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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="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="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> <h4 id="index_theory">Index theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/index+theory">index theory</a>, <a class="existingWikiWord" href="/nlab/show/KK-theory">KK-theory</a></strong></p> <p><a class="existingWikiWord" href="/nlab/show/noncommutative+topology">noncommutative topology</a>, <a class="existingWikiWord" href="/nlab/show/noncommutative+geometry">noncommutative geometry</a></p> <p><a class="existingWikiWord" href="/nlab/show/noncommutative+stable+homotopy+theory">noncommutative stable homotopy theory</a></p> <p><strong><a class="existingWikiWord" href="/nlab/show/partition+function">partition function</a></strong></p> <p><strong><a class="existingWikiWord" href="/nlab/show/genus">genus</a>, <a class="existingWikiWord" href="/nlab/show/orientation+in+generalized+cohomology">orientation in generalized cohomology</a></strong></p> <h2 id="definitions">Definitions</h2> <p><strong><a class="existingWikiWord" href="/nlab/show/operator+K-theory">operator K-theory</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/C%2A-algebra">C*-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hilbert+module">Hilbert module</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/K-homology">K-homology</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Fredholm+operator">Fredholm operator</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+operator">differential operator</a>, <a class="existingWikiWord" href="/nlab/show/pseudodifferential+operator">pseudodifferential operator</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/symbol+of+a+differential+operator">symbol of a differential operator</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/elliptic+operator">elliptic operator</a>, <a class="existingWikiWord" href="/nlab/show/elliptic+complex">elliptic complex</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Dirac+operator">Dirac operator</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Spin%5Ec+Dirac+operator">Spin^c Dirac operator</a></li> </ul> </li> </ul> <h2 id="index_theorems">Index theorems</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+index">topological index</a>, <a class="existingWikiWord" href="/nlab/show/analytical+index">analytical index</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Atiyah-Singer+index+theorem">Atiyah-Singer index theorem</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Gauss-Bonnet+theorem">Gauss-Bonnet theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hirzebruch-Riemann-Roch+theorem">Hirzebruch-Riemann-Roch theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/global+analytic+index+theory">global analytic index theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hirzebruch+signature+theorem">Hirzebruch signature theorem</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Mishchenko-Fomenko+index+theorem">Mishchenko-Fomenko index theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Baum-Connes+conjecture">Baum-Connes conjecture</a></p> </li> </ul> <h2 id="higher_genera">Higher genera</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/elliptic+genus">elliptic genus</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Witten+genus">Witten genus</a></p> </li> </ul> </div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#definitions'>Definitions</a></li> <ul> <li><a href='#abstract_algebras'>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^\ast</annotation></semantics></math>-algebras</a></li> <li><a href='#concrete_algebras_and_representations'>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^\ast</annotation></semantics></math>-algebras and <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>-representations</a></li> <li><a href='#DaggerFormulation'>In <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>†</mo></mrow><annotation encoding="application/x-tex">\dagger</annotation></semantics></math>-compact categories</a></li> </ul> <li><a href='#Properties'>Properties</a></li> <ul> <li><a href='#category_theoretic_properties'>Category theoretic properties</a></li> <li><a href='#PartialOrderAndPositiveElements'>Partial order and positive elements</a></li> <li><a href='#gelfandnaimark_theorem'>Gelfand-Naimark theorem</a></li> <li><a href='#gelfandnaimarksegal_construction'>Gelfand-Naimark-Segal construction</a></li> <li><a href='#gelfand_duality'>Gelfand duality</a></li> <li><a href='#General'>General</a></li> <li><a href='#construction_as_groupoid_convolution_algebras'>Construction as groupoid convolution algebras</a></li> <li><a href='#homotopy_theory'>Homotopy theory</a></li> </ul> <li><a href='#examples'>Examples</a></li> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="definitions">Definitions</h2> <h3 id="abstract_algebras">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^\ast</annotation></semantics></math>-algebras</h3> <div class="num_defn"> <h6 id="definition">Definition</h6> <p>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 <a class="existingWikiWord" href="/nlab/show/Banach+algebra">Banach