1637:
307:
182:
235:
110:
3019:
2802:
2892:
2929:
2968:
2736:
3095:
1024:
418:
if and only if every hereditary C*-subalgebra has an approximate identity consisting of projections. This was known as property (HP) in earlier literature.
1846:
3126:
2180:
1526:
1126:
2825:
248:
123:
759:
2302:
1362:
17:
1834:
3121:
1189:
781:
2235:
1352:
764:
537:
786:
2207:
1479:
1334:
1841:
1673:
1310:
1111:
774:
2554:
1004:
2496:
857:
655:
2312:
852:
195:
70:
2395:
1862:
1202:
414:
One sometimes considers approximate identities consisting of specific types of elements. For example, a C*-algebra has
2036:
1819:
1291:
1182:
1009:
2984:
2631:
1954:
1797:
1561:
827:
2484:
2420:
1971:
1206:
796:
2265:
415:
2758:
2676:
2479:
2158:
1937:
710:
594:
2855:
2517:
2295:
2250:
2240:
1357:
1019:
530:
2352:
2342:
2270:
2197:
2073:
1742:
1640:
1413:
1347:
1175:
645:
333:
of norm ≤ 1 with its natural order is an approximate identity for any C*-algebra. This is called the
2347:
2690:
2680:
2280:
1666:
1377:
1156:
1076:
630:
3049:
2851:
2513:
2327:
2220:
2215:
2110:
2083:
2048:
1900:
1793:
1622:
1576:
1500:
1382:
1131:
1029:
909:
450:
In ring theory, an approximate identity is defined in a similar way, except that the ring is given the
2907:
2807:
2501:
2474:
2457:
2275:
2120:
1789:
1617:
1433:
1136:
999:
832:
817:
625:
589:
3152:
3024:
2317:
2307:
2225:
2163:
2090:
2044:
1959:
1784:
1469:
1367:
1270:
728:
718:
599:
523:
400:
373:
2946:
2709:
2290:
2230:
1566:
1342:
1091:
1066:
884:
873:
584:
3058:
2332:
2245:
2026:
1942:
1778:
1772:
1659:
1597:
1541:
1505:
942:
932:
927:
635:
505:
349:
2572:
3116:
3111:
2586:
2534:
2491:
2415:
2368:
2105:
1767:
1734:
1707:
1304:
687:
2405:
1300:
3054:
2260:
2255:
1966:
1850:
1756:
1580:
1101:
1080:
994:
842:
469:
431:
1167:
8:
2897:
2698:
2655:
2469:
2192:
1922:
1729:
1546:
1484:
1198:
904:
640:
28:
2843:
3131:
3042:
2665:
2635:
2452:
2410:
2017:
1927:
1872:
1719:
1571:
1438:
1034:
963:
894:
738:
700:
510:
345:, the net consisting of finite rank projections would be another approximate identity.
44:
2973:
2943:
2904:
2812:
2285:
2066:
2009:
1989:
1551:
1141:
1116:
801:
723:
451:
329:
elements is the same as an approximate identity. The net of all positive elements in
313:
is a net which is both a right approximate identity and a left approximate identity.
36:
2442:
2437:
2425:
2337:
2322:
2185:
2125:
2100:
2031:
2021:
1884:
1556:
1474:
1443:
1423:
1408:
1403:
1398:
1146:
847:
695:
650:
574:
365:
338:
48:
1829:
1235:
2462:
2447:
2373:
2175:
2168:
2135:
2095:
2061:
2053:
1981:
1949:
1814:
1746:
1418:
1372:
1320:
1315:
1286:
1121:
1106:
1014:
977:
973:
937:
899:
837:
822:
791:
733:
692:
679:
604:
546:
515:
361:
1245:
435:
3032:
2980:
2640:
2506:
2153:
2143:
1762:
1714:
1607:
1459:
1260:
1071:
1050:
968:
958:
769:
676:
609:
569:
439:
369:
40:
3146:
3037:
2650:
2604:
2539:
2390:
2385:
2378:
1999:
1932:
1905:
1724:
1697:
1612:
1536:
1265:
1250:
1240:
342:
430:
algebra plays the same role as a sequence of function approximations to the
302:{\displaystyle \lim _{\lambda \in \Lambda }\lVert e_{\lambda }a-a\rVert =0.}
177:{\displaystyle \lim _{\lambda \in \Lambda }\lVert ae_{\lambda }-a\rVert =0.}
2549:
2544:
2004:
1994:
1867:
1857:
1702:
1682:
1602:
1255:
1225:
889:
743:
684:
326:
377:
2838:
2754:
2660:
