class="existingWikiWord" href="/nlab/show/comonad">comonad</a></li> <li><a class="existingWikiWord" href="/nlab/show/distributive+law">distributive law</a></li> </ul> <h2 id="group_theory">Group theory</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/group">group</a>, <a class="existingWikiWord" href="/nlab/show/normal+subgroup">normal subgroup</a></li> <li><a class="existingWikiWord" href="/nlab/show/action">action</a>, <a class="existingWikiWord" href="/nlab/show/Cayley%27s+theorem">Cayley's theorem</a></li> <li><a class="existingWikiWord" href="/nlab/show/centralizer">centralizer</a>, <a class="existingWikiWord" href="/nlab/show/normalizer">normalizer</a></li> <li><a class="existingWikiWord" href="/nlab/show/abelian+group">abelian group</a>, <a class="existingWikiWord" href="/nlab/show/cyclic+group">cyclic group</a></li> <li><a class="existingWikiWord" href="/nlab/show/group+extension">group extension</a>, <a class="existingWikiWord" href="/nlab/show/Galois+extension">Galois extension</a></li> <li><a class="existingWikiWord" href="/nlab/show/algebraic+group">algebraic group</a>, <a class="existingWikiWord" href="/nlab/show/formal+group">formal group</a></li> <li><a class="existingWikiWord" href="/nlab/show/Lie+group">Lie group</a>, <a class="existingWikiWord" href="/nlab/show/quantum+group">quantum group</a></li> </ul> <h2 id="ring_theory">Ring theory</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/ring">ring</a>, <a class="existingWikiWord" href="/nlab/show/commutative+ring">commutative ring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/local+ring">local ring</a>, <a class="existingWikiWord" href="/nlab/show/Artinian+ring">Artinian ring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Noetherian+ring">Noetherian ring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/skewfield">skewfield</a>, <a class="existingWikiWord" href="/nlab/show/field">field</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/integral+domain">integral domain</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/ideal">ideal</a>, <a class="existingWikiWord" href="/nlab/show/prime+ideal">prime ideal</a>, <a class="existingWikiWord" href="/nlab/show/maximal+ideal">maximal ideal</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Ore+localization">Ore localization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/group+extension">group extension</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/central+simple+algebra">central simple algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/derivation">derivation</a>, <a class="existingWikiWord" href="/nlab/show/Ore+extension">Ore extension</a></p> </li> </ul> <h2 id="module_theory">Module theory</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/module">module</a>, <a class="existingWikiWord" href="/nlab/show/bimodule">bimodule</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/vector+space">vector space</a>, <a class="existingWikiWord" href="/nlab/show/linear+algebra">linear algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/matrix">matrix</a>, <a class="existingWikiWord" href="/nlab/show/eigenvalue">eigenvalue</a>, <a class="existingWikiWord" href="/nlab/show/trace">trace</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/determinant">determinant</a>, <a class="existingWikiWord" href="/nlab/show/quasideterminant">quasideterminant</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/representation+theory">representation theory</a>, <a class="existingWikiWord" href="/nlab/show/Schur+lemma">Schur lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/extension+of+scalars">extension of scalars</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/restriction+of+scalars">restriction of scalars</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Frobenius+reciprocity">Frobenius reciprocity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Morita+equivalence">Morita equivalence</a>, <a class="existingWikiWord" href="/nlab/show/Morita+context">Morita context</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Wedderburn-Artin+theorem">Wedderburn-Artin theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/abelian+category">abelian category</a>, <a class="existingWikiWord" href="/nlab/show/additive+category">additive category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal category</a>, <a class="existingWikiWord" href="/nlab/show/Tannaka+duality">Tannaka duality</a></p> </li> </ul> <h2 id=""><a class="existingWikiWord" href="/nlab/show/gebra+theory">Gebras</a></h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/coalgebra">coalgebra</a>, <a class="existingWikiWord" href="/nlab/show/coring">coring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/bialgebra">bialgebra</a>, <a class="existingWikiWord" href="/nlab/show/Hopf+algebra">Hopf algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/comodule">comodule</a>, <a class="existingWikiWord" href="/nlab/show/Hopf+module">Hopf module</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Yetter-Drinfeld+module">Yetter-Drinfeld module</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/associative+bialgebroid">associative bialgebroid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/dual+gebra">dual gebra</a>, <a class="existingWikiWord" href="/nlab/show/cotensor+product">cotensor product</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hopf-Galois+extension">Hopf-Galois extension</a></p> </li> </ul> </div></div> <h4 id="functional_analysis">Functional analysis</h4> <div class="hide"><div> <ul> <li><strong><a class="existingWikiWord" href="/nlab/show/functional+analysis">Functional Analysis</a></strong></li> </ul> <h2 id="overview_diagrams">Overview diagrams</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/TVS+relationships">topological vector spaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/diagram+of+LCTVS+properties">locally convex topological vector spaces</a></p> </li> </ul> <h2 id="basic_concepts">Basic concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+vector+space">topological vector space</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+convex+topological+vector+space">locally convex topological vector space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Banach+space">Banach Spaces</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/reflexive+Banach+space">reflexive</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Smith+space+%28functional+analysis%29">Smith Spaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hilbert+space">Hilbert Spaces</a>, <a class="existingWikiWord" href="/nlab/show/Fr%C3%A9chet+space">Fréchet Spaces</a>, <a class="existingWikiWord" href="/nlab/show/Sobolev+space">Sobolev spaces</a>, <a class="existingWikiWord" href="/nlab/show/Lebesgue+space">Lebesgue Spaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/bornological+vector+space">Bornological Vector Spaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/barrelled+topological+vector+space">Barrelled Vector Spaces</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/linear+operator">linear operator</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/bounded+linear+operator">bounded</a>, <a class="existingWikiWord" href="/nlab/show/unbounded+linear+operator">unbounded</a>, <a class="existingWikiWord" href="/nlab/show/self-adjoint+operator">self-adjoint</a>, <a class="existingWikiWord" href="/nlab/show/compact+operator">compact</a>, <a class="existingWikiWord" href="/nlab/show/Fredholm+operator">Fredholm</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spectrum+of+an+operator">spectrum of an operator</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/operator+algebras">operator algebras</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/functional+calculus">functional calculus</a></li> </ul> </li> </ul> <h2 id="theorems">Theorems</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Stone-Weierstrass+theorem">Stone-Weierstrass theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spectral+theory">spectral theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/spectral+theorem">spectral theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Gelfand+duality">Gelfand duality</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/functional+calculus">functional calculus</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Riesz+representation+theorem">Riesz representation theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/measure+theory">measure theory</a></p> </li> </ul> <h2 id="topics_in_functional_analysis">Topics in Functional Analysis</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/basis+in+functional+analysis">Bases</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/algebraic+theories+in+functional+analysis">Algebraic Theories in Functional Analysis</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/an+elementary+treatment+of+Hilbert+spaces">An Elementary Treatment of Hilbert Spaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/isomorphism+classes+of+Banach+spaces">When are two Banach spaces isomorphic?