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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> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#definition'>Definition</a></li> <ul> <li><a href='#rings'><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>-Rings</a></li> <li><a href='#banach_algebras'>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>-algebras</a></li> </ul> <li><a href='#examples'>Examples</a></li> <ul> <li><a href='#General'>General</a></li> <li><a href='#InvolutiveHopfAlgebras'>Involutive Hopf algebras</a></li> </ul> <li><a href='#generalizations'>Generalizations</a></li> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>A <strong><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</strong> is an <a class="existingWikiWord" href="/nlab/show/associative+algebra">algebra</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> (<a class="existingWikiWord" href="/nlab/show/associative+algebra">associative</a> or <a class="existingWikiWord" href="/nlab/show/nonassociative+algebra">non-associative</a>) equipped with an <a class="existingWikiWord" href="/nlab/show/anti-involution">anti-involution</a>, meaning a map <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><mo>:</mo><mi>A</mi><mo>→</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">(-)^\ast:A \to A</annotation></semantics></math> such that</p> <ul> <li> <p>for all elements <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>∈</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">a \in A</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi><mo>∈</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">b \in A</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>a</mi><mi>b</mi><msup><mo stretchy="false">)</mo> <mo>*</mo></msup><mo>=</mo><msup><mi>b</mi> <mo>*</mo></msup><msup><mi>a</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">(a b)^\ast = b^\ast a^\ast</annotation></semantics></math></p> </li> <li> <p>for all elements <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>∈</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">a \in A</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msup><mi>a</mi> <mo>*</mo></msup><msup><mo stretchy="false">)</mo> <mo>*</mo></msup><mo>=</mo><mi>a</mi></mrow><annotation encoding="application/x-tex">(a^\ast)^\ast = a</annotation></semantics></math></p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mn>1</mn> <mo>*</mo></msup><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">1^\ast = 1</annotation></semantics></math></p> </li> </ul> <h2 id="definition">Definition</h2> <p>In more detail, begin with a <a class="existingWikiWord" href="/nlab/show/commutative+ring">commutative ring</a> (often a <a class="existingWikiWord" href="/nlab/show/field">field</a>, but possibly just a <a class="existingWikiWord" href="/nlab/show/rig">rig</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> equipped with an <a class="existingWikiWord" href="/nlab/show/involution">involution</a> (a <a class="existingWikiWord" href="/nlab/show/homomorphism">homomorphism</a> whose square is the <a class="existingWikiWord" href="/nlab/show/identity+morphism">identity</a>), written <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>↦</mo><mover><mi>x</mi><mo stretchy="false">¯</mo></mover></mrow><annotation encoding="application/x-tex">x \mapsto \bar{x}</annotation></semantics></math>. (The usual example for <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> is the field of <a class="existingWikiWord" href="/nlab/show/complex+numbers">complex numbers</a> with involution given by <a class="existingWikiWord" href="/nlab/show/complex+conjugation">complex conjugation</a>, but the concept of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>*</mo></mrow><annotation encoding="application/x-tex">*</annotation></semantics></math>-algebra makes sense in more general contexts. (An example is given by <em>any</em> commutative ring <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> with trivial involution <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>x</mi><mo stretchy="false">¯</mo></mover><mo>≔</mo><mi>x</mi></mrow><annotation encoding="application/x-tex">\bar{x} \coloneqq x</annotation></semantics></math>.)