6297:
920:
2332:
4897:
Proof: Since every element of a commutative C*-algebra is normal, the
Gelfand representation is isometric; in particular, it is injective and its image is closed. But the image of the Gelfand representation is dense by the
1457:
is a real Banach algebra, but it is not a complex algebra (and hence not a complex Banach algebra) for the simple reason that the center of the quaternions is the real numbers, which cannot contain a copy of the complex
178:
4060:
4772:
1266:
4465:
765:
3525:
since the kernel of a character is a maximal ideal, which is closed. Moreover, the norm (that is, operator norm) of a character is one. Equipped with the topology of pointwise convergence on
7679:
3686:
2522:
5684:
3875:
3441:
2737:
7462:
4839:
4493:
4271:
2859:
7552:
2944:
3478:
2890:
2044:
1617:
4368:
2224:
760:
4638:
4551:
3186:
1317:
704:
675:
7589:
3134:
1037:
4151:
4220:
3945:
3523:
3280:
3043:
1551:
7786:
6840:
6186:
5786:
4698:
3819:
2986:
2109:
1573:
1455:
7628:
7396:
3977:
3907:
3602:
2457:
7755:
7485:
3797:
3768:
3715:
3309:
3235:
2191:
1990:
1734:
is isomorphic to the reals, the complexes, or the quaternions. Hence, the only complex Banach algebra that is a division algebra is the complexes. (This is known as the
4673:
4578:
4403:
2652:
2402:
1652:
2013:
1129:
577:
509:
446:
4522:
3634:
2831:
2363:
1961:
4297:
5419:
3570:
3359:
2135:
342:
311:
277:
2573:
1351:
1149:
1089:
104:
3336:
3110:
2158:
2067:
1914:
1851:
1722:
1698:
1675:
1400:
218:
7781:
5849:
4178:
4110:
4090:
3739:
3543:
3498:
3379:
3255:
3206:
3154:
3083:
3063:
3014:
2964:
2910:
2802:
2782:
2759:
2672:
2613:
2593:
2544:
2422:
2211:
2087:
1891:
1871:
1828:
1808:
1788:
1726:
The various algebras of functions given in the examples above have very different properties from standard examples of algebras such as the reals. For example:
1371:
1169:
1059:
1002:
974:
632:
612:
529:
469:
400:
377:
246:
67:
3982:
4072:
is zero) if and only if its
Gelfand representation has trivial kernel. An important example of such an algebra is a commutative C*-algebra. In fact, when
6895:
6012:
7822:
5424:
5197:
3188:
is a Banach algebra that is a field, and it follows from the
Gelfand–Mazur theorem that there is a bijection between the set of all maximal ideals of
6867:
6139:
5994:
5771:
4714:
6506:
6501:
6333:
5970:
5664:
7214:
355:
of an element of a nontrivial complex Banach algebra can never be empty, whereas in a real Banach algebra it could be empty for some elements.
7156:
5517:
5315:
5157:
5512:
7055:
6494:
344:
and then applying the outcome in the original algebra. However, this is not the case all the time. For example, one cannot define all the
6522:
5862:
6696:
6479:
5951:
5842:
5669:
5135:
5116:
5085:
5059:
5037:
3638:
2462:
1177:
6614:
6457:
6221:
5487:
2677:
6925:
7144:
7080:
6631:
5866:
5456:
2366:
449:
7139:
6818:
6597:
5446:
5441:
5434:
5370:
5254:
1701:
950:
939:
352:
183:
6910:
4899:
4409:
915:{\displaystyle \left(x_{1},\ldots ,x_{n}\right)\left(y_{1},\ldots ,y_{n}\right)=\left(x_{1}y_{1},\ldots ,x_{n}y_{n}\right).}
7177:
6955:
6900:
6017:
5679:
5190:
6733:
6073:
7012:
7002:
6962:
6930:
6857:
6402:
6300:
6022:
6007:
5835:
5305:
931:
The algebra of all bounded real- or complex-valued functions defined on some set (with pointwise multiplication and the
6940:
6037:
5816:
5290:
7350:
6326:
5736:
1767:
351:
The theory of real Banach algebras can be very different from the theory of complex Banach algebras. For example, the
7709:
7511:
6880:
6875:
6560:
6282:
6042:
5791:
5689:
5569:
7644:
1465:
is a certain kind of Banach algebra over a nonarchimedean field. Affinoid algebras are the basic building blocks in
7173:
6987:
6972:
6770:
6743:
6708:
6453:
6236:
6160:
7467:
7117:
6935:
6277:
5285:
1735:
7817:
7161:
7134:
6780:
6449:
6093:
5796:
5659:
5492:
5477:
5249:
4863:
3824:
3384:
7684:
6977:
6967:
6885:
6027:
5378:
7812:
7418:
6823:
6750:
6704:
6619:
6444:
6129:
5930:
5388:
5259:
5183:
4780:
4300:
1576:
540:
6950:
6890:
6002:
4470:
4228:
2836:
2761:
of a C*-algebra coincides with its spectral radius. This generalizes an analogous fact for normal operators.
1763:
Every commutative real unital
Noetherian Banach algebra (possibly having zero divisors) is finite-dimensional.
7515:
2915:
6226:
5751:
5726:
5544:
5533:
5244:
5152:. Encyclopaedia of Mathematical Sciences. Vol. 124 (1st ed.). Berlin Heidelberg: Springer-Verlag.
3446:
2864:
2018:
1582:
1491:
5602:
5592:
5587:
4308:
1521:, and the inversion operation on this set is continuous (and hence is a homeomorphism), so that it forms a
709:
6992:
6905:
6686:
6602:
6438:
6432:
6319:
6257:
6201:
6165:
5295:
4092:
is a commutative unital C*-algebra, the
Gelfand representation is then an isometric *-isomorphism between
3718:
345:
4584:
928:
form a 4-dimensional real Banach algebra, with the norm being given by the absolute value of quaternions.
317:
that the algebra under consideration is unital because one can develop much of the theory by considering
7776:
7771:
7246:
7194:
7151:
7075:
7028:
6765:
6427:
6394:
6367:
5347:
1930:
Unital Banach algebras over the complex field provide a general setting to develop spectral theory. The
1422:
953:
943:
94:
7065:
4527:
3159:
1271:
680:
651:
7714:
7567:
6920:
6915:
6626:
6510:
6416:
6240:
5761:
5740:
5654:
5539:
5502:
4854:
3115:
2424:
1993:
1499:
1487:
1466:
1007:
634:
98:
78:
4115:
7358:
7315:
7129:
6852:
6582:
6389:
6206:
6144:
5858:
5564:
5300:
4187:
3912:
3503:
3260:
3023:
1531:
536:
47:
35:
5021:
4681:
3802:
2969:
2092:
1556:
1438:
7791:
7702:
7606:
7369:
7340:
7336:
7325:
7295:
7291:
7112:
7070:
6677:
6587:
6532:
6379:
6231:
6098:
5694:
5623:
5554:
5398:
5360:
5145:
4991:
4065:
3950:
3880:
3575:
2430:
1514:
472:
82:
7718:
3773:
3744:
3691:
3285:
3211:
2167:
1966:
4643:
4560:
4373:
2622:
2372:
1622:
7472:
6945:
6726:
6669:
6649:
6211:
5801:
5776:
5461:
5383:
5163:
5153:
5131:
5112:
5081:
5055:
5033:
4983:
4554:
2327:{\displaystyle \sup\{|\lambda |:\lambda \in \sigma (x)\}=\lim _{n\to \infty }\|x^{n}\|^{1/n}.}
1998:
1522:
1479:
1098:
1068:
546:
478:
415:
4498:
3610:
2807:
2339:
1937:
1407:: A Banach algebra that is a closed *-subalgebra of the algebra of bounded operators on some
7102:
7097:
7085:
6997:
6982:
6845:
6785:
6760:
6691:
6681:
6544:
6489:
6216:
6134:
6103:
6083:
6068:
6063:
6058:
5895:
5806:
5507:
5355:
5310:
5234:
5104:
5096:
4975:
4872:
4276:
4069:
1731:
1507:
1503:
1462:
981:
194:
3548:
3344:
2114:
320:
289:
255:
7122:
7107:
7033:
7007:
6835:
6828:
6795:
6755:
6721:
6713:
6641:
6609:
6474:
6406:
6078:
6032:
5980:
5975:
5946:
5827:
5781:
5766:
5674:
5637:
5633:
5597:
5559:
5497:
5482:
5451:
5393:
5352:
5339:
5264:
5206:
5077:
4878:
3312:
2616:
2549:
2218:
1925:
1757:
1746:
1495:
1414:
1376:
1327:
1321:
1134:
1074:
1065:
1062:
382:
283:
5905:
3318:
3092:
2140:
2049:
1896:
1833:
1707:
1680:
1657:
1382:
200:
7692:
7640:
7557:
7300:
7166:
6813:
6422:
6374:
6267:
5920:
5731:
5710:
5628:
5618:
5429:
5336:
5269:
5069:
5026:
4181:
4163:
4095:
4075:
3724:
3528:
3483:
3364:
3240:
3191:
3139:
3068:
3048:
2999:
2949:
2895:
2787:
2767:
2744:
2657:
2598:
2578:
2529:
2407:
2196:
2072:
1876:
1856:
1813:
1793:
1773:
1356:
1154:
1044:
987:
959:
638:
617:
597:
588:
514:
454:
385:
362:
231:
74:
52:
4774:
Some authors include this isometric property in the definition of a Banach *-algebra.
7806:
7697:
7503:
7310:
7264:
7232:
7199:
7050:
7045:
7038:
6659:
6592:
6565:
6384:
6357:
6272:
6196:
5925:
5910:
5900:
5047:
3086:
2161:
1418:
1408:
977:
359:
43:
7209:
7204:
6664:
6654:
6527:
6517:
6362:
6342:
6262:
5915:
5885:
5549:
5403:
5344:
1742:
1483:
1353:
with the supremum norm and that contains the constants and separates the points of
1092:
90:
86:
1760:
Banach algebra with no zero divisors is isomorphic to the real or complex numbers.
5175:
4857: – net in a normed algebra that acts as a substitute for an identity element
7498:
7414:
7320:
7305:
7285:
7259:
7224:
6775:
6738:
6411:
6191:
6181:
6088:
5890:
5746:
5331:
1426:
1172:
642:
587:
The set of real (or complex) numbers is a Banach algebra with norm given by the
532:
225:
70:
31:
17:
7603:
7564:
7254:
7219:
7060:
6808:
6570:
6124:
5964:
5960:
5956:
5239:
4842:
1750:
1433:
1404:
925:
580:
280:
5167:
5108:
4987:
2595:(for example, the algebra of square matrices), the notion of the spectrum in
7330:
6575:
6539:
5224:
5210:
4701:
2214:
173:{\displaystyle \|x\,y\|\ \leq \|x\|\,\|y\|\quad {\text{ for all }}x,y\in A.}
7633:
7596:
7480:
7406:
7366:
7269:
7092:
5811:
5756:
4708:
1518:
1379:: A uniform algebra all of whose characters are evaluations at points of
932:
249:
946:(again with pointwise operations and supremum norm) is a Banach algebra.