2645:
2625:
2599:
2564:
2115:
2078:
1751:
1531:
1521:
1428:
1230:
1086:
671:
427:
337:
of a C*-algebra. Approximate identities are not unique. For example, for
32:
24:
2594:
2575: ((cs, lcs)-closed, (cs, bcs)-complete, (lower) ideally convex, (H
2559:
2400:
2148:
1910:
1464:
1296:
579:
322:
1651:
16:
This article is about the Banach algebra concept. For other uses, see
2670:
1915:
1879:
564:
550:
500:
2936:
2820:
2746:
2706:
2609:
2432:
1151:
1096:
434:(which is the identity element for convolution). For example, the
356:
and a C*-algebra with a sequential approximate identity is called
2741:
1824:
47:(generally without an identity) that acts as a substitute for an
1197:
388:
contains a strictly positive element, i.e. there exists
325:, a right (or left) approximate identity consisting of
3061:
2987:
2949:
2910:
2858:
2761:
2712:
251:
230:{\displaystyle \{e_{\lambda }:\lambda \in \Lambda \}}
198:
126:
105:{\displaystyle \{e_{\lambda }:\lambda \in \Lambda \}}
73:
368:
is false. A commutative C*-algebra is σ-unital
3089:
3013:
2962:
2923:
2886:
2796:
2730:
1527:Spectral theory of ordinary differential equations
1127:Spectral theory of ordinary differential equations
545:
301:
229:
176:
104:
1025:Schröder–Bernstein theorems for operator algebras
3144:
472:over a ring with approximate identity is called
253:
128:
3014:{\displaystyle S\left(\mathbb {R} ^{n}\right)}
3122:Mathematical formulation of quantum mechanics
1667:
1183:
531:
442:theory give rise to an approximate identity.
290:
268:
224:
199:
165:
143:
99:
74:
1674:
1660:
1190:
1176:
538:
524:
2997:
2797:{\displaystyle B_{p,q}^{s}(\mathbb {R} )}
2787:
1480:Group algebra of a locally compact group
364:C*-algebra is σ-unital, though the
2887:{\displaystyle L^{\lambda ,p}(\Omega )}
1681:
421:
3145:
3127:Ordinary Differential Equations (ODEs)
2241:Banach–Steinhaus (Uniform boundedness)
1655:
1171:
858:Spectral theory of normal C*-algebras
656:Spectral theory of normal C*-algebras
519:
853:Spectral theory of compact operators
480:in the module there is some λ with
13:
2955:
2916:
2878:
2722:
1005:Cohen–Hewitt factorization theorem
263:
221:
138:
96:
14:
3164:
2619:Subsets / set operations
2396:Differentiation in Fréchet spaces
1010:Extensions of symmetric operators
1636:
1635:
1562:Topological quantum field theory
828:Positive operator-valued measure
384:is σ-unital if and only if
348:If an approximate identity is a
2924:{\displaystyle \ell ^{\infty }}
1112:Rayleigh–Faber–Krahn inequality
354:sequential approximate identity
35:, an approximate identity is a
3084:
3065:
2881:
2875:
2791:
2783:
2725:
2719:
2313:Lomonosov's invariant subspace
2236:Banach–Schauder (open mapping)
335:canonical approximate identity
316:
1:
1358:Uniform boundedness principle
1020:Limiting absorption principle
426:An approximate identity in a
54:
18:Approximation to the identity
2198:Singular value decomposition
646:Singular value decomposition
237:such that for every element
112:such that for every element
7:
2963:{\displaystyle L^{\infty }}
2731:{\displaystyle ba(\Sigma )}
2600:Radially convex/Star-shaped
1077:Hearing the shape of a drum
760:Decomposition of a spectrum
494:
380:. In general, a C*-algebra
10:
3169:
3090:{\displaystyle W(X,L^{p})}
1501:Invariant subspace problem