</p> <p>Then a <strong><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>-<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</strong> (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>-algebra over <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>) is 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>-<a class="existingWikiWord" href="/nlab/show/module">module</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> equipped with 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>-<a class="existingWikiWord" href="/nlab/show/bilinear+map">bilinear map</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>×</mo><mi>A</mi><mo>→</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">A\times A \to A</annotation></semantics></math>, written as multiplication (and often assumed to be <a class="existingWikiWord" href="/nlab/show/associativity">associative</a>) and 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>-<a class="existingWikiWord" href="/nlab/show/antilinear+map">antilinear map</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>→</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">A \to A</annotation></semantics></math>, written as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>↦</mo><msup><mi>x</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">x \mapsto x^*</annotation></semantics></math>, such that</p> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msup><mi>x</mi> <mo>*</mo></msup><msup><mo stretchy="false">)</mo> <mo>*</mo></msup><mo>=</mo><mi>x</mi></mrow><annotation encoding="application/x-tex"> (x^\ast)^\ast = x</annotation></semantics></math> for all <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> in <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> (so we have an <a class="existingWikiWord" href="/nlab/show/involution">involution</a> on the underlying <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>-module), and</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>x</mi><mi>y</mi><msup><mo stretchy="false">)</mo> <mo>*</mo></msup><mo>=</mo><msup><mi>y</mi> <mo>*</mo></msup><msup><mi>x</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">(x y)^* = y^* x^*</annotation></semantics></math> for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>,</mo><mi>y</mi></mrow><annotation encoding="application/x-tex">x,y</annotation></semantics></math> in <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> (so it is an <a class="existingWikiWord" href="/nlab/show/anti-involution">anti-involution</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> itself)</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mn>1</mn> <mo>*</mo></msup><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">1^\ast = 1</annotation></semantics></math>.</p> </li> </ul> <p>The hypothesis that the anti-involution is <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>-antilinear means that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>r</mi><mi>x</mi><msup><mo stretchy="false">)</mo> <mo>*</mo></msup><mo>=</mo><mover><mi>r</mi><mo>¯</mo></mover><msup><mi>x</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">(r x)^* = \overline{r} x^*</annotation></semantics></math> for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>r</mi></mrow><annotation encoding="application/x-tex">r</annotation></semantics></math> in <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> and all <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> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> (as well as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mi>y</mi><msup><mo stretchy="false">)</mo> <mo>*</mo></msup><mo>=</mo><msup><mi>x</mi> <mo>*</mo></msup><mo lspace="verythinmathspace" rspace="0em">+</mo><msup><mi>y</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">(x + y)^* = x^* + y^*</annotation></semantics></math>).</p> <p>If 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>-<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 <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 itself commutative, then it is in particular a commutative ring with involution, and one can consider <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>-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>*</mo></mrow><annotation encoding="application/x-tex">*</annotation></semantics></math>-algebras as well. On the other hand, a commutative ring with involution is simply a commutative <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 over the ring of <a class="existingWikiWord" href="/nlab/show/integers">integers</a> (with trivial involution), and similarly for rigs and <a class="existingWikiWord" href="/nlab/show/natural+numbers">natural numbers</a>.</p> <h3 id="rings"><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>-Rings</h3> <p>A <strong><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>-ring</strong> is simply 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>-algebra over the ring of <a class="existingWikiWord" href="/nlab/show/integers">integers</a> (with trivial involution). Similarly, a <strong><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>-rig</strong> 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>-algebra over the rig of <a class="existingWikiWord" href="/nlab/show/natural+numbers">natural numbers</a>.