</a></p> </li> </ul> <div> <p> <a href="/nlab/edit/functional+analysis+-+contents">Edit this sidebar</a> </p> </div></div></div> <h4 id="measure_and_probability_theory">Measure and probability theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/measure+theory">measure theory</a></strong></p> <p><strong><a class="existingWikiWord" href="/nlab/show/probability+theory">probability theory</a></strong></p> <p>(<a class="existingWikiWord" href="/nlab/show/quantum+probability">quantum probability</a>)</p> <h2 id="measure_theory">Measure theory</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/measurable+space">measurable space</a>, <a class="existingWikiWord" href="/nlab/show/measurable+locale">measurable locale</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/measure">measure</a>, <a class="existingWikiWord" href="/nlab/show/measure+space">measure space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/von+Neumann+algebra">von Neumann algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+measure+theory">geometric measure theory</a></p> </li> </ul> <h2 id="probability_theory">Probability theory</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/probability+space">probability space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/probability+distribution">probability distribution</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/state">state</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/states+in+AQFT+and+operator+algebra">in AQFT and operator algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/GNS+construction">GNS construction</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Fell%27s+theorem">Fell's theorem</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/entropy">entropy</a>, <a class="existingWikiWord" href="/nlab/show/relative+entropy">relative entropy</a></p> </li> </ul> <h2 id="information_geometry">Information geometry</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/information+geometry">information geometry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/information+metric">information metric</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Wasserstein+metric">Wasserstein metric</a></p> </li> </ul> <h2 id="thermodynamics">Thermodynamics</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/thermodynamics">thermodynamics</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/second+law+of+thermodynamics">second law of thermodynamics</a>, <a class="existingWikiWord" href="/nlab/show/generalized+second+law+of+theormodynamics">generalized second law</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/ergodic+theory">ergodic theory</a></p> </li> </ul> <h2 id="theorems">Theorems</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Riesz+representation+theorem">Riesz representation theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/de+Finetti%27s+theorem">de Finetti's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/law+of+large+numbers">law of large numbers</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kolmogorov+extension+theorem">Kolmogorov extension theorem</a></p> </li> </ul> <h2 id="applications">Applications</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/machine+learning">machine learning</a>, <a class="existingWikiWord" href="/nlab/show/neural+networks">neural networks</a></li> </ul> </div></div> <h4 id="aqft">AQFT</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/algebraic+quantum+field+theory">algebraic quantum field theory</a></strong> (<a class="existingWikiWord" href="/nlab/show/perturbative+AQFT">perturbative</a>, <a class="existingWikiWord" href="/nlab/show/AQFT+on+curved+spacetime">on curved spacetimes</a>, <a class="existingWikiWord" href="/nlab/show/homotopical+algebraic+quantum+field+theory">homotopical</a>)</p> <p><a class="existingWikiWord" href="/nlab/show/A+first+idea+of+quantum+field+theory">Introduction</a></p> <h2 id="concepts">Concepts</h2> <p><strong><a class="existingWikiWord" href="/nlab/show/field+theory">field theory</a></strong>:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/classical+field+theory">classical</a>, <a class="existingWikiWord" href="/nlab/show/prequantum+field+theory">pre-quantum</a>, <a class="existingWikiWord" href="/nlab/show/quantum+field+theory">quantum</a>, <a class="existingWikiWord" href="/nlab/show/perturbative+quantum+field+theory">perturbative quantum</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/relativistic+field+theory">relativistic</a>, <a class="existingWikiWord" href="/nlab/show/Euclidean+field+theory">Euclidean</a>, <a class="existingWikiWord" href="/nlab/show/thermal+quantum+field+theory">thermal</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/Lagrangian+field+theory">Lagrangian field theory</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/field+%28physics%29">field (physics)</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/field+bundle">field bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/field+history">field history</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/space+of+field+histories">space of field histories</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lagrangian+density">Lagrangian density</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Euler-Lagrange+form">Euler-Lagrange form</a>, <a class="existingWikiWord" href="/nlab/show/presymplectic+current">presymplectic current</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Euler-Lagrange+equations">Euler-Lagrange</a><a class="existingWikiWord" href="/nlab/show/equations+of+motion">equations of motion</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+variational+field+theory">locally variational field theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/covariant+phase+space">covariant phase space</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Peierls-Poisson+bracket">Peierls-Poisson bracket</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/advanced+and+retarded+propagator">advanced and retarded propagator</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/causal+propagator">causal propagator</a></p> </li> </ul> </li> </ul> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/quantization">quantization</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+quantization">geometric quantization</a><a class="existingWikiWord" href="/nlab/show/geometric+quantization+of+symplectic+groupoids">of symplectic groupoids</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/algebraic+deformation+quantization">algebraic deformation quantization</a>, <a class="existingWikiWord" href="/nlab/show/star+algebra">star algebra</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/quantum+mechanical+system">quantum mechanical system</a></strong>, <strong><a class="existingWikiWord" href="/nlab/show/quantum+probability">quantum probability</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/subsystem">subsystem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/observables">observables</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/field+observables">field observables</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/local+observables">local observables</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/polynomial+observables">polynomial observables</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/microcausal+observables">microcausal observables</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/operator+algebra">operator algebra</a>, <a class="existingWikiWord" href="/nlab/show/C%2A-algebra">C*-algebra</a>, <a class="existingWikiWord" href="/nlab/show/von+Neumann+algebra">von Neumann algebra</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/local+net+of+observables">local net of observables</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/causal+locality">causal locality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Haag-Kastler+axioms">Haag-Kastler axioms</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Wightman+axioms">Wightman axioms</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/field+net">field net</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/conformal+net">conformal net</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/state+on+a+star-algebra">state on a star-algebra</a>, <a class="existingWikiWord" href="/nlab/show/expectation+value">expectation value</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/pure+state">pure state</a></p> <p><a class="existingWikiWord" href="/nlab/show/wave+function">wave function</a></p> <p><a class="existingWikiWord" href="/nlab/show/collapse+of+the+wave+function">collapse of the wave function</a>/<a class="existingWikiWord" href="/nlab/show/conditional+expectation+value">conditional expectation value</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mixed+state">mixed state</a>, <a class="existingWikiWord" href="/nlab/show/density+matrix">density matrix</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/space+of+quantum+states">space of quantum states</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/vacuum+state">vacuum state</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/quasi-free+state">quasi-free state</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hadamard+state">Hadamard state</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Wightman+propagator">Wightman propagator</a></p> </li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/picture+of+quantum+mechanics">picture of quantum mechanics</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/free+field">free field</a> <a class="existingWikiWord" href="/nlab/show/quantization">quantization</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/star+algebra">star algebra</a>, <a class="existingWikiWord" href="/nlab/show/Moyal+deformation+quantization">Moyal deformation quantization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Wick+algebra">Wick algebra</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/canonical+commutation+relations">canonical commutation relations</a>, <a class="existingWikiWord" href="/nlab/show/Weyl+relations">Weyl relations</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/normal+ordered+product">normal ordered product</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Fock+space">Fock