algebra</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><mrow><mo stretchy="false">‖</mo><mo>−</mo><mo stretchy="false">‖</mo></mrow><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(A, {\|-\|})</annotation></semantics></math> over a <a class="existingWikiWord" href="/nlab/show/topological+field">topological field</a> <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> (often the field <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo>≔</mo><mi>ℂ</mi></mrow><annotation encoding="application/x-tex">K \coloneqq \mathbb{C}</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/complex+numbers">complex numbers</a>) equipped with an <a class="existingWikiWord" href="/nlab/show/anti-involution">anti-involution</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msup><mo stretchy="false">)</mo> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">(-)^\ast</annotation></semantics></math> compatible with <a class="existingWikiWord" href="/nlab/show/complex+conjugation">complex conjugation</a> if appropriate (that is: a Banach <a class="existingWikiWord" href="/nlab/show/star-algebra">star-algebra</a>) that satisfies the <strong><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>-identity</strong></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mo stretchy="false">‖</mo><mrow><msup><mi>A</mi> <mo>*</mo></msup><mi>A</mi></mrow><mo stretchy="false">‖</mo></mrow><mo>=</mo><mrow><mo stretchy="false">‖</mo><mrow><msup><mi>A</mi> <mo>*</mo></msup></mrow><mo stretchy="false">‖</mo></mrow><mspace width="thinmathspace"></mspace><mrow><mo stretchy="false">‖</mo><mi>A</mi><mo stretchy="false">‖</mo></mrow></mrow><annotation encoding="application/x-tex"> {\|{A^* A}\|} = {\|{A^*}\|} \, {\|{A}\|} </annotation></semantics></math></div> <p>or equivalently the <strong><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>B</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">B^*</annotation></semantics></math>-identity</strong></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mo stretchy="false">‖</mo><mrow><msup><mi>A</mi> <mo>*</mo></msup><mi>A</mi></mrow><mo stretchy="false">‖</mo></mrow><mo>=</mo><mrow><mo stretchy="false">‖</mo><mi>A</mi><msup><mo stretchy="false">‖</mo> <mn>2</mn></msup></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> {\|{A^* A}\|} = {\|{A}\|^2} \,. </annotation></semantics></math></div> <p>A <a class="existingWikiWord" href="/nlab/show/homomorphism">homomorphism</a> 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 is a map that preserves all this structure. For this it is sufficient for it to be a <a class="existingWikiWord" href="/nlab/show/star-algebra">star-algebra</a> homomorphism.</p> <p><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 with these homomorphisms form a <a class="existingWikiWord" href="/nlab/show/category">category</a> <a class="existingWikiWord" href="/nlab/show/C%2AAlg">C*Alg</a>.</p> </div> <div class="num_remark"> <h6 id="remark">Remark</h6> <p>Often one sees the definition without the clause (which should be in the definition of Banach <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 the involution is an <a class="existingWikiWord" href="/nlab/show/isometry">isometry</a> (so that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow><mo stretchy="false">‖</mo><msup><mi>A</mi> <mo>*</mo></msup><mo stretchy="false">‖</mo></mrow><mo>=</mo><mrow><mo stretchy="false">‖</mo><mi>A</mi><mo stretchy="false">‖</mo></mrow></mrow><annotation encoding="application/x-tex">{\|A^*\|} = {\|A\|}</annotation></semantics></math>, which is key for the equivalence of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>B</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">B^*</annotation></semantics></math> and <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> identities). This follows easily from the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>B</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">B^*</annotation></semantics></math>-identity, while it follows from 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>-identity after some difficulty.</p> </div> <div class="num_remark"> <h6 id="remark_2">Remark</h6> <p>There are different concepts for the <a class="existingWikiWord" href="/nlab/show/tensor+product">tensor product</a> 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, see for example at <em><a class="existingWikiWord" href="/nlab/show/spatial+tensor+product">spatial tensor product</a></em>.</p> </div> <div class="num_remark"> <h6 id="remark_3">Remark</h6> <p><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 equipped with the <a class="existingWikiWord" href="/nlab/show/action">action</a> of a <a class="existingWikiWord" href="/nlab/show/group">group</a> by <a class="existingWikiWord" href="/nlab/show/automorphisms">automorphisms</a> of the algebra are called <em><a class="existingWikiWord" href="/nlab/show/C-star-systems">C-star-systems</a></em> .