</p> <p>Arguably, when we began this article with a commutative ring <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> equipped with involution, we should have begun it with a ring with <em>anti</em>-involution instead. However, since the ring (or rig) is commutative, there is no difference.</p> <h3 id="banach_algebras">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>-algebras</h3> <p>When <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> is the field <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℂ</mi></mrow><annotation encoding="application/x-tex">\mathbb{C}</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/complex+numbers">complex numbers</a> (or the field <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">\mathbb{R}</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/real+numbers">real numbers</a>, with trivial involution), we can additionally ask that the <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 be a <a class="existingWikiWord" href="/nlab/show/Banach+algebra">Banach algebra</a>; then it is a <strong>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</strong>. Special cases of this are</p> <ul> <li> <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>-<a class="existingWikiWord" href="/nlab/show/C-star-algebra">algebras</a> (aka <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>-algebras)</p> </li> <li> <p>and <a class="existingWikiWord" href="/nlab/show/von+Neumann+algebras">von Neumann algebras</a> (aka <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>W</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">W^*</annotation></semantics></math>-algebras)</p> </li> </ul> <p>Arguably, one should require that the map <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> be an <a class="existingWikiWord" href="/nlab/show/isometry">isometry</a> (which follows already if it is required to be <a class="existingWikiWord" href="/nlab/show/short+map">short</a>); some authors require this and some don't. However, this is automatic in the case 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 (and hence also von Neumann algebras).</p> <h2 id="examples">Examples</h2> <h3 id="General">General</h3> <p> <div class='num_remark' id='TrivialStarStructure'> <h6>Example</h6> <p><strong>(trivial star-structure)</strong><br /> Any plain <a class="existingWikiWord" href="/nlab/show/algebra">algebra</a> becomes a star-algebra by equipping it with the <a class="existingWikiWord" href="/nlab/show/identity+morphism">identity</a> anti-involution.</p> </div> </p> <p> <div class='num_remark' id='ComplexConjugation'> <h6>Example</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/complex+conjugation">complex conjugation</a>)</strong><br /> The <a class="existingWikiWord" href="/nlab/show/complex+numbers">complex numbers</a> – regarded as an <a class="existingWikiWord" href="/nlab/show/associative+algebra">associative algebra</a> over the <a class="existingWikiWord" href="/nlab/show/real+numbers">real numbers</a> – form a star-algebra with anti-involution (which here is an involution, since the product is <a class="existingWikiWord" href="/nlab/show/commutative+algebra">commutative</a>) given by <a class="existingWikiWord" href="/nlab/show/complex+conjugation">complex conjugation</a>.</p> </div> </p> <p> <div class='num_remark'> <h6>Example</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/Cayley-Dickson+construction">Cayley-Dickson construction</a>)</strong> The <a class="existingWikiWord" href="/nlab/show/Cayley-Dickson+construction">Cayley-Dickson construction</a> takes any star-algebra over the <a class="existingWikiWord" href="/nlab/show/real+numbers">real numbers</a> to a new star algebra.</p> <p>Applied to the <a class="existingWikiWord" href="/nlab/show/real+numbers">real numbers</a> trivially regarded as a star-algebra (via Example <a class="maruku-ref" href="#TrivialStarStructure"></a>), this yields, successively, the star-algebras of</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/real+numbers">real numbers</a> (Example <a class="maruku-ref" href="#TrivialStarStructure"></a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/complex+numbers">complex numbers</a> (Example <a class="maruku-ref" href="#ComplexConjugation"></a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quaternions">quaternions</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/octonions">octonions</a></p> </li> </ul> <p>(which are the four <a class="existingWikiWord" href="/nlab/show/real+normed+division+algebras">real normed division algebras</a>) and then further the</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/sedenions">sedenions</a></p> </li> <li> <p>…</p> </li> </ul> <p>In each case the star-operation may be thought of as <a class="existingWikiWord" href="/nlab/show/complex+conjugation">complex conjugation</a>, given by changing the sign of the <a class="existingWikiWord" href="/nlab/show/imaginary+part">imaginary part</a> and keeping the <a class="existingWikiWord" href="/nlab/show/real+part">real part</a> intact.