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A [[Banach algebra|Banach algebra]] <img align="absmiddle"  border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c0200202.png" /> over  the field of complex numbers, with an involution <img  align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c0200203.png" />, <img  align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c0200204.png" />, such that the  norm and the involution are connected by the relation <img  align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c0200205.png" /> for any element  <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c0200206.png" />. <img  align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c0200207.png" />-algebras were  introduced in 1943 [[#References|[1]]] under the name of totally regular  rings; they are also known under the name of <img align="absmiddle"  border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c0200209.png"  />-algebras. The most important examples of <img align="absmiddle"  border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c02002011.png"  />-algebras are:
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1) The algebra <img  align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c02002012.png" /> of continuous  complex-valued functions on a locally compact Hausdorff space <img  align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c02002013.png" /> which tend  towards zero at infinity (i.e. continuous functions <img  align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c02002014.png" /> on <img  align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c02002015.png" /> such that, for  any <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c02002016.png" />, the set of  points <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c02002017.png" /> which satisfy the  condition <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c02002018.png" /> is compact in  <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c02002019.png" />); <img  align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c02002020.png" /> has the uniform  norm
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<table class="eq" style="width:100%;">  <tr><td valign="top"  style="width:94%;text-align:center;"><img align="absmiddle"  border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c02002021.png"  /></td> </tr></table>
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The  involution in <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c02002022.png" /> is defined as  transition to the complex-conjugate function: <img align="absmiddle"  border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c02002023.png" />. Any  commutative <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c02002024.png" />-algebra <img  align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c02002025.png" /> is isometrically  and symmetrically isomorphic (i.e. is isomorphic as a Banach algebra  <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c02002026.png" /> with involution)  to the <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c02002027.png" />-algebra <img  align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c02002028.png" />, where <img  align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c02002029.png" /> is the space of  maximal ideals of <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c02002030.png" /> endowed with the  Gel'fand topology [[#References|[1]]], [[#References|[2]]],  [[#References|[3]]].
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2) The algebra <img  align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c02002031.png" /> of all bounded  linear operators on a Hilbert space <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c02002032.png" />, considered with  respect to the ordinary linear operations and operator multiplication.  The involution in <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c02002033.png" /> is defined as  transition to the adjoint operator, and the norm is defined as the  ordinary operator norm.
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A subset <img  align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c02002034.png" /> is said to be  self-adjoint if <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c02002036.png" />, where <img  align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c02002037.png" />. Any closed  self-adjoint subalgebra <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c02002038.png" /> of a <img  align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c02002039.png" />-algebra <img  align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c02002040.png" /> is a <img  align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c02002041.png" />-algebra with  respect to the linear operations, multiplication, involution, and norm  taken from <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c02002042.png" />; <img  align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c02002043.png" /> is said to be a  <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c02002044.png" />-subalgebra of  <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c02002045.png" />. Any <img  align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c02002046.png" />-algebra is  isometrically and symmetrically isomorphic to a <img  align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c02002047.png" />-subalgebra of  some <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c02002048.png" />-algebra of the  form <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c02002049.png" />. Any closed  two-sided ideal <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c02002050.png" /> in a <img  align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c02002051.png" />-algebra is  self-adjoint (thus <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c02002052.png" /> is a <img  align="absm
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iddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c02002053.png" />-subalgebra of  <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c02002054.png" />), and the  quotient algebra <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c02002055.png" />, endowed with the  natural linear operations, multiplication, involution, and quotient  space norm, is a <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c02002056.png" />-algebra. The set  <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c02002057.png" /> of  completely-continuous linear operators on a Hilbert space <img  align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c02002058.png" /> is a closed  two-sided ideal in <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c02002059.png" />. If <img  align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c02002060.png" /> is a <img  align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c02002061.png" />-algebra and  <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c02002062.png" /> is the algebra  with involution obtained from <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c02002063.png" /> by addition of a  unit element, there exists a unique norm on <img align="absmiddle"  border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c02002064.png" /> which  converts <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c02002065.png" /> into a <img  align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c02002066.png" />-algebra and which  extends the norm on <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c02002067.png" />. Moreover, the  operations of bounded direct sum and tensor product [[#References|[3]]],  [[#References|[4]]] have been defined for <img align="absmiddle"  border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c02002068.png"  />-algebras.
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As in all symmetric Banach algebras  with involution, in a <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c02002069.png" />-algebra <img  align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c02002070.png" /> it is possible to  define the following subsets: the real linear space <img  align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c02002071.png" /> of Hermitian  elements; the set of normal elements; the multiplicative group <img  align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c02002072.png" /> of unitary  elements (if <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c02002073.png" /> contains a unit  element); and the set <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c02002074.png" /> of positive  elements. The set <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c02002075.png" /> is a closed cone  in <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c02002076.png" />, <img  align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c02002077.png" />, <img  align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c02002078.png" />, and the cone  <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c02002079.png" /> converts <img  align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c02002080.png" /> into a real  ordered vector space. If <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c02002081.png" /> contains a unit  element 1, then 1 is an interior point of the cone <img  align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c02002082.png" />. A linear  functional <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c02002083.png" /> on <img  align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c02002084.png" /> is called  positive if <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c02002085.png" /> fo