7401:
7235: ((cs, lcs)-closed, (cs, bcs)-complete, (lower) ideally convex, (H
6484:
4995:
4963:
6311:
3065:
is then a commutative ring with unit, every non-invertible element of
583:. More generally, every C*-algebra is a Banach algebra by definition.
4933:
4931:
4055:{\displaystyle \sigma ({\hat {x}})=\{\chi (x):\chi \in \Delta (A)\}.}
4979:
1830:
have a multiplicative inverse element in a Banach algebra extension
2784:
be a complex unital Banach algebra in which every non-zero element
448:, the space of (complex-valued) continuous functions, defined on a
1486:
may be defined in any unital Banach algebra; examples include the
5130:. Chicago Lectures in Mathematics. University of Chicago Press).
5101:
Introduction to Banach
Algebras, Operators and Harmonic Analysis
3909:
in the formula above, is the spectrum as element of the algebra
1753:, is isomorphic to the reals, the complexes, or the quaternions.
1004:
is a Banach algebra and closed ideal. It is without identity if
6315:
5831:
5179:
4866: – Numerous conjectures by mathematician Irving Kaplansky
1324:: A Banach algebra that is a subalgebra of the complex algebra
4767:{\displaystyle \|x^{*}\|=\|x\|\quad {\text{ for all }}x\in A.}
4707:
In most natural examples, one also has that the involution is
4678:
In other words, a Banach *-algebra is a Banach algebra over
1498:. (In particular, the exponential map can be used to define
1261:{\displaystyle xy(g)=\int x(h)y\left(h^{-1}g\right)d\mu (h)}
4964:"A New Simple Proof of the Gelfand-Mazur-Kaplansky Theorem"
1510:
also holds for two commuting elements of a Banach algebra.
4962:
GarcĂa, Miguel
Cabrera; Palacios, Angel RodrĂguez (1995).
3545:(that is, the topology induced by the weak-* topology of
641:
Banach algebra if we equip it with a sub-multiplicative
4868:
Pages displaying short descriptions of redirect targets
976:(with functional composition as multiplication and the
4460:{\displaystyle (\lambda x)^{*}={\bar {\lambda }}x^{*}}
4064:
As an algebra, a unital commutative Banach algebra is
2575:
of bounded linear operators on a complex Banach space
7721:
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7609:
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4717:
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4311:
4279:
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4118:
4098:
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of complex continuous functions on the compact space
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2075:
2052:
2021:
2001:
1969:
1940:
1899:
1879:
1859:
1836:
1816:
1810:
some elements that are singular in the given algebra
1796:
1776:
1766:
Permanently singular elements in Banach algebras are
1710:
1683:
1660:
1625:
1585:
1559:
1534:
1506:
remains valid in general unital Banach algebras. The
1441:
1385:
1359:
1330:
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1180:
1157:
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1101:
1077:
1047:
1010:
990:
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517:
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365:
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258:
234:
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107:
101:
induced by the norm. The norm is required to satisfy
55:
5095:
Dales, H. G.; Aeina, P.; Eschmeier, J; Laursen, K.;
4859:
Pages displaying wikidata descriptions as a fallback
980:
as norm) is a unital Banach algebra. The set of all
7764:
7349:
7278:
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6866:
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5703:
5647:
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5324:
5278:
5217:
358:Banach algebras can also be defined over fields of
7749:
7673:
7622:
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7456:
7390:
6187:Spectral theory of ordinary differential equations
5787:Spectral theory of ordinary differential equations
5025:
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371:
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305:
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240:
212:
182:This ensures that the multiplication operation is
172:
61:
5685:Schröder–Bernstein theorems for operator algebras
3480:Every character is automatically continuous from
2988:(the complex case of the Gelfand–Mazur theorem).
2459:Furthermore, the spectral mapping theorem holds:
1873:are permanently singular in any Banach extension
4968:Proceedings of the American Mathematical Society
2275:
2228:
1425:, where the product of two measures is given by
726:
412:The prototypical example of a Banach algebra is
5076:. Graduate Texts in Mathematics. Vol. 96.
2804:is invertible (a division algebra). For every
7674:{\displaystyle S\left(\mathbb {R} ^{n}\right)}
3681:{\displaystyle \sigma (x)=\sigma ({\hat {x}})}
248:(whether it is unital or not) can be embedded
7782:Mathematical formulation of quantum mechanics
6327:
5843:
5191:
8:
4828:
4822:
4819:
4806:
4800:
4784:
4743:
4737:
4731:
4718:
4046:
4010:
2723:
2696:
2517:{\displaystyle \sigma (f(x))=f(\sigma (x)).}
2304:
2290:
2268:
2231:
2124:
2118:
719:
713:
143:
137:
133:
127:
118:
108:
3870:{\displaystyle {\hat {x}}(\chi )=\chi (x).}
3436:{\displaystyle \chi (ab)=\chi (a)\chi (b),}
2892:is not invertible (because the spectrum of
2732:{\displaystyle \sigma (f)=\{f(t):t\in X\}.}
6334:
6320:
6312:
5878:
5850:
5836:
5828:
5198:
5184:
5176:
1373:(which must be a compact Hausdorff space).
942:real- or complex-valued functions on some
7738:
7720:
7661:
7657:
7656:
7646:
7614:
7608:
7575:
7569:
7523:
7517:
7457:{\displaystyle B_{p,q}^{s}(\mathbb {R} )}
7447:
7446:
7437:
7426:
7420:
7371:
4834:{\displaystyle \|x^{*}x\|=\|x^{*}\|\|x\|}
4813:
4791:
4782:
4747:
4725:
4716:
4686:
4685:
4683:
4645:
4624:
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4241:
4230:
4194:
4192:
4189:
4165:
4117:
4097:
4077:
3993:
3992:
3984:
3952:
3914:
3885:
3884:
3882:
3829:
3828:
3826:
3807:
3806:
3804:
3775:
3749:
3748:
3746:
3726:
3696:
3695:
3693:
3664:
3663:
3640:
3612:
3577:
3556:
3550:
3530:
3508:
3507:
3505:
3485:
3456:
3448:
3386:
3381:that is at the same time multiplicative,
3366:
3346:
3320:
3287:
3265:
3264:
3262:
3242:
3213:
3193:
3172:
3171:
3166:
3161:
3141:
3120:
3119:
3117:
3094:
3070:
3050:
3028:
3027:
3025:
3001:
2974:
2973:
2971:
2951:
2928:
2917:
2897:
2877:
2866:
2847:
2846:
2838:
2809:
2789:
2769:
2746:
2679:
2659:
2624:
2600:
2580:
2551:
2531:
2464:
2432:
2409:
2374:
2341:
2311:
2307:
2297:
2278:
2242:
2234:
2226:
2198:
2169:
2142:
2116:
2097:
2096:
2094:
2089:is a closed subset of the closed disc in
2074:
2051:
2031:
2020:
2000:
1968:
1939:
1898:
1878:
1858:
1835:
1815:
1795:
1775:
1741:Every unital real Banach algebra with no
1709:
1682:
1659:
1624:
1604:
1584:
1560:
1558:
1535:
1533:
1443:
1442:
1440:
1384:
1358:
1329:
1291:
1273:
1226:
1179:
1156:
1136:
1106:
1100:
1076:
1046:
1009:
989:
961:
898:
888:
869:
859:
836:
817:
797:
778:
767:
762:and define multiplication componentwise:
747:
741:
732:
729:
711:
690:
686:
685:
682:
661:
657:
656:
653:
619:
599:
554:
548:
516:
486:
480:
456:
423:
417:
387:
364:
328:
322:
297:
291:
263:
257:
233:
202:
147:
136:
114:
106:
54:
6140:Group algebra of a locally compact group
5008:
4488:{\displaystyle \lambda \in \mathbb {C} }
4266:{\displaystyle \left(x^{*}\right)^{*}=x}
2854:{\displaystyle \lambda \in \mathbb {C} }
7547:{\displaystyle L^{\lambda ,p}(\Omega )}
4915:
4890:
2939:{\displaystyle a=\lambda \mathbf {1} :}
7787:Ordinary Differential Equations (ODEs)
6901:Banach–Steinhaus (Uniform boundedness)
4949:
4937:
4922:
4180:is a Banach algebra over the field of
3473:{\displaystyle \chi (\mathbf {1} )=1.}
2885:{\displaystyle a-\lambda \mathbf {1} }
2039:{\displaystyle x-\lambda \mathbf {1} }
1612:{\displaystyle xy-yx\neq \mathbf {1} }
348:in a Banach algebra without identity.
27:Particular kind of algebraic structure
5518:Spectral theory of normal C*-algebras
5316:Spectral theory of normal C*-algebras
4875: – Branch of functional analysis
4363:{\displaystyle (x+y)^{*}=x^{*}+y^{*}}
1417:: A Banach algebra consisting of all
755:{\displaystyle \|x\|=\max _{}|x_{i}|}
197:for the multiplication whose norm is
186:with respect to the metric topology.
7:
5513:Spectral theory of compact operators
1730:Every real Banach algebra that is a
4633:{\displaystyle (xy)^{*}=y^{*}x^{*}}
4222:that has the following properties:
3173:
3121:
1517:in any unital Banach algebra is an
1171:becomes a Banach algebra under the
7615:
7576:
7538:
7382:
5665:Cohen–Hewitt factorization theorem
4125:
4034:
3954:
3922:
3777:
3579:
3289:
3237:of all nonzero homomorphisms from
3215:
2285:
1770:, that is, considering extensions
1023:
85:) that at the same time is also a
25:
7279:Subsets / set operations
7056:Differentiation in Fréchet spaces
5670:Extensions of symmetric operators
4546:{\displaystyle {\bar {\lambda }}}
3181:{\displaystyle A/{\mathfrak {m}}}
935:norm) is a unital Banach algebra.
7823:Science and technology in Poland
6296:
6295:
6222:Topological quantum field theory
5488:Positive operator-valued measure
3770:is the continuous function from
3457:
2929:
2878:
2654:(with a compact Hausdorff space
2615:coincides with the usual one in
2032:
1992:, consists of all those complex
1853:Topological divisors of zero in
1605:
1561:
1536:
1312:{\displaystyle x,y\in L^{1}(G).}
699:{\displaystyle \mathbb {C} ^{n}}
670:{\displaystyle \mathbb {R} ^{n}}
7584:{\displaystyle \ell ^{\infty }}
5772:Rayleigh–Faber–Krahn inequality
5074:A Course in Functional Analysis
4746:
3129:{\displaystyle {\mathfrak {m}}}
2367:holomorphic functional calculus
1377:Natural Banach function algebra
1032:{\displaystyle \dim E=\infty .}
594:The set of all real or complex
450:locally compact Hausdorff space
146:
7744:
7725:
7541:
7535:
7451:
7443:
7385:
7379:
6973:Lomonosov's invariant subspace
6896:Banach–Schauder (open mapping)
5103:. Cambridge University Press.
5032:. Cambridge University Press.
4777:A Banach *-algebra satisfying
4598:
4588:
4537:
4441:
4423:
4413:
4325:
4312:
4206:
4146:{\displaystyle C(\Delta (A)).}
4137:
4134:
4128:
4122:
4043:
4037:
4022:
4016:
4004:
3998:
3989:
3963:
3957:
3934:
3931:
3925:
3919:
3890:
3861:
3855:
3846:
3840:
3834:
3786:
3780:
3754:
3701:
3675:
3669:
3660:
3651:
3645:
3604:is a Hausdorff compact space.