</p> </div> <h3 id="concrete_algebras_and_representations">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^\ast</annotation></semantics></math>-algebras and <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>-representations</h3> <div class="num_defn"> <h6 id="definition_2">Definition</h6> <p>Given a <a class="existingWikiWord" href="/nlab/show/complex+numbers">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>, a <strong><a class="existingWikiWord" href="/nlab/show/concrete+structure">concrete</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</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 <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>-<span class="newWikiWord">subalgebra<a href="/nlab/new/subalgebra">?</a></span> of the algebra 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> that is <a class="existingWikiWord" href="/nlab/show/closed+subspace">closed</a> in the <a class="existingWikiWord" href="/nlab/show/norm+topology">norm topology</a>.</p> </div> <div class="num_defn"> <h6 id="definition_3">Definition</h6> <p>A <strong><a class="existingWikiWord" href="/nlab/show/representation+of+a+C-star+algebra">representation</a></strong> of 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 <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> 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> is 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">*</annotation></semantics></math>-homomorphism 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 the algebra 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>.</p> </div> <div class="num_remark"> <h6 id="remark_4">Remark</h6> <p>It is immediate that 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>-algebras correspond precisely to <a class="existingWikiWord" href="/nlab/show/faithful+representations">faithful representations</a> of 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>-algebras. It is an important theorem that <em>every</em> <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 has a faithful representation; that is, every 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>-algebra is <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphic</a> to a 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>-algebra.</p> </div> <div class="num_remark"> <h6 id="remark_5">Remark</h6> <p>The original definition of the term ‘<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’ was in fact the concrete notion; the ‘C’ stood for ‘closed’. Furthermore, the original term for the abstract notion was ‘<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>B</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">B^*</annotation></semantics></math>-algebra’ (where the ‘B’ stood for ‘Banach’). However, we now usually interpret ‘<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’ abstractly. (Compare ‘<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’ and ‘<a class="existingWikiWord" href="/nlab/show/von+Neumann+algebra">von Neumann algebra</a>’.)</p> </div> <h3 id="DaggerFormulation">In <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>†</mo></mrow><annotation encoding="application/x-tex">\dagger</annotation></semantics></math>-compact categories</h3> <p>The notion 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>-algebra can be abstracted to the general context of <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+%E2%80%A0-categories">symmetric monoidal †-categories</a>, which serves to illuminate their role in <em><a class="existingWikiWord" href="/nlab/show/quantum+mechanics+in+terms+of+%E2%80%A0-compact+categories">quantum mechanics in terms of †-compact categories</a></em>.</p> <p>For a discussion of this in the finite-dimensional case see for instance (<a href="#Vicary">Vicary</a>).</p> <h2 id="Properties">Properties</h2> <h3 id="category_theoretic_properties">Category theoretic properties</h3> <p><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 are <a class="existingWikiWord" href="/nlab/show/monadic">monadic</a> over sets. More precisely, the forgetful functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mrow><msup><mi>C</mi> <mo>*</mo></msup><mi>Alg</mi></mrow></mstyle><mo>→</mo><mstyle mathvariant="bold"><mi>Set</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{C^*Alg}\to\mathbf{Set}</annotation></semantics></math> that assigns to each algebra the set of points in its unit ball is monadic. See <a href="#PelletierRosicky93">Pelletier & Rosicky (1993)</a>.</p> <p>See also <em><a class="existingWikiWord" href="/nlab/show/operator+algebras">operator algebras</a></em>.</p> <h3 id="PartialOrderAndPositiveElements">Partial order and positive elements</h3> <p>The <a class="existingWikiWord" href="/nlab/show/self-adjoint+elements">self-adjoint elements</a> in 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^\ast</annotation></semantics></math>-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">\mathcal{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><mi>Herm</mi><mo stretchy="false">(</mo><mi>𝒜</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>≔</mo><mspace width="thickmathspace"></mspace><mo maxsize="1.2em" minsize="1.2em">{</mo><mi>A</mi><mspace width="thinmathspace"></mspace><mo>∈</mo><mspace width="thinmathspace"></mspace><mi>𝒜</mi><mspace width="thinmathspace"></mspace><mo maxsize="1.2em" minsize="1.2em">|</mo><mspace width="thinmathspace"></mspace><msup><mi>A</mi> <mo>*</mo></msup><mspace width="thinmathspace"></mspace><mo>=</mo><mspace width="thinmathspace"></mspace><mi>A</mi><mo maxsize="1.2em" minsize="1.2em">}</mo></mrow><annotation