2932:
700:
2149:
from the category of commutative C*-algebras with unit and unit-preserving continuous *-homomorphisms, to the category of compact
Hausdorff spaces and continuous maps. This functor is one half of a
78:
One of
Gelfand's original applications (and one which historically motivated much of the study of Banach algebras) was to give a much shorter and more conceptual proof of a celebrated lemma of
1820:
In particular, the spectrum of a commutative C*-algebra is a locally compact
Hausdorff space: In the unital case, i.e. where the C*-algebra has a multiplicative unit element 1, all characters
1761:
753:
369:
1408:
131:
1325:
848:
2075:
1587:
1210:
1110:
1642:
1476:
1448:
911:
963:
809:
1669:
1357:
1281:
1232:
331:
1259:
1037:
780:
599:
568:
490:
440:
648:
517:
1498:
is a completely regular
Hausdorff space, then the representation space of the Banach algebra of bounded continuous functions is the StoneâÄech compactification of
3455:
1516:
1496:
1154:
1134:
1057:
1010:
987:
933:
868:
619:
537:
463:
413:
393:
305:
3557:
2821:
1844:âȘ {0}, where 0 is the zero homomorphism, and the removal of a single point from a compact Hausdorff space yields a locally compact Hausdorff space.
3190:
2657:
2484:
3212:
2263:) this is *-isomorphic to an algebra of continuous functions on a locally compact space. This observation leads almost immediately to:
2647:
3195:
2968:
3217:
2774:
2629:
83:
3542:
3205:
2605:
653:
3435:
58:
of an integrable function. In the latter case, the
GelfandâNaimark representation theorem is one avenue in the development of
3288:
3086:
3283:
2497:
1451:
3440:
2586:
2477:
2421:
2395:
2376:
1695:
705:
3258:
2856:
340:
3227:
2501:
3141:
3025:
1362:
96:
1286:
3628:
3450:
2961:
2652:
818:
2223:, which though not quite analogous to the Gelfand representation, does provide a concrete representation of
3618:
3076:
2935:
2708:
2642:
2470:
2032:
1552:
1171:
1066:
1596:
3587:
3507:
3061:
2672:
2212:
1907:
and as such can be given the relative weak-* topology. This is the topology of pointwise convergence. A
1456:
3562:
3460:
3340:
2917:
2871:
2795:
2677:
1420:
571:
1991:
can be regarded as a metric space. So the topology can be characterized via convergence of sequences.
873:
54:
In the former case, one may regard the
Gelfand representation as a far-reaching generalization of the
3567:
3430:
3263:
3248:
3056:
3020:
2912:
2728:
2150:
1060:
939:
785:
3623:
3608:
3159:
3149:
3030:
2954:
2764:
2662:
2565:
1772:
1647:
1333:
1264:
1215:
314:
3522:
3497:
3315:
3304:
3015:
2861:
2637:
540:
1689:, not just as sets but as topological spaces. The Gelfand representation is then an isomorphism
3373:
3363:
3358:
2892:
2836:
2800:
1237:
1015:
758:
577:
546:
468:
418:
3118:
2599:
142:
2595:
627:
3613:
3532:
3511:
3425:
3310:
3273:
2875:
2320:
This allows us to apply continuous functions to bounded normal operators on
Hilbert space.
2122:. (See the earlier remarks for the general, commutative Banach algebra case.) For any such
2119:
495:
334:
2462:
8:
3335:
3071:
2841:
2779:
2493:
1999:
25:
3465:
3394:
3325:
3169:
3131:
2866:
2733:
2451:
1501:
1481:
1157:
1139:
1119:
1042:
995:
972:
918:
853:
812:
604:
522:
448:
398:
378:
290:
170:
3572:
3547:
3232:
3154:
2846:
2417:
2391:
2372:
2145:
In the case of C*-algebras with unit, the spectrum map gives rise to a contravariant
1908:
1328:
148:
55:
181:
is obtained by considering the pointwise operations of addition and multiplication.
3577:
3278:
3126:
3081:
3005:
2851:
2769:
2738:
2718:
2703:
2698:
2693:
2443:
1411:
1113:
2530:
3552:
3537:
3445:
3408:
3404:
3368:
3330:
3268:
3253:
3222:
3164:
3123:
3110:
3035:
2977:
2946:
2713:
2667:
2615:
2610:
2581:
2413:
2364:
2154:
1980:
1896:
204:
145:
63:
59:
2540:
1875:. For unital C*-algebras, the two notions are connected in the following way: Ï(
3502:
3481:
3399:
3389:
3200:
3107:
3040:
3000:
2902:
2754:
2555:
2431:
2405:
2216:
1934:
308:
245:
178:
79:
40:
2107:
The spectrum of a commutative C*-algebra can also be viewed as the set of all
1817:, so that this definition of the term 'character' agrees with the one above.)
1410:, the group algebra of the multiplicative reals, the Gelfand transform is the
3602:
2907:
2831:
2560:
2545:
2535:
2108:
1987:
on bounded subsets. Thus the spectrum of a separable commutative C*-algebra
253:
211:
is the common zero set of a family of continuous complex-valued functions on
67:
29:
3320:
3174:
3115:
2897:
2550:
2520:
2201:
188:
2255:, or equivalently if and only if it generates a commutative C*-algebra C*(
2235:
One of the most significant applications is the existence of a continuous
3517:
3102:
2826:
2816:
2723:
2525:
282:
37:
17:
3010:
2759:
2591:
2455:
2009:
1984:
1828:(1) is the complex number one. This excludes the zero homomorphism. So
47:
2130:
is one-dimensional (by the
Gelfand-Mazur theorem), and therefore any
1679:
are of this form; a more precise analysis shows that we may identify Ί
989:. In general, the representation is neither injective nor surjective.