665:Special Elements/Operators
61:right approximate identity
15:
3104:
2689:
2636:Algebraic interior (core)
2618:
2527:
2361:
2251:Cauchy–Schwarz inequality
2206:
2134:
1980:
1894:Function space Topologies
1893:
1807:
1690:
1631:
1590:
1514:
1493:
1452:
1391:
1333:
1279:
1221:
1214:
1137:Superstrong approximation
1059:
1043:
1000:Banach algebra cohomology
987:
951:
920:
866:
833:Projection-valued measure
818:Borel functional calculus
810:
752:
709:
664:
618:
590:Projection-valued measure
557:
186:left approximate identity
1470:Spectrum of a C*-algebra
729:Spectrum of a C*-algebra
600:Spectrum of a C*-algebra
445:
401:hereditary C*-subalgebra
1567:Noncommutative geometry
1157:Wiener–Khinchin theorem
1092:Kuznetsov trace formula
1067:Almost Mathieu operator
885:Banach function algebra
874:Amenable Banach algebra
631:Gelfand–Naimark theorem
585:Noncommutative topology
3091:
3015:
2964:
2925:
2888:
2798:
2732:
1901:Banach–Mazur compactum
1691:Types of Banach spaces
1623:Tomita–Takesaki theory
1598:Approximation property
1542:Calculus of variations
1132:Sturm–Liouville theory
1030:Sherman–Takeda theorem
910:Tomita–Takesaki theory
685:Hermitian/Self-adjoint
636:Gelfand representation
506:Nascent delta function
303:
231:
178:
106:
3117:Finite element method
3112:Differential operator
3092:
3016:
2965:
2926:
2889:
2799:
2733:
2573:Convex series related
2369:Abstract Wiener space
2296:hyperplane separation
1851:Minkowski functionals
1735:Polarization identity
1618:Banach–Mazur distance
1581:Generalized functions
626:Gelfand–Mazur theorem
304:
232:
179:
107:
3059:
2985:
2947:
2908:
2856:
2759:
2710:
2699:Absolute continuity
2353:Schauder fixed-point
2343:Riesz representation
2303:Kakutani fixed-point
2271:Freudenthal spectral
1757:L-semi-inner product
1363:Kakutani fixed-point
1348:Riesz representation
1102:Proto-value function
1081:Dirichlet eigenvalue
995:Abstract index group
880:Approximate identity
843:Rigged Hilbert space
719:Krein–Rutman theorem
565:Involution/*-algebra
432:Dirac delta function
422:Convolution algebras
311:approximate identity
249:
196:
188:in a Banach algebra
124:
71:
63:in a Banach algebra
2782:
2520:measurable function
2470:Functional calculus
2333:Parseval's identity
2246:Bessel's inequality
2193:Polar decomposition
1972:Uniform convergence
1730:Inner product space
1547:Functional calculus
1506:Mahler's conjecture
1485:Von Neumann algebra
1199:Functional analysis
905:Von Neumann algebra
641:Polar decomposition
29:functional analysis
3132:Validated numerics
3087:
3043:Sobolev inequality
3011:
2960:
2921:
2884:
2813:Bounded variation
2794:
2762:
2747:Banach coordinate
2728:
2666:Minkowski addition
2328:M. Riesz extension
1808:Banach spaces are:
1572:Riemann hypothesis
1271:Topological vector
1035:Unbounded operator
964:Essential spectrum
943:Schur–Horn theorem
933:Bauer–Fike theorem
928:Alon–Boppana bound
921:Finite-Dimensional
895:Nuclear C*-algebra
739:Spectral asymmetry
511:Summability kernel
299:
267:
227:
174:
142:
102:
27:, particularly in
3140:
3139:
2852:Morrey–Campanato
2834:compact Hausdorff
2681:Relative interior
2535:Absolutely convex
2502:Projection-valued
2111:Strictly singular
2037:on Hilbert spaces
1798:of Hilbert spaces
1649:
1648:
1552:Integral operator
1329:
1328:
1165:
1164:
1142:Transfer operator
1117:Spectral geometry
802:Spectral abscissa
782:Approximate point
724:Normal eigenvalue
452:discrete topology
339:compact operators
252:
127:
3160:
3096:
3094:
3093:
3088:
3083:
3082:
3050:Triebel–Lizorkin
3020:
3018:
3017:
3012:
3010:
3006:
3005:
3000:
2969:
2967:
2966:
2961:
2959:
2958:
2930:
2928:
2927:
2922:
2920:
2919:
2893:
2891:
2890:
2885:
2874:
2873:
2803:
2801:
2800:
2795:
2790:
2781:
2776:
2737:
2735:
2734:
2729:
2590:
2568:
2550:Balanced/Circled
2348:Robinson-Ursescu
2266:Eberlein–Šmulian
2186:Spectral theorem
1982:Linear operators
1779:Uniformly smooth
1676:
1669:
1662:
1653:
1652:
1639:
1638:
1557:Jones polynomial
1475:Operator algebra
1219:
1218:
1192:
1185:
1178:
1169:
1168:
1147:Transform theory
867:Special algebras
848:Spectral theorem
811:Spectral Theorem
651:Spectral theorem
540:
533:
526:
517:
516:
308:
306:
305:
300:
280:
279:
266:
236:
234:
233:
228:
211:
210:
183:
181:
180:
175:
158:
157:
141:
111:
109:
108:
103:
86:
85:
49:identity element
3168:
3167:
3163:
3162:
3161:
3159:
3158:
3157:
3153:Banach algebras
3143:
3142:
3141:
3136:
3100:
3078:
3074:
3060:
3057:
3056:
3055:Wiener amalgam
3025:Segal–Bargmann
3001:
2996:
2995:
2991:
2986:
2983:
2982:
2954:
2950:
2948:
2945:
2944:
2915:
2911:
2909:
2906:
2905:
2863:
2859:
2857:
2854:
2853:
2808:Birnbaum–Orlicz
2786:
2777:
2766:
2760:
2757:
2756:
2711:
2708:
2707:
2685:
2641:Bounding points
2614:
2588:
2566:
2523:
2374:Banach manifold
2357:
2281:Gelfand–Naimark
2202:
2176:Spectral theory
2144:Banach algebras
2136:Operator theory
2130:
2091:Pseudo-monotone
2074:Hilbert–Schmidt
2054:Densely defined
1976:
1889:
1803:
1686:
1680:
1650:
1645:
1627:
1591:Advanced topics
1586:
1510:
1489:
1448:
1414:Hilbert–Schmidt
1387:
1378:Gelfand–Naimark
1325:
1275:
1210:
1196:
1166:
1161:
1122:Spectral method
1107:Ramanujan graph
1055:
1039:
1015:Fredholm theory
983:
978:Shilov boundary
974:Structure space
952:Generalizations
947:
938:Numerical range
916:
900:Uniform algebra
862:
838:Riesz projector
823:Min-max theorem
806:
792:Direct integral
748:
734:Spectral radius
705:
660:
614:
605:Spectral radius
553:
547:Spectral theory
544:
497:
490:
464:
448:
424:
398:
352:, we call it a
319:
275:
271:
256:
250:
247:
246:
206:
202:
197:
194:
193:
153:
149:
131:
125:
122:
121:
81:
77:
72:
69:
68:
57:
21:
12:
11:
5:
3166:
3156:
3155:
3138:
3137:
3135:
3134:
3129:
3124:
3119:
3114:
3108:
3106:
3102:
3101:
3099:
3098:
3086:
3081:
3077:
3073:
3070:
3067:
3064:
3052:
3047:
3046:
3045:
3035:
3033:Sequence space
3030:
3022:
3009:
3004:
2999:
2994:
2990:
2978:
2977:
2976:
2971:
2957:
2953:
2934:
2933:
2932:
2918:
2914:
2895:
2883:
2880:
2877:
2872:
2869:
2866:
2862:
2849:
2841:
2836:
2823:
2818:
2810:
2805:
2793:
2789:
2785:
2780:
2775:
2772:
2769:
2765:
2752:
2744:
2739:
2727:
2724:
2721:
2718:
2715:
2704:
2695:
2693:
2687:
2686:
2684:
2683:
2673:
2668:
2663:
2658:
2653:
2648:
2643:
2638:
2628:
2622:
2620:
2616:
2615:
2613:
2612:
2607:
2602:
2597:
2592:
2584:
2570:
2562:
2557:
2552:
2547:
2542:
2537:
2531:
2529:
2525:
2524:
2522:
2521:
2511:
2510:
2509:
2504:
2499:
2489:
2488:
2487:
2482:
2477:
2467:
2466:
2465:
2460:
2455:
2450:
2448:Gelfand–Pettis
2445:
2440:
2430:
2429:
2428:
2423:
2418:
2413:
2408:
2398:
2393:
2388:
2383:
2382:
2381:
2371:
2365:
2363:
2359:
2358:
2356:
2355:
2350:
2345:
2340:
2335:
2330:
2325:
2320:
2315:
2310:
2305:
2300:
2299:
2298:
2288:
2283:
2278:
2273:
2268:
2263:
2258:
2253:
2248:
2243:
2238:
2233:
2228:
2223:
2221:Banach–Alaoglu
2218:
2216:Anderson–Kadec
2212:
2210:
2204:
2203:
2201:
2200:
2195:
2190:
2189:
2188:
2183:
2173:
2172:
2171:
2166:
2156:
2154:Operator space
2151:
2146:
2140:
2138:
2132:
2131:
2129:
2128:
2123:
2118:
2113:
2108:
2103:
2098:
2093:
2088:
2087:
2086:
2076:
2071:
2070:
2069:
2064:
2056:
2051:
2041:
2040:
2039:
2029:
2024:
2014:
2013:
2012:
2007:
2002:
1992:
1986:
1984:
1978:
1977:
1975:
1974:
1969:
1964:
1963:
1962:
1957:
1947:
1946:
1945:
1940:
1930:
1925:
1920:
1919:
1918:
1908:
1903:
1897:
1895:
1891:
1890:
1888:
1887:
1882:
1877:
1876:
1875:
1865:
1860:
1855:
1854:
1853:
1842:Locally convex
1839:
1838:
1837:
1827:
1822:
1817:
1811:
1809:
1805:
1804:
1802:
1801:
1794:Tensor product
1787:
1781:
1776:
1770:
1765:
1759:
1754:
1749:
1739:
1738:
1737:
1732:
1722:
1717:
1715:Banach lattice
1712:
1711:
1710:
1700:
1694:
1692:
1688:
1687:
1679:
1678:
1671:
1664:
1656:
1647:
1646:
1644:
1643:
1632:
1629:
1628:
1626:
1625:
1620:
1615:
1610:
1608:Choquet theory
1605:
1600:
1594:
1592:
1588:
1587:
1585:
1584:
1574:
1569:
1564:
1559:
1554:
1549:
1544:
1539:
1534:
1529:
1524:
1518:
1516:
1512:
1511:
1509:
1508:
1503:
1497:
1495:
1491:
1490:
1488:
1487:
1482:
1477:
1472:
1467:
1462:
1460:Banach algebra
1456:
1454:
1450:
1449:
1447:
1446:
1441:
1436:
1431:
1426:
1421:
1416:
1411:
1406:
1401:
1395:
1393:
1389:
1388:
1386:
1385:
1383:Banach–Alaoglu
1380:
1375:
1370:
1365:
1360:
1355:
1350:
1345:
1339:
1337:
1331:
1330:
1327:
1326:
1324:
1323:
1318:
1313:
1311:Locally convex
1308:
1294:
1289:
1283:
1281:
1277:
1276:
1274:
1273:
1268:
1263:
1258:
1253:
1248:
1243:
1238:
1233:
1228:
1222:
1216:
1212:
1211:
1195:
1194:
1187:
1180:
1172:
1163:
1162:
1160:
1159:
1154:
1149:
1144:
1139:
1134:
1129:
1124:
1119:
1114:
1109:
1104:
1099:
1094:
1089:
1084:
1074:
1072:Corona theorem
1069:
1063:
1061:
1057:
1056:
1054:
1053:
1051:Wiener algebra
1047:
1045:
1041:
1040:
1038:
1037:
1032:
1027:
1022:
1017:
1012:
1007:
1002:
997:
991:
989:
985:
984:
982:
981:
971:
969:Pseudospectrum
966:
961:
959:Dirac spectrum
955:
953:
949:
948:
946:
945:
940:
935:
930:
924:
922:
918:
917:
915:
914:
913:
912:
902:
897:
892:
887:
882:
876:
870:
868:
864:
863:
861:
860:
855:
850:
845:
840:
835:
830:
825:
820:
814:
812:
808:
807:
805:
804:
799:
794:
789:
784:
779:
778:
777:
772:
767:
756:
754:
750:
749:
747:
746:
741:
736:
731:
726:
721:
715:
713:
707:
706:
704:
703:
698:
690:
682:
674:
668:
666:
662:
661:
659:
658:
653:
648:
643:
638:
633:
628:
622:
620:
616:
615:
613:
612:
610:Operator space
607:
602:
597:
592:
587:
582:
577:
572:
570:Banach algebra
567:
561:
559:
558:Basic concepts
555:
554:
543:
542:
535:
528:
520:
514:
513:
508:
503:
496:
493:
488:
474:non-degenerate
462:
447:
444:
440:Fourier series
423:
420:
416:real rank zero
399:such that the
396:
378:σ-compact
370:if and only if
318:
315:
298:
295:
292:
289:
286:
283:
278:
274:
270:
265:
262:
259:
255:
226:
223:
220:
217:
214:
209:
205:
201:
173:
170:
167:
164:
161:
156:
152:
148:
145:
140:
137:
134:
130:
101:
98:
95:
92:
89:
84:
80:
76:
56:
53:
41:Banach algebra
9:
6:
4:
3:
2:
3165:
3154:
3151:
3150:
3148:
3133:
3130:
3128:
3125:
3123:
3120:
3118:
3115:
3113:
3110:
3109:
3107:
3103:
3097:
3079:
3075:
3071:
3068:
3062:
3053:
3051:
3048:
3044:
3041:
3040:
3039:
3036:
3034:
3031:
3029:
3028:
3023:
3021:
3007:
3002:
2992:
2988:
2979:
2975:
2972:
2970:
2951:
2942:
2941:
2940:
2939:
2935:
2931:
2912:
2903:
2902:
2901:
2900:
2896:
2894:
2870:
2867:
2864:
2860:
2850:
2848:
2847:
2842:
2840:
2837:
2835:
2833:
2829:
2824:
2822:
2819:
2817:
2816:
2811:
2809:
2806:
2804:
2778:
2773:
2770:
2767:
2763:
2753:
2751:
2750:
2745:
2743:
2740:
2738:
2716:
2713:
2705:
2703:
2702:
2697:
2696:
2694:
2692:
2688:
2682:
2678:
2674:
2672:
2669:
2667:
2664:
2662:
2659:
2657:
2654:
2652:
2651:Extreme point
2649:
2647:
2644:
2642:
2639:
2637:
2633:
2629:
2627:
2624:
2623:
2621:
2617:
2611:
2608:
2606:
2603:
2601:
2598:
2596:
2593:
2591:
2585:
2582:
2578:
2574:
2571:
2569:
2563:
2561:
2558:
2556:
2553:
2551:
2548:
2546:
2543:
2541:
2538:
2536:
2533:
2532:
2530:
2528:Types of sets
2526:
2519:
2515:
2512:
2508:
2505:
2503:
2500:
2498:
2495:
2494:
2493:
2490:
2486:
2483:
2481:
2478:
2476:
2473:
2472:
2471:
2468:
2464:
2461:
2459:
2456:
2454:
2451:
2449:
2446:
2444:
2441:
2439:
2436:
2435:
2434:
2431:
2427:
2424:
2422:
2419:
2417:
2414:
2412:
2409:
2407:
2404:
2403:
2402:
2399:
2397:
2394:
2392:
2391:Convex series
2389:
2387:
2386:Bochner space
2384:
2380:
2377:
2376:
2375:
2372:
2370:
2367:
2366:
2364:
2360:
2354:
2351:
2349:
2346:
2344:
2341:
2339:
2338:Riesz's lemma
2336:
2334:
2331:
2329:
2326:
2324:
2323:Mazur's lemma
2321:
2319:
2316:
2314:
2311:
2309:
2306:
2304:
2301:
2297:
2294:
2293:
2292:
2289:
2287:
2284:
2282:
2279:
2277:
2276:Gelfand–Mazur
2274:
2272:
2269:
2267:
2264:
2262:
2259:
2257:
2254:
2252:
2249:
2247:
2244:
2242:
2239:
2237:
2234:
2232:
2229:
2227:
2224:
2222:
2219:
2217:
2214:
2213:
2211:
2209:
2205:
2199:
2196:
2194:
2191:
2187:
2184:
2182:
2179:
2178:
2177:
2174:
2170:
2167:
2165:
2162:
2161:
2160:
2157:
2155:
2152:
2150:
2147:
2145:
2142:
2141:
2139:
2137:
2133:
2127:
2124:
2122:
2119:
2117:
2114:
2112:
2109:
2107:
2104:
2102:
2099:
2097:
2094:
2092:
2089:
2085:
2082:
2081:
2080:
2077:
2075:
2072:
2068:
2065:
2063:
2060:
2059:
2057:
2055:
2052:
2050:
2046:
2042:
2038:
2035:
2034:
2033:
2030:
2028:
2025:
2023:
2019:
2015:
2011:
2008:
2006:
2003:
2001:
1998:
1997:
1996:
1993:
1991:
1988:
1987:
1985:
1983:
1979:
1973:
1970:
1968:
1965:
1961:
1958:
1956:
1953:
1952:
1951:
1948:
1944:
1941:
1939:
1936:
1935:
1934:
1931:
1929:
1926:
1924:
1921:
1917:
1914:
1913:
1912:
1909:
1907:
1904:
1902:
1899:
1898:
1896:
1892:
1886:
1883:
1881:
1878:
1874:
1871:
1870:
1869:
1866:
1864:
1861:
1859:
1856:
1852:
1848:
1845:
1844:
1843:
1840:
1836:
1833:
1832:
1831:
1828:
1826:
1823:
1821:
1818:
1816:
1813:
1812:
1810:
1806:
1799:
1795:
1791:
1788:
1786:
1782:
1780:
1777:
1775:) convex
1774:
1771:
1769:
1766:
1764:
1760:
1758:
1755:
1753:
1750:
1748:
1744:
1740:
1736:
1733:
1731:
1728:
1727:
1726:
1723:
1721:
1720:Grothendieck
1718:
1716:
1713:
1709:
1706:
1705:
1704:
1701:
1699:
1696:
1695:
1693:
1689:
1684:
1677:
1672:
1670:
1665:
1663:
1658:
1657:
1654:
1642:
1634:
1633:
1630:
1624:
1621:
1619:
1616:
1614:
1613:Weak topology
1611:
1609:
1606:
1604:
1601:
1599:
1596:
1595:
1593:
1589:
1582:
1578:
1575:
1573:
1570:
1568:
1565:
1563:
1560:
1558:
1555:
1553:
1550:
1548:
1545:
1543:
1540:
1538:
1537:Index theorem
1535:
1533:
1530:
1528:
1525:
1523:
1520:
1519:
1517:
1513:
1507:
1504:
1502:
1499:
1498:
1496:
1494:Open problems
1492:
1486:
1483:
1481:
1478:
1476:
1473:
1471:
1468:
1466:
1463:
1461:
1458:
1457:
1455:
1451:
1445:
1442:
1440:
1437:
1435:
1432:
1430:
1427:
1425:
1422:
1420:
1417:
1415:
1412:
1410:
1407:
1405:
1402:
1400:
1397:
1396:
1394:
1390:
1384:
1381:
1379:
1376:
1374:
1371:
1369:
1366:
1364:
1361:
1359:
1356:
1354:
1351:
1349:
1346:
1344:
1341:
1340:
1338:
1336:
1332:
1322:
1319:
1317:
1314:
1312:
1309:
1306:
1302:
1298:
1295:
1293:
1290:
1288:
1285:
1284:
1282:
1278:
1272:
1269:
1267:
1264:
1262:
1259:
1257:
1254:
1252:
1249:
1247:
1244:
1242:
1239:
1237:
1234:
1232:
1229:
1227:
1224:
1223:
1220:
1217:
1213:
1208:
1204:
1200:
1193:
1188:
1186:
1181:
1179:
1174:
1173:
1170:
1158:
1155:
1153:
1150:
1148:
1145:
1143:
1140:
1138:
1135:
1133:
1130:
1128:
1125:
1123:
1120:
1118:
1115:
1113:
1110:
1108:
1105:
1103:
1100:
1098:
1095:
1093:
1090:
1088:
1085:
1082:
1078:
1075:
1073:
1070:
1068:
1065:
1064:
1062:
1058:
1052:
1049:
1048:
1046:
1042:
1036:
1033:
1031:
1028:
1026:
1023:
1021:
1018:
1016:
1013:
1011:
1008:
1006:
1003:
1001:
998:
996:
993:
992:
990:
988:Miscellaneous
986:
979:
975:
972:
970:
967:
965:
962:
960:
957:
956:
954:
950:
944:
941:
939:
936:
934:
931:
929:
926:
925:
923:
919:
911:
908:
907:
906:
903:
901:
898:
896:
893:
891:
888:
886:
883:
881:
877:
875:
872:
871:
869:
865:
859:
856:
854:
851:
849:
846:
844:
841:
839:
836:
834:
831:
829:
826:
824:
821:
819:
816:
815:
813:
809:
803:
800:
798:
795:
793:
790:
788:
785:
783:
780:
776:
773:
771:
768:
766:
763:
762:
761:
758:
757:
755:
753:Decomposition
751:
745:
742:
740:
737:
735:
732:
730:
727:
725:
722:
720:
717:
716:
714:
712:
708:
702:
699:
697:
694:
691:
689:
686:
683:
681:
678:
675:
673:
670:
669:
667:
663:
657:
654:
652:
649:
647:
644:
642:
639:
637:
634:
632:
629:
627:
624:
623:
621:
617:
611:
608:
606:
603:
601:
598:
596:
593:
591:
588:
586:
583:
581:
578:
576:
573:
571:
568:
566:
563:
562:
560:
556:
552:
548:
541:
536:
534:
529:
527:
522:
521:
518:
512:
509:
507:
504:
502:
499:
498:
492:
487:
483:
479:
476:if for every
475:
471:
466:
461:
457:
453:
443:
441:
437:
436:Fejér kernels
433:
429:
419:
417:
412:
410:
406:
403:generated by
402:
395:
391:
387:
383:
379:
375:
371:
367:
363:
359:
358:σ-unital
355:
351:
346:
344:
343:Hilbert space
340:
336:
332:
328:
324:
314:
312:
296:
293:
287:
284:
281:
276:
272:
260:
257:
244:
240:
218:
215:
212:
207:
203:
191:
187:
184:Similarly, a
171:
168:
162:
159:
154:
150:
146:
135:
132:
119:
115:
93:
90:
87:
82:
78:
66:
62:
52:
50:
46:
42:
38:
34:
30:
26:
19:
3105:Applications
3026:
2937:
2898:
2845:
2831:
2827:
2814:
2748:
2700:
2587:Linear cone
2580:
2576:
2565:Convex cone
2458:Paley–Wiener
2318:Mackey–Arens
2308:Krein–Milman
2261:Closed range
2256:Closed graph
2226:Banach–Mazur
2106:Self-adjoint
2010:sesquilinear
1743:Polynomially
1683:Banach space
1603:Balanced set
1577:Distribution
1515:Applications
1368:Krein–Milman
1353:Closed graph
1060:Applications
890:Disk algebra
879:
744:Spectral gap
619:Main results
485:
481:
477:
473:
467:
465:for some λ.
