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float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <html xmlns="http://www.w3.org/1999/xhtml" xmlns:svg="http://www.w3.org/2000/svg" xml:lang="en" lang="en"> <head><meta http-equiv="Content-type" content="application/xhtml+xml;charset=utf-8" /><title>-theory</title></head> <body> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <h4 id="cohomology">Cohomology</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cocycle">cocycle</a>, <a class="existingWikiWord" href="/nlab/show/coboundary">coboundary</a>, <a class="existingWikiWord" href="/nlab/show/coefficient">coefficient</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homology">homology</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/chain">chain</a>, <a class="existingWikiWord" href="/nlab/show/cycle">cycle</a>, <a class="existingWikiWord" href="/nlab/show/boundary">boundary</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/characteristic+class">characteristic class</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/universal+characteristic+class">universal characteristic class</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/secondary+characteristic+class">secondary characteristic class</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+characteristic+class">differential characteristic class</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fiber+sequence">fiber sequence</a>/<a class="existingWikiWord" href="/nlab/show/long+exact+sequence+in+cohomology">long exact sequence in cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fiber+%E2%88%9E-bundle">fiber ∞-bundle</a>, <a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-bundle">principal ∞-bundle</a>, <a class="existingWikiWord" href="/nlab/show/associated+%E2%88%9E-bundle">associated ∞-bundle</a>,</p> <p><a class="existingWikiWord" href="/nlab/show/twisted+%E2%88%9E-bundle">twisted ∞-bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group+extension">∞-group extension</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/obstruction">obstruction</a></p> </li> </ul> <h3 id="special_and_general_types">Special and general types</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cochain+cohomology">cochain cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/ordinary+cohomology">ordinary cohomology</a>, <a class="existingWikiWord" href="/nlab/show/singular+cohomology">singular cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/group+cohomology">group cohomology</a>, <a class="existingWikiWord" href="/nlab/show/nonabelian+group+cohomology">nonabelian group cohomology</a>, <a class="existingWikiWord" href="/nlab/show/Lie+group+cohomology">Lie group cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Galois+cohomology">Galois cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/groupoid+cohomology">groupoid cohomology</a>, <a class="existingWikiWord" href="/nlab/show/nonabelian+groupoid+cohomology">nonabelian groupoid cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/generalized+%28Eilenberg-Steenrod%29+cohomology">generalized (Eilenberg-Steenrod) cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cobordism+cohomology+theory">cobordism cohomology theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/integral+cohomology">integral cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/K-theory">K-theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/elliptic+cohomology">elliptic cohomology</a>, <a class="existingWikiWord" href="/nlab/show/tmf">tmf</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/taf">taf</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/abelian+sheaf+cohomology">abelian sheaf cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Deligne+cohomology">Deligne cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/de+Rham+cohomology">de Rham cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Dolbeault+cohomology">Dolbeault cohomology</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/etale+cohomology">etale cohomology</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/group+of+units">group of units</a>, <a class="existingWikiWord" href="/nlab/show/Picard+group">Picard group</a>, <a class="existingWikiWord" href="/nlab/show/Brauer+group">Brauer group</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/crystalline+cohomology">crystalline cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/syntomic+cohomology">syntomic cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/motivic+cohomology">motivic cohomology</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cohomology+of+operads">cohomology of