570:
turns out to be locally compact and
Hausdorff. (This follows from the
2995:
2981:
1814:
850:
defines a norm-decreasing, unit-preserving algebra homomorphism from
601:
is compact (in the topology just defined) if and only if the algebra
2447:
3582:
3527:
133:
whose translates span dense subspaces in the respective algebras.
2146:
1832:
is closed under weak-* convergence and the spectrum is actually
1212:
is a Banach algebra under the convolution, the group algebra of
2157:
being the functor that assigns to each compact
Hausdorff space
1859:
of an algebra with unit 1, that is the set of complex numbers
1136:. Thus the Gelfand representation is injective if and only if
1063:). As a consequence, the kernel of the Gelfand representation
2293:) from the algebra of continuous functions on the spectrum Ï(
275:), the algebra of all continuous complex-valued functions on
82:(see the citation below), characterizing the elements of the
1801:
to the complex numbers. Elements of the spectrum are called
2492:
1766:
695:{\displaystyle {\widehat {a}}:\Phi _{A}\to {\mathbb {C} }}
1851:
is an overloaded word. It also refers to the spectrum Ï(
1809:. (It can be shown that every algebra homomorphism from
2086:. The Gelfand map Îł is an isometric *-isomorphism from
199:
being locally compact and Hausdorff is that this turns
1012:
has an identity element, there is a bijection between
283:
Gelfand representation of a commutative Banach algebra
2035:
1698:
1650:
1599:
1555:
1504:
1484:
1459:
1423:
1365:
1336:
1289:
1267:
1240:
1218:
1174:
1142:
1122:
1069:
1045:
1018:
998:
975:
942:
921:
876:
856:
821:
788:
761:
708:
656:
630:
607:
580:
549:
525:
498:
471:
451:
421:
401:
381:
343:
317:
293:
99:
2010:
Statement of the commutative GelfandâNaimark theorem
3558:Spectral theory of ordinary differential equations
2976:
2822:Spectral theory of ordinary differential equations
2172:)). In particular, given compact Hausdorff spaces
2069:
1755:
1663:
1636:
1581:
1510:
1490:
1470:
1442:
1402:
1351:
1319:
1275:
1253:
1226:
1204:
1148:
1128:
1104:
1051:
1031:
1004:
981:
957:
927:
905:
862:
842:
803:
774:
747:
694:
642:
613:
593:
562:
531:
511:
484:
457:
434:
407:
387:
363:
325:
299:
125:
50:, this representation is an isometric isomorphism.
3456:SchröderâBernstein theorems for operator algebras
2312:It maps the identity function on the spectrum to
3600:
1836:. In the non-unital case, the weak-* closure of
184:The involution is pointwise complex conjugation.
66:, and generalizes the notion of diagonalizing a
2412:. Graduate Texts in Mathematics. Vol. 96.
2336:
1756:{\displaystyle C_{0}(X)\to C_{0}(\Phi _{A}).\ }
748:{\displaystyle {\widehat {a}}(\phi )=\phi (a)}
173:is in a natural way a commutative C*-algebra:
2962:
2478:
2385:
2080:be the Gelfand representation defined above.
1675:, and it can be shown that all characters of
364:{\displaystyle \Phi \colon A\to \mathbb {C} }
2305:It maps 1 to the multiplicative identity of
2259:). By the Gelfand isomorphism applied to C*(
539:; moreover, when equipped with the relative
165:) of continuous complex-valued functions on
2138:gives rise to a complex-valued function on
2969:
2955:
2485:
2471:
2006:), where Îł is the Gelfand representation.
1813:to the complex numbers is automatically a
215:, allowing one to recover the topology of
1526:As motivation, consider the special case
1464:
1403:{\displaystyle A=L^{1}(\mathbb {R} _{+})}
1387:
1310:
1269:
1220:
1195:
687:
357:
319:
207:. In such a space every closed subset of
126:{\displaystyle \ell ^{1}({\mathbf {Z} })}
2775:Group algebra of a locally compact group
1767:The spectrum of a commutative C*-algebra
1320:{\displaystyle f\in L^{1}(\mathbb {R} )}
445:It can be shown that every character on
2363:
2353:A course in commutative Banach algebras
843:{\displaystyle a\mapsto {\widehat {a}}}
465:is automatically continuous, and hence
3601:
2430:
2404:
1521:
3289:Spectral theory of normal C*-algebras
3087:Spectral theory of normal C*-algebras
2950:
2466:
2215:is a result for arbitrary (abstract)
2070:{\displaystyle \gamma :A\to C_{0}(X)}
1582:{\displaystyle \varphi _{x}\in A^{*}}
1205:{\displaystyle A=L^{1}(\mathbb {R} )}
1105:{\displaystyle A\to C_{0}(\Phi _{A})}
337:(a multiplicative linear functional)
73:
3284:Spectral theory of compact operators
2018:be a commutative C*-algebra and let
1637:{\displaystyle \varphi _{x}(f)=f(x)}
519:of continuous linear functionals on
136:
43:as algebras of continuous functions;
2442:(1). Annals of Mathematics: 1â100.
2386:Bonsall, F. F.; Duncan, J. (1973).