3588:
3582:
3461:
3453:
3427:
3421:
3415:
3409:
3400:
3391:
3338:and its members "characters".
3298:
3292:
3224:
3218:
2708:
2702:
2690:
2684:
2641:
2635:
2562:
2556:
2508:
2505:
2499:
2493:
2484:
2481:
2475:
2469:
2443:
2437:
2385:
2379:
2282:
2265:
2259:
2243:
2235:
2180:
2174:
1979:
1973:
1756:Every commutative real unital
1340:
1334:
1303:
1297:
1255:
1249:
1211:
1205:
1193:
1187:
1118:
1112:
748:
733:
566:
560:
498:
492:
435:
429:
1:
6018:Uniform boundedness principle
5680:Limiting absorption principle
5150:Theory of Operator Algebras I
5054:. New York: Springer-Verlag.
4215:{\displaystyle {}^{*}:A\to A}
3940:{\displaystyle C(\Delta (A))}
3518:{\displaystyle \mathbb {C} ,}
3275:{\displaystyle \mathbb {C} .}
3038:{\displaystyle \mathbb {C} .}
2741:The norm of a normal element
1546:{\displaystyle \mathbf {1} ,}
1528:If a Banach algebra has unit
252:into a unital Banach algebra
6858:Singular value decomposition
5306:Singular value decomposition
4693:{\displaystyle \mathbb {C} }
3814:{\displaystyle \mathbb {C} }
2981:{\displaystyle \mathbb {C} }
2104:{\displaystyle \mathbb {C} }
2069:The spectrum of any element
1768:topological divisors of zero
1568:{\displaystyle \mathbf {1} }
1450:{\displaystyle \mathbb {H} }
956:operators on a Banach space
7623:{\displaystyle L^{\infty }}
7391:{\displaystyle ba(\Sigma )}
7260:Radially convex/Star-shaped
5737:Hearing the shape of a drum
5420:Decomposition of a spectrum
3972:{\displaystyle \Delta (A).}
3902:{\displaystyle {\hat {x}},}
3597:{\displaystyle \Delta (A),}
2966:is naturally isomorphic to
2452:{\displaystyle \sigma (x).}
938:The algebra of all bounded
189:A Banach algebra is called
7839:
7750:{\displaystyle W(X,L^{p})}
6161:Invariant subspace problem
5325:Special Elements/Operators
3792:{\displaystyle \Delta (A)}
3763:{\displaystyle {\hat {x}}}
3710:{\displaystyle {\hat {x}}}
3361:is a linear functional on
3315:" or "character space" of
3304:{\displaystyle \Delta (A)}
3230:{\displaystyle \Delta (A)}
2186:{\displaystyle \sigma (x)}
1985:{\displaystyle \sigma (x)}
1923:
7296:Algebraic interior (core)
6911:Cauchy–Schwarz inequality
6554:Function space Topologies
6291:
5881:
5797:Superstrong approximation
5660:Banach algebra cohomology
5493:Projection-valued measure
5478:Borel functional calculus
5250:Projection-valued measure
4900:Stone–Weierstrass theorem
4668:{\displaystyle x,y\in A.}
4573:{\displaystyle \lambda .}
4398:{\displaystyle x,y\in A.}
2647:{\displaystyle f\in C(X)}
2397:{\displaystyle f(x)\in A}
2164:. Moreover, the spectrum
1647:{\displaystyle x,y\in A.}
1494:, and more generally any
1151:-integrable functions on
511:is unital if and only if
224:if its multiplication is
6130:Spectrum of a C*-algebra
5389:Spectrum of a C*-algebra
5260:Spectrum of a C*-algebra
5109:10.1017/CBO9780511615429
5052:Complete Normed Algebras
3572:), the character space,
2526:When the Banach algebra
2008:{\displaystyle \lambda }
1124:{\displaystyle L^{1}(G)}
1095:, then the Banach space
572:{\displaystyle C_{0}(X)}
504:{\displaystyle C_{0}(X)}
441:{\displaystyle C_{0}(X)}
6227:Noncommutative geometry
5817:Wiener–Khinchin theorem
5752:Kuznetsov trace formula
5727:Almost Mathieu operator
5545:Banach function algebra
5534:Amenable Banach algebra
5291:Gelfand–Naimark theorem
5245:Noncommutative topology
4517:{\displaystyle x\in A;}
3629:{\displaystyle x\in A,}
2826:{\displaystyle a\in A,}
2358:{\displaystyle x\in A,}
1956:{\displaystyle x\in A,}
1502:.) The formula for the
1492:trigonometric functions
1467:rigid analytic geometry
1427:convolution of measures
346:trigonometric functions
7751:
7675:
7624:
7585:
7548:
7458:
7392:
6561:Banach–Mazur compactum
6351:Types of Banach spaces
6283:Tomita–Takesaki theory
6258:Approximation property
6202:Calculus of variations
5792:Sturm–Liouville theory
5690:Sherman–Takeda theorem
5570:Tomita–Takesaki theory
5345:Hermitian/Self-adjoint
5296:Gelfand representation
4864:Kaplansky's conjecture
4835:
4768:
4694:
4669:
4634:
4574:
4547:
4518:
4489:
4461:
4399:
4364:
4293:
4292:{\displaystyle x\in A}
4267:
4216:
4184:, together with a map
4174:
4147:
4106:
4086:
4056:
3973:
3941:
3903:
3871:
3815:
3793:
3764:
3735:
3719:Gelfand representation
3711:
3682:
3630:
3598:
3566:
3539:
3519:
3494:
3474:
3437:
3375:
3355:
3332:
3305:
3276:
3251:
3231:
3202:
3182:
3150:
3130:
3112:Since a maximal ideal
3106:
3079:
3059:
3039:
3010:
2982:
2960:
2940:
2906:
2886:
2855:
2827:
2798:
2778:
2755:
2733:
2668:
2648:
2609:
2589:
2569:
2540:
2518:
2453:
2418:
2398:
2359:
2328:
2207:
2187:
2154:
2131:
2105:
2083:
2063:
2040:
2009:
1986:
1957:
1910:
1887:
1867:
1847:
1824:
1804:
1784:
1718:
1694:
1671:
1648:
1613:
1569:
1547:
1525:under multiplication.
1451:
1396:
1367:
1347:
1313:
1262:
1165:
1145:
1125:
1085:
1055:
1033:
998:
970:
916:
756:
700:
671:
648:Take the Banach space
628:
608:
573:
525:
505:
465:
442:
396:
373:
338:
307:
273:
242:
214:
174:
63:
7777:Finite element method
7772:Differential operator
7752:
7676:
7625:
7586:
7549:
7459:
7393:
7233:Convex series related
7029:Abstract Wiener space
6956:hyperplane separation
6511:Minkowski functionals
6395:Polarization identity
6278:Banach–Mazur distance
6241:Generalized functions
5286:Gelfand–Mazur theorem
5126:Mosak, R. D. (1975).
5050:; Duncan, J. (1973).
4836:
4769:
4695:
4670:
4635:
4575:
4548:
4519:
4490:
4462:
4400:
4365:
4294:
4268:
4217:
4175:
4148:
4107:
4087:
4057:
3974:
3942:
3904:
3872:
3816:
3794:
3765:
3736:
3712:
3683:
3631:
3599:
3567:
3565:{\displaystyle A^{*}}
3540:
3520:
3495:
3475:
3438:
3376:
3356:
3354:{\displaystyle \chi }
3333:
3306:
3277:
3252:
3232:
3203:
3183:
3151:
3131:
3107:
3080:
3060:
3040:
3011:
2992:Ideals and characters
2983:
2961:
2941:
2907:
2887:
2856:
2828:
2799:
2779:
2756:
2734:
2669:
2649:
2610:
2590:
2570:
2541:
2519:
2454:
2427:in a neighborhood of
2419:
2399:
2360:
2329:
2208:
2188:
2155:
2132:
2130:{\displaystyle \|x\|}
2106:
2084:
2064:
2046:is not invertible in
2041:
2010:
1987:
1958:
1911:
1888:
1868:
1848:
1825:
1805:
1785:
1745:, and in which every
1736:Gelfand–Mazur theorem
1719:
1695:
1672:
1649:
1614:
1570:
1548:
1500:abstract index groups
1482:that are defined via
1452:
1423:locally compact group
1397:
1368:
1348:
1314:
1263:
1166:
1146:
1126:
1086:
1056:
1034:
999:
971:
944:locally compact space
917:
757:
701:
672:
629:
609:
574:
526:
506:
466:
443:
397:
374:
339:
337:{\displaystyle A_{e}}
308:
306:{\displaystyle A_{e}}
274:
272:{\displaystyle A_{e}}
243:
228:. Any Banach algebra
215:
175:
64:
7719:
7645:
7607:
7568:
7516:
7419:
7370:
7359:Absolute continuity
7013:Schauder fixed-point
7003:Riesz representation
6963:Kakutani fixed-point
6931:Freudenthal spectral
6417:L-semi-inner product
6023:Kakutani fixed-point
6008:Riesz representation
5762:Proto-value function
5741:Dirichlet eigenvalue
5655:Abstract index group
5540:Approximate identity
5503:Rigged Hilbert space
5379:Krein–Rutman theorem
5225:Involution/*-algebra
4855:Approximate identity
4781:
4715:
4682:
4644:
4585:
4561:
4528:
4499:
4471:
4410:
4374:
4309:
4277:
4229:
4188:
4164:
4116:
4096:
4076:
3983:
3951:
3913:
3881:
3825:
3803:
3774:
3745:
3741:defined as follows:
3725:
3692:
3639:
3611:
3576:
3549:
3529:
3504:
3484:
3447:
3385:
3365:
3345:
3319:
3286:
3261:
3241:
3212:
3192:
3160:
3140:
3116:
3093:
3069:
3049:
3024:
3020:Banach algebra over
3000:
2970:
2950:
2916:
2912:is not empty) hence
2896:
2865:
2837:
2808:
2788:
2768:
2745:
2678:
2658:
2623:
2599:
2579:
2568:{\displaystyle L(X)}
2550:
2530:
2463:
2431:
2408:
2373:
2340:
2225:
2197:
2168:
2141:
2115:
2093:
2073:
2050:
2019:
1999:
1967:
1938:
1897:
1877:
1857:
1834:
1814:
1794:
1774:
1708:
1681:
1658:
1623:
1583:
1557:
1532:
1488:exponential function
1480:elementary functions
1439:
1383:
1357:
1346:{\displaystyle C(X)}
1328:
1272:
1178:
1155:
1144:{\displaystyle \mu }
1135:
1099:
1084:{\displaystyle \mu }
1075:
1045:
1008:
988:
960:
766:
710:
681:
652:
618:
598:
547:
515:
479:
455:
416:
386:
363:
321:
313:. Often one assumes
290:
256:
232:
201:
105:
53:
7442:
7180:measurable function
7130:Functional calculus
6993:Parseval's identity
6906:Bessel's inequality
6853:Polar decomposition
6632:Uniform convergence
6390:Inner product space
6207:Functional calculus
6166:Mahler's conjecture
6145:Von Neumann algebra
5859:Functional analysis
5565:Von Neumann algebra
5301:Polar decomposition
4749: for all
4160:A Banach *-algebra
1790:of Banach algebras
1515:invertible elements
1432:The algebra of the
949:The algebra of all
537:complex conjugation
149: for all
77:numbers (or over a
48:associative algebra
36:functional analysis
7792:Validated numerics
7747:
7703:Sobolev inequality
7671:
7620:
7581:
7544:
7473:Bounded variation
7454:
7422:
7407:Banach coordinate
7388:
7326:Minkowski addition
6988:M. Riesz extension
6468:Banach spaces are:
6232:Riemann hypothesis
5931:Topological vector
5695:Unbounded operator
5624:Essential spectrum
5603:Schur–Horn theorem
5593:Bauer–Fike theorem
5588:Alon–Boppana bound
5581:Finite-Dimensional
5555:Nuclear C*-algebra
5399:Spectral asymmetry
5011:, Proposition 2.8.