operads</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hochschild+cohomology">Hochschild cohomology</a>, <a class="existingWikiWord" href="/nlab/show/cyclic+cohomology">cyclic cohomology</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/string+topology">string topology</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/nonabelian+cohomology">nonabelian cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-bundle">principal ∞-bundle</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/universal+principal+%E2%88%9E-bundle">universal principal ∞-bundle</a>, <a class="existingWikiWord" href="/nlab/show/groupal+model+for+universal+principal+%E2%88%9E-bundles">groupal model for universal principal ∞-bundles</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/principal+bundle">principal bundle</a>, <a class="existingWikiWord" href="/nlab/show/Atiyah+Lie+groupoid">Atiyah Lie groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/principal+2-bundle">principal 2-bundle</a>/<a class="existingWikiWord" href="/nlab/show/gerbe">gerbe</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+constant+%E2%88%9E-stack">covering ∞-bundle</a>/<a class="existingWikiWord" href="/nlab/show/local+system">local system</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-vector+bundle">(∞,1)-vector bundle</a> / <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2Cn%29-vector+bundle">(∞,n)-vector bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+anomaly">quantum anomaly</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/orientation">orientation</a>, <a class="existingWikiWord" href="/nlab/show/Spin+structure">Spin structure</a>, <a class="existingWikiWord" href="/nlab/show/Spin%5Ec+structure">Spin^c structure</a>, <a class="existingWikiWord" href="/nlab/show/String+structure">String structure</a>, <a class="existingWikiWord" href="/nlab/show/Fivebrane+structure">Fivebrane structure</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cohomology+with+constant+coefficients">cohomology with constant coefficients</a> / <a class="existingWikiWord" href="/nlab/show/cohomology+with+a+local+system+of+coefficients">with a local system of coefficients</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+algebra+cohomology">∞-Lie algebra cohomology</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Lie+algebra+cohomology">Lie algebra cohomology</a>, <a class="existingWikiWord" href="/nlab/show/nonabelian+Lie+algebra+cohomology">nonabelian Lie algebra cohomology</a>, <a class="existingWikiWord" href="/nlab/show/Lie+algebra+extensions">Lie algebra extensions</a>, <a class="existingWikiWord" href="/nlab/show/Gelfand-Fuks+cohomology">Gelfand-Fuks cohomology</a>,</li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Gerstenhaber-Schack+cohomology">bialgebra cohomology</a></p> </li> </ul> <h3 id="special_notions">Special notions</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%C4%8Cech+cohomology">Čech cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/hypercohomology">hypercohomology</a></p> </li> </ul> <h3 id="variants">Variants</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/equivariant+cohomology">equivariant cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/equivariant+homotopy+theory">equivariant homotopy theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bredon+cohomology">Bredon cohomology</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/twisted+cohomology">twisted cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/twisted+bundle">twisted bundle</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/twisted+K-theory">twisted K-theory</a>, <a class="existingWikiWord" href="/nlab/show/twisted+spin+structure">twisted spin structure</a>, <a class="existingWikiWord" href="/nlab/show/twisted+spin%5Ec+structure">twisted spin^c structure</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/twisted+differential+c-structures">twisted differential c-structures</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/twisted+differential+string+structure">twisted differential string structure</a>, <a class="existingWikiWord" href="/nlab/show/twisted+differential+fivebrane+structure">twisted differential fivebrane structure</a></li> </ul> </li> </ul> </li> <li> <p>differential cohomology</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+cohomology">differential generalized (Eilenberg-Steenrod) cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+cobordism+cohomology">differential cobordism cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Deligne+cohomology">Deligne cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+K-theory">differential