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r all <img  align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c02002086.png" />; such a  functional is continuous. If <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c02002087.png" />, where <img  align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c02002088.png" /> is a <img  align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c02002089.png" />-subalgebra of  <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c02002090.png" />, the spectrum of  <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c02002091.png" /> in <img  align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c02002092.png" /> coincides with  the spectrum of <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c02002093.png" /> in <img  align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c02002094.png" />. The spectrum of a  Hermitian element is real, the spectrum of a unitary element lies on  the unit circle, and the spectrum of a positive element is non-negative.  A functional calculus for the normal elements of a <img  align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c02002095.png" />-algebra has been  constructed. Any <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c02002096.png" />-algebra <img  align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c02002097.png" /> has an  approximate unit, located in the unit ball of <img align="absmiddle"  border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c02002098.png" /> and  formed by positive elements of <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c02002099.png" />. If <img  align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c020020100.png" /> are closed  two-sided ideals in <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c020020101.png" />, then <img  align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c020020102.
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png" /> is a closed  two-sided ideal in <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c020020103.png" /> and <img  align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c020020104.png" />. If <img  align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c020020105.png" /> is a closed  two-sided ideal in <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c020020106.png" /> and <img  align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c020020107.png" /> is a closed  two-sided ideal in <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c020020108.png" />, then <img  align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c020020109.png" /> is a closed  two-sided ideal in <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c020020110.png" />. Any closed  two-sided ideal is the intersection of the primitive two-sided ideals in  which it is contained; any closed left ideal in <img  align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c020020111.png" /> is the  intersection of the maximal regular left ideals in which it is  contained.
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Any *-isomorphism of a <img  align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c020020112.png" />-algebra is  isometric. Any *-isomorphism <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c020020113.png" /> of a Banach  algebra <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c020020114.png" /> with involution  into a <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c020020115.png" />-algebra <img  align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c020020116.png" /> is continuous,  and <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c020020117.png" /> for all <img  align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c020020118.png" />. In particular,  all representations of a Banach algebra with involution (i.e. all  *-homomorphism of <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c020020119.png" /> into a <img  align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c020020120.png" />-algebra of the  form <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c020020121.png" />) are continuous.  The theory of representations of <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c020020122.png" />-algebras forms a  significant part of the theory of <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c020020123.png" />-algebras, and  the applications of the theory of <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c020020124.png" />-algebras are  related to the theory of representations of <img align="absmiddle"  border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c020020125.png"  />-algebras. The properties of representations of <img  align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c020020126.png" />-algebras make it  possible to construct for each <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c020020127.png" />-algeb
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ra <img  align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c020020128.png" /> a topological  space <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c020020129.png" />, called the  spectrum of the <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c020020131.png" />-algebra <img  align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c020020132.png" />, and to endow  this space with a [[Mackey–Borel structure|Mackey–Borel structure]]. In  the general case, the spectrum of a <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c020020133.png" />-algebra does  not satisfy any separation axiom, but is a locally compact [[Baire  space|Baire space]].
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A <img align="absmiddle"  border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c020020134.png"  />-algebra <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c020020135.png" /> is said to be a  CCR-algebra (respectively, a GCR-algebra) if the relation <img  align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c020020136.png" /> (respectively,  <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c020020137.png" />) is satisfied  for any non-null irreducible representation <img align="absmiddle"  border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c020020138.png" /> of  the <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c020020139.png" />-algebra <img  align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c020020140.png" /> in a Hilbert  space <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c020020141.png" />.
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A  <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c020020142.png" />-algebra <img  align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c020020143.png" /> is said to be an  NGCR-algebra if <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c020020144.png" /> does not contain  non-zero closed two-sided <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c020020145.png" />-ideals (i.e.  ideals which are <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c020020146.png" />-algebras). Any  <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c020020147.png" />-algebra contains  a maximal two-sided <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c020020148.png" />-ideal <img  align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c020020149.png" />, and the  quotient algebra <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c020020150.png" /> is an <img  align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c020020151.png" />-algebra. Any  <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c020020152.png" />-algebra contains  an increasing family of closed two-sided ideals <img  align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c020020153.png" />, indexed by  ordinals <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c020020154.png" />, <img  align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c020020155.png" />, such that  <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c020020156.png" />, <img  align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c020020157.png" />, <img  align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c020020158.png" /> is a <img  align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c020020159.png" />-algebra for all  <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/
 +
c020/c020020/c020020160.png" />, and <img  align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c020020161.png" /> for limit  ordinals <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c020020162.png" />. The spectrum of  a <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c020020163.png" />-algebra contains  an open, everywhere-dense, separable, locally compact subset.
 +
 
