534:
2451:
1536:
316:
1908:
on a complex vector space is algebraically equivalent to a topologically irreducible *-representation on a
Hilbert space. Topologically irreducible *-representations on a Hilbert space are algebraically isomorphic if and only if they are unitarily
1657:
645:
137:
locally compact groups of type I and a decomposition theorem for arbitrary representations of separable locally compact groups of type I. The resulting duality theory for locally compact groups is however much weaker than the
737:
459:
1419:
1339:
1007:
1209:
374:
1083:
801:
537:
3-dimensional commutative C*-algebra and its ideals. Each of 8 ideals corresponds to a closed subset of discrete 3-points space (or to an open complement). Primitive ideals correspond to closed
229:
2974:
1843:
The result implies a far-reaching generalization of the structure of representations of separable type I C*-algebras and correspondingly of separable locally compact groups of type I.
3076:
2340:
1563:
1752:
locally compact group is of type I if and only if the Borel space is standard, i.e. is isomorphic (in the category of Borel spaces) to the underlying Borel space of a
2709:
154:
groups, both of which are complete invariants. That the dual is not a complete invariant is easily seen as the dual of any finite-dimensional full matrix algebra M
2003:
591:
2166:
1368:
which arises by considering the space of representations as a topological space with an appropriate pointwise convergence topology. More precisely, let
2714:
2487:
3061:
2293:
2148:
1921:
2124:
656:
2954:
408:
2807:
2605:
1531:{\displaystyle \langle \pi _{i}(x)\xi \mid \eta \rangle \to \langle \pi (x)\xi \mid \eta \rangle \quad \forall \xi ,\eta \in H_{n}\ x\in A.}
2802:
1291:
952:
2016:
1148:
2959:
2105:
1996:
331:
2777:
2375:
2746:
2020:
1038:
756:
2736:
2731:
2724:
2660:
2544:
2969:
2480:
2171:
2227:
311:{\displaystyle {\overline {X}}=\left\{\rho \in \operatorname {Prim} (A):\rho \supseteq \bigcap _{\pi \in X}\pi \right\}.}
2595:
2454:
2176:
2161:
1989:
3106:
2580:
2191:
3026:
513:
3081:
2979:
2859:
2436:
2196:
203:, where a primitive ideal is the kernel of a non-zero irreducible *-representation. The set of primitive ideals is a
2390:
2314:
42:
2575:
2431:
139:
3086:
2949:
2782:
2767:
2539:
2247:
1878:
1100:, other characterizations of the topology on the spectrum in terms of positive definite functions are desirable.
380:
143:
2668:
2181:
533:
3132:
2473:
2084:
750:) so is in fact primitive. For details of the proof, see the Dixmier reference. For a commutative C*-algebra,
2156:
3041:
3016:
2834:
2823:
2534:
2380:
538:
2892:
2882:
2877:
2585:
2411:
2355:
2319:
580:
550:
38:
922:) on the other hand is much larger. There are many inequivalent irreducible representations with kernel
2637:
119:
115:
1093:
The hull-kernel topology is easy to describe abstractly, but in practice for C*-algebras associated to
1108:
398:
Since unitarily equivalent representations have the same kernel, the map Ï âŠ ker(Ï) factors through a
3127:
3051:
3030:
2944:
2829:
2792:
2394:
1652:{\displaystyle \pi _{i}(x)\xi \to \pi (x)\xi \quad {\mbox{ normwise }}\forall \xi \in H_{n}\ x\in A.}
2854:
2590:
2360:
2298:
2012:
1677:
consisting of equivalence classes of representations whose underlying
Hilbert space has dimension
96:
2984:
2913:
2844:
2688:
2650:
2385:
2252:
1862:
1852:
147:
127:
108:
92:
irreducible representation, thus excluding trivial (i.e. identically 0) representations on one-
3091:
3066:
2751:
2673:
2365:
1097:
1029:
204:
123:
104:
1407:
with the point-weak topology. In terms of convergence of nets, this topology is defined by Ï
3096:
2797:
2645:
2600:
2524:
2370:
2288:
2257:
2237:
2222:
2217:
2212:
2049:
1817:
1233:
502:
131:
50:
3071:
3056:
2964:
2927:
2923:
2887:
2849:
2787:
2772:
2741:
2683:
2642:
2629:
2554:
2496:
2232:
2186:
2134:
2129:
2100:
1981:
1866:
1802:
1094:
576:
192:
134:
2059:
3021:
3000:
2918:
2908:
2719:
2626:
2559:
2519:
2421:
2273:
2074:
1890:
640:{\displaystyle \operatorname {I} :X\cong \operatorname {Prim} (\operatorname {C} (X)).}
1877:
is regarded algebraically. For a ring an ideal is primitive if and only if it is the
3121:
2426:
2350:
2079:
2064:
2054:
1937:
1882:
816:
583:
573:
58:
2839:
2693:
2634:
2416:
2069:
2039:
1753:
558:
2465:
114:
One of the most important applications of this concept is to provide a notion of
3036:
2621:
2345:
2335:
2242:
2044:
1813:
1761:
180:
can be defined in several equivalent ways. We first define it in terms of the
2529:
2278:
2118:
2114:
2110:
1249:
512:
induced from the hull-kernel topology has other characterizations in terms of
399:
383:. As a consequence, it can be shown that there is a unique topology Ï on Prim(
322:
22:
1215:
with ||Ο|| = 1, is the weak limit of states associated to representations in
2514:
2500:
1809:
1806:
1764:
for separable C*-algebras in the 1961 paper listed in the references below.