K-theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+elliptic+cohomology">differential elliptic cohomology</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/schreiber/show/differential+cohomology+in+a+cohesive+topos">differential cohomology in a cohesive topos</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Chern-Weil+theory">Chern-Weil theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Chern-Weil+theory">∞-Chern-Weil theory</a></p> </li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/relative+cohomology">relative cohomology</a></p> </li> </ul> <h3 id="extra_structure">Extra structure</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Hodge+structure">Hodge structure</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/orientation">orientation</a>, <a class="existingWikiWord" href="/nlab/show/orientation+in+generalized+cohomology">in generalized cohomology</a></p> </li> </ul> <h3 id="operations">Operations</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cohomology+operations">cohomology operations</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cup+product">cup product</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/connecting+homomorphism">connecting homomorphism</a>, <a class="existingWikiWord" href="/nlab/show/Bockstein+homomorphism">Bockstein homomorphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fiber+integration">fiber integration</a>, <a class="existingWikiWord" href="/nlab/show/transgression">transgression</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cohomology+localization">cohomology localization</a></p> </li> </ul> <h3 id="theorems">Theorems</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/universal+coefficient+theorem">universal coefficient theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/K%C3%BCnneth+theorem">Künneth theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/de+Rham+theorem">de Rham theorem</a>, <a class="existingWikiWord" href="/nlab/show/Poincare+lemma">Poincare lemma</a>, <a class="existingWikiWord" href="/nlab/show/Stokes+theorem">Stokes theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hodge+theory">Hodge theory</a>, <a class="existingWikiWord" href="/nlab/show/Hodge+theorem">Hodge theorem</a></p> <p><a class="existingWikiWord" href="/nlab/show/nonabelian+Hodge+theory">nonabelian Hodge theory</a>, <a class="existingWikiWord" href="/nlab/show/noncommutative+Hodge+theory">noncommutative Hodge theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Brown+representability+theorem">Brown representability theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/abelian+sheaf+cohomology">hypercovering theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Eckmann-Hilton+duality">Eckmann-Hilton-Fuks duality</a></p> </li> </ul> <div> <p> <a href="/nlab/edit/cohomology+-+contents">Edit this sidebar</a> </p> </div></div></div> <h4 id="operator_algebra">Operator algebra</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> <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="noncommutative_geometry">Noncommutative geometry</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/noncommutative+geometry">noncommutative geometry</a></strong></p> <p>(<a class="existingWikiWord" href="/nlab/show/geometry">geometry</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>←</mo></mrow><annotation encoding="application/x-tex">\leftarrow</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/Isbell+duality">Isbell duality</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>→</mo></mrow><annotation encoding="application/x-tex">\to</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/algebra">algebra</a>)</p> <h2 id="topology">Topology</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/noncommutative+topology">noncommutative topology</a></li> <li><a class="existingWikiWord" href="/nlab/show/C%2A-algebra">C*-algebra</a>, <a class="existingWikiWord" href="/nlab/show/Gelfand+duality">Gelfand duality</a></li> <li><a class="existingWikiWord" href="/nlab/show/quantale">quantale</a></li> </ul> <h2 id="smooth_and_riemannian_geometry">Smooth and Riemannian geometry</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/dense+subalgebra">dense subalgebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spectral+triple">spectral triple</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/2-spectral+triple">2-spectral triple</a></p> </li> </ul> <h2 id="algebraic_geometry">Algebraic geometry</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/noncommutative+algebraic+geometry">noncommutative algebraic geometry</a></li> <li><a class="existingWikiWord" href="/nlab/show/noncommutative+scheme">noncommutative scheme</a></li> <li><a