 +
A  <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c020020164.png" />-algebra <img  align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c020020165.png" /> is said to be a  <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c020020168.png" />-algebra of type I  if, for any representation <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c020020169.png" /> of the <img  align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c020020170.png" />-algebra <img  align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c020020171.png" /> in a Hilbert  space <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c020020172.png" />, the [[Von  Neumann algebra|von Neumann algebra]] generated by the family <img  align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c020020173.png" /> in <img  align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c020020174.png" /> is a type I von  Neumann algebra. For a <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c020020175.png" />-algebra, the  following conditions are equivalent: a) <img align="absmiddle"  border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c020020176.png" /> is a  <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c020020177.png" />-algebra of type  I; b) <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c020020178.png" /> is a <img  align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c020020179.png" />-algebra; and c)  any quotient representation of the <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c020020180.png" />-algebra <img  align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c020020181.png" /> is a multiple of  the irreducible representation. If <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c020020182.png" /> satisfies these  conditions, then: 1) two irreducible representations
 +
of the <img  align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c020020183.png" />-algebra <img  align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c020020184.png" /> are equivalent  if and only if their kernels are identical; and 2) the spectrum of the  <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c020020185.png" />-algebra <img  align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c020020186.png" /> is a <img  align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c020020187.png" />-space. If  <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c020020188.png" /> is a separable  <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c020020189.png" />-algebra, each of  the conditions 1) and 2) is equivalent to the conditions a)–c). In  particular, each separable <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c020020190.png" />-algebra with a  unique (up to equivalence) irreducible representation, is isomorphic to  the <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c020020191.png" />-algebra <img  align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c020020192.png" /> for some Hilbert  space <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c020020193.png" />.
 +
 