space</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/gauge+theories">gauge theories</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/gauge+symmetry">gauge symmetry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/BRST+complex">BRST complex</a>, <a class="existingWikiWord" href="/nlab/show/BV-BRST+formalism">BV-BRST formalism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/local+BV-BRST+complex">local BV-BRST complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/BV-operator">BV-operator</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+master+equation">quantum master equation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/master+Ward+identity">master Ward identity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/gauge+anomaly">gauge anomaly</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/interacting+field+theory">interacting field</a> <a class="existingWikiWord" href="/nlab/show/quantization">quantization</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/causal+perturbation+theory">causal perturbation theory</a>, <a class="existingWikiWord" href="/nlab/show/perturbative+AQFT">perturbative AQFT</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/interaction">interaction</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/S-matrix">S-matrix</a>, <a class="existingWikiWord" href="/nlab/show/scattering+amplitude">scattering amplitude</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/causal+additivity">causal additivity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/time-ordered+product">time-ordered product</a>, <a class="existingWikiWord" href="/nlab/show/Feynman+propagator">Feynman propagator</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Feynman+diagram">Feynman diagram</a>, <a class="existingWikiWord" href="/nlab/show/Feynman+perturbation+series">Feynman perturbation series</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/effective+action">effective action</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/vacuum+stability">vacuum stability</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/interacting+field+algebra">interacting field algebra</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Bogoliubov%27s+formula">Bogoliubov's formula</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+M%C3%B8ller+operator">quantum Møller operator</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/adiabatic+limit">adiabatic limit</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/infrared+divergence">infrared divergence</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/interacting+vacuum">interacting vacuum</a></p> </li> </ul> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/renormalization">renormalization</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/renormalization+scheme">("re-")normalization scheme</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/extension+of+distributions">extension of distributions</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/renormalization+condition">("re"-)normalization condition</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+anomaly">quantum anomaly</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/renormalization+group">renormalization group</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/interaction+vertex+redefinition">interaction vertex redefinition</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/St%C3%BCckelberg-Petermann+renormalization+group">Stückelberg-Petermann renormalization group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/renormalization+group+flow">renormalization group flow</a>/<a class="existingWikiWord" href="/nlab/show/running+coupling+constants">running coupling constants</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/effective+quantum+field+theory">effective quantum field theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/UV+cutoff">UV cutoff</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/counterterms">counterterms</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/relative+effective+action">relative effective action</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Wilsonian+RG">Wilsonian RG</a>, <a class="existingWikiWord" href="/nlab/show/Polchinski+flow+equation">Polchinski flow equation</a></p> </li> </ul> </li> </ul> <h2 id="Theorems">Theorems</h2> <h3 id="states_and_observables">States and observables</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/order-theoretic+structure+in+quantum+mechanics">order-theoretic structure in quantum mechanics</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Alfsen-Shultz+theorem">Alfsen-Shultz theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Harding-D%C3%B6ring-Hamhalter+theorem">Harding-Döring-Hamhalter theorem</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kochen-Specker+theorem">Kochen-Specker theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bell%27s+theorem">Bell's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Fell%27s+theorem">Fell's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Gleason%27s+theorem">Gleason's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Wigner+theorem">Wigner theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bub-Clifton+theorem">Bub-Clifton theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kadison-Singer+problem">Kadison-Singer problem</a></p> </li> </ul> <h3 id="operator_algebra">Operator algebra</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Wick%27s+theorem">Wick's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/GNS+construction">GNS construction</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/cyclic+vector">cyclic vector</a>, <a class="existingWikiWord" href="/nlab/show/separating+vector">separating vector</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/modular+theory">modular theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Fell%27s+theorem">Fell's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Stone-von+Neumann+theorem">Stone-von Neumann theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Haag%27s+theorem">Haag's theorem</a></p> </li> </ul> <h3 id="local_qft">Local QFT</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Reeh-Schlieder+theorem">Reeh-Schlieder theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bisognano-Wichmann+theorem">Bisognano-Wichmann theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/PCT+theorem">PCT theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spin-statistics+theorem">spin-statistics theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/DHR+superselection+theory">DHR superselection theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Osterwalder-Schrader+theorem">Osterwalder-Schrader theorem</a> (<a class="existingWikiWord" href="/nlab/show/Wick+rotation">Wick rotation</a>)</p> </li> </ul> <h3 id="perturbative_qft">Perturbative QFT</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Schwinger-Dyson+equation">Schwinger-Dyson equation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/main+theorem+of+perturbative+renormalization">main theorem of perturbative renormalization</a></p> </li> </ul> </div></div> </div> </div> <h1 id='section_table_of_contents'>Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#history_and_terminology'>History and terminology</a></li> <li><a href='#definitions'>Definitions</a></li> <ul> <li><a href='#abstractdefn'>Abstract von Neumann algebras</a></li> <li><a href='#concrete_von_neumann_algebras'>Concrete von Neumann algebras</a></li> </ul> <li><a href='#preduals'>Sakai’s theorem and properties of preduals</a></li> <li><a href='#elementary_examples'>Elementary examples</a></li> <li><a href='#properties_of_morphisms_of_von_neumann_algebras'>Properties of morphisms of von Neumann algebras</a></li> <li><a href='#the_category_of_von_neumann_algebras'>The category of von Neumann algebras</a></li> <li><a href='#monoidal_structures'>Monoidal structures</a></li> <li><a href='#categories'><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>W</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">W^*</annotation></semantics></math>-categories</a></li> <li><a href='#modules_over_von_neumann_algebras'>Modules over von Neumann algebras</a></li> <li><a href='#bimodules_over_von_neumann_algebras_and_connes_fusion'>Bimodules over von Neumann algebras and Connes fusion</a></li> <li><a href='#modular_algebra_and_tomitatakesaki_theory'>Modular algebra and Tomita–Takesaki theory</a></li> <li><a href='#gelfand_duality_for_commutative_von_neumann_algebras'>Gelfand duality for commutative von Neumann algebras</a></li> <li><a href='#relevance'>Relevance</a></li> <li><a href='#general'>General</a></li> <li><a href='#RelationToMeasureSpaces'>Relation to measurable spaces</a></li> <li><a href='#topics_of_interest_for_the_understanding_of_aqft'>Topics of interest for the understanding of AQFT</a></li> <ul> <li><a href='#vectors'>Vectors</a></li> <li><a href='#projections_in_von_neumann_algebras'>Projections in von Neumann algebras</a></li> <ul> <li><a href='#projections_are_norm_dense'>Projections are norm dense</a></li> <li><a href='#GleasonsTheorem'>Gleason’s theorem</a></li> <li><a href='#murrayvon_neumann_classification_of_factors'>Murray–von Neumann classification of factors</a></li> </ul> <li><a href='#miscellaneous'>Miscellaneous</a></li> </ul> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#References'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>A <em>von Neumann algebra</em> or <em><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>W</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">W^*</annotation></semantics></math>-algebra</em> is an important and special kind of <a class="existingWikiWord" href="/nlab/show/operator+algebra">operator algebra</a>, relevant in particular to <a class="existingWikiWord" href="/nlab/show/measure+theory">measure theory</a> and <a class="existingWikiWord" href="/nlab/show/quantum+mechanics">quantum mechanics</a>/<a class="existingWikiWord" href="/nlab/show/quantum+field+theory">quantum field theory</a> in its algebraic formulation as <a class="existingWikiWord" href="/nlab/show/AQFT">AQFT</a>. Specifically, (non-commutative) von Neumann algebras can be understood as the <a class="existingWikiWord" href="/nlab/show/Isbell+duality">formal duals</a> of (<a class="existingWikiWord" href="/nlab/show/non-commutative+geometry">non-commutative</a>) <a class="existingWikiWord" href="/nlab/show/localizable+measurable+spaces">localizable measurable spaces</a> (or <a class="existingWikiWord" href="/nlab/show/measurable+locales">measurable locales</a>); see the section <em><a href="#RelationToMeasureSpaces">Relation to measurable spaces</a></em> below.