1745:
1741:
732:{\displaystyle \operatorname {I} (x)=\{f\in \operatorname {C} (X):f(x)=0\}.}
93:
1244:. By one of the basic theorems associated to the GNS construction, a state
454:{\displaystyle \operatorname {k} :{\hat {A}}\to \operatorname {Prim} (A).}
3101:
3046:
1126:. Then the following are equivalent for an irreducible representation Ï;
501:
The hull-kernel topology is an analogue for non-commutative rings of the
173:
1778:
if and only if the center of the von
Neumann algebra generated by Ï(
1032:. For finite-dimensional C*-algebras, we also have the isomorphism
1786:
is of type I if and only if any separable factor representation of
1334:{\displaystyle \kappa :\operatorname {PureState} (A)\to {\hat {A}}}
1002:{\displaystyle A\cong \bigoplus _{e\in \operatorname {min} (A)}Ae,}
1904:
be a C*-algebra. Any algebraically irreducible representation of
1820:
are all of type I. Compact and abelian groups are also of type I.
532:
1913:
This is the
Corollary of Theorem 2.9.5 of the Dixmier reference.
1770:. A non-degenerate *-representation Ï of a separable C*-algebra
1225:
The second condition means exactly that Ï is weakly contained in
1204:{\displaystyle f_{\xi }(x)=\langle \xi \mid \pi (x)\xi \rangle }
2469:
1985:
1920:
is a locally compact group, the topology on dual space of the
1279:
From the previous theorem one can easily prove the following;
946:
is isomorphic to a finite direct sum of full matrix algebras:
391:
with respect to Ï is identical to the hull-kernel closure of
88:. We implicitly assume that irreducible representation means
369:{\displaystyle {\overline {\overline {X}}}={\overline {X}},}
1740:
is a topological space and thus can also be regarded as a
1790:
is a finite or countable multiple of an irreducible one.
1364:
There is yet another characterization of the topology on
1258:
is irreducible. Moreover, the mapping Îș : PureState(
1078:{\displaystyle {\hat {A}}\cong \operatorname {Prim} (A).}
796:{\displaystyle {\hat {A}}\cong \operatorname {Prim} (A).}
1705:
can be regarded as the quotient topological space of Irr
1609:
1566:
1422:
1345:
given by the GNS construction is continuous and open.
1294:
1151:
1041:
955:
759:
659:
594:
411:
334:
232:
1139:
Every state associated to Ï, that is one of the form
3009:
2993:
2937:
2901:
2870:
2816:
2760:
2702:
2659:
2614:
2568:
2507:
2404:
2328:
2307:
2266:
2205:
2147:
2093:
2028:
1396:) is the space of irreducible *-representations of
1236:is a recipe for associating states of a C*-algebra
3077:Spectral theory of ordinary differential equations
2341:Spectral theory of ordinary differential equations
1651:
1551:) is the same as the point-strong topology, i.e. Ï
1530:
1333:
1203:
1077:
1001:
795:
731:
639:
453:
368:
310:
122:. This dual object is suitable for formulating a
2975:SchröderâBernstein theorems for operator algebras
1756:. Mackey called Borel spaces with this property
1111:of representations as is shown by the following:
942:is a finite-dimensional C*-algebra. It is known
76:and {0} which is invariant under all operators Ï(
1252:if and only if the associated representation Ï
2481:
1997:
1793:Examples of separable locally compact groups
846:) of compact operators. Thus as a set, Prim(
321:Hull-kernel closure is easily shown to be an
68:if, and only if, there is no closed subspace
8:
1893:it is primitive in the sense defined above.