encoding="application/x-tex"> Herm(\mathcal{A}) \;\coloneqq\; \big\{ A \,\in\, \mathcal{A} \,\big\vert\, A^\ast \,=\, A \big\} </annotation></semantics></math></div> <p>form a <a class="existingWikiWord" href="/nlab/show/partial+order">partially ordered</a> <a class="existingWikiWord" href="/nlab/show/real+vector+space">real vector space</a> by declaring an 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> to be “larger” than some <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> if the difference is a <a class="existingWikiWord" href="/nlab/show/normal+operator">normal operator</a></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><mi>B</mi><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>⇔</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><munder><mo>∃</mo><mrow><mi>C</mi><mo>∈</mo><mi>𝒜</mi></mrow></munder><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mi>A</mi><mo>−</mo><mi>B</mi><mspace width="thinmathspace"></mspace><mo>=</mo><mspace width="thinmathspace"></mspace><msup><mi>C</mi> <mo>*</mo></msup><mi>C</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> A \geq B \;\;\;\;\; \Leftrightarrow \;\;\;\;\; \underset{C \in \mathcal{A} }{\exists} \;\; A - B \,=\, C^\ast C \,. </annotation></semantics></math></div> <p>In particular, the positive elements are exactly the <a class="existingWikiWord" href="/nlab/show/normal+operators">normal operators</a></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><mn>0</mn><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>⇔</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><munder><mo>∃</mo><mrow><mi>C</mi><mo>∈</mo><mi>𝒜</mi></mrow></munder><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mi>A</mi><mspace width="thinmathspace"></mspace><mo>=</mo><mspace width="thinmathspace"></mspace><msup><mi>C</mi> <mo>*</mo></msup><mi>C</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> A \geq 0 \;\;\;\;\; \Leftrightarrow \;\;\;\;\; \underset{C \in \mathcal{A} }{\exists} \;\; A \,=\, C^\ast C \,. </annotation></semantics></math></div> <p>(It is (only) this <a class="existingWikiWord" href="/nlab/show/partial+order">partial order</a> on the <a class="existingWikiWord" href="/nlab/show/underlying">underlying</a> <a class="existingWikiWord" href="/nlab/show/real+vector+space">real vector space</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">\mathcal{A}</annotation></semantics></math> that determines which <a class="existingWikiWord" href="/nlab/show/linear+functions">linear functions</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒜</mi><mo>→</mo><mi>ℂ</mi></mrow><annotation encoding="application/x-tex">\mathcal{A} \to \mathbb{C}</annotation></semantics></math> count as <em><a class="existingWikiWord" href="/nlab/show/states+on+a+C-star+algebra">states</a></em>.)</p> <p>E.g. <a href="#Murphy90">Murphy (1990) §2.2</a>, <a href="#Blackadar06">Blackadar (2006) §II.3.1</a></p> <p>Discussion in the context of <a class="existingWikiWord" href="/nlab/show/algebraic+quantum+field+theory">algebraic quantum field theory</a>: <a href="AQFT#BratteliRobinson79">Bratteli & Robinson (1979) §2.2.2</a>, <a href="AQFT#Fredenhagen03">Fredenhagen (2003) p. 6</a>.</p> <h3 id="gelfandnaimark_theorem">Gelfand-Naimark theorem</h3> <p>The <em><a class="existingWikiWord" href="/nlab/show/Gelfand-Naimark+theorem">Gelfand-Naimark theorem</a></em> says that every <a class="existingWikiWord" href="/nlab/show/C%2A-algebra">C*-algebra</a> is <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphic</a> to 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^\ast</annotation></semantics></math>-algebra of <a class="existingWikiWord" href="/nlab/show/bounded+linear+operators">bounded linear operators</a> on a <a class="existingWikiWord" href="/nlab/show/Hilbert+space">Hilbert space</a>. In other words, every 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>-algebra may be made into a 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>-algebra.</p> <h3 id="gelfandnaimarksegal_construction">Gelfand-Naimark-Segal construction</h3> <p>The <a class="existingWikiWord" href="/nlab/show/Gelfand-Naimark-Segal+construction">Gelfand-Naimark-Segal construction</a> (<a class="existingWikiWord" href="/nlab/show/GNS+construction">GNS construction</a>) establishes a correspondence between cyclic <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>-<a class="existingWikiWord" href="/nlab/show/representation">representation</a>s 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>-<a class="existingWikiWord" href="/nlab/show/C%2A-algebra">algebras</a> and certain linear functionals (usually called <em><a class="existingWikiWord" href="/nlab/show/state+on+an+operator+algebra">states</a></em>) on those same <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. The correspondence comes about from an explicit construction of the <a class="existingWikiWord" href="/nlab/show/star-representation">*-representation</a> from one of the <a class="existingWikiWord" href="/nlab/show/linear+functionals">linear functionals</a> (states).