</p> <h2 id="history_and_terminology">History and terminology</h2> <p>Since terminology varies in the literature, we will say something about this first. There are no precise definitions here; see below for those.</p> <p>(Of course, <em><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>W</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">W^*</annotation></semantics></math>-algebras</em> should not be confused with <em><a class="existingWikiWord" href="/nlab/show/W-algebras">W-algebras</a></em> in (logarithmic) <a class="existingWikiWord" href="/nlab/show/conformal+field+theory">conformal field theory</a>.)</p> <p><a class="existingWikiWord" href="/nlab/show/John+von+Neumann">John von Neumann</a> originally studied certain <a class="existingWikiWord" href="/nlab/show/operator+algebras">operator algebras</a> (back then they were called <em><a class="existingWikiWord" href="/nlab/show/rings">rings</a> of <a class="existingWikiWord" href="/nlab/show/linear+operator">operators</a></em>), defined as unital <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>*</mo></mrow><annotation encoding="application/x-tex">*</annotation></semantics></math>-subalgebras of the algebra <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><mo stretchy="false">(</mo><mi>H</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">B(H)</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/bounded+operators">bounded operators</a> on some <a class="existingWikiWord" href="/nlab/show/Hilbert+space">Hilbert space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi></mrow><annotation encoding="application/x-tex">H</annotation></semantics></math> that are <a class="existingWikiWord" href="/nlab/show/closed+subspace">closed</a> in any of the several <a class="existingWikiWord" href="/nlab/show/operator+topology">operator topologies</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><mo stretchy="false">(</mo><mi>H</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">B(H)</annotation></semantics></math> (except for the norm topology, which gives <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">C^*</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/C-star-algebra">algebras</a>); the ultraweak topology is most convenient for our purposes.</p> <p>One disadvantage of such a definition is that it makes it difficult to separate properties of von Neumann algebras from properties of their <a class="existingWikiWord" href="/nlab/show/representations">representations</a> on Hilbert spaces. For example, all faithful representations induce the same ultraweak topology on a von Neumann algebra, but different representations induce different weak topologies. Furthermore, not all von Neumann algebras come automatically equipped with a representation on a Hilbert space, such as the <a class="existingWikiWord" href="/nlab/show/coproduct">coproduct</a> of two von Neumann algebras (although such a representation can always be constructed). Finally, this definition unnecessarily confuses two very distinct notions: algebras and <a class="existingWikiWord" href="/nlab/show/modules">modules</a> (or representations).</p> <p>Therefore, we may use the modern abstract terminology in which a von Neumann algebra is defined as an <a class="existingWikiWord" href="/nlab/show/associative+algebra">algebra</a> with certain <a class="existingWikiWord" href="/nlab/show/extra+structure">structures</a> and <a class="existingWikiWord" href="/nlab/show/extra+property">properties</a>. It then becomes a theorem that every von Neumann algebra has a free representation on a Hilbert space (such as Haagerup's standard form), so we may study von Neumann algebras in the historical concrete sense if we wish; but now we think of these as particular <em>representations</em> of algebras.</p> <p>In the old terminology, morphisms of representations of von Neumann algebras (von Neumann algebras in the historical concrete sense) are sometimes called <em>spatial morphisms of von Neumann algebras</em> (as opposed to the <em>abstract morphisms</em> that we will define below). Similarly, the concrete von Neumann algebras themselves are sometimes called <em>von Neumann algebras</em>, whereas the abstract von Neumann algebras are called <em><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>W</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">W^*</annotation></semantics></math>-algebras</em>. Compare the historic definitions of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">C^*</annotation></semantics></math>-algebras, as well as other examples of <a class="existingWikiWord" href="/nlab/show/concrete+and+abstract+structures">concrete and abstract structures</a> such as <a class="existingWikiWord" href="/nlab/show/manifolds">manifolds</a>.</p> <p>The <a class="existingWikiWord" href="/nlab/show/nPOV">nPOV</a> dictates that a clear distinction between the categories of algebras and modules must be maintained, in particular, modules should not be confused with algebras. Hence we stick to the modern terminology, which also seems to be preferred in new papers on von Neumann algebras, see for example <a href="http://arxiv.org/abs/1110.5671">arXiv:1110.5671v1</a>.</p> <h2 id="definitions">Definitions</h2> <p>For completeness, we give both the modern abstract and historical concrete definitions.</p> <h3 id="abstractdefn">Abstract von Neumann algebras</h3> <p>We build on the concepts of <a class="existingWikiWord" href="/nlab/show/Banach+space">Banach space</a> and (abstract) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">C^*</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/C-star-algebra">algebra</a>. In this definition, a Banach space is a <a class="existingWikiWord" href="/nlab/show/complex+number">complex</a> Banach space and a <a class="existingWikiWord" href="/nlab/show/morphism">morphism</a> of Banach spaces is a <a class="existingWikiWord" href="/nlab/show/short+linear+map">short linear map</a> (a complex-linear map of norm at most <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>1</mn></mrow><annotation encoding="application/x-tex">1</annotation></semantics></math>); a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">C^*</annotation></semantics></math>-algebra is a complex unital <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">C^*</annotation></semantics></math>-algebra, and a morphism of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">C^*</annotation></semantics></math>-algebras is a unital <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>*</mo></mrow><annotation encoding="application/x-tex">*</annotation></semantics></math>-homomorphism (which is necessarily also a short linear map). Note in particular that an <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a> of either must be an <a class="existingWikiWord" href="/nlab/show/isometry">isometry</a>.</p> <p>Given a Banach space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>, a <strong><a class="existingWikiWord" href="/nlab/show/predual">predual</a></strong> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> is a Banach space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math> whose <a class="existingWikiWord" href="/nlab/show/dual+Banach+space">dual Banach space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>V</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">V^*</annotation></semantics></math> is isomorphic to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>V</mi> <mo>*</mo></msup><mover><mo>→</mo><mi>i</mi></mover><mi>A</mi></mrow><annotation encoding="application/x-tex"> V^* \overset{i}\to A </annotation></semantics></math></div> <p>(or more properly, equipped with such an isomorphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math>). Similarly, given a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><mi>A</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">f\colon A \to B</annotation></semantics></math> (properly, with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math> so equipped), a <strong>predual</strong> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> is a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>t</mi><mo lspace="verythinmathspace">:</mo><mi>W</mi><mo>→</mo><mi>V</mi></mrow><annotation encoding="application/x-tex">t\colon W \to V</annotation></semantics></math> whose dual is isomorphic to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msup><mi>V</mi> <mo>*</mo></msup></mtd> <mtd><mover><mo>→</mo><mi>i</mi></mover></mtd> <mtd><mi>A</mi></mtd></mtr> <mtr><mtd><mpadded width="0" lspace="-100%width"><mrow><msup><mi>t</mi> <mo>*</mo></msup></mrow></mpadded><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo><mpadded