class="existingWikiWord" href="/nlab/show/noncommutative+motive">noncommutative motive</a></li> <li><a class="existingWikiWord" href="/nlab/show/quantum+flag+variety">quantum flag variety</a></li> <li><a class="existingWikiWord" href="/nlab/show/quantum+Schubert+cell">quantum Schubert cell</a></li> </ul> <h2 id="homotopy_theory">Homotopy theory</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/noncommutative+stable+homotopy+theory">noncommutative stable homotopy theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+operator+algebras">model structure on operator algebras</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopical+structure+on+C%2A-algebras">homotopical structure on C*-algebras</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/KK-theory">KK-theory</a>, <a class="existingWikiWord" href="/nlab/show/E-theory">E-theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cyclic+homology">cyclic homology</a></p> </li> </ul> <h2 id="relation_to_physics">Relation to physics</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/deformation+quantization">deformation quantization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/algebra+of+observables">algebra of observables</a>, <a class="existingWikiWord" href="/nlab/show/local+net+of+observables">local net of observables</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bohr+topos">Bohr topos</a></p> </li> </ul> <div> <p> <a href="/nlab/edit/noncommutative+geometry+-+contents">Edit this sidebar</a> </p> </div></div></div> </div> </div> <h1 id="theory"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math>-theory</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#definition'>Definition</a></li> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#universal_characterization'>Universal characterization</a></li> <li><a href='#RelationToKKTheory'>Relation to KK-theory</a></li> </ul> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> <ul> <li><a href='#general'>General</a></li> <li><a href='#relation_to_shape_theory'>Relation to shape theory</a></li> </ul> </ul> </div> <h2 id="idea">Idea</h2> <p><em><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math>-Theory</em> is the name of a <a class="existingWikiWord" href="/nlab/show/category">category</a> whose <a class="existingWikiWord" href="/nlab/show/objects">objects</a> are <a class="existingWikiWord" href="/nlab/show/C%2A-algebras">C*-algebras</a> and whose <a class="existingWikiWord" href="/nlab/show/hom-sets">hom-sets</a> are homotopy classes of slightly generalized <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>*</mo></mrow><annotation encoding="application/x-tex">C*</annotation></semantics></math>-homomorphisms, called <em><a class="existingWikiWord" href="/nlab/show/asymptotic+C%2A-homomorphisms">asymptotic C*-homomorphisms</a></em>. These hom-sets have the structure of an <a class="existingWikiWord" href="/nlab/show/abelian+group">abelian group</a> and are also called the <em>E-groups</em> of their arguments. Since under <a class="existingWikiWord" href="/nlab/show/Gelfand+duality">Gelfand duality</a> <a class="existingWikiWord" href="/nlab/show/C%2A-algebras">C*-algebras</a> may be thought of as exhibiting <a class="existingWikiWord" href="/nlab/show/noncommutative+topology">noncommutative topology</a>, one also speaks of <em><a class="existingWikiWord" href="/nlab/show/noncommutative+stable+homotopy+theory">noncommutative stable homotopy theory</a></em>.</p> <p>This construction may be understood as the universal improvement of <a class="existingWikiWord" href="/nlab/show/KK-theory">KK-theory</a> under <a class="existingWikiWord" href="/nlab/show/excision">excision</a> (<a href="#Higson90">Higson 90</a>). Accordingly, the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math>-groups behave like groups of a <a class="existingWikiWord" href="/nlab/show/K-theory">K-theory</a>-like <a class="existingWikiWord" href="/nlab/show/generalized+cohomology+theory">generalized cohomology theory</a>.</p> <p>In terms of <a class="existingWikiWord" href="/nlab/show/noncommutative+topology">noncommutative topology</a> (regarding, in view of <a class="existingWikiWord" href="/nlab/show/Gelfand+duality">Gelfand duality</a>, noncommutative <a class="existingWikiWord" href="/nlab/show/C%2A-algebras">C*-algebras</a> as <a class="existingWikiWord" href="/nlab/show/algebras+of+functions">algebras of functions</a> on “noncommutative topological spaces”) one may understand this as dealing with “locally badly behaved space” such as certain <a class="existingWikiWord" href="/nlab/show/quotients">quotients</a> of <a class="existingWikiWord" href="/nlab/show/foliations">foliations</a> (<a href="ConnesHigson90">Connes-Higson 90</a>) in a way that resembles a noncommutative version of <a class="existingWikiWord" href="/nlab/show/shape+theory">shape theory</a> (<a href="#Dadarlat94">Dādārlat 94</a>).