459:
455:
449:
425:
413:
408:
404:
393:
389:
385:
381:
357:
353:
347:
341:acting on a
334:
330:
327:self-adjoint
320:
310:
242:
238:
189:
185:
117:
113:
64:
60:
58:
22:
2826:Continuous
2661:Linear span
2646:Convex hull
2626:Affine hull
2485:holomorphic
2421:holomorphic
2401:Derivatives
2291:Hahn–Banach
2231:Banach–Saks
2149:C*-algebras
2116:Trace class
2079:Functionals
1967:Ultrastrong
1880:Quasinormed
1532:Heat kernel
1522:Hardy space
1429:Trace class
1343:Hahn–Banach
1305:Topological
1087:Heat kernel
787:Compression
672:Isospectral
428:convolution
323:C*-algebras
317:C*-algebras
33:ring theory
25:mathematics
2579:), and (Hw
2480:continuous
2416:functional
2164:C*-algebra
2049:Continuous
1911:Dual space
1885:Stereotype
1863:Metrizable
1790:Projective
1465:C*-algebra
1280:Properties
765:Continuous
580:C*-algebra
575:B*-algebra
55:Definition
3038:Sobolev W
2981:Schwartz
2956:∞
2917:∞
2913:ℓ
2879:Ω
2865:λ
2723:Σ
2605:Symmetric
2540:Absorbing
2453:regulated
2433:Integrals
2286:Goldstine
2121:Transpose
2058:Fredholm
1928:Ultraweak
1916:Dual norm
1847:Seminorms
1815:Barrelled
1785:Injective
1773:Uniformly
1747:Reflexive
1439:Unbounded
1434:Transpose
1392:Operators
1321:Separable
1316:Reflexive
1301:Algebraic
1287:Barrelled
551:-algebras
501:Mollifier
362:separable
291:‖
285:−
277:λ
269:‖
264:Λ
261:∈
258:λ
222:Λ
219:∈
216:λ
208:λ
192:is a net
166:‖
160:−
155:λ
144:‖
139:Λ
136:∈
133:λ
97:Λ
94:∈
91:λ
83:λ
67:is a net
3147:Category
2974:weighted
2844:Hilbert
2821:Bs space
2691:Examples
2656:Interior
2632:Relative
2610:Zonotope
2589:(subset)
2567:(subset)
2518:Strongly
2497:Lebesgue
2492:Measures
2362:Analysis
2208:Theorems
2159:Spectrum
2084:positive
2067:operator
2005:operator
1995:Bilinear
1960:operator
1943:operator
1923:Operator
1820:Complete
1768:Strictly
1641:Category
1453:Algebras
1335:Theorems
1292:Complete
1261:Schwartz
1207:glossary
1152:Weyl law
1097:Lax pair
1044:Examples
878:With an
797:Discrete
775:Residual
711:Spectrum
696:operator
688:operator
680:operator
595:Spectrum
495:See also
454:so that
374:spectrum
366:converse
360:. Every
350:sequence
2839:Hardy H
2742:c space
2679:)
2634:)
2555:Bounded
2443:Dunford
2438:Bochner
2411:Gateaux
2406:Fréchet
2181:of ODEs
2126:Unitary
2101:Nuclear
2032:Compact
2022:Bounded
1990:Adjoint
1830:Fréchet
1825:F-space
1796: (
1792:)
1745:)
1725:Hilbert
1698:Asplund
1444:Unitary
1424:Nuclear
1409:Compact
1404:Bounded
1399:Adjoint
1373:Min–max
1266:Sobolev
1251:Nuclear
1241:Hilbert
1236:Fréchet
1201: (
693:Unitary
2755:Besov
2595:Radial
2560:Convex
2545:Affine
2514:Weakly
2507:Vector
2379:bundle
2169:radius
2096:Normal
2062:kernel
2027:Closed
1950:Strong
1868:Normed
1858:Mackey
1703:Banach
1685:topics
1419:Normal
1256:Orlicz
1246:Hölder
1226:Banach
1215:Spaces
1203:topics
677:Normal
470:module
2830:with
2677:Quasi
2671:Polar
2475:Borel
2426:quasi
1955:polar
1938:polar
1752:Riesz
1231:Besov
770:Point
446:Rings
39:in a
2828:C(K)
2463:weak
2000:form
1933:Weak
1906:Dual
1873:norm
1835:tame
1708:list
1579:(or
1297:Dual
701:Unit
549:and
372:its
321:For
45:ring
31:and
2045:Dis
438:of
407:is
392:in
376:is
309:An
254:lim
241:of
129:lim
116:of
43:or
37:net
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