4952:, Theorem VII.2.2.
4940:, Example VII.1.9.
4925:, Example VII.1.8.
4831:
4764:
4690:
4665:
4630:
4570:
4543:
4514:
4485:
4457:
4395:
4360:
4299:(so the map is an
4289:
4263:
4212:
4170:
4143:
4102:
4082:
4052:
3969:
3937:
3899:
3867:
3811:
3789:
3760:
3731:
3707:
3678:
3626:
3594:
3562:
3535:
3515:
3490:
3470:
3433:
3371:
3351:
3331:{\displaystyle A,}
3328:
3301:
3272:
3247:
3227:
3198:
3178:
3146:
3126:
3105:{\displaystyle A.}
3102:
3075:
3055:
3035:
3006:
2978:
2956:
2936:
2902:
2882:
2851:
2823:
2794:
2774:
2751:
2729:
2674:), one sees that:
2664:
2644:
2605:
2585:
2565:
2536:
2514:
2449:
2414:
2394:
2355:
2324:
2289:
2217:and satisfies the
2203:
2183:
2153:{\displaystyle 0,}
2150:
2127:
2101:
2079:
2062:{\displaystyle A.}
2059:
2036:
2005:
1982:
1953:
1909:{\displaystyle A.}
1906:
1883:
1863:
1846:{\displaystyle B.}
1843:
1820:
1800:
1780:
1717:{\displaystyle 0.}
1714:
1693:{\displaystyle yx}
1690:
1670:{\displaystyle xy}
1667:
1644:
1609:
1565:
1543:
1447:
1395:{\displaystyle X.}
1392:
1363:
1343:
1309:
1258:
1161:
1141:
1121:
1081:
1051:
1029:
994:
966:
912:
752:
731:
696:
667:
624:
604:
569:
521:
501:
473:vanish at infinity
461:
438:
392:
381:. This is part of
369:
334:
303:
269:
238:
213:{\displaystyle 1,}
210:
170:
59:
7800:
7799:
7512:Morrey–Campanato
7494:compact Hausdorff
7341:Relative interior
7195:Absolutely convex
7162:Projection-valued
6771:Strictly singular
6697:on Hilbert spaces
6458:of Hilbert spaces
6309:
6308:
6212:Integral operator
5989:
5988:
5825:
5824:
5802:Transfer operator
5777:Spectral geometry
5462:Spectral abscissa
5442:Approximate point
5384:Normal eigenvalue
5159:978-3-540-42248-8
4750:
4555:complex conjugate
4540:
4444:
4173:{\displaystyle A}
4156:Banach *-algebras
4105:{\displaystyle A}
4085:{\displaystyle A}
4001:
3893:
3837:
3757:
3734:{\displaystyle x}
3704:
3672:
3538:{\displaystyle A}
3493:{\displaystyle A}
3374:{\displaystyle A}
3250:{\displaystyle A}
3201:{\displaystyle A}
3149:{\displaystyle A}
3078:{\displaystyle A}
3058:{\displaystyle A}
3009:{\displaystyle A}
2959:{\displaystyle A}
2905:{\displaystyle a}
2797:{\displaystyle x}
2777:{\displaystyle A}
2754:{\displaystyle x}
2667:{\displaystyle X}
2608:{\displaystyle A}
2588:{\displaystyle X}
2539:{\displaystyle A}
2417:{\displaystyle f}
2404:for any function
2369:allows to define
2274:
2206:{\displaystyle x}
2082:{\displaystyle x}
1886:{\displaystyle B}
1866:{\displaystyle A}
1823:{\displaystyle A}
1803:{\displaystyle A}
1783:{\displaystyle B}
1523:topological group
1366:{\displaystyle X}
1164:{\displaystyle G}
1069:topological group
1054:{\displaystyle G}
997:{\displaystyle E}
982:compact operators
969:{\displaystyle E}
725:
627:{\displaystyle n}
607:{\displaystyle n}
524:{\displaystyle X}
464:{\displaystyle X}
395:{\displaystyle p}
372:{\displaystyle p}
241:{\displaystyle A}
150:
123:
62:{\displaystyle A}
16:(Redirected from
7830:
7818:Fourier analysis
7756:
7754:
7753:
7748:
7743:
7742:
7710:Triebel–Lizorkin
7680:
7678:
7677:
7672:
7670:
7666:
7665:
7660:
7629:
7627:
7626:
7621:
7619:
7618:
7590:
7588:
7587:
7582:
7580:
7579:
7553:
7551:
7550:
7545:
7534:
7533:
7463:
7461:
7460:
7455:
7450:
7441:
7436:
7397:
7395:
7394:
7389:
7250:
7228:
7210:Balanced/Circled
7008:Robinson-Ursescu
6926:Eberlein–Šmulian
6846:Spectral theorem
6642:Linear operators
6439:Uniformly smooth
6336:
6329:
6322:
6313:
6299:
6298:
6217:Jones polynomial
6135:Operator algebra
5879:
5852:
5845:
5838:
5829:
5807:Transform theory
5527:Special algebras
5508:Spectral theorem
5471:Spectral Theorem
5311:Spectral theorem
5200:
5193:
5186:
5177:
5171:
5141:
5122:
5091:
5065:
5043:
5031:
5012:
5006:
5000:
4999:
4974:(9): 2663–2666.
4959:
4953:
4947:
4941:
4935:
4926:
4920:
4903:
4895:
4873:Operator algebra
4869:
4860:
4840:
4838:
4837:
4832:
4818:
4817:
4796:
4795:
4773:
4771:
4770:
4765:
4751:
4748:
4730:
4729:
4699:
4697:
4696:
4691:
4689:
4674:
4672:
4671:
4666:
4639:
4637:
4636:
4631:
4629:
4628:
4619:
4618:
4606:
4605:
4579:
4577:
4576:
4571:
4552:
4550:
4549:
4544:
4542:
4541:
4533:
4523:
4521:
4520:
4515:
4494:
4492:
4491:
4486:
4484:
4466:
4464:
4463:
4458:
4456:
4455:
4446:
4445:
4437:
4431:
4430:
4404:
4402:
4401:
4396:
4369:
4367:
4366:
4361:
4359:
4358:
4346:
4345:
4333:
4332:
4298:
4296:
4295:
4290:
4272:
4270:
4269:
4264:
4256:
4255:
4250:
4246:
4245:
4221:
4219:
4218:
4213:
4199:
4198:
4193:
4179:
4177:
4176:
4171:
4152:
4150:
4149:
4144:
4111:
4109:
4108:
4103:
4091:
4089:
4088:
4083:
4070:Jacobson radical
4061:
4059:
4058:
4053:
4003:
4002:
3994:
3978:
3976:
3975:
3970:
3946:
3944:
3943:
3938:
3908:
3906:
3905:
3900:
3895:
3894:
3886:
3877:The spectrum of
3876:
3874:
3873:
3868:
3839:
3838:
3830:
3820:
3818:
3817:
3812:
3810:
3798:
3796:
3795:
3790:
3769:
3767:
3766:
3761:
3759:
3758:
3750:
3740:
3738:
3737:
3732:
3716:
3714:
3713:
3708:
3706:
3705:
3697:
3687:
3685:
3684:
3679:
3674:
3673:
3665:
3635:
3633:
3632:
3627:
3603:
3601:
3600:
3595:
3571:
3569:
3568:
3563:
3561:
3560:
3544:
3542:
3541:
3536:
3524:
3522:
3521:
3516:
3511:
3499:
3497:
3496:
3491:
3479:
3477:
3476:
3471:
3460:
3442:
3440:
3439:
3434:
3380:
3378:
3377:
3372:
3360:
3358:
3357:
3352:
3337:
3335:
3334:
3329:
3310:
3308:
3307:
3302:
3281:
3279:
3278:
3273:
3268:
3256:
3254:
3253:
3248:
3236:
3234:
3233:
3228:
3207:
3205:
3204:
3199:
3187:
3185:
3184:
3179:
3177:
3176:
3170:
3155:
3153:
3152:
3147:
3135:
3133:
3132:
3127:
3125:
3124:
3111:
3109:
3108:
3103:
3085:belongs to some
3084:
3082:
3081:
3076:
3064:
3062:
3061:
3056:
3044:
3042:
3041:
3036:
3031:
3015:
3013:
3012:
3007:
2987:
2985:
2984:
2979:
2977:
2965:
2963:
2962:
2957:
2945:
2943:
2942:
2937:
2932:
2911:
2909:
2908:
2903:
2891:
2889:
2888:
2883:
2881:
2860:
2858:
2857:
2852:
2850:
2832:
2830:
2829:
2824:
2803:
2801:
2800:
2795:
2783:
2781:
2780:
2775:
2760:
2758:
2757:
2752:
2738:
2736:
2735:
2730:
2673:
2671:
2670:
2665:
2653:
2651:
2650:
2645:
2614:
2612:
2611:
2606:
2594:
2592:
2591:
2586:
2574:
2572:
2571:
2566:
2545:
2543:
2542:
2537:
2523:
2521:
2520:
2515:
2458:
2456:
2455:
2450:
2423:
2421:
2420:
2415:
2403:
2401:
2400:
2395:
2364:
2362:
2361:
2356:
2333:
2331:
2330:
2325:
2320:
2319:
2315:
2302:
2301:
2288:
2246:
2238:
2212:
2210:
2209:
2204:
2192:
2190:
2189:
2184:
2159:
2157:
2156:
2151:
2136:
2134:
2133:
2128:
2110:
2108:
2107:
2102:
2100:
2088:
2086:
2085:
2080:
2068:
2066:
2065:
2060:
2045:
2043:
2042:
2037:
2035:
2014:
2012:
2011:
2006:
1991:
1989:
1988:
1983:
1962:
1960:
1959:
1954:
1915:
1913:
1912:
1907:
1892:
1890:
1889:
1884:
1872:
1870:
1869:
1864:
1852:
1850:
1849:
1844:
1829:
1827:
1826:
1821:
1809:
1807:
1806:
1801:
1789:
1787:
1786:
1781:
1732:division algebra
1723:
1721:
1720:
1715:
1704:except possibly
1699:
1697:
1696:
1691:
1676:
1674:
1673:
1668:
1654:This is because
1653:
1651:
1650:
1645:
1619: for any
1618:
1616:
1615:
1610:
1608:
1574:
1572:
1571:
1566:
1564:
1552:
1550:
1549:
1544:
1539:
1508:binomial theorem
1504:geometric series
1463:affinoid algebra
1456:
1454:
1453:
1448:
1446:
1401:
1399:
1398:
1393:
1372:
1370:
1369:
1364:
1352:
1350:
1349:
1344:
1318:
1316:
1315:
1310:
1296:
1295:
1267:
1265:
1264:
1259:
1242:
1238:
1234:
1233:
1170:
1168:
1167:
1162:
1150:
1148:
1147:
1142:
1130:
1128:
1127:
1122:
1111:
1110:
1090:
1088:
1087:
1082:
1060:
1058:
1057:
1052:
1038:
1036:
1035:
1030:
1003:
1001:
1000:
995:
975:
973:
972:
967:
921:
919:
918:
913:
908:
904:
903:
902:
893:
892:
874:
873:
864:
863:
846:
842:
841:
840:
822:
821:
807:
803:
802:
801:
783:
782:
761:
759:
758:
753:
751:
746:
745:
736:
730:
705:
703:
702:
697:
695:
694:
689:
676:
674:
673:
668:
666:
665:
660:
633:
631:
630:
625:
613:
611:
610:
605:
578:
576:
575:
570:
559:
558:
530:
528:
527:
522:
510:
508:
507:
502:
491:
490:
470:
468:
467:
462:
447:
445:
444:
439:
428:
427:
401:
399:
398:
393:
378:
376:
375:
370:
343:
341:
340:
335:
333:
332:
312:
310:
309:
304:
302:
301:
279:so as to form a
278:
276:
275:
270:
268:
267:
247:
245:
244:
239:
219:
217:
216:
211:
195:identity element
179:
177:
176:
171:
151:
148:
121:
68:
66:
65:
60:
21:
7838:
7837:
7833:
7832:
7831:
7829:
7828:
7827:
7813:Banach algebras
7803:
7802:
7801:
7796:
7760:
7734:
7717:
7716:
7715:Wiener amalgam
7685:Segal–Bargmann
7655:
7651:
7643:
7642:
7610:
7605:
7604:
7571:
7566:
7565:
7519:
7514:
7513:
7468:Birnbaum–Orlicz
7417:
7416:
7368:
7367:
7345:
7301:Bounding points
7274:
7248:
7226:
7183:
7034:Banach manifold
7017:
6941:Gelfand–Naimark
6862:
6836:Spectral theory
6804:Banach algebras
6796:Operator theory
6790:
6751:Pseudo-monotone
6734:Hilbert–Schmidt
6714:Densely defined
6636:
6549:
6463:
6346:
6340:
6310:
6305:
6287:
6251:Advanced topics
6246:
6170:
6149:
6108:
6074:Hilbert–Schmidt
6047:
6038:Gelfand–Naimark
5985:
5935:
5870:
5856:
5826:
5821:
5782:Spectral method
5767:Ramanujan graph
5715:
5699:
5675:Fredholm theory
5643:
5638:Shilov boundary
5634:Structure space
5612:Generalizations
5607:
5598:Numerical range
5576:
5560:Uniform algebra
5522:
5498:Riesz projector
5483:Min-max theorem
5466:
5452:Direct integral
5408:
5394:Spectral radius
5365:
5320:
5274:
5265:Spectral radius
5213:
5207:Spectral theory
5204:
5174:
5160:
5144:
5138:
5128:Banach algebras
5125:
5119:
5094:
5088:
5078:Springer Verlag
5068:
5062:
5046:
5040:
5028:Linear Analysis
5020:
5016:
5015:
5007:
5003:
4980:10.2307/2160559
4961:
4960:
4956:
4948:
4944:
4936:
4929:
4921:
4917:
4912:
4907:
4906:
4896:
4892:
4887:
4879:Shilov boundary
4867:
4858:
4851:
4809:
4787:
4779:
4778:
4721:
4713:
4712:
4700:that is also a
4680:
4679:
4642:
4641:
4620:
4610:
4597:
4583:
4582:
4559:
4558:
4526:
4525:
4497:
4496:
4469:
4468:
4447:
4422:
4408:
4407:
4372:
4371:
4350:
4337:
4324:
4307:
4306:
4275:
4274:
4237:
4233:
4232:
4227:
4226:
4191:
4186:
4185:
4182:complex numbers
4162:
4161:
4158:
4114:
4113:
4094:
4093:
4074:
4073:
3981:
3980:
3949:
3948:
3911:
3910:
3879:
3878:
3823:
3822:
3801:
3800:
3772:
3771:
3743:
3742:
3723:
3722:
3690:
3689:
3637:
3636:
3609:
3608:
3574:
3573:
3552:
3547:
3546:
3527:
3526:
3502:
3501:
3482:
3481:
3445:
3444:
3383:
3382:
3363:
3362:
3343:
3342:
3317:
3316:
3313:structure space
3311:is called the "
3284:
3283:
3259:
3258:
3239:
3238:
3210:
3209:
3190:
3189:
3158:
3157:
3138:
3137:
3114:
3113:
3091:
3090:
3067:
3066:
3047:
3046:
3022:
3021:
2998:
2997:
2994:
2968:
2967:
2948:
2947:
2914:
2913:
2894:
2893:
2863:
2862:
2835:
2834:
2806:
2805:
2786:
2785:
2766:
2765:
2743:
2742:
2676:
2675:
2656:
2655:
2621:
2620:
2617:operator theory
2597:
2596:
2577:
2576:
2548:
2547:
2546:is the algebra
2528:
2527:
2461:
2460:
2429:
2428:
2406:
2405:
2371:
2370:
2338:
2337:
2303:
2293:
2223:
2222:
2219:spectral radius
2195:
2194:
2166:
2165:
2139:
2138:
2113:
2112:
2091:
2090:
2071:
2070:
2048:
2047:
2017:
2016:
1997:
1996:
1965:
1964:
1936:
1935:
1928:
1926:Spectral theory
1922:
1920:Spectral theory
1895:
1894:
1875:
1874:
1855:
1854:
1832:
1831:
1812:
1811:
1792:
1791:
1772:
1771:
1747:principal ideal
1706:
1705:
1679:
1678:
1656:
1655:
1621:
1620:
1581:
1580:
1555:
1554:
1530:
1529:
1496:entire function
1476:
1437:
1436:
1415:Measure algebra
1381:
1380:
1355:
1354:
1326:
1325:
1322:Uniform algebra
1287:
1270:
1269:
1222:
1221:
1217:
1176:
1175:
1153:
1152:
1133:
1132:
1102:
1097:
1096:
1073:
1072:
1063:locally compact
1043:
1042:
1006:
1005:
986:
985:
958:
957:
894:
884:
865:
855:
854:
850:
832:
813:
812:
808:
793:
774:
773:
769:
764:
763:
737:
708:
707:
684:
679:
678:
655:
650:
649:
616:
615:
596:
595:
550:
545:
544:
513:
512:
482:
477:
476:
453:
452:
419:
414:
413:
410:
384:
383:
361:
360:
324:
319:
318:
293:
288:
287:
259:
254:
253:
230:
229:
199:
198:
103:
102:
79:non-Archimedean
51:
50:
28:
23:
22:
18:Structure space
15:
12:
11:
5:
7836:
7834:
7826:
7825:
7820:
7815:
7805:
7804:
7798:
7797:
7795:
7794:
7789:
7784:
7779:
7774:
7768:
7766:
7762:
7761:
7759:
7758:
7746:
7741:
7737:
7733:
7730:
7727:
7724:
7712:
7707:
7706:
7705:
7695:
7693:Sequence space
7690:
7682:
7669:
7664:
7659:
7654:
7650:
7638:
7637:
7636:
7631:
7617:
7613:
7594:
7593:
7592:
7578:
7574:
7555:
7543:
7540:
7537:
7532:
7529:
7526:
7522:
7509:
7501:
7496:
7483:
7478:
7470:
7465:
7453:
7449:
7445:
7440:
7435:
7432:
7429:
7425:
7412:
7404:
7399:
7387:
7384:
7381:
7378:
7375:
7364:
7355:
7353:
7347:
7346:
7344:
7343:
7333:
7328:
7323:
7318:
7313:
7308:
7303:
7298:
7288:
7282:
7280:
7276:
7275:
7273:
7272:
7267:
7262:
7257:
7252:
7244:
7230:
7222:
7217:
7212:
7207:
7202:
7197:
7191:
7189:
7185:
7184:
7182:
7181:
7171:
7170:
7169:
7164:
7159:
7149:
7148:
7147:
7142:
7137:
7127:
7126:
7125:
7120:
7115:
7110:
7108:Gelfand–Pettis
7105:
7100:
7090:
7089:
7088:
7083:
7078:
7073:
7068:
7058:
7053:
7048:
7043:
7042:
7041:
7031:
7025:
7023:
7019:
7018:
7016:
7015:
7010:
7005:
7000:
6995:
6990:
6985:
6980:
6975:
6970:
6965:
6960:
6959:
6958:
6948:
6943:
6938:
6933:
6928:
6923:
6918:
6913:
6908:
6903:
6898:
6893:
6888:
6883:
6881:Banach–Alaoglu
6878:
6876:Anderson–Kadec
6872:
6870:
6864:
6863:
6861:
6860:
6855:
6850:
6849:
6848:
6843:
6833:
6832:
6831:
6826:
6816:
6814:Operator space
6811:
6806:
6800:
6798:
6792:
6791:
6789:
6788:
6783:
6778:
6773:
6768:
6763:
6758:
6753:
6748:
6747:
6746:
6736:
6731:
6730:
6729:
6724:
6716:
6711:
6701:
6700:
6699:
6689:
6684:
6674:
6673:
6672:
6667:
6662:
6652:
6646:
6644:
6638:
6637:
6635:
6634:
6629:
6624:
6623:
6622:
6617:
6607:
6606:
6605:
6600:
6590:
6585:
6580:
6579:
6578:
6568:
6563:
6557:
6555:
6551:
6550:
6548:
6547:
6542:
6537:
6536:
6535:
6525:
6520:
6515:
6514:
6513:
6502:Locally convex
6499:
6498:
6497:
6487:
6482:
6477:
6471:
6469:
6465:
6464:
6462:
6461:
6454:Tensor product
6447:
6441:
6436:
6430:
6425:
6419:
6414:
6409:
6399:
6398:
6397:
6392:
6382:
6377:
6375:Banach lattice
6372:
6371:
6370:
6360:
6354:
6352:
6348:
6347:
6341:
6339:
6338:
6331:
6324:
6316:
6307:
6306:
6304:
6303:
6292:
6289:
6288:
6286:
6285:
6280:
6275:
6270:
6268:Choquet theory
6265:
6260:
6254:
6252:
6248:
6247:
6245:
6244:
6234:
6229:
6224:
6219:
6214:
6209:
6204:
6199:
6194:
6189:
6184:
6178:
6176:
6172:
6171:
6169:
6168:
6163:
6157:
6155:
6151:
6150:
6148:
6147:
6142:
6137:
6132:
6127:
6122:
6120:Banach algebra
6116:
6114:
6110:
6109:
6107:
6106:
6101:
6096:
6091:
6086:
6081:
6076:
6071:
6066:
6061:
6055:
6053:
6049:
6048:
6046:
6045:
6043:Banach–Alaoglu
6040:
6035:
6030:
6025:
6020:
6015:
6010:
6005:
5999:
5997:
5991:
5990:
5987:
5986:
5984:
5983:
5978:
5973:
5971:Locally convex
5968:
5954:
5949:
5943:
5941:
5937:
5936:
5934:
5933:
5928:
5923:
5918:
5913:
5908:
5903:
5898:
5893:
5888:
5882:
5876:
5872:
5871:
5857:
5855:
5854:
5847:
5840:
5832:
5823:
5822:
5820:
5819:
5814:
5809:
5804:
5799:
5794:
5789:
5784:
5779:
5774:
5769:
5764:
5759:
5754:
5749:
5744:
5734:
5732:Corona theorem
5729:
5723:
5721:
5717:
5716:
5714:
5713:
5711:Wiener algebra
5707:
5705:
5701:
5700:
5698:
5697:
5692:
5687:
5682:
5677:
5672:
5667:
5662:
5657:
5651:
5649:
5645:
5644:
5642:
5641:
5631:
5629:Pseudospectrum
5626:
5621:
5619:Dirac spectrum
5615:
5613:
5609:
5608:
5606:
5605:
5600:
5595:
5590:
5584:
5582:
5578:
5577:
5575:
5574:
5573:
5572:
5562:
5557:
5552:
5547:
5542:
5536:
5530:
5528:
5524:
5523:
5521:
5520:
5515:
5510:
5505:
5500:
5495:
5490:
5485:
5480:
5474:
5472:
5468:
5467:
5465:
5464:
5459:
5454:
5449:
5444:
5439:
5438:
5437:
5432:
5427:
5416:
5414:
5410:
5409:
5407:
5406:
5401:
5396:
5391:
5386:
5381:
5375:
5373:
5367:
5366:
5364:
5363:
5358:
5350:
5342:
5334:
5328:
5326:
5322:
5321:
5319:
5318:
5313:
5308:
5303:
5298:
5293:
5288:
5282:
5280:
5276:
5275:
5273:
5272:
5270:Operator space
5267:
5262:
5257:
5252:
5247:
5242:
5237:
5232:
5230:Banach algebra
5227:
5221:
5219:
5218:Basic concepts
5215:
5214:
5205:
5203:
5202:
5195:
5188:
5180:
5173:
5172:
5158:
5142:
5136:
5123:
5117:
5092:
5086:
5066:
5060:
5048:Bonsall, F. F.