1484:
1460:
1454:
1423:
1379:be the canonical Hilbert space of dimension
1198:
1174:
1107:is intimately connected with the concept of
723:
678:
541:. See details at the image description page.
2488:
2474:
2466:
2032:
2004:
1990:
1982:
1628:
1608:
1571:
1565:
1507:
1430:
1421:
1320:
1319:
1293:
1156:
1150:
1043:
1042:
1040:
1016:) are the minimal central projections of
966:
954:
761:
760:
758:
658:
593:
545:The spectrum of a commutative C*-algebra
419:
418:
410:
353:
335:
333:
285:
233:
231:
2294:Group algebra of a locally compact group
1978:, The University of Chicago Press, 1955.
1962:Les C*-algÚbres et leurs représentations
1955:Les C*-algÚbres et leurs représentations
1953:, North-Holland, 1977 (a translation of
1722:. The piecing together of the various
1698:is continuous and open. In particular,
1089:Other characterizations of the spectrum
107:; this is similar to the notion of the
1971:, Annals of Mathematics, vol 73, 1961.
1889:, an ideal is algebraically primitive
1885:. It turns out that for a C*-algebra
1541:It turns out that this topology on Irr
2808:Spectral theory of normal C*-algebras
2606:Spectral theory of normal C*-algebras
7:
2803:Spectral theory of compact operators
1782:) is one-dimensional. A C*-algebra
815:be a separable infinite-dimensional
1976:The Theory of Group Representations
807:The C*-algebra of bounded operators
379:and it can be shown to satisfy the
99:. As explained below, the spectrum
2955:CohenâHewitt factorization theorem
1865:, we can also consider the set of
1615:
1488:
687:
660:
616:
595:
412:
219:is a set of primitive ideals, its
215:). This is defined as follows: If
14:
2960:Extensions of symmetric operators
1024:is canonically isomorphic to min(
746:) is a closed maximal ideal in C(
387:) such that the closure of a set
2778:Positive operator-valued measure
2450:
2449:
2376:Topological quantum field theory
1760:. This conjecture was proved by
827:) has two norm-closed *-ideals:
3062:RayleighâFaberâKrahn inequality
1754:complete separable metric space
1607:
1487:
834: = {0} and the ideal
1601:
1595:
1589:
1583:
1577:
1472:
1466:
1457:
1442:
1436:
1325:
1316:
1313:
1307:
1192:
1186:
1168:
1162:
1129:The equivalence class of Ï in
1069:
1063:
1048:
985:
979:
934:Finite-dimensional C*-algebras
787:
781:
766:
714:
708:
699:
693:
672:
666:
631:
628:
622:
613:
498:). This is indeed a topology.
445:
439:
430:
424:
269:
263:
164:) consists of a single point.
1:
2970:Limiting absorption principle
2172:Uniform boundedness principle
1372:be a cardinal number and let
911:)) is a non-Hausdorff space.
873:} is a closed subset of Prim(
557:(not to be confused with the
2596:Singular value decomposition
1715:) under unitary equivalence.
358:
345:
341:
238:
3027:Hearing the shape of a drum
2710:Decomposition of a spectrum
1847:Algebraic primitive spectra
930:) or with kernel {0}.
650:This mapping is defined by
568:). In particular, suppose
3149:
2615:Special Elements/Operators
2315:Invariant subspace problem
1850:
1744:. A famous conjecture of
1729:can be quite complicated.
468:to define the topology on
187:The primitive spectrum of
144:compact topological groups
3087:Superstrong approximation
2950:Banach algebra cohomology
2783:Projection-valued measure
2768:Borel functional calculus
2540:Projection-valued measure
2445:
2035:
1964:, Gauthier-Villars, 1969.
1835:is smooth if and only if
1103:In fact, the topology on
381:Kuratowski closure axioms
2679:Spectrum of a C*-algebra
2550:Spectrum of a C*-algebra
2284:Spectrum of a C*-algebra
3107:WienerâKhinchin theorem
3042:Kuznetsov trace formula
3017:Almost Mathieu operator
2835:Banach function algebra
2824:Amenable Banach algebra
2581:GelfandâNaimark theorem
2535:Noncommutative topology
2381:Noncommutative geometry
1681:. The canonical map Irr
529:Commutative C*-algebras
505:for commutative rings.