 +
Let  <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c020020194.png" /> be a <img  align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c020020195.png" />-algebra, and let  <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c020020196.png" /> be a set of  elements <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c020020197.png" /> such that the  function <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c020020198.png" /> is finite and  continuous on the spectrum of <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c020020199.png" />. If the linear  envelope of <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c020020200.png" /> is everywhere  dense in <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c020020201.png" />, then <img  align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c020020202.png" /> is said to be a  <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c020020204.png" />-algebra with  continuous trace. The spectrum of such a <img align="absmiddle"  border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c020020205.png"  />-algebra is separable and, under certain additional conditions, a  <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c020020206.png" />-algebra with a  continuous trace may be represented as the algebra of vector functions  on its spectrum <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c020020207.png" />  [[#References|[3]]].
 +
 
 +
Let <img align="absmiddle"  border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c020020208.png" /> be a  <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c020020209.png" />-algebra, let  <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c020020210.png" /> be the set of  positive linear functionals on <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c020020211.png" /> with norm  <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c020020212.png" /> and let <img  align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c020020213.png" /> be the set of  non-zero boundary points of the convex set <img align="absmiddle"  border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c020020214.png" />. Then  <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c020020215.png" /> will be the set  of pure states of <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c020020216.png" />. Let <img  align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c020020217.png" /> be a <img  align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c020020218.png" />-subalgebra of  <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c020020219.png" />. If <img  align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c020020220.png" /> is a <img  align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c020020221.png" />-algebra and if  <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c020020222.png" /> separates the  points of the set <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c020020223.png" />, i.e. for any  <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c020020224.png" />, <img  align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c020020225.png" />, there exists an  <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c020020226.png" /> such that  <img
 +
align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c020020227.png" />, then <img  align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c020020228.png" /> (the  Stone–Weierstrass theorem). If <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c020020229.png" /> is any <img  align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c020020230.png" />-algebra and  <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c020020231.png" /> separates the  points of the set <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c020020232.png" />, then <img  align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c020020233.png" />.
 +
 
 +
The  second dual space <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c020020234.png" /> of a <img  align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c020020235.png" />-algebra <img  align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c020020236.png" /> is obviously  provided with a multiplication converting <img align="absmiddle"  border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c020020237.png" /> into a  <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c020020238.png" />-algebra  isomorphic to some von Neumann algebra; this algebra is named the von  Neumann algebra enveloping the <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c020020240.png" />-algebra  [[#References|[3]]], [[#References|[4]]].
 +
 
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The theory of  <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c020020241.png" />-algebras has  numerous applications in the theory of representations of groups and  symmetric algebras [[#References|[3]]], the theory of dynamical systems  [[#References|[4]]], statistical physics and quantum field theory  [[#References|[5]]], and also in the theory of operators on a Hilbert  space [[#References|[6]]].
 +
 