5044:
5038:
5017:
5014:
5013:
5001:
4954:
4942:
4927:
4914:
4913:
4911:
4908:
4905:
4904:
4889:
4888:
4886:
4883:
4882:
4881:
4876:
4870:
4861:
4850:
4847:
4830:
4827:
4824:
4821:
4816:
4812:
4808:
4805:
4802:
4799:
4794:
4790:
4786:
4763:
4760:
4757:
4754:
4745:
4742:
4739:
4736:
4733:
4728:
4724:
4720:
4688:
4676:
4675:
4664:
4661:
4658:
4655:
4652:
4649:
4627:
4623:
4617:
4613:
4609:
4604:
4600:
4596:
4593:
4590:
4580:
4569:
4566:
4539:
4536:
4513:
4510:
4507:
4504:
4483:
4479:
4476:
4454:
4450:
4443:
4440:
4434:
4429:
4425:
4421:
4418:
4415:
4405:
4394:
4391:
4388:
4385:
4382:
4379:
4357:
4353:
4349:
4344:
4340:
4336:
4331:
4327:
4323:
4320:
4317:
4314:
4304:
4288:
4285:
4282:
4262:
4259:
4254:
4249:
4244:
4240:
4236:
4211:
4208:
4205:
4202:
4197:
4169:
4157:
4154:
4142:
4139:
4136:
4133:
4130:
4127:
4124:
4121:
4101:
4081:
4068:(that is, its
4051:
4048:
4045:
4042:
4039:
4036:
4033:
4030:
4027:
4024:
4021:
4018:
4015:
4012:
4009:
4006:
4000:
3997:
3991:
3988:
3968:
3965:
3962:
3959:
3956:
3936:
3933:
3930:
3927:
3924:
3921:
3918:
3898:
3892:
3889:
3866:
3863:
3860:
3857:
3854:
3851:
3848:
3845:
3842:
3836:
3833:
3809:
3788:
3785:
3782:
3779:
3756:
3753:
3730:
3703:
3700:
3677:
3671:
3668:
3662:
3659:
3656:
3653:
3650:
3647:
3644:
3625:
3622:
3619:
3616:
3593:
3590:
3587:
3584:
3581:
3559:
3555:
3534:
3514:
3510:
3489:
3469:
3466:
3463:
3459:
3455:
3452:
3443:and satisfies
3432:
3429:
3426:
3423:
3420:
3417:
3414:
3411:
3408:
3405:
3402:
3399:
3396:
3393:
3390:
3370:
3350:
3327:
3324:
3300:
3297:
3294:
3291:
3271:
3267:
3246:
3226:
3223:
3220:
3217:
3197:
3175:
3169:
3165:
3145:
3123:
3101:
3098:
3074:
3054:
3034:
3030:
3005:
2993:
2990:
2976:
2955:
2935:
2931:
2927:
2924:
2921:
2901:
2880:
2876:
2873:
2870:
2849:
2845:
2842:
2822:
2819:
2816:
2813:
2793:
2773:
2750:
2728:
2725:
2722:
2719:
2716:
2713:
2710:
2707:
2704:
2701:
2698:
2695:
2692:
2689:
2686:
2683:
2663:
2643:
2640:
2637:
2634:
2631:
2628:
2604:
2584:
2564:
2561:
2558:
2555:
2535:
2513:
2510:
2507:
2504:
2501:
2498:
2495:
2492:
2489:
2486:
2483:
2480:
2477:
2474:
2471:
2468:
2448:
2445:
2442:
2439:
2436:
2413:
2393:
2390:
2387:
2384:
2381:
2378:
2354:
2351:
2348:
2345:
2323:
2318:
2314:
2310:
2306:
2300:
2296:
2292:
2287:
2284:
2281:
2277:
2273:
2270:
2267:
2264:
2261:
2258:
2255:
2252:
2249:
2245:
2241:
2237:
2233:
2230:
2202:
2193:of an element
2182:
2179:
2176:
2173:
2149:
2146:
2126:
2123:
2120:
2099:
2078:
2058:
2055:
2034:
2030:
2027:
2024:
2004:
1981:
1978:
1975:
1972:
1952:
1949:
1946:
1943:
1934:of an element
1924:Main article:
1921:
1918:
1917:
1916:
1905:
1902:
1882:
1862:
1842:
1839:
1819:
1799:
1779:
1764:
1761:
1754:
1739:
1713:
1700:have the same
1689:
1686:
1666:
1663:
1643:
1640:
1637:
1634:
1631:
1628:
1607:
1603:
1600:
1597:
1594:
1591:
1588:
1563:
1542:
1538:
1475:
1472:
1471:
1470:
1459:
1445:
1430:
1419:Radon measures
1412:
1402:
1391:
1388:
1374:
1362:
1342:
1339:
1336:
1333:
1319:
1308:
1305:
1302:
1299:
1294:
1290:
1286:
1283:
1280:
1277:
1257:
1254:
1251:
1248:
1245:
1241:
1237:
1232:
1229:
1225:
1220:
1216:
1213:
1210:
1207:
1204:
1201:
1198:
1195:
1192:
1189:
1186:
1183:
1160:
1140:
1120:
1117:
1114:
1109:
1105:
1080:
1050:
1039:
1028:
1025:
1022:
1019:
1016:
1013:
993:
965:
947:
936:
929:
922:
911:
907:
901:
897:
891:
887:
883:
880:
877:
872:
868:
862:
858:
853:
849:
845:
839:
835:
831:
828:
825:
820:
816:
811:
806:
800:
796:
792:
789:
786:
781:
777:
772:
750:
744:
740:
735:
728:
724:
721:
718:
715:
693:
688:
664:
659:
646:
623:
603:
592:
589:absolute value
568:
565:
562:
557:
553:
520:
500:
497:
494:
489:
485:
460:
437:
434:
431:
426:
422:
409:
406:
402:-adic analysis
391:
368:
331:
327:
300:
296:
266:
262:
237:
209:
206:
169:
166:
163:
160:
157:
154:
145:
142:
139:
135:
132:
129:
126:
120:
117:
113:
110:
58:
42:, named after
40:Banach algebra
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
7835:
7824:
7821:
7819:
7816:
7814:
7811:
7810:
7808:
7793:
7790:
7788:
7785:
7783:
7780:
7778:
7775:
7773:
7770:
7769:
7767:
7763:
7757:
7739:
7735:
7731:
7728:
7722:
7713:
7711:
7708:
7704:
7701:
7700:
7699:
7696:
7694:
7691:
7689:
7688:
7683:
7681:
7667:
7662:
7652:
7648:
7639:
7635:
7632:
7630:
7611:
7602:
7601:
7600:
7599:
7595:
7591:
7572:
7563:
7562:
7561:
7560:
7556:
7554:
7530:
7527:
7524:
7520:
7510:
7508:
7507:
7502:
7500:
7497:
7495:
7493:
7489:
7484:
7482:
7479:
7477:
7476:
7471:
7469:
7466:
7464:
7438:
7433:
7430:
7427:
7423:
7413:
7411:
7410:
7405:
7403:
7400:
7398:
7376:
7373:
7365:
7363:
7362:
7357:
7356:
7354:
7352:
7348:
7342:
7338:
7334:
7332:
7329:
7327:
7324:
7322:
7319:
7317:
7314:
7312:
7311:Extreme point
7309:
7307:
7304:
7302:
7299:
7297:
7293:
7289:
7287:
7284:
7283:
7281:
7277:
7271:
7268:
7266:
7263:
7261:
7258:
7256:
7253:
7251:
7245:
7242:
7238:
7234:
7231:
7229:
7223:
7221:
7218:
7216:
7213:
7211:
7208:
7206:
7203:
7201:
7198:
7196:
7193:
7192:
7190:
7188:Types of sets
7186:
7179:
7175:
7172:
7168:
7165:
7163:
7160:
7158:
7155:
7154:
7153:
7150:
7146:
7143:
7141:
7138:
7136:
7133:
7132:
7131:
7128:
7124:
7121:
7119:
7116:
7114:
7111:
7109:
7106:
7104:
7101:
7099:
7096:
7095:
7094:
7091:
7087:
7084:
7082:
7079:
7077:
7074:
7072:
7069:
7067:
7064:
7063:
7062:
7059:
7057:
7054:
7052:
7051:Convex series
7049:
7047:
7046:Bochner space
7044:
7040:
7037:
7036:
7035:
7032:
7030:
7027:
7026:
7024:
7020:
7014:
7011:
7009:
7006:
7004:
7001:
6999:
6998:Riesz's lemma
6996:
6994:
6991:
6989:
6986:
6984:
6983:Mazur's lemma
6981:
6979:
6976:
6974:
6971:
6969:
6966:
6964:
6961:
6957:
6954:
6953:
6952:
6949:
6947:
6944:
6942:
6939:
6937:
6936:Gelfand–Mazur
6934:
6932:
6929:
6927:
6924:
6922:
6919:
6917:
6914:
6912:
6909:
6907:
6904:
6902:
6899:
6897:
6894:
6892:
6889:
6887:
6884:
6882:
6879:
6877:
6874:
6873:
6871:
6869:
6865:
6859:
6856:
6854:
6851:
6847:
6844:
6842:
6839:
6838:
6837:
6834:
6830:
6827:
6825:
6822:
6821:
6820:
6817:
6815:
6812:
6810:
6807:
6805:
6802:
6801:
6799:
6797:
6793:
6787:
6784:
6782:
6779:
6777:
6774:
6772:
6769:
6767:
6764:
6762:
6759:
6757:
6754:
6752:
6749:
6745:
6742:
6741:
6740:
6737:
6735:
6732:
6728:
6725:
6723:
6720:
6719:
6717:
6715:
6712:
6710:
6706:
6702:
6698:
6695:
6694:
6693:
6690:
6688:
6685:
6683:
6679:
6675:
6671:
6668:
6666:
6663:
6661:
6658:
6657:
6656:
6653:
6651:
6648:
6647:
6645:
6643:
6639:
6633:
6630:
6628:
6625:
6621:
6618:
6616:
6613:
6612:
6611:
6608:
6604:
6601:
6599:
6596:
6595:
6594:
6591:
6589:
6586:
6584:
6581:
6577:
6574:
6573:
6572:
6569:
6567:
6564:
6562:
6559:
6558:
6556:
6552:
6546:
6543:
6541:
6538:
6534:
6531:
6530:
6529:
6526:
6524:
6521:
6519:
6516:
6512:
6508:
6505:
6504:
6503:
6500:
6496:
6493:
6492:
6491:
6488:
6486:
6483:
6481:
6478:
6476:
6473:
6472:
6470:
6466:
6459:
6455:
6451:
6448:
6446:
6442:
6440:
6437:
6435:) convex
6434:
6431:
6429:
6426:
6424:
6420:
6418:
6415:
6413:
6410:
6408:
6404:
6400:
6396:
6393:
6391:
6388:
6387:
6386:
6383:
6381:
6380:Grothendieck
6378:
6376:
6373:
6369:
6366:
6365:
6364:
6361:
6359:
6356:
6355:
6353:
6349:
6344:
6337:
6332:
6330:
6325:
6323:
6318:
6317:
6314:
6302:
6294:
6293:
6290:
6284:
6281:
6279:
6276:
6274:
6273:Weak topology
6271:
6269:
6266:
6264:
6261:
6259:
6256:
6255:
6253:
6249:
6242:
6238:
6235:
6233:
6230:
6228:
6225:
6223:
6220:
6218:
6215:
6213:
6210:
6208:
6205:
6203:
6200:
6198:
6197:Index theorem
6195:
6193:
6190:
6188:
6185:
6183:
6180:
6179:
6177:
6173:
6167:
6164:
6162:
6159:
6158:
6156:
6154:Open problems
6152:
6146:
6143:
6141:
6138:
6136:
6133:
6131:
6128:
6126:
6123:
6121:
6118:
6117:
6115:
6111:
6105:
6102:
6100:
6097:
6095:
6092:
6090:
6087:
6085:
6082:
6080:
6077:
6075:
6072:
6070:
6067:
6065:
6062:
6060:
6057:
6056:
6054:
6050:
6044:
6041:
6039:
6036:
6034:
6031:
6029:
6026:
6024:
6021:
6019:
6016:
6014:
6011:
6009:
6006:
6004:
6001:
6000:
5998:
5996:
5992:
5982:
5979:
5977:
5974:
5972:
5969:
5966:
5962:
5958:
5955:
5953:
5950:
5948:
5945:
5944:
5942:
5938:
5932:
5929:
5927:
5924:
5922:
5919:
5917:
5914:
5912:
5909:
5907:
5904:
5902:
5899:
5897:
5894:
5892:
5889:
5887:
5884:
5883:
5880:
5877:
5873:
5868:
5864:
5860:
5853:
5848:
5846:
5841:
5839:
5834:
5833:
5830:
5818:
5815:
5813:
5810:
5808:
5805:
5803:
5800:
5798:
5795:
5793:
5790:
5788:
5785:
5783:
5780:
5778:
5775:
5773:
5770:
5768:
5765:
5763:
5760:
5758:
5755:
5753:
5750:
5748:
5745:
5742:
5738:
5735:
5733:
5730:
5728:
5725:
5724:
5722:
5718:
5712:
5709:
5708:
5706:
5702:
5696:
5693:
5691:
5688:
5686:
5683:
5681:
5678:
5676:
5673:
5671:
5668:
5666:
5663:
5661:
5658:
5656:
5653:
5652:
5650:
5648:Miscellaneous
5646:
5639:
5635:
5632:
5630:
5627:
5625:
5622:
5620:
5617:
5616:
5614:
5610:
5604:
5601:
5599:
5596:
5594:
5591:
5589:
5586:
5585:
5583:
5579:
5571:
5568:
5567:
5566:
5563:
5561:
5558:
5556:
5553:
5551:
5548:
5546:
5543:
5541:
5537:
5535:
5532:
5531:
5529:
5525:
5519:
5516:
5514:
5511:
5509:
5506:
5504:
5501:
5499:
5496:
5494:
5491:
5489:
5486:
5484:
5481:
5479:
5476:
5475:
5473:
5469:
5463:
5460:
5458:
5455:
5453:
5450:
5448:
5445:
5443:
5440:
5436:
5433:
5431:
5428:
5426:
5423:
5422:
5421:
5418:
5417:
5415:
5413:Decomposition
5411:
5405:
5402:
5400:
5397:
5395:
5392:
5390:
5387:
5385:
5382:
5380:
5377:
5376:
5374:
5372:
5368:
5362:
5359:
5357:
5354:
5351:
5349:
5346:
5343:
5341:
5338:
5335:
5333:
5330:
5329:
5327:
5323:
5317:
5314:
5312:
5309:
5307:
5304:
5302:
5299:
5297:
5294:
5292:
5289:
5287:
5284:
5283:
5281:
5277:
5271:
5268:
5266:
5263:
5261:
5258:
5256:
5253:
5251:
5248:
5246:
5243:
5241:
5238:
5236:
5233:
5231:
5228:
5226:
5223:
5222:
5220:
5216:
5212:
5208:
5201:
5196:
5194:
5189:
5187:
5182:
5181:
5178:
5169:
5165:
5161:
5155:
5151:
5147:
5143:
5139:
5137:0-226-54203-3
5133:
5129:
5124:
5120:
5118:0-521-53584-0
5114:
5110:
5106:
5102:
5098:
5097:Willis, G. A.
5093:
5089:
5087:0-387-97245-5
5083:
5079:
5075:
5071:
5070:Conway, J. B.
5067:
5063:
5061:0-387-06386-2
5057:
5053:
5049:
5045:
5041:
5039:0-521-38729-9
5035:
5030:
5029:
5023:
5019:
5018:
5010:
5009:Takesaki 1979
5005:
5002:
4997:
4993:
4989:
4985:
4981:
4977:
4973:
4969:
4965:
4958:
4955:
4951:
4946:
4943:
4939:
4934:
4932:
4928:
4924:
4919:
4916:
4909:
4901:
4894:
4891:
4884:
4880:
4877:
4874:
4871:
4865:
4862:
4856:
4853:
4852:
4848:
4846:
4844:
4825:
4814:
4810:
4803:
4797:
4792:
4788:
4775:
4761:
4758:
4755:
4752:
4740:
4734:
4726:
4722:
4710:
4705:
4703:
4662:
4659:
4656:
4653:
4650:
4647:
4625:
4621:
4615:
4611:
4607:
4602:
4594:
4591:
4581:
4567:
4564:
4556:
4534:
4511:
4508:
4505:
4502:
4477:
4474:
4452:
4448:
4438:
4432:
4427:
4419:
4416:
4406:
4392:
4389:
4386:
4383:
4380:
4377:
4355:
4351:
4347:
4342:
4338:
4334:
4329:
4321:
4318:
4315:
4305:
4302:
4286:
4283:
4280:
4260:
4257:
4252:
4247:
4242:
4238:
4234:
4225:
4224:
4223:
4209:
4203:
4200:
4195:
4183:
4167:
4155:
4153:
4140:
4131:
4119:
4099:
4079:
4071:
4067:
4062:
4049:
4040:
4031:
4028:
4025:
4019:
4013:
4007:
3995:
3986:
3966:
3960:
3928:
3916:
3896:
3887:
3864:
3858:
3852:
3849:
3843:
3831:
3783:
3751:
3728:
3720:
3698:
3666:
3657:
3654:
3648:
3642:
3623:
3620:
3617:
3614:
3605:
3591:
3585:
3557:
3553:
3532:
3512:
3487:
3467:
3464:
3450:
3430:
3424:
3418:
3412:
3406:
3403:
3397:
3394:
3388:
3368:
3348:
3339:
3325:
3322:
3314:
3295:
3269:
3244:
3221:
3195:
3167:
3163:
3143:
3099:
3096:
3088:
3087:maximal ideal
3072:
3052:
3032:
3019:
3003:
2991:
2989:
2953:
2946:this algebra
2933:
2925:
2922:
2919:
2899:
2874:
2871:
2868:
2843:
2840:
2820:
2817:
2814:
2811:
2791:
2771:
2762:
2748:
2739:
2726:
2720:
2717:
2714:
2711:
2705:
2699:
2693:
2687:
2681:
2661:
2638:
2632:
2629:
2626:
2618:
2602:
2582:
2559:
2553:
2533:
2524:
2511:
2502:
2496:
2490:
2487:
2478:
2472:
2466:
2446:
2440:
2434:
2426:
2411:
2391:
2388:
2382:
2376:
2368:
2352:
2349:
2346:
2343:
2334:
2321:
2316:
2312:
2308:
2298:
2294:
2279:
2271:
2262:
2256:
2253:
2250:
2247:
2239:
2220:
2216:
2200:
2177:
2171:
2163:
2160:and thus is
2147:
2144:
2121:
2076:
2056:
2053:
2028:
2025:
2022:
2002:
1995:
1976:
1970:
1950:
1947:
1944:
1941:
1933:
1927:
1919:
1903:
1900:
1880:
1860:
1840:
1837:
1817:
1797:
1777:
1769:
1765:
1762:
1759:
1755:
1752:
1748:
1744:
1743:zero divisors
1740:
1737:
1733:
1729:
1728:
1727:
1724:
1711:
1703:
1687:
1684:
1664:
1661:
1641:
1638:
1635:
1632:
1629:
1626:
1601:
1598:
1595:
1592:
1589:
1586:
1578:
1540:
1526:
1524:
1520:
1516:
1511:
1509:
1505:
1501:
1497:
1493:
1489:
1485:
1481:
1473:
1468:
1464:
1460:
1435:
1431:
1428:
1424:
1420:
1416:
1413:
1410:
1409:Hilbert space
1406:
1403:
1389:
1386:
1378:
1375:
1360:
1337:
1331:
1323:
1320:
1306:
1300:
1292:
1288:
1284:
1281:
1278:
1275:
1252:
1246:
1243:
1239:
1235:
1230:
1227:
1223:
1218:
1214:
1208:
1202:
1199:
1196:
1190:
1184:
1181:
1174:
1158:
1138:
1115:
1107:
1103:
1094:
1078:
1070:
1067:
1064:
1048:
1040:
1026:
1020:
1017:
1014:
1011:
991:
983:
979:
978:operator norm
963:
955:
952:
948:
945:
941:
937:
934:
930:
927:
923:
909:
905:
899:
895:
889:
885:
881:
878:
875:
870:
866:
860:
856:
851:
847:
843:
837:
833:
829:
826:
823:
818:
814:
809:
804:
798:
794:
790:
787:
784:
779:
775:
770:
742:
738:
722:
716:
691:
662:
647:
644:
640:
636:
621:
601:
593:
590:
586:
585:
584:
582:
579:is in fact a
563:
555:
551:
542:
538:
534:
518:
495:
487:
483:
474:
458:
451:
432:
424:
420:
407:
405:
403:
389:
380:
379:-adic numbers
366:
356:
354:
349:
347:
329:
325:
316:
298:
294:
285:
282:
264:
260:
251:
250:isometrically
235:
227:
223:
207:
204:
196:
193:if it has an
192:
187:
185:
180:
167:
164:
161:
158:
155:
152:
140:
130:
124:
115:
111:
100:
96:
92:
89:, that is, a
88:
84:
80:
76:
72:
56:
49:
45:
44:Stefan Banach
41:
37:
34:, especially
33:
19:
7765:Applications
7686:
7597:
7558:
7505:
7491:
7487:
7474:
7408:
7360:
7247:Linear cone
7240:
7236:
7225:Convex cone
7118:Paley–Wiener
6978:Mackey–Arens
6968:Krein–Milman
6921:Closed range
6916:Closed graph
6886:Banach–Mazur
6803:
6766:Self-adjoint
6670:sesquilinear
6403:Polynomially
6343:Banach space
6263:Balanced set
6237:Distribution
6175:Applications
6119:
6028:Krein–Milman
6013:Closed graph
5720:Applications
5550:Disk algebra
5404:Spectral gap
5279:Main results
5229:
5149:
5146:Takesaki, M.