</p> <h2 id="definition">Definition</h2> <p>First some notation and terminology.</p> <p>For <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 \in </annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/C%2AAlg">C*Alg</a>, we write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Σ</mi><mi>A</mi><mo>≔</mo><msub><mi>C</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \Sigma A \coloneqq C_0((0,1),A) </annotation></semantics></math></div> <p>for 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^\ast</annotation></semantics></math>-algebra of continuous <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>-valued functions on the open inverval <a class="existingWikiWord" href="/nlab/show/vanishing+at+infinity">vanishing at infinity</a>. This is also called the <em>suspension</em> 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>For <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></mrow><annotation encoding="application/x-tex">A,B \in </annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/C%2AAlg">C*Alg</a>, write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[A,B]</annotation></semantics></math> for the set of <a class="existingWikiWord" href="/nlab/show/homotopy">homotopy</a>-<a class="existingWikiWord" href="/nlab/show/equivalence+classes">equivalence classes</a> of <a class="existingWikiWord" href="/nlab/show/asymptotic+C%2A-homomorphisms">asymptotic C*-homomorphisms</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">A \to B</annotation></semantics></math>. As discussed there</p> <ol> <li> <p>there is a natural <a class="existingWikiWord" href="/nlab/show/composition+operation">composition operation</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo stretchy="false">]</mo><mo>×</mo><mo stretchy="false">[</mo><mi>B</mi><mo>,</mo><mi>C</mi><mo stretchy="false">]</mo><mo>→</mo><mo stretchy="false">[</mo><mi>A</mi><mo>,</mo><mi>C</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[A,B] \times [B,C] \to [A,C]</annotation></semantics></math>;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>A</mi><mo>,</mo><mi>Σ</mi><mi>B</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[A,\Sigma B]</annotation></semantics></math> is naturally an <a class="existingWikiWord" href="/nlab/show/abelian+group">abelian group</a>.</p> </li> </ol> <p>Finally, write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒦</mi><mo>∈</mo></mrow><annotation encoding="application/x-tex">\mathcal{K} \in </annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/C%2AAlg">C*Alg</a> for 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^\ast</annotation></semantics></math>-algebra of <a class="existingWikiWord" href="/nlab/show/compact+operators">compact operators</a> on an infinite-dimensional <a class="existingWikiWord" href="/nlab/show/separable+Hilbert+space">separable Hilbert space</a>. For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>∈</mo><msup><mi>C</mi> <mo>*</mo></msup><mi>Alg</mi></mrow><annotation encoding="application/x-tex">A \in C^\ast Alg</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/tensor+product+of+C%2A-algebras">tensor product of C*-algebras</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>⊗</mo><mi>𝒦</mi></mrow><annotation encoding="application/x-tex">A \otimes \mathcal{K}</annotation></semantics></math> is also called the <em>stabilization</em> 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> <div class="num_defn"> <h6 id="definition_2">Definition</h6> <p>For <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></mrow><annotation encoding="application/x-tex">A,B \in </annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/C%2AAlg">C*Alg</a>, the <strong>E-group</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> with <a class="existingWikiWord" href="/nlab/show/coefficients">coefficients</a> in <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> is</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>E</mi><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo stretchy="false">)</mo><mo>≔</mo><mo stretchy="false">[</mo><mo stretchy="false">(</mo><mi>Σ</mi><mi>A</mi><mo stretchy="false">)</mo><mo>⊗</mo><mi>𝒦</mi><mo>,</mo><mo stretchy="false">(</mo><mi>Σ</mi><mi>B</mi><mo stretchy="false">)</mo><mo>⊗</mo><mi>𝒦</mi><mo stretchy="false">]</mo><mo>∈</mo><mi>Ab</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> E(A,B) \coloneqq [(\Sigma A )\otimes \mathcal{K}, (\Sigma B) \otimes \mathcal{K}] \in Ab \,. </annotation></semantics></math></div> <p>Under the induced <a class="existingWikiWord" href="/nlab/show/composition">composition</a> operation this yields an <a class="existingWikiWord" href="/nlab/show/additive+category">additive category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math> whose <a class="existingWikiWord" href="/nlab/show/objects">objects</a> are <a class="existingWikiWord" href="/nlab/show/C%2A-algebras">C*-algebras</a>, and whose <a class="existingWikiWord" href="/nlab/show/hom-objects">hom-objects</a> are <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">E(-,-)</annotation></semantics></math>.