width="0"><mi>f</mi></mpadded></mtd></mtr> <mtr><mtd><msup><mi>W</mi> <mo>*</mo></msup></mtd> <mtd><munder><mo>→</mo><mi>j</mi></munder></mtd> <mtd><mi>B</mi></mtd></mtr></mtable></mrow><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array { V^* & \overset{i}\to & A \\ \mathllap{t^*}\downarrow & & \downarrow\mathrlap{f} \\ W^* & \underset{j}\to & B } . </annotation></semantics></math></div> <p>With these preliminaries, a <strong><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>W</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">W^*</annotation></semantics></math>-algebra</strong> or (“abstract”) <em>von Neumann algebra</em> is a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">C^*</annotation></semantics></math>-algebra that admits a predual (or more properly, equipped with one), and a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>W</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">W^*</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/homomorphism">homomorphism</a> is a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">C^*</annotation></semantics></math>-homomorphism that admits a predual. In this way, the <a class="existingWikiWord" href="/nlab/show/category">category</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>W</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">W^*</annotation></semantics></math>-algebras becomes a <a class="existingWikiWord" href="/nlab/show/subcategory">subcategory</a> of the category of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">C^*</annotation></semantics></math>-algebras.</p> <p>It is a theorem (see <a href="#preduals">below</a>) that the predual of a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>W</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">W^*</annotation></semantics></math>-algebra or <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>W</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">W^*</annotation></semantics></math>-homomorphism is essentially unique; we speak of <a class="existingWikiWord" href="/nlab/show/generalized+the">the</a> predual of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>, write it <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mo>*</mo></msub></mrow><annotation encoding="application/x-tex">A_*</annotation></semantics></math>, and identify <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>A</mi> <mo>*</mo></msub><msup><mo stretchy="false">)</mo> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">(A_*)^*</annotation></semantics></math> (and similarly for morphisms). (So in fact we don't need the word ‘equipped’; being a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>W</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">W^*</annotation></semantics></math>-algebra is an <a class="existingWikiWord" href="/nlab/show/extra+property">extra property</a>, not an <a class="existingWikiWord" href="/nlab/show/extra+structure">extra structure</a>, on a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">C^*</annotation></semantics></math>-algebra.)</p> <h3 id="concrete_von_neumann_algebras">Concrete von Neumann algebras</h3> <p>Fix a <a class="existingWikiWord" href="/nlab/show/complex+number">complex</a> <a class="existingWikiWord" href="/nlab/show/Hilbert+space">Hilbert space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi></mrow><annotation encoding="application/x-tex">H</annotation></semantics></math> and consider the <a class="existingWikiWord" href="/nlab/show/associative+algebra">algebra</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><mo stretchy="false">(</mo><mi>H</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">B(H)</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/bounded+operators">bounded operators</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi></mrow><annotation encoding="application/x-tex">H</annotation></semantics></math>. A (“concrete”) <strong>von Neumann algebra</strong> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi></mrow><annotation encoding="application/x-tex">H</annotation></semantics></math> is a unital <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>*</mo></mrow><annotation encoding="application/x-tex">*</annotation></semantics></math>-subalgebra of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><mo stretchy="false">(</mo><mi>H</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">B(H)</annotation></semantics></math> that is <a class="existingWikiWord" href="/nlab/show/closed+subspace">closed</a> in the <a class="existingWikiWord" href="/nlab/show/weak+operator+topology">weak operator topology</a>, or equivalently in the <a class="existingWikiWord" href="/nlab/show/ultraweak+topology">ultraweak topology</a> or in the <a class="existingWikiWord" href="/nlab/show/strong+topology">strong topology</a>. As such is automatically <a class="existingWikiWord" href="/nlab/show/closed+subspace">closed</a> in the <a class="existingWikiWord" href="/nlab/show/norm+topology">norm topology</a>, the von Neumann algebras form a (particularly nice) class of concrete <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">C^*</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/C-star-algebra">algebras</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi></mrow><annotation encoding="application/x-tex">H</annotation></semantics></math>, where the latter are defined as unital <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>*</mo></mrow><annotation encoding="application/x-tex">*</annotation></semantics></math>-subalgebras of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><mo stretchy="false">(</mo><mi>H</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">B(H)</annotation></semantics></math> closed under the norm topology.</p> <p>We equip a von Neumann algebra with the topology induced by its inclusion into <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><mo stretchy="false">(</mo><mi>H</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">B(H)</annotation></semantics></math> equipped with the ultraweak topology. An <strong>abstract morphism</strong> of von Neumann algebras can then be defined as a unital <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>*</mo></mrow><annotation encoding="application/x-tex">*</annotation></semantics></math>-homomorphism that is continous in the ultraweak topology. Here we are disregarding the data of the inclusion of a von Neumann algebra into <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><mo stretchy="false">(</mo><mi>H</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">B(H)</annotation></semantics></math> and treating it as an algebra on its own.</p> <p>Alternatively, we can define a von Neumann algebra <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> as a unital <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>*</mo></mrow><annotation encoding="application/x-tex">*</annotation></semantics></math>-algebra that admits an injective morphism into <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><mo stretchy="false">(</mo><mi>H</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">B(H)</annotation></semantics></math> for some Hilbert space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi></mrow><annotation encoding="application/x-tex">H</annotation></semantics></math> such that the image of the inclusion is closed in the ultraweak topology on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><mo stretchy="false">(</mo><mi>H</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">B(H)</annotation></semantics></math>. One can then prove that the topology induced on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> by the ultraweak topology on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><mo stretchy="false">(</mo><mi>H</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">B(H)</annotation></semantics></math> does not depend on the choice of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi></mrow><annotation encoding="application/x-tex">H</annotation></semantics></math> or the particular inclusion of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> into <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><mo stretchy="false">(</mo><mi>H</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">B(H)</annotation></semantics></math>. Hence one can define an abstract morphism of von Neumann algebras as a unital morphism of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>*</mo></mrow><annotation encoding="application/x-tex">*</annotation></semantics></math>-algebras that is continuous in the ultraweak topology.</p> <p>It is a theorem that the category of (concrete) von Neumann algebras and abstract morphisms is <a class="existingWikiWord" href="/nlab/show/equivalence+of+categories">equivalent</a> to the category of (abstract) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>W</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">W^*</annotation></semantics></math>-algebras and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>W</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">W^*</annotation></semantics></math>-homomorphisms. Similarly, we get the category of <a class="existingWikiWord" href="/nlab/show/representations">representations</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>W</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">W^*</annotation></semantics></math>-algebras on Hilbert spaces using instead the <strong>spatial morphisms</strong> of concrete von Neumann algebras.