2196:) (as a C*-algebra) if and only if
1983:C*-algebra, the weak-* topology is
782:and the topology on it ensure that
13:
3436:CohenâHewitt factorization theorem
2273:be a C*-algebra with identity and
2239:for normal elements in C*-algebra
2153:between these two categories (its
1735:
1471:{\displaystyle \beta \mathbb {N} }
1450:, the representation space is the
1435:
1242:
1090:
1020:
891:
763:
673:
582:
551:
473:
423:
344:
14:
3640:
3441:Extensions of symmetric operators
2339:General theory of Banach algebras
2104:See the Arveson reference below.
1443:{\displaystyle A=\ell ^{\infty }}
1039:and the set of maximal ideals in
3259:Positive operator-valued measure
2931:
2930:
2857:Topological quantum field theory
1903:is a subset of the unit ball of
1879:) is the set of complex numbers
906:{\displaystyle C_{0}(\Phi _{A})}
115:
3543:RayleighâFaberâKrahn inequality
2410:A Course in Functional Analysis
2230:
1926:of elements of the spectrum of
395:; the set of all characters of
333:of complex numbers. A non-zero
177:The algebra structure over the
2434:(1932). "Tauberian theorems".
2345:
2330:
2281:. Then there is a *-morphism
2064:
2058:
2045:
1945:, the net of complex numbers {
1744:
1731:
1718:
1715:
1709:
1631:
1625:
1616:
1610:
1397:
1382:
1343:
1314:
1306:
1199:
1191:
1099:
1086:
1073:
958:{\displaystyle {\widehat {a}}}
900:
887:
825:
804:{\displaystyle {\widehat {a}}}
742:
736:
727:
721:
682:
353:
120:
110:
46:the fact that for commutative
1:
3451:Limiting absorption principle
2653:Uniform boundedness principle
2390:. New York: Springer-Verlag.
2323:
1891:ranges over Gelfand space of
1283:and the Gelfand transform of
3077:Singular value decomposition
2369:An Invitation to C*-Algebras
2227:as an algebra of operators.
1785:of a commutative C*-algebra
1664:{\displaystyle \varphi _{x}}
1352:{\displaystyle {\tilde {f}}}
1276:{\displaystyle \mathbb {R} }
1227:{\displaystyle \mathbb {R} }
326:{\displaystyle \mathbb {C} }
7:
3508:Hearing the shape of a drum
3191:Decomposition of a spectrum
2341:, van Nostrand, p. 114
1589:be pointwise evaluation at
1452:StoneâÄech compactification
1163:
1112:may be identified with the
913:. This homomorphism is the
650:, one defines the function
32:) is either of two things:
10:
3645:
3096:Special Elements/Operators
2796:Invariant subspace problem
2251:commutes with its adjoint
1770:
915:Gelfand representation of
3568:Superstrong approximation
3490:
3474:
3431:Banach algebra cohomology
3418:
3382:
3351:
3297:
3264:Projection-valued measure
3249:Borel functional calculus
3241:
3183:
3140:
3095:
3049:
3021:Projection-valued measure
2988:
2926:
2885:
2809:
2788:
2747:
2686:
2628:
2574:
2516:
2509:
2247:is normal if and only if
2151:contravariant equivalence
1793:, consists of the set of
1254:{\displaystyle \Phi _{A}}
1032:{\displaystyle \Phi _{A}}
775:{\displaystyle \Phi _{A}}
621:has an identity element.
594:{\displaystyle \Phi _{A}}
563:{\displaystyle \Phi _{A}}
492:is a subset of the space
485:{\displaystyle \Phi _{A}}
435:{\displaystyle \Phi _{A}}
311:, defined over the field
3160:Spectrum of a C*-algebra
3031:Spectrum of a C*-algebra
2765:Spectrum of a C*-algebra
2388:Complete Normed Algebras
2337:Charles Rickart (1974),
1773:Spectrum of a C*-algebra
205:completely regular space
3588:WienerâKhinchin theorem
3523:Kuznetsov trace formula
3498:Almost Mathieu operator
3316:Banach function algebra
3305:Amenable Banach algebra
3062:GelfandâNaimark theorem
3016:Noncommutative topology
2862:Noncommutative geometry
2213:GelfandâNaimark theorem
1897:spectral radius formula
1871:1 is not invertible in
3563:SturmâLiouville theory
3461:ShermanâTakeda theorem
3341:TomitaâTakesaki theory
3116:Hermitian/Self-adjoint
3067:Gelfand representation
2918:TomitaâTakesaki theory
2893:Approximation property
2837:Calculus of variations
2071:
1757:
1665:
1638:
1583:
1512:
1492:
1478:. More generally, if
1472:
1444:
1404:
1353:
1321:
1277:
1255:
1228:
1206:
1150:
1130:
1106:
1053:
1033:
1006:
983:
959:
929:
907:
864:
844:
805:
776:
749:
696:
644:
643:{\displaystyle a\in A}
615:
595:
572:BanachâAlaoglu theorem
564:
533:
513:
486:
459:
436:
409:
389:
365:
327:
301:
127:
36:a way of representing
22:Gelfand representation
3057:GelfandâMazur theorem
2913:BanachâMazur distance
2876:Generalized functions
2126:the quotient algebra
2072:
1824:must be unital, i.e.