3082:SturmâLiouville theory
2980:ShermanâTakeda theorem
2860:TomitaâTakesaki theory
2635:Hermitian/Self-adjoint
2586:Gelfand representation
2437:TomitaâTakesaki theory
2412:Approximation property
2356:Calculus of variations
1816:Lie groups. Thus the
1733:MackeyâBorel structure
1653:
1532:
1335:
1240:to representations of
1205:
1079:
1003:
797:
733:
641:
542:
455:
370:
312:
2576:GelfandâMazur theorem
2432:BanachâMazur distance
2395:Generalized functions
1776:factor representation
1654:
1533:
1336:
1276:is a surjective map.
1206:
1133:is in the closure of
1080:
1004:
798:
734:
642:
536:
456:
371:
313:
140:TannakaâKrein duality
120:locally compact group
45:*-representations of
3052:Proto-value function
3031:Dirichlet eigenvalue
2945:Abstract index group
2830:Approximate identity
2793:Rigged Hilbert space
2669:KreinâRutman theorem
2515:Involution/*-algebra
2177:Kakutani fixed-point
2162:Riesz representation
1611: normwise
1564:
1420:
1413:â Ï; if and only if
1292:
1149:
1039:
953:
757:
657:
592:
564:of the Banach space
409:
332:
230:
209:hull-kernel topology
150:for locally compact
103:is also naturally a
28:dual of a C*-algebra
19:In mathematics, the
16:Mathematical concept
2855:Von Neumann algebra
2591:Polar decomposition
2361:Functional calculus
2320:Mahler's conjecture
2299:Von Neumann algebra
2013:Functional analysis
1857:Since a C*-algebra
1812:and connected real
1801:) is of type I are
1557:â Ï if and only if
1020:. The spectrum of
579:. Then there is a
549:coincides with the
482:are inverse images
478:. The open sets of
325:operation, that is
221:hull-kernel closure
39:unitary equivalence
2985:Unbounded operator
2914:Essential spectrum
2893:SchurâHorn theorem
2883:BauerâFike theorem
2878:AlonâBoppana bound
2871:Finite-Dimensional
2845:Nuclear C*-algebra
2689:Spectral asymmetry
2386:Riemann hypothesis
2085:Topological vector
1969:Type I C*-algebras
1853:Primitive spectrum
1649:
1613:
1528:
1331:
1201:
1098:topological groups
1075:
999:
989:
793:
729:
637:
543:
490:) of open subsets
451:
366:
308:
296:
182:primitive spectrum
168:Primitive spectrum
148:Pontryagin duality
128:Plancherel theorem
109:spectrum of a ring
3115:
3114:
3092:Transfer operator
3067:Spectral geometry
2752:Spectral abscissa
2732:Approximate point
2674:Normal eigenvalue
2463:
2462:
2366:Integral operator
2143:
2142:
1818:Heisenberg groups
1673:be the subset of
1636:
1612:
1515:
1328:
1051:
1030:discrete topology
962:
769:
427:
361:
348:
344:
281:
241:
213:Jacobson topology
205:topological space
124:Fourier transform
105:topological space
3140:
3097:Transform theory
2817:Special algebras
2798:Spectral theorem
2761:Spectral Theorem
2601:Spectral theorem
2490:
2483:
2476:
2467:
2453:
2452:
2371:Jones polynomial
2289:Operator algebra