</p> <h3 id="gelfand_duality">Gelfand duality</h3> <p><a class="existingWikiWord" href="/nlab/show/Gelfand+duality">Gelfand duality</a> says that every (<a class="existingWikiWord" href="/nlab/show/unital+algebra">unital</a>) <em><a class="existingWikiWord" href="/nlab/show/commutative+C-star-algebra">commutative</a></em> <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 over the <a class="existingWikiWord" href="/nlab/show/complex+numbers">complex numbers</a> is that of complex-valued <a class="existingWikiWord" href="/nlab/show/continuous+functions">continuous functions</a> from some <a class="existingWikiWord" href="/nlab/show/compactum">compact Hausdorff topological space</a>: there is an <a class="existingWikiWord" href="/nlab/show/equivalence+of+categories">equivalence of categories</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mo>*</mo></msup><mi>CAlg</mi><mo>≃</mo></mrow><annotation encoding="application/x-tex">C^* CAlg \simeq </annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/Top">Top</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mrow></mrow> <mi>cpt</mi></msub></mrow><annotation encoding="application/x-tex">{}_{cpt}</annotation></semantics></math>.</p> <p>Accordingly one may think of the study of non-commutative <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 as <em><a class="existingWikiWord" href="/nlab/show/non-commutative+topology">non-commutative topology</a></em>.</p> <h3 id="General">General</h3> <div class="num_prop"> <h6 id="proposition">Proposition</h6> <p>For <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> two <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 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>A</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">f : A \to B</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/star-algebra">star-algebra</a> <a class="existingWikiWord" href="/nlab/show/homomorphism">homomorphism</a> the set-theoretic <a class="existingWikiWord" href="/nlab/show/image">image</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>⊂</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">f(A) \subset B</annotation></semantics></math> 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^\ast</annotation></semantics></math>-subalgebra of <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>, hence is also the <a class="existingWikiWord" href="/nlab/show/image">image</a> 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> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mo>*</mo></msup><mi>Alg</mi></mrow><annotation encoding="application/x-tex">C^\ast Alg</annotation></semantics></math>.</p> </div> <p>This is (<a href="#KadisonRingrose">KadisonRingrose, theorem 4.1.9</a>).</p> <div class="num_cor"> <h6 id="corollary">Corollary</h6> <p>There is a <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>𝒞</mi><mo>:</mo><msup><mi>C</mi> <mo>*</mo></msup><mi>Alg</mi><mo>→</mo><mi>Poset</mi></mrow><annotation encoding="application/x-tex"> \mathcal{C} : C^\ast Alg \to Poset </annotation></semantics></math></div> <p>to the <a class="existingWikiWord" href="/nlab/show/category">category</a> <a class="existingWikiWord" href="/nlab/show/Poset">Poset</a> of <a class="existingWikiWord" href="/nlab/show/posets">posets</a>, which sends each <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> to its <a class="existingWikiWord" href="/nlab/show/poset+of+commutative+subalgebras">poset of commutative subalgebras</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{C}(A)</annotation></semantics></math> and sends each morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>A</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">f : A \to B</annotation></semantics></math> to the <a class="existingWikiWord" href="/nlab/show/functor">functor</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo>:</mo><mi>𝒞</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>→</mo><mi>𝒞</mi><mo stretchy="false">(</mo><mi>B</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{C}(f) : \mathcal{C}(A) \to \mathcal{C}(B)</annotation></semantics></math> which sends a <a class="existingWikiWord" href="/nlab/show/commutative+C-star-algebra">commutative</a> subalgebra <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>⊂</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">C \subset A</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo><mo>⊂</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">f(C) \subset B</annotation></semantics></math>.</p> </div> <h3 id="construction_as_groupoid_convolution_algebras">Construction as groupoid convolution algebras</h3> <p>Many <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 arise as <a class="existingWikiWord" href="/nlab/show/groupoid+algebras">groupoid algebras</a> of <a class="existingWikiWord" href="/nlab/show/Lie+groupoids">Lie groupoids</a>. See at <em><a href="category+algebra#ReferencesForSmoothGeometry">groupoid algebra - References - For smooth geometry</a></em></p> <h3 id="homotopy_theory">Homotopy theory</h3> <p>There is <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a> 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, being a non-commutative generalization of that of <a class="existingWikiWord" href="/nlab/show/Top">Top</a>. (e.g. <a href="#Uuye">Uuye 12</a>). For more see at <em><a class="existingWikiWord" href="/nlab/show/homotopical+structure+on+C%2A-algebras">homotopical structure on C*-algebras</a></em>.