</p> <p></p> </div> </p> <p> <div class='num_remark'> <h6>Example</h6> <p><strong>(operators on a <a class="existingWikiWord" href="/nlab/show/Hilbert+space">Hilbert space</a>)</strong> The algebra of <a class="existingWikiWord" href="/nlab/show/continuous+operators">continuous operators</a> on a <a class="existingWikiWord" href="/nlab/show/Hilbert+space">Hilbert space</a> is a star-algebra, with the <a class="existingWikiWord" href="/nlab/show/Hermitian+adjoint">Hermitian adjoint</a> as the anti-involution.</p> </div> </p> <p> <div class='num_remark'> <h6>Example</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/C-star+algebras">C-star algebras</a>)</strong> A <em><a class="existingWikiWord" href="/nlab/show/C-star+algebra">C-star algebra</a></em> is a star-algebra where the anti-involution is compatible with the <a class="existingWikiWord" href="/nlab/show/norm">norm</a> of the underlying <a class="existingWikiWord" href="/nlab/show/Banach+algebra">Banach algebra</a>.</p> </div> </p> <h3 id="InvolutiveHopfAlgebras">Involutive Hopf algebras</h3> <p> <div class='num_remark' id='InvolutiveHopfAlgebrasAreStarAlgebras'> <h6>Example</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/involutive+Hopf+algebras">involutive Hopf algebras</a> are <a class="existingWikiWord" href="/nlab/show/star-algebras">star-algebras</a>)</strong> <br /> Any <a class="existingWikiWord" href="/nlab/show/involutive+Hopf+algebra">involutive Hopf algebra</a> is a star-algebra, with star-involution given by the <a class="existingWikiWord" href="/nlab/show/antipode">antipode</a> (by <a href="Hopf+algebra#AntipodeIsAnAntihomomorphism">this Prop.</a>).</p> </div> </p> <p> <div class='num_remark'> <h6>Example</h6> <p><strong>(groupoid algebras are star-algebras)</strong> A <a class="existingWikiWord" href="/nlab/show/group+algebra">group algebra</a> and, more generally, a <a class="existingWikiWord" href="/nlab/show/groupoid+convolution+algebra">groupoid convolution algebra</a>, is a star-algebra, with the star-<a class="existingWikiWord" href="/nlab/show/involution">involution</a> given by pullback along the <a class="existingWikiWord" href="/nlab/show/inverse">inversion</a> operation of the groupoid.</p> <p>Yet more generally, the <a class="existingWikiWord" href="/nlab/show/category+convolution+algebra">category convolution algebra</a> of a <a class="existingWikiWord" href="/nlab/show/dagger-category">dagger-category</a> 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>-algebra, with the involution being the pullback along the <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> operation.</p> <p>All these are <a class="existingWikiWord" href="/nlab/show/involutive+Hopf+algebras">involutive Hopf algebras</a> (since taking <a class="existingWikiWord" href="/nlab/show/inverses">inverses</a> and taking <a class="existingWikiWord" href="/nlab/show/dagger-category">dagger-operations</a> squares to the identity) and as such are special cases of Example <a class="maruku-ref" href="#InvolutiveHopfAlgebrasAreStarAlgebras"></a></p> </div> </p> <p> <div class='num_remark' id='StarAlgebraOfHorizontalChordDiagrams'> <h6>Example</h6> <p><strong>(star-algebra of <a class="existingWikiWord" href="/nlab/show/horizontal+chord+diagrams">horizontal chord diagrams</a>)</strong> <br /> The algebra of <a class="existingWikiWord" href="/nlab/show/horizontal+chord+diagrams">horizontal chord diagrams</a> is a star-algebra under reversal of orientation of strands (see <a href="horizontal+chord+diagram#StarAlgebraStructure">here</a>, <a href="#CSS21">CSS 21, Prop. 2.9</a>).</p> <p>Since <a class="existingWikiWord" href="/nlab/show/horizontal+chord+diagrams+are+the+homology+of+the+loop+space+of+configuration+space">horizontal chord diagrams are the homology of the loop space of configuration space</a> and the <a class="existingWikiWord" href="/nlab/show/homology+of+a+loop+space">homology of a loop space</a> is an <a class="existingWikiWord" href="/nlab/show/involutive+Hopf+algebra">involutive Hopf algebra</a>, this is a special case of Example <a class="maruku-ref" href="#InvolutiveHopfAlgebrasAreStarAlgebras"></a>.