</p> <h2 id="preduals">Sakai’s theorem and properties of preduals</h2> <p>Sakai’s theorem states that preduals considered in the abstract definition are necessarily unique. More precisely, given a von Neumann algebra <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> we consider the category whose objects are pairs <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>V</mi><mo>,</mo><mi>f</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(V,f)</annotation></semantics></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math> is a Banach space and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><msup><mi>V</mi> <mo>*</mo></msup><mo>→</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">f\colon V^*\to A</annotation></semantics></math> is an isomorphism of Banach spaces. A morphism from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>V</mi><mo>,</mo><mi>f</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(V,f)</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>W</mi><mo>,</mo><mi>g</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(W,g)</annotation></semantics></math> is a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>h</mi><mo lspace="verythinmathspace">:</mo><mi>V</mi><mo>→</mo><mi>W</mi></mrow><annotation encoding="application/x-tex">h\colon V\to W</annotation></semantics></math> of Banach spaces such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><msup><mi>h</mi> <mo>*</mo></msup><mo>=</mo><mi>g</mi></mrow><annotation encoding="application/x-tex">f h^* = g</annotation></semantics></math>.</p> <p>Sakai’s theorem then states that in the above category there is exactly one morphism between any pair of objects, which is necessarily an isomorphism. In particular, the category of preduals is canonically isomorphic to the <a class="existingWikiWord" href="/nlab/show/terminal+category">terminal category</a>.</p> <p>Sakai’s theorem can be extended to morphisms of von Neumann algebras. Thus preduals of von Neumann algebras and their morphisms are unique up to a unique isomorphism, in particular we can talk about <a class="existingWikiWord" href="/nlab/show/generalized+the">the</a> predual of a von Neumann algebra and <a class="existingWikiWord" href="/nlab/show/generalized+the">the</a> predual of a morphism of von Neumann algebras.</p> <p>The weak topology induced on a von Neumann algebra by its predual is called the <strong>ultraweak topology</strong>. The role of the ultraweak topology for von Neumann algebras is analogous to the role of the norm topology for C*-algebras. In particular, morphisms of von Neumann algebras are precisely those morphisms of C*-algebras that are continuous in the ultraweak topology.</p> <p>Consider the <a class="existingWikiWord" href="/nlab/show/dual+vector+space">dual space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math> of a von Neumann algebra <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> equipped with the ultraweak topology. The <a class="existingWikiWord" href="/nlab/show/topological+vector+space">topological vector space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math> canonically embeds into the dual of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> as a Banach space, the embedding map being induced by the canonical continuous map from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> equipped with the norm topology to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> equipped with the ultraweak topology. Thus <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math> is also a Banach space. There is a canonical morphism of Banach spaces from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>V</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">V^*</annotation></semantics></math> given by evaluating an element of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math> on the given element of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>. This morphism is in fact an isomorphism, hence <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math> is the predual of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>. In other words, the predual of a von Neumann algebra is canonically isomorphic to its dual in the ultraweak topology. Similarly, the predual of a morphism of von Neumann algebras is canonically isomorphic to its dual in the ultraweak topology.</p> <h2 id="elementary_examples">Elementary examples</h2> <p>The easiest example of a von Neumann algebra is given by the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">C^*</annotation></semantics></math>-algebra <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><mo stretchy="false">(</mo><mi>H</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">B(H)</annotation></semantics></math> of bounded operators on a complex <a class="existingWikiWord" href="/nlab/show/Hilbert+space">Hilbert space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi></mrow><annotation encoding="application/x-tex">H</annotation></semantics></math>. The predual can be canonically identified with the Banach space of <a class="existingWikiWord" href="/nlab/show/trace+class+operator">trace class operators</a>.</p> <p>Any <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">C^*</annotation></semantics></math>-subalgebra of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><mo stretchy="false">(</mo><mi>H</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">B(H)</annotation></semantics></math> closed in the ultraweak topology is again a von Neumann algebra.</p> <p>Another example is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>L</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">L^\infty(X)</annotation></semantics></math> under pointwise almost everywhere multiplication, where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is a σ-finite <a class="existingWikiWord" href="/nlab/show/measure+space">measure space</a> or a <a class="existingWikiWord" href="/nlab/show/localizable+measurable+space">localizable measurable space</a>. These are (up to isomorphism) all of the <em>commutative</em> von Neumann algebras, according to a specialized version of the <a class="existingWikiWord" href="/nlab/show/Gelfand+duality+theorem">Gelfand duality theorem</a>.</p> <p>A faithful representation (in fact, the standard from) of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>L</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>↪</mo><mi>B</mi><mo stretchy="false">(</mo><msup><mi>L</mi> <mn>2</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">L^\infty(X) \hookrightarrow B(L^2(X))</annotation></semantics></math> is given by considering <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>L</mi> <mn>2</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">L^2(X)</annotation></semantics></math> as an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>L</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">L^\infty(X)</annotation></semantics></math>-module given by pointwise almost everywhere multiplication.</p> <h2 id="properties_of_morphisms_of_von_neumann_algebras">Properties of morphisms of von Neumann algebras</h2> <h2 id="the_category_of_von_neumann_algebras">The category of von Neumann algebras</h2> <p>The category of von Neumann algebras is not a <a class="existingWikiWord" href="/nlab/show/locally+presentable+category">locally presentable category</a> since its <a class="existingWikiWord" href="/nlab/show/small+objects">small objects</a> are precisely von Neumann algebras of dimension 0 and 1. See Theorem 4.2 in <a href="#CK">Chirvasitu–Ko</a>.</p> <p>The <a class="existingWikiWord" href="/nlab/show/forgetful+functor">forgetful functor</a> from von Neumann algebras to <a class="existingWikiWord" href="/nlab/show/sets">sets</a> that sends a von Neumann algebra to its <a class="existingWikiWord" href="/nlab/show/unit+ball">unit ball</a> is a <a class="existingWikiWord" href="/nlab/show/right+adjoint+functor">right adjoint functor</a>. In fact, it is a <a class="existingWikiWord" href="/nlab/show/monadic+functor">monadic functor</a> and preserves all <a class="existingWikiWord" href="/nlab/show/sifted+colimits">sifted colimits</a>.</p> <p>Thus, <a class="existingWikiWord" href="/nlab/show/limits">limits</a> and <a class="existingWikiWord" href="/nlab/show/sifted+colimits">sifted colimits</a> of von Neumann algebras can be computed on the level of underlying unit balls.</p> <p>Small <a class="existingWikiWord" href="/nlab/show/coproducts">coproducts</a> of von Neumann algebras exist. There is also a “reduced” version of small <a class="existingWikiWord" href="/nlab/show/coproducts">coproducts</a>, known as <a class="existingWikiWord" href="/nlab/show/free+products">free products</a>, which can be defined in a manner analogous to the <a class="existingWikiWord" href="/nlab/show/spatial+tensor+product">spatial tensor product</a>.</p> <h2 id="monoidal_structures">Monoidal structures</h2> <p>There are two different <a class="existingWikiWord" href="/nlab/show/tensor+products">tensor products</a> one can define on von Neumann algebras.</p> <p>First, one can use the usual universal property of tensor products and postulate that morphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>⊗</mo><mi>B</mi><mo>→</mo><mi>M</mi></mrow><annotation encoding="application/x-tex">A\otimes B\to M</annotation></semantics></math> are in a natural bijection with pairs of morphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>→</mo><mi>M</mi></mrow><annotation encoding="application/x-tex">A\to M</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><mo>→</mo><mi>M</mi></mrow><annotation encoding="application/x-tex">B\to M</annotation></semantics></math> whose images commute in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math>. This yields a <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+structure">symmetric monoidal structure</a> on von Neumann algebras. This monoidal structure is not <span class="newWikiWord">closed<a href="/nlab/new/closed">?