1797:*-homomorphisms from
1758:
1666:
1639:
1584:
1513:
1493:
1473:
1445:
1405:
1354:
1322:
1278:
1256:
1229:
1207:
1158:(Jacobson) semisimple
1151:
1131:
1107:
1061:GelfandâMazur theorem
1054:
1034:
1007:
984:
960:
930:
908:
865:
845:
806:
777:
750:
697:
645:
616:
596:
565:
534:
514:
512:{\displaystyle A^{*}}
487:
460:
437:
410:
390:
366:
328:
302:
128:
3629:Von Neumann algebras
3533:Proto-value function
3512:Dirichlet eigenvalue
3426:Abstract index group
3311:Approximate identity
3274:Rigged Hilbert space
3150:KreinâRutman theorem
2996:Involution/*-algebra
2658:Kakutani fixed-point
2643:Riesz representation
2277:a normal element of
2120:hull-kernel topology
2033:
1895:. Together with the
1696:
1648:
1597:
1553:
1502:
1482:
1457:
1421:
1363:
1334:
1287:
1265:
1238:
1216:
1172:
1140:
1120:
1067:
1059:(this relies on the
1043:
1016:
996:
973:
940:
919:
874:
854:
819:
813:vanishes at infinity
786:
759:
755:. The definition of
706:
654:
628:
605:
578:
547:
523:
496:
469:
449:
419:
399:
379:
341:
335:algebra homomorphism
315:
291:
97:
3619:Functional analysis
3336:Von Neumann algebra
3072:Polar decomposition
2842:Functional calculus
2801:Mahler's conjecture
2780:Von Neumann algebra
2494:Functional analysis
2371:. Springer-Verlag.
2237:functional calculus
2188:) is isomorphic to
2022:be the spectrum of
1522:The C*-algebra case
1359:. Similarly, with
1261:is homeomorphic to
815:, and that the map
26:functional analysis
3466:Unbounded operator
3395:Essential spectrum
3374:SchurâHorn theorem
3364:BauerâFike theorem
3359:AlonâBoppana bound
3352:Finite-Dimensional
3326:Nuclear C*-algebra
3170:Spectral asymmetry
2867:Riemann hypothesis
2566:Topological vector
2067:
1899:, this shows that
1753:
1671:is a character on
1661:
1634:
1579:
1508:
1488:
1468:
1440:
1400:
1349:
1317:
1273:
1251:
1224:
1202:
1146:
1126:
1102:
1049:
1029:
1002:
992:In the case where
979:
955:
925:
903:
860:
840:
811:is continuous and
801:
772:
745:
692:
640:
611:
591:
560:
529:
509:
482:
455:
432:
405:
385:
361:
323:
297:
195:The importance of
171:vanish at infinity
123:
74:Historical remarks
3596:
3595:
3573:Transfer operator
3548:Spectral geometry
3233:Spectral abscissa
3213:Approximate point
3155:Normal eigenvalue
2944:
2943:
2847:Integral operator
2624:
2623:
1752:
1511:{\displaystyle X}
1491:{\displaystyle X}
1346:
1329:Fourier transform
1168:The Banach space
1149:{\displaystyle A}
1129:{\displaystyle A}
1052:{\displaystyle A}
1005:{\displaystyle A}
982:{\displaystyle a}
967:Gelfand transform
952:
928:{\displaystyle A}
863:{\displaystyle A}
837:
798:
718:
666:
614:{\displaystyle A}
532:{\displaystyle A}
458:{\displaystyle A}
408:{\displaystyle A}
388:{\displaystyle A}
307:be a commutative
300:{\displaystyle A}
256:, in which case
149:topological space
137:The model algebra
56:Fourier transform
3636:
3578:Transform theory
3298:Special algebras
3279:Spectral theorem
3242:Spectral Theorem
3082:Spectral theorem
2971:
2964:
2957:
2948:
2947:
2934:
2933:
2852:Jones polynomial
2770:Operator algebra
2514:
2513:
2487:
2480:
2473:
2464:
2463:
2459:
2427:
2401:
2382:
2356:
2349:
2343:
2342:
2334:
2076:
2074:
2073:
2068:
2057:
2056:
1994:Equivalently, Ï(
1855:) of an element
1762:
1760:
1759:
1754:
1750:
1743:
1742:
1730:
1729:
1708:
1707:
1670:
1668:
1667:
1662:
1660:
1659:
1643:
1641:
1640:
1635:
1609:
1608:
1588:
1586:
1585:
1580:
1578:
1577:
1565:
1564:
1517:
1515:
1514:
1509:
1497:
1495:
1494:
1489:
1477:
1475:
1474:
1469:
1467:
1449:
1447:
1446:
1441:
1439:
1438:
1412:Mellin transform
1409:
1407:
1406:
1401:
1396:
1395:
1390:
1381:
1380:
1358:
1356:
1355:
1350:
1348:
1347:
1339:
1326:
1324:
1323:
1318:
1313:
1305:
1304:
1282:
1280:
1279:
1274:
1272:
1260:
1258:
1257:
1252:
1250:
1249:
1233:
1231:
1230:
1225:
1223:
1211:
1209:
1208:
1203:
1198:
1190:
1189:
1155:
1153:
1152:
1147:
1135:
1133:
1132:
1127:
1114:Jacobson radical
1111:
1109:
1108:
1103:
1098:
1097:
1085:
1084:
1058:
1056:
1055:
1050:
1038:
1036:
1035:
1030:
1028:
1027:
1011:
1009:
1008:
1003:
988:
986:
985:
980:
964:
962:
961:
956:
954:
953:
945:
934:
932:
931:
926:
912:
910:
909:
904:
899:
898:
886:
885:
869:
867:
866:
861:
849:
847:
846:
841:
839:
838:
830:
810:
808:
807:
802:
800:
799:
791:
781:
779:
778:
773:
771:
770:
754:
752:
751:
746:
720:
719:
711:
701:
699:
698:
693:
691:
690:
681:
680:
668:
667:
659:
649:
647:
646:
641:
620:
618:
617:
612:
600:
598:
597:
592:
590:
589:
569:
567:
566:
561:
559:
558:
538:
536:
535:
530:
518:
516:
515:
510:
508:
507:
491:
489:
488:
483:
481:
480:
464:
462:
461:
456:
441:
439:
438:
433:
431:
430:
414:
412:
411:
406:
394:
392:
391:
386:
370:
368:
367:
362:
360:
332:
330:
329:
324:
322:
306:
304:
303:
298:
187:The norm is the
132:
130:
129:
124:
119:
118:
109:
108:
64:normal operators
3644:
3643:
3639:
3638:
3637:
3635:
3634:
3633:
3624:Operator theory
3609:Banach algebras
3599:
3598:
3597:
3592:
3553:Spectral method
3538:Ramanujan graph
3486:
3470:
3446:Fredholm theory
3414:
3409:Shilov boundary
3405:Structure space
3383:Generalizations
3378:
3369:Numerical range
3347:
3331:Uniform algebra
3293:
3269:Riesz projector
3254:Min-max theorem
3237:
3223:Direct integral
3179:
3165:Spectral radius
3136:
3091:
3045:
3036:Spectral radius
2984:
2978:Spectral theory
2975:
2945:
2940:
2922:
2886:Advanced topics
2881:
2805:
2784:
2743:
2709:HilbertâSchmidt
2682:
2673:GelfandâNaimark
2620:
2570:
2505:
2491:
2448:10.2307/1968102
2424:
2414:Springer Verlag
2398:
2379:
2360:
2359:
2351:Kainuth (2009)
2350:
2346:
2335:
2331:
2326:
2233:
2167:
2161:the C*-algebra
2096:
2052:
2048:
2034:
2031:
2030:
2012:
1963:
1953:
1925:
1919:
1775:
1769:
1738:
1734:
1725:
1721:
1703:
1699:
1697:
1694:
1693:
1684:
1655:
1651:
1649:
1646:
1645:
1604:
1600:
1598:
1595:
1594:
1573:
1569:
1560:
1556:
1554:
1551:
1550:
1536:
1524:
1503:
1500:
1499:
1483:
1480:
1479:
1463:
1458:
1455:
1454:
1434:
1430:
1422:
1419:
1418:
1391:
1386:
1385:
1376:
1372:
1364:
1361:
1360:
1338:
1337:
1335:
1332:
1331:
1309:
1300:
1296:
1288:
1285:
1284:
1268:
1266:
1263:
1262:
1245:
1241:
1239:
1236:
1235:
1219:
1217:
1214:
1213:
1194:
1185:
1181:
1173:
1170:
1169:
1166:
1141:
1138:
1137:
1121:
1118:
1117:
1093:
1089:
1080:
1076:
1068:
1065:
1064:
1044:
1041:
1040:
1023:
1019:
1017:
1014:
1013:
997:
994:
993:
974:
971:
970:
969:of the element
944:
943:
941:
938:
937:
920:
917:
916:
894:
890:
881:
877:
875:
872:
871:
855:
852:
851:
829:
828:
820:
817:
816:
790:
789:
787:
784:
783:
766:
762:
760:
757:
756:
710:
709:
707:
704:
703:
686:
685:
676:
672:
658:
657:
655:
652:
651:
629:
626:
625:
606:
603:
602:
585:
581:
579:
576:
575:
554:
550:
548:
545:
544:
541:weak-* topology
524:
521:
520:
503:
499:
497:
494:
493:
476:
472:
470:
467:
466:
450:
447:
446:
426:
422:
420:
417:
416:
400:
397:
396:
380:
377:
376:
356:
342:
339:
338:
318:
316:
313:
312:
292:
289:
288:
285:
262:
248:if and only if
239:
225:
179:complex numbers
160:
143:locally compact
139:
114:
113:
104:
100:
98:
95:
94:
76:
60:spectral theory
41:Banach algebras
12:
11:
5:
3642:
3632:
3631:
3626:
3621:
3616:
3611:
3594:
3593:
3591:
3590:
3585:
3580:
3575:
3570:
3565:
3560:
3555:
3550:
3545:
3540:
3535:
3530:
3525:
3520:
3515:
3505:
3503:Corona theorem
3500:
3494:
3492:
3488:
3487:
3485:
3484:
3482:Wiener algebra
3478:
3476:
3472:
3471:
3469:
3468:
3463:
3458:
3453:
3448:
3443:
3438:
3433:
3428:
3422:
3420:
3416:
3415:
3413:
3412:
3402:
3400:Pseudospectrum
3397:
3392:
3390:Dirac spectrum
3386:
3384:
3380:
3379:
3377:
3376:
3371:
3366:
3361:
3355:
3353:
3349:
3348:
3346:
3345:
3344:
3343:
3333:
3328:
3323:
3318:
3313:
3307:
3301:
3299:
3295:
3294:
3292:
3291:
3286:
3281:
3276:
3271:
3266:
3261:
3256:
3251:
3245:
3243:
3239:
3238:
3236:
3235:
3230:
3225:
3220:
3215:
3210:
3209:
3208:
3203:
3198:
3187:
3185:
3181:
3180:
3178:
3177:
3172:
3167:
3162:
3157:
3152:
3146:
3144:
3138:
3137:
3135:
3134:
3129:
3121:
3113:
3105:
3099:
3097:
3093:
3092:
3090:
3089:
3084:
3079:
3074:
3069:
3064:
3059:
3053:
3051:
3047:
3046:
3044:
3043:
3041:Operator space
3038:
3033:
3028:
3023:
3018:
3013:
3008:
3003:
3001:Banach algebra
2998:
2992:
2990:
2989:Basic concepts
2986:
2985:
2974:
2973:
2966:
2959:
2951:
2942:
2941:
2939:
2938:
2927:
2924:
2923:
2921:
2920:
2915:
2910:
2905:
2903:Choquet theory
2900:
2895:
2889:
2887:
2883:
2882:
2880:
2879:
2869:
2864:
2859:
2854:
2849:
2844:
2839:
2834:
2829:
2824:
2819:
2813:
2811:
2807:
2806:
2804:
2803:
2798:
2792:
2790:
2786:
2785:
2783:
2782:
2777:
2772:
2767:
2762:
2757:
2755:Banach algebra
2751:
2749:
2745:
2744:
2742:
2741:
2736:
2731:
2726:
2721:
2716:
2711:
2706:
2701:
2696:
2690:
2688:
2684:
2683:
2681:
2680:
2678:BanachâAlaoglu
2675:
2670:
2665:
2660:
2655:
2650:
2645:
2640:
2634:
2632:
2626:
2625:
2622:
2621:
2619:
2618:
2613:
2608:
2606:Locally convex
2603:
2589:
2584:
2578:
2576:
2572:
2571:
2569:
2568:
2563:
2558:
2553:
2548:
2543:
2538:
2533:
2528:
2523:
2517:
2511:
2507:
2506:
2490:
2489:
2482:
2475:
2467:
2461:
2460:
2428:
2422:
2402:
2396:
2383:
2377:
2358:
2357:
2344:
2328:
2327:
2325:
2322:
2318:
2317:
2310:
2232:
2229:
2217:noncommutative
2165:
2109:maximal ideals
2094:
2078:
2077:
2066:
2063:
2060:
2055:
2051:
2047:
2044:
2041:
2038:
2011:
2008:
1959:
1949:
1935:if and only if
1921:
1915:
1815:*-homomorphism
1768:
1765:
1764:
1763:
1749:
1746:
1741:
1737:
1733:
1728:
1724:
1720:
1717:
1714:
1711:
1706:
1702:
1680:
1658:
1654:
1633:
1630:
1627:
1624:
1621:
1618:
1615:
1612:
1607:
1603:
1576:
1572:
1568:
1563:
1559:
1534:
1523:
1520:
1507:
1487:
1466:
1462:
1437:
1433:
1429:
1426:
1399:
1394:
1389:
1384:
1379:
1375:
1371:
1368:
1345:
1342:
1316:
1312:
1308:
1303:
1299:
1295:
1292:
1271:
1248:
1244:
1222:
1201:
1197:
1193:
1188:
1184:
1180:
1177:
1165:
1162:
1145:
1125:
1101:
1096:
1092:
1088:
1083:
1079:
1075:
1072:
1048:
1026:
1022:
1001:
978:
951:
948:
924:
902:
897:
893:
889:
884:
880:
859:
836:
833:
827:
824:
797:
794:
769:
765:
744:
741:
738:
735:
732:
729:
726:
723:
717:
714:
689:
684:
679:
675:
671:
665:
662:
639:
636:
633:
610:
588:
584:
557:
553:
528:
506:
502:
479:
475:
454:
429:
425:
415:is denoted by
404:
384:
359:
355:
352:
349:
346:
321:
309:Banach algebra
296:
284:
281:
267:) is equal to
260:
237:
223:
193:
192:
185:
182:
158:
138:
135:
122:
117:
112:
107:
103:
84:group algebras
80:Norbert Wiener
75:
72:
52:
51:
44:
9:
6:
4:
3:
2:
3641:
3630:
3627:
3625:
3622:
3620:
3617:
3615:
3612:
3610:
3607:
3606:
3604:
3589:
3586:
3584:
3581:
3579:
3576:
3574:
3571:
3569:
3566:
3564:
3561:
3559:
3556:
3554:
3551:
3549:
3546:
3544:
3541:
3539:
3536:
3534:
3531:
3529:
3526:
3524:
3521:
3519:
3516:
3513:
3509:
3506:
3504:
3501:
3499:
3496:
3495:
3493:
3489:
3483:
3480:
3479:
3477:
3473:
3467:
3464:
3462:
3459:
3457:
3454:
3452:
3449:
3447:
3444:
3442:
3439:
3437:
3434:
3432:
3429:
3427:
3424:
3423:
3421:
3419:Miscellaneous
3417:
3410:
3406:
3403:
3401:
3398:
3396:
3393:
3391:
3388:
3387:
3385:
3381:
3375:
3372:
3370:
3367:
3365:
3362:
3360:
3357:
3356:
3354:
3350:
3342:
3339:
3338:
3337:
3334:
3332:
3329:
3327:
3324:
3322:
3319:
3317:
3314:
3312:
3308:
3306:
3303:
3302:
3300:
3296:
3290:
3287:
3285:
3282:
3280:
3277:
3275:
3272:
3270:
3267:
3265:
3262:
3260:
3257:
3255:
3252:
3250:
3247:
3246:
3244:
3240:
3234:
3231:
3229:
3226:
3224:
3221:
3219:
3216:
3214:
3211:
3207:
3204:
3202:
3199:
3197:
3194:
3193:
3192:
3189:
3188:
3186:
3184:Decomposition
3182:
3176:
3173:
3171:
3168:
3166:
3163:
3161:
3158:
3156:
3153:
3151:
3148:
3147:
3145:
3143:
3139:
3133:
3130:
3128:
3125:
3122:
3120:
3117:
3114:
3112:
3109:
3106:
3104:
3101:
3100:
3098:
3094:
3088:
3085:
3083:
3080:
3078:
3075:
3073:
3070:
3068:
3065:
3063:
3060:
3058:
3055:
3054:
3052:
3048:
3042:
3039:
3037:
3034:
3032:
3029:
3027:
3024:
3022:
3019:
3017:
3014:
3012:
3009:
3007:
3004:
3002:
2999:
2997:
2994:
2993:
2991:
2987:
2983:
2979:
2972:
2967:
2965:
2960:
2958:
2953:
2952:
2949:
2937:
2929:
2928:
2925:
2919:
2916:
2914:
2911:
2909:
2908:Weak topology
2906:
2904:
2901:
2899:
2896:
2894:
2891:
2890:
2888:
2884:
2877:
2873:
2870:
2868:
2865:
2863:
2860:
2858:
2855:
2853:
2850:
2848:
2845:
2843:
2840:
2838:
2835:
2833:
2832:Index theorem
2830:
2828:
2825:
2823:
2820:
2818:
2815:
2814:
2812:
2808:
2802:
2799:
2797:
2794:
2793:
2791:
2789:Open problems
2787:
2781:
2778:
2776:
2773:
2771:
2768:
2766:
2763:
2761:
2758:
2756:
2753:
2752:
2750:
2746:
2740:
2737:
2735:
2732:
2730:
2727:
2725:
2722:
2720:
2717:
2715:
2712:
2710:
2707:
2705:
2702:
2700:
2697:
2695:
2692:
2691:
2689:
2685:
2679:
2676:
2674:
2671:
2669:
2666:
2664:
2661:
2659:
2656:
2654:
2651:
2649:
2646:
2644:
2641:
2639:
2636:
2635:
2633:
2631:
2627:
2617:
2614:
2612:
2609:
2607:
2604:
2601:
2597:
2593:
2590:
2588:
2585:
2583:
2580:
2579:
2577:
2573:
2567:
2564:
2562:
2559:
2557:
2554:
2552:
2549:
2547:
2544:
2542:
2539:
2537:
2534:
2532:
2529:
2527:
2524:
2522:
2519:
2518:
2515:
2512:
2508:
2503:
2499:
2495:
2488:
2483:
2481:
2476:
2474:
2469:
2468:
2465:
2457:
2453:
2449:
2445:
2441:
2437:
2433:
2429:
2425:
2423:0-387-97245-5
2419:
2415:
2411:
2407:
2406:Conway, J. B.
2403:
2399:
2397:0-387-06386-2
2393:
2389:
2384:
2380:
2378:0-387-90176-0
2374:
2370:
2366:
2362:
2361:
2354:
2348:
2340:
2333:
2329:
2321:
2315:
2311:
2308:
2304:
2303:
2302:
2300:
2296:
2292:
2288:
2284:
2280:
2276:
2272:
2268:
2264:
2262:
2258:
2254:
2250:
2246:
2243:: An element
2242:
2238:
2228:
2226:
2222:
2218:
2214:
2209:
2207:
2203:
2199:
2195:
2191:
2187:
2183:
2179:
2175:
2171:
2164:
2160:
2156:
2152:
2148:
2143:
2141:
2137:
2133:
2129:
2125:
2121:
2117:
2113:
2110:
2105:
2102:
2100:
2093:
2089:
2085:
2081:
2061:
2053:
2049:
2042:
2039:
2036:
2029:
2028:
2027:
2025:
2021:
2017:
2007:
2005:
2001:
1997:
1992:
1990:
1986:
1982:
1978:
1973:
1971:
1967:
1964:converges to
1962:
1957:
1952:
1948:
1944:
1940:
1936:
1933:
1930:converges to
1929:
1924:
1918:
1914:
1910:
1906:
1902:
1898:
1894:
1890:
1886:
1882:
1878:
1874:
1870:
1867: â
1866:
1862:
1858:
1854:
1850:
1845:
1843:
1839:
1835:
1831:
1827:
1823:
1818:
1816:
1812:
1808:
1804:
1800:
1796:
1792:
1788:
1784:
1783:Gelfand space
1780:
1774:
1747:
1739:
1726:
1722:
1712:
1704:
1700:
1692:
1691:
1690:
1688:
1683:
1678:
1674:
1656:
1652:
1628:
1622:
1619:
1613:
1605:
1601:
1592:
1574:
1570:
1566:
1561:
1557:
1548:
1544:
1540:
1533:
1529:
1519:
1505:
1485:
1460:
1453:
1431:
1427:
1424:
1415:
1413:
1392:
1377:
1373:
1369:
1366:
1340:
1330:
1301:
1297:
1293:
1290:
1246:
1186:
1182:
1178:
1175:
1161:
1159:
1143:
1123:
1115:
1094:
1081:
1077:
1070:
1062:
1046:
1024:
999:
990:
976:
968:
949:
946:
935:
922:
895:
882:
878:
857:
834:
831:
822:
814:
795:
792:
767:
739:
733:
730:
724:
715:
712:
677:
669:
663:
660:
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608:
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574:.) The space
573:
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504:
500:
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427:
402:
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347:
336:
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280:
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259:
255:
251:
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218:
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210:
206:
202:
198:
191:on functions.