5127:
5100:
5073:
5051:
5027:
5004:
4971:
4967:
4957:
4945:
4918:
4893:
4776:
4711:, that is,
4706:
4677:
4553:denotes the
4159:
4063:
3979:Explicitly,
3606:
3341:A character
3340:
3208:and the set
3017:
3016:be a unital
2995:
2763:
2740:
2525:
2335:
2111:with radius
1931:
1929:
1725:
1575:cannot be a
1527:
1512:
1484:power series
1477:
1093:Haar measure
706:) with norm
411:
357:
350:
314:
221:
190:
188:
181:
91:normed space
87:Banach space
83:normed field
39:
29:
7486:Continuous
7321:Linear span
7306:Convex hull
7286:Affine hull
7145:holomorphic
7081:holomorphic
7061:Derivatives
6951:Hahn–Banach
6891:Banach–Saks
6809:C*-algebras
6776:Trace class
6739:Functionals
6627:Ultrastrong
6540:Quasinormed
6192:Heat kernel
6182:Hardy space
6089:Trace class
6003:Hahn–Banach
5965:Topological
5747:Heat kernel
5447:Compression
5332:Isospectral
5022:Bollobás, B
4950:Conway 1990
4938:Conway 1990
4923:Conway 1990
3156:is closed,
3018:commutative
2425:holomorphic
2137:and center
1963:denoted by
1579:; that is,
1513:The set of
1434:quaternions
1173:convolution
926:quaternions
643:matrix norm
226:commutative
222:commutative
32:mathematics
7807:Categories
7239:), and (Hw
7140:continuous
7076:functional
6824:C*-algebra
6709:Continuous
6571:Dual space
6545:Stereotype
6523:Metrizable
6450:Projective
6125:C*-algebra
5940:Properties
5425:Continuous
5240:C*-algebra
5235:B*-algebra
4910:References
4843:C*-algebra
4495:and every
4467:for every
4301:involution
4066:semisimple
2861:such that
2015:such that
1758:Noetherian
1577:commutator
1474:Properties
1405:C*-algebra
951:continuous
940:continuous
637:becomes a
581:C*-algebra
541:involution
184:continuous
7698:Sobolev W
7641:Schwartz
7616:∞
7577:∞
7573:ℓ
7539:Ω
7525:λ
7383:Σ
7265:Symmetric
7200:Absorbing
7113:regulated
7093:Integrals
6946:Goldstine
6781:Transpose
6718:Fredholm
6588:Ultraweak
6576:Dual norm
6507:Seminorms
6475:Barrelled
6445:Injective
6433:Uniformly
6407:Reflexive
6099:Unbounded
6094:Transpose
6052:Operators
5981:Separable
5976:Reflexive
5961:Algebraic
5947:Barrelled
5211:-algebras
5168:0938-0396
4988:0002-9939
4829:‖
4823:‖
4820:‖
4815:∗
4807:‖
4801:‖
4793:∗
4785:‖
4756:∈
4744:‖
4738:‖
4732:‖
4727:∗
4719:‖
4709:isometric
4702:*-algebra
4657:∈
4626:∗
4616:∗
4603:∗
4565:λ
4538:¯
4535:λ
4506:∈
4478:∈
4475:λ
4453:∗
4442:¯
4439:λ
4428:∗
4417:λ
4387:∈
4356:∗
4343:∗
4330:∗
4284:∈
4253:∗
4243:∗
4207:→
4196:∗
4126:Δ
4035:Δ
4032:∈
4029:χ
4014:χ
3999:^
3987:σ
3955:Δ
3923:Δ
3891:^
3853:χ
3844:χ
3835:^
3821:given by
3778:Δ
3755:^
3702:^
3670:^
3658:σ
3643:σ
3618:∈
3580:Δ
3558:∗
3451:χ
3419:χ
3407:χ
3389:χ
3349:χ
3290:Δ
3216:Δ
2926:λ
2875:λ
2872:−
2844:∈
2841:λ
2833:there is
2815:∈
2718:∈
2682:σ
2630:∈
2497:σ
2467:σ
2435:σ
2389:∈
2347:∈
2305:‖
2291:‖
2286:∞
2283:→
2257:σ
2254:∈
2251:λ
2240:λ
2221:formula:
2215:non-empty
2172:σ
2125:‖
2119:‖
2029:λ
2026:−
2003:λ
1971:σ
1945:∈
1636:∈
1602:≠
1593:−
1285:∈
1247:μ
1228:−
1200:∫
1139:μ
1079:μ
1066:Hausdorff
1024:∞
1015:
879:…
827:…
788:…
720:‖
714:‖
539:being an
162:∈
144:‖
138:‖
134:‖
128:‖
125:≤
119:‖
109:‖
81:complete
69:over the
7634:weighted
7504:Hilbert
7481:Bs space
7351:Examples
7316:Interior
7292:Relative
7270:Zonotope
7249:(subset)
7227:(subset)
7178:Strongly
7157:Lebesgue
7152:Measures
7022:Analysis
6868:Theorems
6819:Spectrum
6744:positive
6727:operator
6665:operator
6655:Bilinear
6620:operator
6603:operator
6583:Operator
6480:Complete
6428:Strictly
6301:Category
6113:Algebras
5995:Theorems
5952:Complete
5921:Schwartz
5867:glossary
5812:Weyl law
5757:Lax pair
5704:Examples
5538:With an
5457:Discrete
5435:Residual
5371:Spectrum
5356:operator
5348:operator
5340:operator
5255:Spectrum
5148:(1979).
5099:(2003).
5072:(1990).
5024:(1990).
4849:See also
4640:for all
4370:for all
4273:for all
3607:For any
3282:The set
1932:spectrum
1702:spectrum
1519:open set
1490:and the
1478:Several
1458:numbers.
1421:on some
933:supremum
635:matrices
408:Examples
353:spectrum
315:a priori
95:complete
93:that is
46:, is an
7499:Hardy H
7402:c space
7339:)
7294:)
7215:Bounded
7103:Dunford
7098:Bochner
7071:Gateaux
7066:Fréchet
6841:of ODEs
6786:Unitary
6761:Nuclear
6692:Compact
6682:Bounded
6650:Adjoint
6490:Fréchet
6485:F-space
6456: (
6452:)
6405:)
6385:Hilbert
6358:Asplund
6104:Unitary
6084:Nuclear
6069:Compact
6064:Bounded
6059:Adjoint
6033:Min–max
5926:Sobolev
5911:Nuclear
5901:Hilbert
5896:Fréchet
5861: (
5353:Unitary
4996:2160559
3717:is the
2162:compact
1994:scalars
1131:of all
1091:is its
533:compact
471:, that
97:in the
75:complex
7415:Besov
7255:Radial
7220:Convex
7205:Affine
7174:Weakly
7167:Vector
7039:bundle
6829:radius
6756:Normal
6722:kernel
6687:Closed
6610:Strong
6528:Normed
6518:Mackey
6363:Banach
6345:topics
6079:Normal
5916:Orlicz
5906:Hölder
5886:Banach
5875:Spaces
5863:topics
5337:Normal
5166:
5156:
5134:
5115:
5084:
5058:
5036:
4994:
4986:
4524:here,
3688:where
3045:Since
2619:. For
2336:Given
1751:closed
954:linear
639:unital
535:. The
281:closed
191:unital
122:
99:metric
7490:with
7337:Quasi
7331:Polar
7135:Borel
7086:quasi
6615:polar
6598:polar
6412:Riesz
5891:Besov
5430:Point
4992:JSTOR
4885:Notes
4841:is a
1553:then
1061:is a
284:ideal
7488:C(K)
7123:weak
6660:form
6593:Weak
6566:Dual
6533:norm
6495:tame
6368:list
6239:(or
5957:Dual
5361:Unit
5209:and
5164:ISSN
5154:ISBN
5132:ISBN
5113:ISBN
5082:ISBN
5056:ISBN
5034:ISBN
4984:ISSN
4112:and
2996:Let
2764:Let
2365:the
1677:and
1268:for
1071:and
924:The
677:(or
614:-by-
220:and
71:real
38:, a
6705:Dis
5105:doi
4976:doi
4972:123
4557:of
3799:to
3721:of
3500:to
3257:to
3136:in
3089:of
2276:lim
2229:sup
2213:is
1893:of
1749:is
1461:An
1041:If
1012:dim
984:on
727:max
531:is
286:of
73:or
30:In
7809::
7475:BV
7409:BK
7361:AC
7243:))
7176:/
6678:Un
5865:–
5162:.
5111:.
5080:.
4990:.
4982:.
4970:.
4966:.
4930:^
4845:.
4704:.
4303:).
3468:1.
1738:.)
1712:0.
543:,
475:.
404:.
7745:)
7740:p
7736:L
7732:,
7729:X
7726:(
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