</a></span>.</p> <p>Secondly, one can also define a “reduced” version, known as the <em>spatial tensor product</em>. Given two von Neumann algebras <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math>, their spatial tensor product is the von Neumann algebra generated by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>⊗</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">A\otimes 1</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>1</mn><mo>⊗</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">1\otimes B</annotation></semantics></math> in the von Neumann algebra <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><mo stretchy="false">(</mo><msup><mi>L</mi> <mn>2</mn></msup><mi>A</mi><mo>⊗</mo><msup><mi>L</mi> <mn>2</mn></msup><mi>B</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">B(L^2 A\otimes L^2 B)</annotation></semantics></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>L</mi> <mn>2</mn></msup><mi>A</mi></mrow><annotation encoding="application/x-tex">L^2 A</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>L</mi> <mn>2</mn></msup><mi>B</mi></mrow><annotation encoding="application/x-tex">L^2 B</annotation></semantics></math> are the Haagerup standard form of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math> respectively. This also results in a <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+structure">symmetric monoidal structure</a>. Furthermore, passing to the <a class="existingWikiWord" href="/nlab/show/opposite+category">opposite category</a> yields a <a class="existingWikiWord" href="/nlab/show/closed+monoidal+structure">closed monoidal structure</a>.</p> <h2 id="categories"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>W</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">W^*</annotation></semantics></math>-categories</h2> <h2 id="modules_over_von_neumann_algebras">Modules over von Neumann algebras</h2> <h2 id="bimodules_over_von_neumann_algebras_and_connes_fusion">Bimodules over von Neumann algebras and Connes fusion</h2> <h2 id="modular_algebra_and_tomitatakesaki_theory">Modular algebra and Tomita–Takesaki theory</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/modular+theory">modular theory</a></li> </ul> <h2 id="gelfand_duality_for_commutative_von_neumann_algebras">Gelfand duality for commutative von Neumann algebras</h2> <p>See the article <a class="existingWikiWord" href="/nlab/show/commutative+von+Neumann+algebra">commutative von Neumann algebra</a>.</p> <h2 id="relevance">Relevance</h2> <p>The <a class="existingWikiWord" href="/nlab/show/Gelfand+duality+theorem">Gelfand duality theorem</a> states that there is a contravariant <a class="existingWikiWord" href="/nlab/show/equivalence+of+categories">equivalence</a> between the <a class="existingWikiWord" href="/nlab/show/category">category</a> of commutative von Neumann algebras and the category of compact strictly localizable enhanced <a class="existingWikiWord" href="/nlab/show/measurable+space">measurable spaces</a>; that is, the <a class="existingWikiWord" href="/nlab/show/opposite+category">opposite category</a> of one is equivalent to the other. See <a href="#RelationToMeasureSpaces">Relation to Measurable Spaces</a> below. General von Neumann algebras are seen then as a ‘noncommutative’ measurable spaces in a sense analogous to <a class="existingWikiWord" href="/nlab/show/noncommutative+geometry">noncommutative geometry</a>.</p> <p>The importance of von Neumann algebras for (<a class="existingWikiWord" href="/nlab/show/higher+category+theory">higher</a>) <a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a> and topology lays in the evidence that von Neumann algebras are deeply connected with the low dimensional <a class="existingWikiWord" href="/nlab/show/quantum+field+theory">quantum field theory</a> (2d <a class="existingWikiWord" href="/nlab/show/CFT">CFT</a>, <a class="existingWikiWord" href="/nlab/show/TQFT">TQFT</a> in low dimensions, inclusions of <a class="existingWikiWord" href="/nlab/show/von+Neumann+algebra+factor">factor</a>s, <a class="existingWikiWord" href="/nlab/show/quantum+group">quantum group</a>s and <a class="existingWikiWord" href="/nlab/show/knot+theory">knot theory</a>; <a class="existingWikiWord" href="/nlab/show/elliptic+cohomology">elliptic cohomology</a>: works of Wenzl, Vaughan Jones, Anthony Wasserman, Kerler, Kawahigashi, Ocneanu, Szlachanyi etc.).</p> <p>The highlights of their structure theory include the results on classification of <a class="existingWikiWord" href="/nlab/show/von+Neumann+algebra+factor">factors</a> (<a class="existingWikiWord" href="/nlab/show/Alain+Connes">Alain Connes</a>, 1970s) and theory of inclusions of subfactors (V. Jones). (Hilbert) <a class="existingWikiWord" href="/nlab/show/bimodule">bimodule</a>s over von Neumann algebras have a remarkable tensor product due Connes (<a class="existingWikiWord" href="/nlab/show/Connes+fusion">Connes fusion</a>). Following Segal’s manifesto</p> <ul> <li>Graeme Segal, <em>Elliptic cohomology (after Landweber-Stong, Ochanine, Witten, and others)</em>. Séminaire Bourbaki, Vol. 1987/88. Astérisque No. 161-162 (1988), Exp. No. 695, 4, 187–201 (1989).</li> </ul> <p>and its update</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Graeme+Segal">Graeme Segal</a>, <em>What is an elliptic object?</em> Elliptic cohomology, 306–317, London Math. Soc. Lecture Note Ser., 342, Cambridge Univ. Press, Cambridge, 2007.</li> </ul> <p>on hypothetical connections between <a class="existingWikiWord" href="/nlab/show/CFT">CFT</a> and <a class="existingWikiWord" href="/nlab/show/elliptic+cohomology">elliptic cohomology</a>, Stolz and Teichner have made a case for a role von Neumann algebras seem to play in a partial realization of that program:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Stefan+Stolz">Stefan Stolz</a> and <a class="existingWikiWord" href="/nlab/show/Peter+Teichner">Peter Teichner</a>, <em>What is an elliptic object?</em> (<a href="http://math.berkeley.edu/~teichner/Papers/Oxford.pdf">pdf</a>)</li> </ul> <p>See also the <a href="http://en.wikipedia.org/wiki/Von_Neumann_algebra">Wikipedia entry</a> entry for more on von Neumann algebras and a list of references and links.</p> <h2 id="general">General</h2> <p>The <em><a class="existingWikiWord" href="/nlab/show/bicommutant+theorem">bicommutant theorem</a></em> (as known as the <em>double commutant theorem</em> , or <em>von Neumann’s double commutant theorem</em> ) is the following result.</p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>⊆</mo><mi>B</mi><mo stretchy="false">(</mo><mi>H</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">A \subseteq B(H)</annotation></semantics></math> be a sub-<a class="existingWikiWord" href="/nlab/show/star-algebra">star-algebra</a> of the <a class="existingWikiWord" href="/nlab/show/C-star+algebra">C-star algebra</a> of <a class="existingWikiWord" href="/nlab/show/bounded+linear+operator">bounded linear operators</a> on a <a class="existingWikiWord" href="/nlab/show/Hilbert+space">Hilbert space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi></mrow><annotation encoding="application/x-tex">H</annotation></semantics></math>. Then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/von+Neumann+algebra">von Neumann algebra</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi></mrow><annotation encoding="application/x-tex">H</annotation></semantics></math> if and only if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>=</mo><mi>A</mi><mo>″</mo></mrow><annotation encoding="application/x-tex">A = A''</annotation></semantics></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">A'</annotation></semantics></math> denotes the <a class="existingWikiWord" href="/nlab/show/commutant">commutant</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>.</p> <p>Notice that the condition of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> being a von Neumann algebra (being closed in the <a class="existingWikiWord" href="/nlab/show/weak+operator+topology">weak operator topology</a>; “weak” here can be replaced by “strong”, “ultrastrong”, or “ultraweak” as described in <a class="existingWikiWord" href="/nlab/show/operator+topology">operator topology</a>), which is a <a class="existingWikiWord" href="/nlab/show/topology">topological</a> condition, is by this result equivalent to an algebraic condition (being equal to its bicommutant).</p> <h2 id="RelationToMeasureSpaces">Relation to measurable spaces</h2> <p>The <a class="existingWikiWord" href="/nlab/show/Gelfand-Naimark+theorem">Gel'fand–Naimark theorem</a> states that the category of <a class="existingWikiWord" href="/nlab/show/localizable+measurable+spaces">localizable measurable spaces</a> is contravariantly <a class="existingWikiWord" href="/nlab/show/equivalence+of+categories">equivalent</a> to (that is equivalent to the <a class="existingWikiWord" href="/nlab/show/opposite+category">opposite</a> of) the category of commutative von Neumann algebras. As such, arbitrary von Neumann algebras may be interpreted as ‘noncommutative’ measurable spaces in a sense analogous to <a class="existingWikiWord" href="/nlab/show/noncommutative+geometry">noncommutative geometry</a>. See at <em><a class="existingWikiWord" href="/nlab/show/noncommutative+probability+space">noncommutative probability space</a></em>.</p> <h2 id="topics_of_interest_for_the_understanding_of_aqft">Topics of interest for the understanding of AQFT</h2> <p>This paragraph will collect some facts of interest for the aspects of <a class="existingWikiWord" href="/nlab/show/AQFT">AQFT</a>.