 +
====References====
 +
<table><TR><TD  valign="top">[1]</TD> <TD valign="top"> I.M. Gel'fand,  M.A. [M.A. Naimark] Neumark, "On the imbedding of normed rings in the  rings of operators in Hilbert space" ''Mat. Sb.'' , '''12 (54)''' : 2  (1943) pp. 197–213 {{MR|9426}} {{ZBL|}}  </TD></TR><TR><TD valign="top">[2]</TD>  <TD valign="top"> M.A. Naimark, "Normed rings" , Reidel (1984)  (Translated from Russian) {{MR|1292007}} {{MR|0355601}} {{MR|0355602}}  {{MR|0205093}} {{MR|0110956}} {{MR|0090786}} {{MR|0026763}}  {{ZBL|0218.46042}} {{ZBL|0137.31703}} {{ZBL|0089.10102}}  {{ZBL|0073.08902}} </TD></TR><TR><TD  valign="top">[3]</TD> <TD valign="top"> J. Dixmier,  "<img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c020020242.png" /> algebras" ,  North-Holland (1977) (Translated from French) {{MR|0498740}}  {{MR|0458185}} {{ZBL|0372.46058}} {{ZBL|0346.17010}} {{ZBL|0339.17007}}  </TD></TR><TR><TD valign="top">[4]</TD>  <TD valign="top"> S. Sakai, "<img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c020020243.png" />-algebras and  <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c020020244.png" />-algebras" ,  Springer (1971) {{MR|0442701}} {{MR|0399878}} {{MR|0318902}}  {{MR|0293415}} {{MR|0293414}} {{ZBL|}}  </TD></TR><TR><TD valign="top">[5]</TD>  <TD valign="top"> D. Ruelle, "Statistical mechanics: rigorous  results" , Benjamin (1974) {{MR|0289084}} {{ZBL|0997.82506}}  {{ZBL|1016.82500}} {{ZBL|0177.57301}}  </TD></TR><TR><TD valign="top">[6]</TD>  <TD valign="top"> R.G. Douglas, "Banach algebra techniques in  operator theory" , Acad. Press (1972) {{MR|0361893}} {{ZBL|0247.47001}}  </TD></TR></table>
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====Comments====
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If  <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c020020245.png" /> over <img  align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c020020246.png" /> is an algebra  with involution, i.e. if there is an operation <img align="absmiddle"  border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c020020247.png" />  satisfying <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c020020248.png" />, <img  align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c020020249.png" />, <img  align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c020020250.png" />, the Hermitian,  normal and positive elements are defined as follows. The element <img  align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c020020251.png" /> is a Hermitian  element if <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c020020252.png" />; it is a normal  element if <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c020020253.png" /> and it is a  positive element if <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c020020254.png" /> for some <img  align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c020020255.png" />. An element  <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c020020256.png" /> is a unitary  element if <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c020020257.png" />. An algebra with  involution is also sometimes called a symmetric algebra (or symmetric  ring), cf., e.g., [[#References|[2]]]. However, this usage conflicts  with the concept of a symmetric algebra as a special kind of Frobenius  algebra, cf. [[Frobenius algebra|Frobenius algebra]].
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Recent  discoveries have revealed connections with, and applications to,  [[Algebraic topology|algebraic topology]]. If <img align="absmiddle"  border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c020020258.png" /> is a  compact metrizable space, a group, <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c020020259.png" />, can be formed  from <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c020020260.png" />-extensions of  the compact operators by <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c020020261.png" />,
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<table  class="eq" style="width:100%;"> <tr><td valign="top"  style="width:94%;text-align:center;"><img align="absmiddle"  border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c020020262.png"  /></td> </tr></table>
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In  [[#References|[a3]]], <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c020020263.png" /> is shown to be a  homotopy invariant functor of <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c020020264.png" /> which may be  identified with the topological <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c020020265.png" />-homology group,  <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c020020266.png" />. In  [[#References|[a1]]] M.F. Atiyah attempted to make a description of  <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c020020267.png" />-homology,  <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c020020268.png" />, in terms of  elliptic operators [[#References|[a5]]], p. 58. In [[#References|[a7]]],  [[#References|[a8]]] G.G. Kasparov developed a solution to this  problem. Kasparov and others have used the equivariant version of  Kasparov <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c020020270.png" />-theory to prove  the strong Novikov conjecture on higher signatures in many cases (see  [[#References|[a2]]], pp. 309-314).
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In addition, deep  and novel connections between [[K-theory|<img align="absmiddle"  border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c020020271.png"  />-theory]] and operator algebras (cf. [[Operator ring|Operator  ring]]) were recently discovered by A. Connes [[#References|[a4]]].  Finally, V.F.R. Jones [[#References|[a6]]] has exploited operator  algebras to provide invariants of topological knots (cf. [[Knot  theory|Knot theory]]).
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Further details on recent developments may be found in [[#References|[a2]]], [[#References|[a5]]].
 +
 