2033:
2006:
1999:
1992:
1983:
1922:group C*-algebra
1867:primitive ideals
1748:proposed that a
1658:
1656:
1655:
1650:
1634:
1633:
1632:
1614:
1610:
1576:
1575:
1537:
1535:
1534:
1529:
1513:
1512:
1511:
1435:
1434:
1340:
1338:
1337:
1332:
1330:
1329:
1321:
1234:GNS construction
1210:
1208:
1207:
1202:
1161:
1160:
1109:weak containment
1084:
1082:
1081:
1076:
1053:
1052:
1044:
1008:
1006:
1005:
1000:
988:
914:The spectrum of
884:The closure of {
802:
800:
799:
794:
771:
770:
762:
738:
736:
735:
730:
646:
644:
643:
638:
508:The topology on
503:Zariski topology
460:
458:
457:
452:
429:
428:
420:
375:
373:
372:
367:
362:
354:
349:
337:
336:
317:
315:
314:
309:
304:
300:
295:
242:
234:
193:primitive ideals
51:*-representation
37:, is the set of
3148:
3147:
3143:
3142:
3141:
3139:
3138:
3137:
3133:Spectral theory
3118:
3117:
3116:
3111:
3072:Spectral method
3057:Ramanujan graph
3005:
2989:
2965:Fredholm theory
2933:
2928:Shilov boundary
2924:Structure space
2902:Generalizations
2897:
2888:Numerical range
2866:
2850:Uniform algebra
2812:
2788:Riesz projector
2773:Min-max theorem
2756:
2742:Direct integral
2698:
2684:Spectral radius
2655:
2610:
2564:
2555:Spectral radius
2503:
2497:Spectral theory
2494:
2464:
2459:
2441:
2405:Advanced topics
2400:
2324:
2303:
2262:
2228:HilbertâSchmidt
2201:
2192:GelfandâNaimark
2139:
2089:
2024:
2010:
1946:
1855:
1849:
1735:
1727:
1710:
1703:
1696:
1686:
1671:
1624:
1567:
1562:
1561:
1556:
1546:
1503:
1426:
1418:
1417:
1412:
1405:
1391:
1377:
1362:
1355:
1290:
1289:
1275:
1257:
1152:
1147:
1146:
1122:be a subset of
1095:locally compact
1091:
1037:
1036:
951:
950:
936:
890:
860:
833:
809:
755:
754:
655:
654:
590:
589:
577:Hausdorff space
531:
526:
464:We use the map
407:
406:
330:
329:
250:
246:
228:
227:
170:
159:
118:object for any
72:different from
17:
12:
11:
5:
3146:
3144:
3136:
3135:
3130:
3120:
3119:
3113:
3112:
3110:
3109:
3104:
3099:
3094:
3089:
3084:
3079:
3074:
3069:
3064:
3059:
3054:
3049:
3044:
3039:
3034:
3024:
3022:Corona theorem
3019:
3013:
3011:
3007:
3006:
3004:
3003:
3001:Wiener algebra
2997:
2995:
2991:
2990:
2988:
2987:
2982:
2977:
2972:
2967:
2962:
2957:
2952:
2947:
2941:
2939:
2935:
2934:
2932:
2931:
2921:
2919:Pseudospectrum
2916:
2911:
2909:Dirac spectrum
2905:
2903:
2899:
2898:
2896:
2895:
2890:
2885:
2880:
2874:
2872:
2868:
2867:
2865:
2864:
2863:
2862:
2852:
2847:
2842:
2837:
2832:
2826:
2820:
2818:
2814:
2813:
2811:
2810:
2805:
2800:
2795:
2790:
2785:
2780:
2775:
2770:
2764:
2762:
2758:
2757:
2755:
2754:
2749:
2744:
2739:
2734:
2729:
2728:
2727:
2722:
2717:
2706:
2704:
2700:
2699:
2697:
2696:
2691:
2686:
2681:
2676:
2671:
2665:
2663:
2657:
2656:
2654:
2653:
2648:
2640:
2632:
2624:
2618:
2616:
2612:
2611:
2609:
2608:
2603:
2598:
2593:
2588:
2583:
2578:
2572:
2570:
2566:
2565:
2563:
2562:
2560:Operator space
2557:
2552:
2547:
2542:
2537:
2532:
2527:
2522:
2520:Banach algebra
2517:
2511:
2509:
2508:Basic concepts
2505:
2504:
2495:
2493:
2492:
2485:
2478:
2470:
2461:
2460:
2458:
2457:
2446:
2443:
2442:
2440:
2439:
2434:
2429:
2424:
2422:Choquet theory
2419:
2414:
2408:
2406:
2402:
2401:
2399:
2398:
2388:
2383:
2378:
2373:
2368:
2363:
2358:
2353:
2348:
2343:
2338:
2332:
2330:
2326:
2325:
2323:
2322:
2317:
2311:
2309:
2305:
2304:
2302:
2301:
2296:
2291:
2286:
2281:
2276:
2274:Banach algebra
2270:
2268:
2264:
2263:
2261:
2260:
2255:
2250:
2245:
2240:
2235:
2230:
2225:
2220:
2215:
2209:
2207:
2203:
2202:
2200:
2199:
2197:BanachâAlaoglu
2194:
2189:
2184:
2179:
2174:
2169:
2164:
2159:
2153:
2151:
2145:
2144:
2141:
2140:
2138:
2137:
2132:
2127:
2125:Locally convex
2122:
2108:
2103:
2097:
2095:
2091:
2090:
2088:
2087:
2082:
2077:
2072:
2067:
2062:
2057:
2052:
2047:
2042:
2036:
2030:
2026:
2025:
2011:
2009:
2008:
2001:
1994:
1986:
1980:
1979:
1972:
1965:
1958:
1945:
1942:
1936:, named after
1932:is called the
1911:
1910:
1891:if and only if
1848:
1845:
1841:
1840:
1831:is separable,
1734:
1731:
1725:
1717:
1716:
1706:
1701:
1694:
1682:
1669:
1660:
1659:
1648:
1645:
1642:
1639:
1631:
1627:
1623:
1620:
1617:
1606:
1603:
1600:
1597:
1594:
1591:
1588:
1585:
1582:
1579:
1574:
1570:
1552:
1542:
1539:
1538:
1527:
1524:
1521:
1518:
1510:
1506:
1502:
1499:
1496:
1493:
1490:
1486:
1483:
1480:
1477:
1474:
1471:
1468:
1465:
1462:
1459:
1456:
1453:
1450:
1447:
1444:
1441:
1438:
1433:
1429:
1425:
1408:
1403:
1387:
1375:
1361:
1351:
1348:
1347:
1346:
1343:
1342:
1341:
1327:
1324:
1318:
1315:
1312:
1309:
1306:
1303:
1300:
1297:
1271:
1253:
1223:
1222:
1221:
1220:
1213:
1212:
1211:
1200:
1197:
1194:
1191:
1188:
1185:
1182:
1179:
1176:
1173:
1170:
1167:
1164:
1159:
1155:
1141:
1140:
1137:
1090:
1087:
1086:
1085:
1074:
1071:
1068:
1065:
1062:
1059:
1056:
1050:
1047:
1010:
1009:
998:
995:
992:
987:
984:
981:
978:
975:
972:
969:
965:
961:
958:
935:
932:
901:
900:
888:
882:
858:
831:
808:
805:
804:
803:
792:
789:
786:
783:
780:
777:
774:
768:
765:
740:
739:
728:
725:
722:
719:
716:
713:
710:
707:
704:
701:
698:
695:
692:
689:
686:
683:
680:
677:
674:
671:
668:
665:
662:
648:
647:
636:
633:
630:
627:
624:
621:
618:
615:
612:
609:
606:
603:
600:
597:
530:
527:
525:
522:
462:
461:
450:
447:
444:
441:
438:
435:
432:
426:
423:
417:
414:
377:
376:
365:
360:
357:
352:
347:
343:
340:
319:
318:
307:
303:
299:
294:
291:
288:
284:
280:
277:
274:
271:
268:
265:
262:
259:
256:
253:
249:
245:
240:
237:
191:is the set of
169:
166:
155:
21:spectrum of a
15:
13:
10:
9:
6:
4:
3:
2:
3145:
3134:
3131:
3129:
3126:
3125:
3123:
3108:
3105:
3103:
3100:
3098:
3095:
3093:
3090:
3088:
3085:
3083:
3080:
3078:
3075:
3073:
3070:
3068:
3065:
3063:
3060:
3058:
3055:
3053:
3050:
3048:
3045:
3043:
3040:
3038:
3035:
3032:
3028:
3025:
3023:
3020:
3018:
3015:
3014:
3012:
3008:
3002:
2999:
2998:
2996:
2992:
2986:
2983:
2981:
2978:
2976:
2973:
2971:
2968:
2966:
2963:
2961:
2958:
2956:
2953:
2951:
2948:
2946:
2943:
2942:
2940:
2938:Miscellaneous
2936:
2929:
2925:
2922:
2920:
2917:
2915:
2912:
2910:
2907:
2906:
2904:
2900:
2894:
2891:
2889:
2886:
2884:
2881:
2879:
2876:
2875:
2873:
2869:
2861:
2858:
2857:
2856:
2853:
2851:
2848:
2846:
2843:
2841:
2838:
2836:
2833:
2831:
2827:
2825:
2822:
2821:
2819:
2815:
2809:
2806:
2804:
2801:
2799:
2796:
2794:
2791:
2789:
2786:
2784:
2781:
2779:
2776:
2774:
2771:
2769:
2766:
2765:
2763:
2759:
2753:
2750:
2748:
2745:
2743:
2740:
2738:
2735:
2733:
2730:
2726:
2723:
2721:
2718:
2716:
2713:
2712:
2711:
2708:
2707:
2705:
2703:Decomposition
2701:
2695:
2692:
2690:
2687:
2685:
2682:
2680:
2677:
2675:
2672:
2670:
2667:
2666:
2664:
2662:
2658:
2652:
2649:
2647:
2644:
2641:
2639:
2636:
2633:
2631:
2628:
2625:
2623:
2620:
2619:
2617:
2613:
2607:
2604:
2602:
2599:
2597:
2594:
2592:
2589:
2587:
2584:
2582:
2579:
2577:
2574:
2573:
2571:
2567:
2561:
2558:
2556:
2553:
2551:
2548:
2546:
2543:
2541:
2538:
2536:
2533:
2531:
2528:
2526:
2523:
2521:
2518:
2516:
2513:
2512:
2510:
2506:
2502:
2498:
2491:
2486:
2484:
2479:
2477:
2472:
2471:
2468:
2456:
2448:
2447:
2444:
2438:
2435:
2433:
2430:
2428:
2427:Weak topology
2425:
2423:
2420:
2418:
2415:
2413:
2410:
2409:
2407:
2403:
2396:
2392:
2389:
2387:
2384:
2382:
2379:
2377:
2374:
2372:
2369:
2367:
2364:
2362:
2359:
2357:
2354:
2352:
2351:Index theorem
2349:
2347:
2344:
2342:
2339:
2337:
2334:
2333:
2331:
2327:
2321:
2318:
2316:
2313:
2312:
2310:
2308:Open problems
2306:
2300:
2297:
2295:
2292:
2290:
2287:
2285:
2282:
2280:
2277:
2275:
2272:
2271:
2269:
2265:
2259:
2256:
2254:
2251:
2249:
2246:
2244:
2241:
2239:
2236:
2234:
2231:
2229:
2226:
2224:
2221:
2219:
2216:
2214:
2211:
2210:
2208:
2204:
2198:
2195:
2193:
2190:
2188:
2185:
2183:
2180:
2178:
2175:
2173:
2170:
2168:
2165:
2163:
2160:
2158:
2155:
2154:
2152:
2150:
2146:
2136:
2133:
2131:
2128:
2126:
2123:
2120:
2116:
2112:
2109:
2107:
2104:
2102:
2099:
2098:
2096:
2092:
2086:
2083:
2081:
2078:
2076:
2073:
2071:
2068:
2066:
2063:
2061:
2058:
2056:
2053:
2051:
2048:
2046:
2043:
2041:
2038:
2037:
2034:
2031:
2027:
2022:
2018:
2014:
2007:
2002:
2000:
1995:
1993:
1988:
1987:
1984:
1977:
1973:
1970:
1966:
1963:
1960:J. Dixmier,
1959:
1956:
1952:
1948:
1947:
1943:
1941:
1939:
1938:J. M. G. Fell
1935:
1934:Fell topology
1931:
1927:
1923:
1919:
1914:
1907:
1903:
1899:
1896:
1895:
1894:
1892:
1888:
1884:
1883:simple module
1880:
1876:
1872:
1868:
1864:
1860:
1854:
1846:
1844:
1839:is of type I.