</p> </div> <h2 id="properties">Properties</h2> <h3 id="universal_characterization">Universal characterization</h3> <p>E-theory is the <a class="existingWikiWord" href="/nlab/show/universal+construction">universal</a> localization <a class="existingWikiWord" href="/nlab/show/C%2AAlg">C*Alg</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>→</mo><mi>E</mi></mrow><annotation encoding="application/x-tex">\to E</annotation></semantics></math> which is homotopy-invariant, stable and preserves <a class="existingWikiWord" href="/nlab/show/exact+sequences">exact sequences</a> in the middle.</p> <p>(…)</p> <h3 id="RelationToKKTheory">Relation to KK-theory</h3> <p>Because KK-theory is the universal <em>split exact</em> (stable and homotopy-invariant) localization of <a class="existingWikiWord" href="/nlab/show/C%2AAlg">C*Alg</a>, and E-theory the universal half-exact localization, and since every <a class="existingWikiWord" href="/nlab/show/split+exact+sequence">split exact sequence</a> is in particular exact, there is a universal <a class="existingWikiWord" href="/nlab/show/functor">functor</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>KK</mi><mo>→</mo><mi>E</mi></mrow><annotation encoding="application/x-tex"> KK \to E </annotation></semantics></math></div> <p>from the <a class="existingWikiWord" href="/nlab/show/KK-theory">KK-theory</a> <a class="existingWikiWord" href="/nlab/show/homotopy+category">homotopy category</a> to that of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math>-theory.</p> <p>Restricted to <a class="existingWikiWord" href="/nlab/show/nuclear+C%2A-algebras">nuclear C*-algebras</a> this is a <a class="existingWikiWord" href="/nlab/show/full+and+faithful+functor">full and faithful functor</a>. (<a href="#Higson90">Higson 90</a>) (…)</p> <p>If in the definition of E-theory by <a class="existingWikiWord" href="/nlab/show/asymptotic+C%2A-homomorphisms">asymptotic C*-homomorphisms</a> one restricts to those which take values in contractive <a class="existingWikiWord" href="/nlab/show/completely+positive+maps">completely positive maps</a>, then the results is isomorphic to KK-theory again. (K. Thomsen, <a href="#Introduction">Introduction, p. 34</a>). The above universal functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>KK</mi><mo>→</mo><mi>E</mi></mrow><annotation encoding="application/x-tex">KK \to E</annotation></semantics></math> is then just the corresponding <a class="existingWikiWord" href="/nlab/show/forgetful+functor">forgetful functor</a>.</p> <p>It follows that the Kasparov product in <a class="existingWikiWord" href="/nlab/show/KK-theory">KK-theory</a> is equivalently given by the composition of the corresponding completely positive <a class="existingWikiWord" href="/nlab/show/asymptotic+C%2A-homomorphisms">asymptotic C*-homomorphisms</a>.</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/KK-bootstrap+category">KK-bootstrap category</a></li> </ul> <div> <table><thead><tr><th>geometric context</th><th>universal additive <a class="existingWikiWord" href="/nlab/show/bivariant+cohomology+theory">bivariant</a> (preserves <a class="existingWikiWord" href="/nlab/show/split+exact+sequences">split exact sequences</a>)</th><th>universal localizing <a class="existingWikiWord" href="/nlab/show/bivariant+cohomology">bivariant</a> (preserves all <a class="existingWikiWord" href="/nlab/show/exact+sequences">exact sequences</a> in the middle)</th><th>universal additive invariant</th><th>universal localizing invariant</th></tr></thead><tbody><tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/noncommutative+algebraic+geometry">noncommutative algebraic geometry</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/noncommutative+motives">noncommutative motives</a> <math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_55f3094059c9fbdb4ade37c5d893a16a5acbd98f_1"><semantics><mrow><msub><mi>Mot</mi> <mi>add</mi></msub></mrow><annotation encoding="application/x-tex">Mot_{add}</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/noncommutative+motives">noncommutative