 +
====References====
 +
<table><TR><TD  valign="top">[a1]</TD> <TD valign="top"> M.F. Atiyah,  "Global theory of elliptic operators" , ''Proc. Internat. Conf. Funct.  Anal. Related Topics'' , Univ. Tokyo Press (1970) {{MR|0266247}}  {{ZBL|0193.43601}} </TD></TR><TR><TD  valign="top">[a2]</TD> <TD valign="top"> B. Blackadar,  "<img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c020020272.png" />-theory for  operator algebras" , Springer (1986) {{MR|0859867}} {{ZBL|0597.46072}}  </TD></TR><TR><TD valign="top">[a3]</TD>  <TD valign="top"> L.G. Brown, R.G. Douglas, P.A. Filmore,  "Extensions of <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c020020273.png" />-algebras and  <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c020020274.png" />-homology" ''Ann.  of Math. (2)'' , '''105''' (1977) pp.  265–324</TD></TR><TR><TD  valign="top">[a4]</TD> <TD valign="top"> A. Connes,  "Non-commutative differential geometry" ''Publ. Math. IHES'' , '''62'''  (1986) pp. 257–360 {{MR|}} {{ZBL|0657.55006}} {{ZBL|0592.46056}}  {{ZBL|0564.58002}} </TD></TR><TR><TD  valign="top">[a5]</TD> <TD valign="top"> R.G. Douglas,  "<img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c020020275.png" />-algebra  extensions and <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c020020276.png" />-homology" ,  Princeton Univ. Press (1980) {{MR|0571362}} {{ZBL|}}  </TD></TR><TR><TD valign="top">[a6]</TD>  <TD valign="top"> V.F.R. Jones, "A polynomial invariant for knots  via von Neumann algebras" ''Bull. Amer. Math. Soc.'' , '''12''' (1985)  pp. 103–111 {{MR|0766964}} {{ZBL|0564.57006}}  </TD></TR><TR><TD valign="top">[a7]</TD>  <TD valign="top"> G.G. Kasparov, "The generalized index of  elliptic operators" ''Funct. Anal. and Its Appl.'' , '''7''' (1973) pp.  238–240 ''Funkt. Anal. i Prilozhen.'' , '''7''' (1973) pp. 82–83  {{MR|445561}} {{ZBL|0305.58017}} </TD></TR><TR><TD  valign="top">[a8]</TD> <TD valign="top"> G.G. Kasparov,  "To
 +
pological invariants of elliptic operators I. <img  align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020020/c020020277.png" />-homology"  ''Math. USSR-Izv.'' , '''9''' (1975) pp. 751–792 ''Izv. Akad. Nauk  SSSR'' , '''4''' (1975) pp. 796–838 {{MR|488027}} {{ZBL|}}  </TD></TR><TR><TD valign="top">[a9]</TD>  <TD valign="top"> M. Takesaki, "Theory of operator algebras" ,  '''1''' , Springer (1979) {{MR|0548728}} {{ZBL|0436.46043}}  </TD></TR></table>

Revision as of 16:38, 20 April 2012

A Banach algebra over the field of complex numbers, with an involution , , such that the norm and the involution are connected by the relation for any element . -algebras were introduced in 1943 [1] under the name of totally regular rings; they are also known under the name of -algebras. The most important examples of -algebras are:

1) The algebra of continuous complex-valued functions on a locally compact Hausdorff space which tend towards zero at infinity (i.e. continuous functions on such that, for any , the set of points which satisfy the condition is compact in ); has the uniform norm

The involution in is defined as transition to the complex-conjugate function: . Any commutative -algebra is isometrically and symmetrically isomorphic (i.e. is isomorphic as a Banach algebra with involution) to the -algebra , where is the space of maximal ideals of endowed with the Gel'fand topology [1], [2], [3].

2) The algebra of all bounded linear operators on a Hilbert space , considered with respect to the ordinary linear operations and operator multiplication. The involution in is defined as transition to the adjoint operator, and the norm is defined as the ordinary operator norm.

A subset is said to be self-adjoint if , where . Any closed self-adjoint subalgebra of a -algebra is a -algebra with respect to the linear operations, multiplication, involution, and norm taken from ; is said to be a -subalgebra of . Any -algebra is isometrically and symmetrically isomorphic to a -subalgebra of some -algebra of the form . Any closed two-sided ideal in a -algebra is self-adjoint (thus is a -subalgebra of ), and the quotient algebra , endowed with the natural linear operations, multiplication, involution, and quotient space norm, is a -algebra. The set of completely-continuous linear operators on a Hilbert space is a closed two-sided ideal in . If is a -algebra and is the algebra with involution obtained from by addition of a unit element, there exists a unique norm on which converts into a -algebra and which extends the norm on . Moreover, the operations of bounded direct sum and tensor product [3], [4] have been defined for -algebras.

