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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/
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| − | 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.
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| − | 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
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| − | 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" />.
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| − | 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]]].
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| − | 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
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| − | 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" />.
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| − | 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]]].
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| − | ====References====
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| − | <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]]].
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| − | ====References====
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| − | <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
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| − | 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>
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