</p> <h2 id="examples">Examples</h2> <div class="num_example"> <h6 id="example">Example</h6> <p>Any algebra <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>M</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">M_n(A)</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/matrices">matrices</a> with <a class="existingWikiWord" href="/nlab/show/coefficients">coefficients</a> in 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^\ast</annotation></semantics></math>-algebra is again 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^\ast</annotation></semantics></math>-algebra. In particular <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>M</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mi>ℂ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">M_n(\mathbb{C})</annotation></semantics></math> 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^\ast</annotation></semantics></math>-algebra for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n \in \mathbb{N}</annotation></semantics></math>.</p> </div> <p> <div class='num_remark' id='ContinuousFunctionsVanishingAtInfinity'> <h6>Example</h6> <p>For <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 <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 and for <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> a <a class="existingWikiWord" href="/nlab/show/locally+compact+topological+space">locally compact</a> <a class="existingWikiWord" href="/nlab/show/Hausdorff+topological+space">Hausdorff topological space</a>, the set of <a class="existingWikiWord" href="/nlab/show/continuous+functions">continuous functions</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>→</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">X \to A</annotation></semantics></math> which <a class="existingWikiWord" href="/nlab/show/vanish+at+infinity">vanish at infinity</a> is again 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^\ast</annotation></semantics></math>-algebra by extending all operations pointwise. (This algebra is <a class="existingWikiWord" href="/nlab/show/unital+algebra">unital</a> precisely if <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 and if <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 <a class="existingWikiWord" href="/nlab/show/compact+topological+space">compact topological space</a>.)</p> <p>This algebra is denoted</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>C</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo><mo>∈</mo><msup><mi>C</mi> <mo>*</mo></msup><mi>Alg</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> C_0(X,A) \in C^\ast Alg \,. </annotation></semantics></math></div> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>=</mo><mi>ℂ</mi></mrow><annotation encoding="application/x-tex">A = \mathbb{C}</annotation></semantics></math> then one usually just writes</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>C</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>≔</mo><msub><mi>C</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>ℂ</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> C_0(X) \coloneqq C_0(X, \mathbb{C}) \,. </annotation></semantics></math></div> <p>This are 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>-algebras to which the <a class="existingWikiWord" href="/nlab/show/Gelfand+duality">Gelfand duality</a> theorem applies and which are the default <a class="existingWikiWord" href="/nlab/show/algebras+of+observables">algebras of observables</a> in <a class="existingWikiWord" href="/nlab/show/classical+physics">classical physics</a> (for <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> a <a class="existingWikiWord" href="/nlab/show/phase+space">phase space</a>, cf. eg. <a href="#Landsman17">Landsman (2017), §3</a>).</p> </div> </p> <p> <div class='num_remark' id='CompactlySupportedContinuousFunctions'> <h6>Remark</h6> <p>The subalgebra <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>C</mi> <mn>00</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>⊂</mo><msub><mi>C</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C_{00}(X) \subset C_0(X)</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/compact+support">compactly supported</a> among all <a class="existingWikiWord" href="/nlab/show/vanishing+at+infinity">vanishing at infinity</a>-functions (Exp. <a class="maruku-ref" href="#ContinuousFunctionsVanishingAtInfinity"></a>) is not in general itself 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^\ast</annotation></semantics></math>-algebra, but is a very well-behaved <a class="existingWikiWord" href="/nlab/show/ideal">ideal</a> inside <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>C</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C_0(X)</annotation></semantics></math>, cf. <a href="#Amini04">Amini (2004)</a>.</p> </div> </p> <div class="num_example"> <h6 id="example_2">Example</h6> <p>A <a class="existingWikiWord" href="/nlab/show/uniformly+hyperfinite+algebra">uniformly hyperfinite algebra</a> is in particular 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^\ast</annotation></semantics></math>-algebra, by definition.</p> </div> <div class="num_example"> <h6 id="example_3">Example</h6> <p>A <a class="existingWikiWord" href="/nlab/show/von+Neumann+algebra">von Neumann algebra</a> is in particular 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^\ast</annotation></semantics></math>-algebra, by definition.