</p> <p>In this paragraph <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℳ</mi></mrow><annotation encoding="application/x-tex">\mathcal{M}</annotation></semantics></math> will always be a von Neumann algebra acting on a <a class="existingWikiWord" href="/nlab/show/Hilbert+space">Hilbert space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℋ</mi></mrow><annotation encoding="application/x-tex">\mathcal{H}</annotation></semantics></math> with <a class="existingWikiWord" href="/nlab/show/commutant">commutant</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℳ</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">\mathcal{M}'</annotation></semantics></math>.</p> <h3 id="vectors">Vectors</h3> <div class="un_defn"> <h6 id="definition">Definition</h6> <p>A <a class="existingWikiWord" href="/nlab/show/vector">vector</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><mi>ℋ</mi></mrow><annotation encoding="application/x-tex">x \in \mathcal{H}</annotation></semantics></math> is a <strong><a class="existingWikiWord" href="/nlab/show/cyclic+vector">cyclic vector</a></strong> if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℳ</mi><mi>x</mi></mrow><annotation encoding="application/x-tex">\mathcal{M}x</annotation></semantics></math> is dense in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℋ</mi></mrow><annotation encoding="application/x-tex">\mathcal{H}</annotation></semantics></math>.</p> </div> <div class="un_defn"> <h6 id="definition_2">Definition</h6> <p>A <a class="existingWikiWord" href="/nlab/show/vector">vector</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><mi>ℋ</mi></mrow><annotation encoding="application/x-tex">x \in \mathcal{H}</annotation></semantics></math> is a <strong><a class="existingWikiWord" href="/nlab/show/separating+vector">separating vector</a></strong> if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">M(x) = 0</annotation></semantics></math> implies <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">M = 0</annotation></semantics></math> for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi><mo>∈</mo><mi>ℳ</mi></mrow><annotation encoding="application/x-tex">M \in \mathcal{M}</annotation></semantics></math>.</p> </div> <div class="un_theorem"> <h6 id="theorem">Theorem</h6> <p>The notions of cyclic and separating are dual with respect to the commutant, that is a vector is cyclic for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℳ</mi></mrow><annotation encoding="application/x-tex">\mathcal{M}</annotation></semantics></math> iff it is separating for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℳ</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">\mathcal{M}'</annotation></semantics></math>.</p> </div> <h3 id="projections_in_von_neumann_algebras">Projections in von Neumann algebras</h3> <p>One crucial feature of von Neumann algebras is that they contain “every projection one could wish for”: there are three points that make this statement precise:</p> <ul> <li> <p>the linear combinations of projections are norm dense in a von Neumann algebra</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Gleason%27s+theorem">Gleason's theorem</a></p> </li> <li> <p>Murray–von Neumann classification of <a class="existingWikiWord" href="/nlab/show/von+Neumann+algebra+factor">factors</a></p> </li> </ul> <h4 id="projections_are_norm_dense">Projections are norm dense</h4> <p>First let us note that every element <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> of a von Neumann algebra can trivially be written as a linear combination of two selfadjoint elements:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo stretchy="false">(</mo><mi>A</mi><mo>+</mo><msup><mi>A</mi> <mo>*</mo></msup><mo stretchy="false">)</mo><mo>+</mo><mi>i</mi><mfrac><mn>1</mn><mrow><mn>2</mn><mi>i</mi></mrow></mfrac><mo stretchy="false">(</mo><mi>A</mi><mo>−</mo><msup><mi>A</mi> <mo>*</mo></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> A = \frac{1}{2} (A + A^*) + i\frac{1}{2i} (A - A^*) </annotation></semantics></math></div> <p>Then, by the <a class="existingWikiWord" href="/nlab/show/spectral+theorem">spectral theorem</a> every selfadjoint element A is represented by it’s spectral measure E via</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>=</mo><msubsup><mo>∫</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">‖</mo><mi>A</mi><mo stretchy="false">‖</mo></mrow> <mrow><mo stretchy="false">‖</mo><mi>A</mi><mo stretchy="false">‖</mo></mrow></msubsup><mi>λ</mi><mi>E</mi><mo stretchy="false">(</mo><mi>d</mi><mi>λ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> A = \integral_{-\|A\|}^{\|A\|} \lambda E(d\lambda) </annotation></semantics></math></div> <p>The integral converges in norm to A and all spectral projections are elements of the von Neumann algebra. It immediatly follows that the set of finite sums of multiples of projections is norm dense in every von Neumann algebra.</p> <h4 id="GleasonsTheorem">Gleason’s theorem</h4> <p>See <a class="existingWikiWord" href="/nlab/show/Gleason%27s+theorem">Gleason's theorem</a>.</p> <h4 id="murrayvon_neumann_classification_of_factors">Murray–von Neumann classification of factors</h4> <p>To be done…</p> <h3 id="miscellaneous">Miscellaneous</h3> <ul> <li><a class="existingWikiWord" href="/nlab/show/split+inclusion+of+von+Neumann+algebras">split inclusion of von Neumann algebras</a></li> </ul> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/enveloping+von+Neumann+algebra">enveloping von Neumann algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/von+Neumann+algebra+factor">von Neumann algebra factor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/W-star+category">W-star category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+relation">quantum relation</a></p> </li> </ul> <h2 id="References">References</h2> <p>The original definition is due to <a class="existingWikiWord" href="/nlab/show/John+von+Neumann">John von Neumann</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/J.+v.+Neumann">J. v. Neumann</a>, <em>Zur Algebra der Funktionaloperationen und Theorie der normalen Operatoren</em>, Mathematische Annalen 102:1 (1930), 370-427, <a href="http://dx.doi.org/10.1007/bf01782352">doi</a>.</li> </ul> <p>Monographs:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Masamichi+Takesaki">Masamichi Takesaki</a><em>,</em>Theory of Operator Algebras_, Encyclopaedia of Mathematical Sciences <strong>124</strong>, <strong>125</strong>, <strong>127</strong> (1979, 2002, 2003)</p> </li> <li id="Blackadar06"> <p><a class="existingWikiWord" href="/nlab/show/Bruce+Blackadar">Bruce Blackadar</a>, <em>Operator Algebras – Theory of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">C^\ast</annotation></semantics></math>-Algebras and von Neumann Algebras</em>, Encyclopaedia of Mathematical Sciences <strong>122</strong>, Springer (2006) [<a href="https://doi.org/10.1007/3-540-28517-2">doi:10.1007/3-540-28517-2</a>]</p> </li> </ul> <p>On <a class="existingWikiWord" href="/nlab/show/von+Neumann+algebras">von Neumann algebras</a> in <a class="existingWikiWord" href="/nlab/show/algebraic+quantum+field+theory">algebraic quantum field theory</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Hans-J%C3%BCrgen+Borchers">Hans-Jürgen Borchers</a>, <a class="existingWikiWord" href="/nlab/show/Jakob+Yngvason">Jakob Yngvason</a>, <em>From quantum fields to local von Neumann algebras</em>, Rev. Math. Phys. <strong>4</strong> spec01 (1992) 15-47 [<a href="https://doi.org/10.1142/S0129055X92000145">doi:10.1142/S0129055X92000145</a>]</li> </ul> <p>See also:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Jacob+Lurie">Jacob Lurie</a>, <em>von Neumann algebras</em>, lecture series (2011) (<a href="http://www.math.harvard.edu/~lurie/261y.html">web</a>)</p> </li> <li> <p>Abraham Westerbaan, <em>The Category of von Neumann Algebras</em>, <a href="https://arxiv.org/abs/1804.02203">1804.02203</a> 2018</p> </li> <li id="CK"> <p><a class="existingWikiWord" href="/nlab/show/Alexandru+Chirvasitu">Alexandru Chirvasitu</a>, Joanna Ko, <em>Monadic forgetful functors and (non-)presentability for C∗- and W<em>-algebras_, <a href="https://arxiv.org/abs/2203.12087">arXiv:2203.12087</a>.</em></em></p> </li> </ul> <p>In <a class="existingWikiWord" href="/nlab/show/quantum+physics">quantum physics</a>/<a class="existingWikiWord" href="/nlab/show/quantum+computing">quantum computing</a>:</p> <p>as <a class="existingWikiWord" href="/nlab/show/categorical+semantics">categorical semantics</a></p> <p>for the <a class="existingWikiWord" href="/nlab/show/quantum+lambda-calculus">quantum lambda-calculus</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Kenta+Cho">Kenta Cho</a>, <a class="existingWikiWord" href="/nlab/show/Abraham+Westerbaan">Abraham Westerbaan</a>, <em>Von Neumann Algebras form a Model for the Quantum Lambda Calculus</em>, QPL 2016 [<a href="https://arxiv.org/abs/1603.02133">arXiv:1603.02133</a>, <a href="https://www.cs.ru.nl/K.Cho/papers/model-qlc.pdf">pdf</a>, slides:<a href="http://qpl2016.cis.strath.ac.uk/pdfs/2cho-final.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/Cho-vNQuantumCalculus.pdf" title="pdf">pdf</a>]</li> </ul> <p>and for <a class="existingWikiWord" href="/nlab/show/QPL">QPL</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Kenta+Cho">Kenta Cho</a>, <em>Semantics for a Quantum Programming Language by Operator Algebras</em>, EPTCS <strong>172</strong> (2014) 165-190 [<a href="https://arxiv.org/abs/1412.8545">arXiv:1412.8545</a>, <a href="https://doi.org/10.4204/EPTCS.172.12">doi:10.4204/EPTCS.172.12</a>]</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on August 21, 2024 at 02:03:42. 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