</p> </div> </p> <h2 id="generalizations">Generalizations</h2> <p>This concept could be generalized from the category of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>-modules to any <a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal category</a>:</p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>C</mi><mo>,</mo><mi>I</mi><mo>,</mo><mo>⊗</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(C, I, \otimes)</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal category</a> and let <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>e</mi><mo>:</mo><mi>I</mi><mo>→</mo><mi>A</mi><mo>,</mo><mi>π</mi><mo>:</mo><mi>A</mi><mo>⊗</mo><mi>A</mi><mo>→</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(A, e:I \to A, \pi:A \otimes A \to A)</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/unital+algebra+object">unital algebra object</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>. Then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> is a <strong><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 object</strong> <a class="existingWikiWord" href="/nlab/show/internalization">in</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> if it comes with a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ι</mi><mo>:</mo><mi>A</mi><mo>→</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">\iota:A \to A</annotation></semantics></math> such that</p> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ι</mi><mo>∘</mo><mi>ι</mi><mo>=</mo><msub><mi mathvariant="normal">id</mi> <mi>A</mi></msub></mrow><annotation encoding="application/x-tex">\iota \circ \iota = \mathrm{id}_A</annotation></semantics></math></p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ι</mi><mo>∘</mo><mi>e</mi><mo>=</mo><mi>e</mi></mrow><annotation encoding="application/x-tex">\iota \circ e = e</annotation></semantics></math></p> </li> <li> <p>for all morphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>:</mo><mi>I</mi><mo>→</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">a:I \to A</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi><mo>:</mo><mi>I</mi><mo>→</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">b:I \to A</annotation></semantics></math>,</p> </li> </ul> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ι</mi><mo>∘</mo><mi>π</mi><mo>∘</mo><mo stretchy="false">(</mo><mi>a</mi><mo>⊗</mo><mi>b</mi><mo stretchy="false">)</mo><mo>=</mo><mi>π</mi><mo>∘</mo><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mi>ι</mi><mo>∘</mo><mi>b</mi><mo stretchy="false">)</mo><mo>⊗</mo><mo stretchy="false">(</mo><mi>ι</mi><mo>∘</mo><mi>a</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\iota \circ \pi \circ (a \otimes b) = \pi \circ ((\iota \circ b) \otimes (\iota \circ a))</annotation></semantics></math></div> <p>In the <a class="existingWikiWord" href="/nlab/show/category+of+sets">category of sets</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>-algebra objects are called <strong><a class="existingWikiWord" href="/nlab/show/anti-involution">anti-involutive</a> <a class="existingWikiWord" href="/nlab/show/unital+magmas">unital magmas</a></strong>, and in the <a class="existingWikiWord" href="/nlab/show/category+of+abelian+groups">category of abelian groups</a>, associative <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 objects are called <strong><a class="existingWikiWord" href="/nlab/show/%2A-rings">*-rings</a></strong>.</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/real+part">real part</a>, <a class="existingWikiWord" href="/nlab/show/imaginary+part">imaginary part</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/anti-linear+map">anti-linear map</a>, <a class="existingWikiWord" href="/nlab/show/anti-dual+linear+space">anti-dual linear space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hermitian+form">Hermitian form</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/state+on+a+star-algebra">state on a star-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/star-representation">star-representation</a></p> </li> </ul> <div> <p><strong><a class="existingWikiWord" href="/nlab/show/quantum+probability+theory">quantum probability theory</a> – <a class="existingWikiWord" href="/nlab/show/observables">observables</a> and <a class="existingWikiWord" href="/nlab/show/states">states</a></strong></p> <ul> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/states">states</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/classical+state">classical state</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+state">quantum state</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/space+of+states+%28in+geometric+quantization%29">space of states (in geometric quantization)</a></p> </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/quasi-state">quasi-state</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/qbit">qbit</a>, <a class="existingWikiWord" href="/nlab/show/Bell+state">Bell state</a></p> <p><a class="existingWikiWord" href="/nlab/show/dimer">dimer</a>, <a class="existingWikiWord" href="/nlab/show/tensor+network+state">tensor network state</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+state+preparation">quantum state preparation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/probability+amplitude">probability amplitude</a>, <a