</p> </div> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/commutative+C-star-algebra">commutative C-star-algebra</a>, <a class="existingWikiWord" href="/nlab/show/Gelfand+duality">Gelfand duality</a>, <a class="existingWikiWord" href="/nlab/show/noncommutative+topology">noncommutative topology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/separable+C%2A-algebra">separable C*-algebra</a>, <a class="existingWikiWord" href="/nlab/show/homogeneous+C%2A-algebra">homogeneous C*-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/nuclear+C%2A-algebra">nuclear C*-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/unitisation+of+C%2A-algebras">unitisation of C*-algebras</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/von+Neumann+algebra">von Neumann algebra</a>, <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/JB-algebra">JB-algebra</a>, <a class="existingWikiWord" href="/nlab/show/JLB-algebra">JLB-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/dense+subalgebra">dense subalgebra</a>, <a class="existingWikiWord" href="/nlab/show/F-star-algebra">F-star-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/multiplier+algebra">multiplier algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/graded+C%2A-algebra">graded C*-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/tensor+product+of+C%2A-algebras">tensor product of C*-algebras</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/crossed+product+C%2A-algebra">crossed product C*-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Cuntz+algebra">Cuntz algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Poincar%C3%A9+duality+C%2A-algebra">Poincaré duality C*-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hilbert+C%2A-module">Hilbert C*-module</a>, <a class="existingWikiWord" href="/nlab/show/Hilbert+C%2A-bimodule">Hilbert C*-bimodule</a>, <a class="existingWikiWord" href="/nlab/show/amplimorphism">amplimorphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/C%2A-category">C*-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/C%2A-coalgebra">C*-coalgebra</a>, <a class="existingWikiWord" href="/nlab/show/Hopf+C%2A-algebra">Hopf C*-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/continuous+field+of+C%2A-algebras">continuous field of C*-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> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/asymptotic+C%2A-homomorphism">asymptotic C*-homomorphism</a>, <a class="existingWikiWord" href="/nlab/show/l.m.c.-C%2A-algebra">l.m.c.-C*-algebra</a></p> </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> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/AQFT">AQFT</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/state+on+an+operator+algebra">state on an operator algebra</a></li> </ul> </li> </ul> <h2 id="references">References</h2> <p>Monographs:</p> <ul> <li id="Dixmier77"> <p><a class="existingWikiWord" href="/nlab/show/Jacques+Dixmier">Jacques Dixmier</a>, Chapter 2 of: <em><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>, North Holland (1977) [ch2:<a class="existingWikiWord" href="/nlab/files/Dixmier-CStarAlgebras-PosForms.pdf" title="pdf">pdf</a>, ch13:<a class="existingWikiWord" href="/nlab/files/Dixmier-CStarAlgebras-UnitaryReps.pdf" title="pdf">pdf</a>]</p> </li> <li id="KadisonRingrose"> <p><a class="existingWikiWord" href="/nlab/show/Richard+V.+Kadison">Richard V. Kadison</a>, <a class="existingWikiWord" href="/nlab/show/John+R.+Ringrose">John R. Ringrose</a>, <em>Fundamentals of the theory of operator algebras</em>, chapter 4 in: Vol I <em>Elementary Theory</em>, Graduate Studies in Mathematics <strong>15</strong>, AMS 1997 (<a href="https://bookstore.ams.org/gsm-15">ISBN:978-0-8218-0819-1</a>, <a href="http://www.zentralblatt-math.org/zmath/en/advanced/?q=an:0888.46039&format=complete">ZMATH</a>)</p> </li> <li id="Murphy90"> <p><a class="existingWikiWord" href="/nlab/show/Gerard+Murphy">Gerard Murphy</a>, <em><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 Operator Theory</em>, Academic Press (1990) [<a href="https://doi.org/10.1016/C2009-0-22289-6">doi:10.1016/C2009-0-22289-6</a>]</p> </li> <li id="Warner10"> <p><a class="existingWikiWord" href="/nlab/show/Garth+Warner">Garth Warner</a>, <em><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>, EPrint Collection, University of Washington (2010) [<a href="http://hdl.handle.net/1773/16302">hdl:1773/16302</a>, <a href="https://sites.math.washington.edu//~warner/C-star.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/Waner-CStarAlgebras.pdf" title="pdf">pdf</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Ian+Putnam">Ian Putnam</a>, <em>Lecture notes on <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> (2019) [<a href="https://www.math.uvic.ca/faculty/putnam/ln/C*-algebras.