As in all symmetric Banach algebras with involution, in a -algebra it is possible to define the following subsets: the real linear space of Hermitian elements; the set of normal elements; the multiplicative group of unitary elements (if contains a unit element); and the set of positive elements. The set is a closed cone in , , , and the cone converts into a real ordered vector space. If contains a unit element 1, then 1 is an interior point of the cone . A linear functional on is called positive if fo r all ; such a functional is continuous. If , where is a -subalgebra of , the spectrum of in coincides with the spectrum of in . The spectrum of a Hermitian element is real, the spectrum of a unitary element lies on the unit circle, and the spectrum of a positive element is non-negative. A functional calculus for the normal elements of a -algebra has been constructed. Any -algebra has an approximate unit, located in the unit ball of and formed by positive elements of . If are closed two-sided ideals in , then is a closed two-sided ideal in and . If is a closed two-sided ideal in and is a closed two-sided ideal in , then is a closed two-sided ideal in . Any closed two-sided ideal is the intersection of the primitive two-sided ideals in which it is contained; any closed left ideal in is the intersection of the maximal regular left ideals in which it is contained.

Any *-isomorphism of a -algebra is isometric. Any *-isomorphism of a Banach algebra with involution into a -algebra is continuous, and for all . In particular, all representations of a Banach algebra with involution (i.e. all *-homomorphism of into a -algebra of the form ) are continuous. The theory of representations of -algebras forms a significant part of the theory of -algebras, and the applications of the theory of -algebras are related to the theory of representations of -algebras. The properties of representations of -algebras make it possible to construct for each -algeb ra a topological space , called the spectrum of the -algebra , and to endow this space with a Mackey–Borel structure. In the general case, the spectrum of a -algebra does not satisfy any separation axiom, but is a locally compact Baire space.

A -algebra is said to be a CCR-algebra (respectively, a GCR-algebra) if the relation (respectively, ) is satisfied for any non-null irreducible representation of the -algebra in a Hilbert space .

A -algebra is said to be an NGCR-algebra if does not contain non-zero closed two-sided -ideals (i.e. ideals which are -algebras). Any -algebra contains a maximal two-sided -ideal , and the quotient algebra is an -algebra. Any -algebra contains an increasing family of closed two-sided ideals , indexed by ordinals , , such that , , is a -algebra for all , and for limit ordinals . The spectrum of a -algebra contains an open, everywhere-dense, separable, locally compact subset.

A -algebra is said to be a -algebra of type I if, for any representation of the -algebra in a Hilbert space , the von Neumann algebra generated by the family in is a type I von Neumann algebra. For a -algebra, the following conditions are equivalent: a) is a -algebra of type I; b) is a -algebra; and c) any quotient representation of the -algebra is a multiple of the irreducible representation. If satisfies these conditions, then: 1) two irreducible representations of the -algebra are equivalent if and only if their kernels are identical; and 2) the spectrum of the -algebra is a -space. If is a separable -algebra, each of the conditions 1) and 2) is equivalent to the conditions a)–c). In particular, each separable -algebra with a unique (up to equivalence) irreducible representation, is isomorphic to the -algebra for some Hilbert space .

Let be a -algebra, and let be a set of elements such that the function is finite and continuous on the spectrum of . If the linear envelope of is everywhere dense in , then is said to be a -algebra with continuous trace. The spectrum of such a -algebra is separable and, under certain additional conditions, a -algebra with a continuous trace may be represented as the algebra of vector functions on its spectrum [3].

Let be a -algebra, let be the set of positive linear functionals on with norm and let be the set of non-zero boundary points of the convex set . Then will be the set of pure states of . Let be a -subalgebra of . If is a -algebra and if separates the points of the set , i.e. for any , , there exists an such that , then (the Stone–Weierstrass theorem). If is any -algebra and separates the points of the set , then .

The second dual space of a -algebra is obviously provided with a multiplication converting into a -algebra isomorphic to some von Neumann algebra; this algebra is named the von Neumann algebra enveloping the -algebra [3], [4].

The theory of -algebras has numerous applications in the theory of representations of groups and symmetric algebras [3], the theory of dynamical systems [4], statistical physics and quantum field theory [5], and also in the theory of operators on a Hilbert space [6].