motives</a> <math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_55f3094059c9fbdb4ade37c5d893a16a5acbd98f_2"><semantics><mrow><msub><mi>Mot</mi> <mi>loc</mi></msub></mrow><annotation encoding="application/x-tex">Mot_{loc}</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/algebraic+K-theory">algebraic K-theory</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/non-connective+algebraic+K-theory">non-connective algebraic K-theory</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/noncommutative+topology">noncommutative topology</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/KK-theory">KK-theory</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/E-theory">E-theory</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/operator+K-theory">operator K-theory</a></td><td style="text-align: left;">…</td></tr> </tbody></table> </div> <h2 id="references">References</h2> <h3 id="general">General</h3> <p>The idea of E-theory was introduced in</p> <ul id="ConnesHigson90"> <li><a class="existingWikiWord" href="/nlab/show/Alain+Connes">Alain Connes</a>, <a class="existingWikiWord" href="/nlab/show/Nigel+Higson">Nigel Higson</a>, <em>Déformations, morphismes asymptotiques et <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math>-théorie bivariante</em>, C. R. Acad. Sci. Paris Sér. I Math. <strong>311</strong> (1990), no. 2, 101–106, <a href="http://www.ams.org/mathscinet-getitem?mr=1065438">MR91m:46114</a>, <a href="ftp://ftp.bnf.fr/578/N5781521_PDF_107_112DM.pdf">pdf</a></li> </ul> <ul id="Higson90"> <li><a class="existingWikiWord" href="/nlab/show/Nigel+Higson">Nigel Higson</a>, <em>Categories of fractions and excision in KK-theory</em> J. Pure Appl. Algebra, 65(2):119–138, (1990) (<a href="http://www.math.psu.edu/higson/math/Papers_files/Higson%20-%201990%20-%20Categories%20of%20fractions%20and%20excision%20in%20KK-theory.pdf">pdf</a>)</li> </ul> <p>Reviews and surveys include</p> <ul id="Introduction"> <li><em>Introduction to KK-theory and E-theory</em>, Lecture notes (Lisbon 2009) (<a href="http://oaa.ist.utl.pt/files/cursos/courseD_Lecture4_KK_and_E1.pdf">pdf slides</a>)</li> </ul> <p>A standard textbook account is in section 25 of</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Bruce+Blackadar">Bruce Blackadar</a>, <em><a class="existingWikiWord" href="/nlab/show/K-Theory+for+Operator+Algebras">K-Theory for Operator Algebras</a></em></li> </ul> <p>The <a class="existingWikiWord" href="/nlab/show/stable+homotopy+theory">stable</a> <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a> aspects are further discussed in</p> <ul> <li> <p>Martin Grensing, <em>Noncommutative stable homotopy theory</em> (<a href="http://arxiv.org/abs/1302.4751">arXiv:1302.4751</a>)</p> </li> <li> <p>Rasmus Bentmann, <em>Homotopy-theoretic E-theory and n-order</em> (<a href="http://arxiv.org/abs/1302.6924">arXiv:1302.6924</a>)</p> </li> </ul> <p>See also</p> <ul id="Mahanta"> <li> <p>web page of a project <a href="http://www.math.ku.dk/~jg/papers/etheory.html">Noncommutative topology - homotopy functors and E-theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Snigdhayan+Mahanta">Snigdhayan Mahanta</a>, <em>Higher nonunital Quillen <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">K'</annotation></semantics></math>-theory, KK-dualities and applications to topological <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝕋</mi></mrow><annotation encoding="application/x-tex">\mathbb{T}</annotation></semantics></math>-duality</em> <a href="http://wwwmath.uni-muenster.de/u/snigdhayan.mahanta/papers/KQ.pdf">pdf</a></p> </li> </ul> <ul> <li><a href="http://faculty.tcu.edu/epark/papers/ETheory_JFA.pdf">pdf</a></li> </ul> <h3 id="relation_to_shape_theory">Relation to shape theory</h3> <p>Relation to <a class="existingWikiWord" href="/nlab/show/shape+theory">shape theory</a> is discussed in</p> <ul id="Dadarlat94"> <li><a class="existingWikiWord" href="/nlab/show/Marius+D%C4%81d%C4%81rlat">Marius Dādārlat</a>, <em>Shape theory and asymptotic morphisms for <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</em>, Duke Math. J. <strong>73</strong> (3):687-711, 1994, <a href="http://www.ams.org/mathscinet-getitem?mr=1262931">MR95c:46117</a>, <a href="http://www.math.purdue.edu/~mdd/Publications/shape.pdf">pdf</a></li> </ul> <ul> <li>Vladimir Manuilov, Klaus Thomsen, <em>Shape theory and extensions 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</em>, (<a href="http://arxiv.org/abs/1007.1663">arxiv/1007.1663</a>)</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on August 14, 2013 at 14:02:06. 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