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/Putnam-CStarAlgebras.pdf" title="pdf">pdf</a>]</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>An exposition that explicitly gives <a class="existingWikiWord" href="/nlab/show/Gelfand+duality">Gelfand duality</a> as an <a class="existingWikiWord" href="/nlab/show/equivalence+of+categories">equivalence of categories</a> and introduces all the notions of <a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a> necessary for this statement is in</p> <ul> <li>Ivo Dell’Ambrogio, <em>Categories 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://www.math.ethz.ch/u/ambrogio/exercise_C_-algebras.pdf">pdf</a>)</li> </ul> <p>See also:</p> <ul> <li>Wikipedia, <em><a href="https://en.wikipedia.org/wiki/C*-algebra"><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</a></em></li> </ul> <p>For <a class="existingWikiWord" href="/nlab/show/operator+algebra">operator algebra</a>-theory see there and see</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Stanis%C5%82aw+Woronowicz">Stanisław Woronowicz</a>, <em>Unbounded elements affiliated with <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 non-compact quantum groups. Commun. Math. Phys. 136, 399–432 (1991)</em></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Stanis%C5%82aw+Woronowicz">Stanisław Woronowicz</a>, K. Napiórkowski, <em><a class="existingWikiWord" href="/nlab/show/Operator+theory+in+the+C%2A-algebra+framework">Operator theory in the C*-algebra framework</a></em>, Reports on Mathematical Physics Volume 31, Issue 3, June 1992, Pages 353–371 (<a href="http://www.sciencedirect.com/science/article/pii/003448779290025V">publisher</a>, <a href="http://www.fuw.edu.pl/~slworono/PDF-y/OP.pdf">pdf</a>)</p> </li> </ul> <p>On category-theoretic properties:</p> <ul> <li id="PelletierRosicky93">J Wick Pelletier, J Rosicky, <em>On the equational 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^*</annotation></semantics></math>-algebras</em>, Algebra Universalis <strong>30</strong> (1993) 275-284</li> </ul> <p>A characterizations of injections of commutative sub-<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 – hence of the <a class="existingWikiWord" href="/nlab/show/poset+of+commutative+subalgebras">poset of commutative subalgebras</a> of 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 in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Chris+Heunen">Chris Heunen</a>, <em>Characterizations of categories of commutative <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</em> (<a href="http://arxiv.org/abs/1106.5942">arXiv:1106.5942</a>)</li> </ul> <p>General properties of 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>C</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">C^\ast</annotation></semantics></math>-algebras are discussed in</p> <ul id="Meyer"> <li><a class="existingWikiWord" href="/nlab/show/Ralf+Meyer">Ralf Meyer</a>, <em>Categorical aspects of bivariant K-theory</em>, (<a href="http://arxiv.org/abs/math/0702145">arXiv:math/0702145</a>)</li> </ul> <p>Specifically <a class="existingWikiWord" href="/nlab/show/pullback">pullback</a> and <a class="existingWikiWord" href="/nlab/show/pushout">pushout</a> 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 is discussed in</p> <ul> <li>Gerd Petersen, <em>Pullback and pushout constructions in <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 theory</em> (<a href="http://www.math.ru.nl/~mueger/ped2.pdf">pdf</a>)</li> </ul> <p>See also</p> <ul> <li id="Amini04"><a class="existingWikiWord" href="/nlab/show/Massoud+Amini">Massoud Amini</a>, <em>Locally Compact Pro-<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>, Canadian Journal of Mathematics <strong>56</strong> 1 (2004) 3-22 [<a href="https://arxiv.org/abs/math/0205253">arXiv:math/0205253</a>, <a href="https://doi.org/10.4153/CJM-2004-001-6">doi:10.4153/CJM-2004-001-6</a>]</li> </ul> <p>The <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a> 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 (a <a class="existingWikiWord" href="/nlab/show/category+of+fibrant+objects">category of fibrant objects</a>-structure on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mo>*</mo></msup><mi>Alg</mi></mrow><annotation encoding="application/x-tex">C^\ast Alg</annotation></semantics></math>) is discussed in</p> <ul id="Uuye"> <li>Otgonbayar Uuye, <em>Homotopy theory 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> (<a href="http://arxiv.org/abs/1011.2926">arXiv:1011.2926</a>)</li> </ul> <p>For more along such lines see the references at <em><a class="existingWikiWord" href="/nlab/show/KK-theory">KK-theory</a></em> and <em><a class="existingWikiWord" href="/nlab/show/E-theory">E-theory</a></em>.</p> <p>Discussion 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 as <a class="existingWikiWord" href="/nlab/show/algebras+of+observables">algebras of observables</a> in <a class="existingWikiWord" href="/nlab/show/quantum+physics">quantum physics</a>/<a class="existingWikiWord" href="/nlab/show/quantum+probability+theory">quantum probability theory</a>:</p> <ul> <li id="Landsman17"><a class="existingWikiWord" href="/nlab/show/Klaas+Landsman">Klaas Landsman</a>, <em>Foundations of quantum theory – From classical concepts to Operator algebras</em>, Springer Open (2017) [<a href="https://link.springer.com/book/10.1007/978-3-319-51777-3">doi:10.1007/978-3-319-51777-3</a>, <a href="https://link.springer.com/content/pdf/10.1007%2F978-3-319-51777-3.pdf">pdf</a>]</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on November 9, 2024 at 04:24:04. 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