References

[1] I.M. Gel'fand, M.A. [M.A. Naimark] Neumark, "On the imbedding of normed rings in the rings of operators in Hilbert space" Mat. Sb. , 12 (54) : 2 (1943) pp. 197–213 MR9426
[2] M.A. Naimark, "Normed rings" , Reidel (1984) (Translated from Russian) MR1292007 MR0355601 MR0355602 MR0205093 MR0110956 MR0090786 MR0026763 Zbl 0218.46042 Zbl 0137.31703 Zbl 0089.10102 Zbl 0073.08902
[3] J. Dixmier, " algebras" , North-Holland (1977) (Translated from French) MR0498740 MR0458185 Zbl 0372.46058 Zbl 0346.17010 Zbl 0339.17007
[4] S. Sakai, "-algebras and -algebras" , Springer (1971) MR0442701 MR0399878 MR0318902 MR0293415 MR0293414
[5] D. Ruelle, "Statistical mechanics: rigorous results" , Benjamin (1974) MR0289084 Zbl 0997.82506 Zbl 1016.82500 Zbl 0177.57301
[6] R.G. Douglas, "Banach algebra techniques in operator theory" , Acad. Press (1972) MR0361893 Zbl 0247.47001


Comments

If over is an algebra with involution, i.e. if there is an operation satisfying , , , the Hermitian, normal and positive elements are defined as follows. The element is a Hermitian element if ; it is a normal element if and it is a positive element if for some . An element is a unitary element if . An algebra with involution is also sometimes called a symmetric algebra (or symmetric ring), cf., e.g., [2]. However, this usage conflicts with the concept of a symmetric algebra as a special kind of Frobenius algebra, cf. Frobenius algebra.

Recent discoveries have revealed connections with, and applications to, algebraic topology. If is a compact metrizable space, a group, , can be formed from -extensions of the compact operators by ,

In [a3], is shown to be a homotopy invariant functor of which may be identified with the topological -homology group, . In [a1] M.F. Atiyah attempted to make a description of -homology, , in terms of elliptic operators [a5], p. 58. In [a7], [a8] G.G. Kasparov developed a solution to this problem. Kasparov and others have used the equivariant version of Kasparov -theory to prove the strong Novikov conjecture on higher signatures in many cases (see [a2], pp. 309-314).

In addition, deep and novel connections between -theory and operator algebras (cf. Operator ring) were recently discovered by A. Connes [a4]. Finally, V.F.R. Jones [a6] has exploited operator algebras to provide invariants of topological knots (cf. Knot theory).

Further details on recent developments may be found in [a2], [a5].

References

[a1] M.F. Atiyah, "Global theory of elliptic operators" , Proc. Internat. Conf. Funct. Anal. Related Topics , Univ. Tokyo Press (1970) MR0266247 Zbl 0193.43601
[a2] B. Blackadar, "-theory for operator algebras" , Springer (1986) MR0859867 Zbl 0597.46072
[a3] L.G. Brown, R.G. Douglas, P.A. Filmore, "Extensions of -algebras and -homology" Ann. of Math. (2) , 105 (1977) pp. 265–324
[a4] A. Connes, "Non-commutative differential geometry" Publ. Math. IHES , 62 (1986) pp. 257–360 Zbl 0657.55006 Zbl 0592.46056 Zbl 0564.58002
[a5] R.G. Douglas, "-algebra extensions and -homology" , Princeton Univ. Press (1980) MR0571362
[a6] V.F.R. Jones, "A polynomial invariant for knots via von Neumann algebras" Bull. Amer. Math. Soc. , 12 (1985) pp. 103–111 MR0766964 Zbl 0564.57006
[a7] G.G. Kasparov, "The generalized index of elliptic operators" Funct. Anal. and Its Appl. , 7 (1973) pp. 238–240 Funkt. Anal. i Prilozhen. , 7 (1973) pp. 82–83 MR445561 Zbl 0305.58017
[a8] G.G. Kasparov, "To pological invariants of elliptic operators I. -homology" Math. USSR-Izv. , 9 (1975) pp. 751–792 Izv. Akad. Nauk SSSR , 4 (1975) pp. 796–838 MR488027
[a9] M. Takesaki, "Theory of operator algebras" , 1 , Springer (1979) MR0548728 Zbl 0436.46043
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