190:
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168:
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157:
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150:
147:
144:
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105:
101:
92:
88:
85:
81:
71:
69:
68:normal matrix
65:
61:
57:
49:
45:
42:
39:
35:
34:
33:
31:
30:I. M. Gelfand
28:(named after
27:
23:
19:
3491:Applications
3321:Disk algebra
3175:Spectral gap
3066:
3050:Main results
2898:Balanced set
2872:Distribution
2810:Applications
2663:KreinâMilman
2648:Closed graph
2439:
2436:Ann. of Math
2435:
2409:
2387:
2368:
2352:
2347:
2338:
2332:
2319:
2313:
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2274:
2270:
2266:
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2244:
2240:
2236:
2234:
2231:Applications
2224:
2220:
2219:C*-algebras
2210:
2205:
2202:homeomorphic
2197:
2193:
2189:
2185:
2181:
2177:
2173:
2169:
2162:
2158:
2144:
2139:
2135:
2131:
2127:
2123:
2115:
2111:
2106:
2103:
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2087:
2083:
2082:
2079:
2023:
2019:
2015:
2013:
2003:
1995:
1993:
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1976:
1974:
1969:
1965:
1960:
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1946:
1942:
1938:
1931:
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1904:
1900:
1892:
1888:
1884:
1880:
1876:
1872:
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1856:
1852:
1848:
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1841:
1837:
1833:
1829:
1825:
1821:
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1806:
1802:
1798:
1794:
1790:
1786:
1782:
1778:
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1538:
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1416:
1167:
991:
966:
914:
623:
444:
372:
371:is called a
286:
276:
272:
268:
264:
257:
249:
241:
234:
232:
227:
220:
216:
212:
208:
200:
196:
194:
189:uniform norm
166:
162:
155:
154:, the space
151:
140:
90:
86:
77:
53:
21:
15:
3614:C*-algebras
3518:Heat kernel
3218:Compression
3103:Isospectral
2827:Heat kernel
2817:Hardy space
2724:Trace class
2638:HahnâBanach
2600:Topological
2365:Arveson, W.
2211:The 'full'
2118:, with the
233:Note that
48:C*-algebras
38:commutative
18:mathematics
3603:Categories
3196:Continuous
3011:C*-algebra
3006:B*-algebra
2760:C*-algebra
2575:Properties
2432:Wiener, N.
2324:References
2301:such that
1985:metrizable
1863:for which
1847:Note that
1803:characters
1789:, denoted
1771:See also:
2982:-algebras
2734:Unbounded
2729:Transpose
2687:Operators
2616:Separable
2611:Reflexive
2596:Algebraic
2582:Barrelled
2046:→
2037:γ
1998:) is the
1981:separable
1937:for each
1736:Φ
1719:→
1653:φ
1602:φ
1575:∗
1567:∈
1558:φ
1541:). Given
1461:β
1436:∞
1432:ℓ
1344:~
1294:∈
1243:Φ
1091:Φ
1074:→
1021:Φ
950:^
892:Φ
835:^
826:↦
796:^
764:Φ
734:ϕ
725:ϕ
716:^
683:→
674:Φ
664:^
635:∈
583:Φ
552:Φ
505:∗
474:Φ
424:Φ
373:character
354:→
348::
345:Φ
146:Hausdorff
102:ℓ
3583:Weyl law
3528:Lax pair
3475:Examples
3309:With an
3228:Discrete
3206:Residual
3142:Spectrum
3127:operator
3119:operator
3111:operator
3026:Spectrum
2936:Category
2748:Algebras
2630:Theorems
2587:Complete
2556:Schwartz
2502:glossary
2408:(1990).
2367:(1981).
1887:) where
1849:spectrum
1795:non-zero
1779:spectrum
1234:. Then
1164:Examples
141:For any
3124:Unitary
2739:Unitary
2719:Nuclear
2704:Compact
2699:Bounded
2694:Adjoint
2668:Minâmax
2561:Sobolev
2546:Nuclear
2536:Hilbert
2531:Fréchet
2496: (
2456:1968102
2297:) into
2269:. Let
2267:Theorem
2180:, then
2155:adjoint
2147:functor
2084:Theorem
2026:. Let
1834:compact
1644:. Then
1593:, i.e.
1327:is the
965:is the
254:compact
203:into a
3108:Normal
2714:Normal
2551:Orlicz
2541:Hölder
2521:Banach
2510:Spaces
2498:topics
2454:
2438:. II.
2420:
2394:
2375:
2355:, p 72
1751:
1549:, let
936:, and
624:Given
246:unital
244:) is
169:which
93:) and
20:, the
3201:Point
2526:Besov
2452:JSTOR
2090:onto
2002:of Îł(
2000:range
1979:is a
1685:with
219:from
3132:Unit
2980:and
2874:(or
2592:Dual
2418:ISBN
2392:ISBN
2373:ISBN
2176:and
2014:Let
1777:The
1417:For
287:Let
62:for
2444:doi
2204:to
2200:is
2134:in
2128:A/m
2114:of
2101:).
1975:If
1972:).
1941:in
1909:net
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1805:on
1781:or
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702:by
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2059:(
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1996:x
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1977:A
1970:x
1968:(
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1961:k
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1920:}
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1748:.
1745:)
1740:A
1732:(
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1611:(
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1425:A
1398:)
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896:A
888:(
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678:A
670::
661:a
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277:X
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271:(
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265:X
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