class="existingWikiWord" href="/nlab/show/quantum+fluctuation">quantum fluctuation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pure+state">pure state</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/wave+function">wave function</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/bra-ket">bra-ket</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bell+state">Bell state</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+superposition">quantum superposition</a>, <a class="existingWikiWord" href="/nlab/show/quantum+interference">quantum interference</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+entanglement">quantum entanglement</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+measurement">quantum measurement</a></p> <p><a class="existingWikiWord" href="/nlab/show/wave+function+collapse">wave function collapse</a></p> <p><a class="existingWikiWord" href="/nlab/show/Born+rule">Born rule</a></p> <p><a class="existingWikiWord" href="/nlab/show/deferred+measurement+principle">deferred measurement principle</a></p> <p><a class="existingWikiWord" href="/nlab/show/quantum+reader+monad">quantum reader monad</a></p> <p><a class="existingWikiWord" href="/nlab/show/measurement+problem">measurement problem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/superselection+sector">superselection sector</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> <p><a class="existingWikiWord" href="/nlab/show/entanglement+entropy">entanglement entropy</a></p> <p><a class="existingWikiWord" href="/nlab/show/holographic+entanglement+entropy">holographic entanglement entropy</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/coherent+quantum+state">coherent quantum state</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/ground+state">ground state</a>, <a class="existingWikiWord" href="/nlab/show/excited+state">excited state</a></p> </li> <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/Fock+space">Fock space</a>, <a class="existingWikiWord" href="/nlab/show/second+quantization">second quantization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/vacuum">vacuum</a>, <a class="existingWikiWord" href="/nlab/show/vacuum+state">vacuum state</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Hadamard+state">Hadamard state</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/vacuum+diagram">vacuum diagram</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/vacuum+expectation+value">vacuum expectation value</a>, <a class="existingWikiWord" href="/nlab/show/vacuum+amplitude">vacuum amplitude</a>, <a class="existingWikiWord" href="/nlab/show/vacuum+fluctuation">vacuum fluctuation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/vacuum+energy">vacuum energy</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/vacuum+polarization">vacuum polarization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/interacting+vacuum">interacting vacuum</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/thermal+vacuum">thermal vacuum</a>, <a class="existingWikiWord" href="/nlab/show/KMS+state">KMS state</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/vacuum+stability">vacuum stability</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/false+vacuum">false vacuum</a>, <a class="existingWikiWord" href="/nlab/show/tachyon">tachyon</a>, <a class="existingWikiWord" href="/nlab/show/Coleman-De+Luccia+instanton">Coleman-De Luccia instanton</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/theta+vacuum">theta vacuum</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/perturbative+string+theory+vacuum">perturbative string theory vacuum</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/non-geometric+string+theory+vacuum">non-geometric string theory vacuum</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/landscape+of+string+theory+vacua">landscape of string theory vacua</a></p> </li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/entangled+state">entangled state</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/tensor+network+state">tensor network state</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/matrix+product+state">matrix product state</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/tree+tensor+network+state">tree tensor network state</a></p> </li> </ul> </li> </ul> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/observables">observables</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+observable">quantum observable</a>, <a class="existingWikiWord" href="/nlab/show/beable">beable</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/algebra+of+observables">algebra of observables</a>, <a class="existingWikiWord" href="/nlab/show/star-algebra">star-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bohr+topos">Bohr topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+operator+%28in+geometric+quantization%29">quantum operator (in geometric quantization)</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+operation">quantum operation</a>, <a class="existingWikiWord" href="/nlab/show/quantum+effect">quantum effect</a>, <a class="existingWikiWord" href="/nlab/show/effect+algebra">effect algebra</a></p> </li> <li> <p>in <a class="existingWikiWord" href="/nlab/show/quantum+field+theory">quantum field theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/local+observable">local observable</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/polynomial+observable">polynomial observable</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/linear+observable">linear observable</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/field+observable">field observable</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/regular+observable">regular observable</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/microcausal+observable">microcausal observable</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/normal-ordered+product">normal-ordered product</a>, <a class="existingWikiWord" href="/nlab/show/time-ordered+products">time-ordered products</a>, <a class="existingWikiWord" href="/nlab/show/retarded+product">retarded product</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Wick+algebra">Wick algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/scattering+amplitude">scattering amplitude</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/interacting+field+algebra+of+observables">interacting field algebra of observables</a>, <a class="existingWikiWord" href="/nlab/show/Bogoliubov%27s+formula">Bogoliubov's formula</a></p> </li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/GNS+construction">GNS construction</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/theorems">theorems</a></p> <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/Gleason%27s+theorem">Gleason's theorem</a></p> </li> <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> <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/Nuiten%27s+lemma">Nuiten's lemma</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Wigner%27s+theorem">Wigner's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/no-cloning+theorem">no-cloning theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bell%27s+theorem">Bell's theorem</a></p> </li> </ul> </li> </ul> </div> <h2 id="references">References</h2> <p>See also:</p> <ul> <li>Wikipedia, <em><a href="https://en.wikipedia.org/wiki/*-algebra">star-algebra</a></em></li> </ul> <p>Discussion for <a class="existingWikiWord" href="/nlab/show/group+algebras">group algebras</a>:</p> <ul> <li>Antonio Giambruno, Cesar Polcino Milies and Sudarshan K. Sehgal, <em>Star-group identities on units of group algebras: The non-torsion case</em>, Forum Mathematicum Volume 30 Issue 1 (<a href="https://doi.org/10.1515/forum-2016-0266">doi:10.1515/forum-2016-0266</a>)</li> </ul> <p>Discussion of the Example <a class="maruku-ref" href="#StarAlgebraOfHorizontalChordDiagrams"></a> of <a class="existingWikiWord" href="/nlab/show/horizontal+chord+diagrams">horizontal chord diagrams</a>:</p> <ul> <li id="CSS21"><a class="existingWikiWord" href="/nlab/show/David+Corfield">David Corfield</a>, <a class="existingWikiWord" href="/nlab/show/Hisham+Sati">Hisham Sati</a>, <a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>: <em><a class="existingWikiWord" href="/schreiber/show/Fundamental+weight+systems+are+quantum+states">Fundamental weight systems are quantum states</a></em> (<a href="https://arxiv.org/abs/2105.02871">arXiv:2105.02871</a>)</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on August 21, 2024 at 01:59:59. See the <a href="/nlab/history/star-algebra" style="color: #005c19">history</a> of this page for a list of all contributions to it. </p> </div> <div class="navigation navfoot"> <a href="/nlab/edit/star-algebra" accesskey="E" class="navlink" id="edit" rel="nofollow">Edit</a><a href="https://nforum.ncatlab.org/discussion/12868/#Item_5">Discuss</a><span class="backintime"><a href="/nlab/revision/star-algebra/23" accesskey="B" class="navlinkbackintime" id="to_previous_revision" rel="nofollow">Previous revision</a></span><a href="/nlab/show/diff/star-algebra" accesskey="C" class="navlink" id="see_changes" rel="nofollow">Changes from previous revision</a><a href="/nlab/history/star-algebra" accesskey="S" class="navlink" id="history" rel="nofollow">History (23 revisions)</a> <a href="/nlab/show/star-algebra/cite" style="color: black">Cite</a> <a href="/nlab/print/star-algebra" accesskey="p" id="view_print" rel="nofollow">Print</a> <a href="/nlab/source/star-algebra" id="view_source" rel="nofollow">Source</a> </div> </div>  </div>  </body> </html>