1838:
1834:
1830:
1826:
1823:
1822:
1821:
1819:
1815:
1811:
1808:
1804:
1800:
1797:such that C*(
1796:
1791:
1789:
1785:
1781:
1777:
1773:
1769:
1765:
1763:
1759:
1755:
1751:
1747:
1743:
1739:
1732:
1730:
1728:
1721:
1714:
1709:
1704:
1697:
1690:
1685:
1680:
1676:
1672:
1665:
1662:
1661:
1646:
1643:
1640:
1637:
1629:
1625:
1621:
1618:
1604:
1598:
1592:
1586:
1580:
1572:
1568:
1560:
1559:
1558:
1555:
1550:
1545:
1525:
1522:
1519:
1516:
1508:
1504:
1500:
1497:
1494:
1491:
1481:
1478:
1475:
1469:
1463:
1451:
1448:
1445:
1439:
1431:
1427:
1416:
1415:
1414:
1411:
1406:
1399:
1395:
1390:
1384:
1382:
1378:
1371:
1367:
1359:
1354:
1350:The space Irr
1349:
1344:
1322:
1310:
1304:
1301:
1298:
1295:
1288:
1287:
1285:
1282:
1281:
1280:
1277:
1274:
1269:
1265:
1261:
1256:
1251:
1247:
1243:
1239:
1235:
1230:
1228:
1218:
1214:
1195:
1189:
1183:
1180:
1177:
1171:
1165:
1157:
1153:
1145:
1144:
1143:
1142:
1138:
1136:
1132:
1128:
1127:
1125:
1121:
1117:
1114:
1113:
1112:
1110:
1106:
1101:
1099:
1096:
1088:
1072:
1066:
1060:
1057:
1054:
1045:
1035:
1034:
1033:
1031:
1027:
1023:
1019:
1015:
996:
993:
990:
982:
976:
973:
970:
967:
963:
959:
956:
949:
948:
947:
945:
941:
933:
931:
929:
925:
921:
917:
912:
910:
906:
898:
894:
887:
883:
880:
876:
872:
868:
867:
866:
864:
857:
853:
849:
845:
841:
838: =
837:
830:
826:
822:
818:
817:Hilbert space
814:
806:
790:
784:
778:
775:
772:
763:
753:
752:
751:
749:
745:
726:
720:
717:
711:
705:
702:
696:
690:
684:
681:
675:
669:
663:
653:
652:
651:
634:
625:
619:
610:
607:
604:
601:
598:
588:
587:
586:
585:
584:homeomorphism
582:
578:
575:
571:
567:
563:
560:
556:
552:
548:
540:
535:
528:
523:
521:
519:
515:
511:
506:
504:
499:
497:
493:
489:
485:
481:
477:
473:
471:
467:
448:
442:
436:
433:
421:
415:
405:
404:
403:
401:
396:
394:
390:
386:
382:
363:
355:
350:
338:
328:
327:
326:
324:
305:
301:
297:
292:
289:
286:
282:
278:
275:
272:
266:
260:
257:
254:
251:
247:
243:
235:
226:
225:
224:
222:
218:
214:
210:
206:
202:
198:
194:
190:
185:
183:
179:
175:
167:
165:
163:
158:
153:
149:
145:
141:
136:
133:
129:
125:
121:
117:
112:
110:
106:
102:
98:
95:
91:
87:
83:
79:
75:
71:
67:
63:
60:
59:Hilbert space
56:
52:
48:
44:
40:
36:
32:
29:
25:
24:
3010:Applications
2840:Disk algebra
2694:Spectral gap
2678:
2569:Main results
2549:
2417:Balanced set
2391:Distribution
2329:Applications
2283:
2182:KreinâMilman
2167:Closed graph
1975:
1968:
1961:
1954:
1950:
1949:J. Dixmier,
1933:
1929:
1925:
1917:
1915:
1912:
1905:
1901:
1897:
1886:
1874:
1870:
1858:
1856:
1842:
1836:
1832:
1828:
1824:
1798:
1794:
1792:
1787:
1783:
1779:
1775:
1771:
1767:
1766:
1757:
1749:
1737:
1736:
1723:
1719:
1718:
1712:
1707:
1699:
1692:
1688:
1683:
1678:
1674:
1667:
1663:
1553:
1548:
1543:
1540:
1409:
1401:
1397:
1393:
1388:
1385:
1380:
1373:
1369:
1365:
1363:
1357:
1352:
1286:The mapping
1283:
1278:
1272:
1267:
1263:
1259:
1254:
1245:
1241:
1237:
1231:
1226:
1224:
1216:
1134:
1130:
1123:
1119:
1115:
1104:
1102:
1092:
1025:
1021:
1017:
1013:
1011:
943:
939:
937:
927:
923:
919:
915:
913:
908:
904:
902:
896:
892:
885:
878:
874:
870:
862:
855:
851:
847:
843:
839:
835:
828:
824:
820:
812:
810:
747:
743:
741:
649:
569:
565:
561:
554:
551:Gelfand dual
546:
544:
517:
509:
507:
500:
495:
491:
487:
483:
479:
475:
474:
472:as follows:
469:
465:
463:
397:
392:
388:
384:
378:
320:
220:
216:
212:
208:
200:
196:
188:
186:
181:
177:
171:
161:
156:
151:
113:
100:
89:
85:
81:
77:
73:
69:
65:
61:
54:
46:
41:classes of
34:
30:
27:
20:
18:
3128:C*-algebras
3037:Heat kernel
2737:Compression
2622:Isospectral
2346:Heat kernel
2336:Hardy space
2243:Trace class
2157:HahnâBanach
2119:Topological
1974:G. Mackey,
1951:C*-Algebras
1909:equivalent.
1879:annihilator
1814:semi-simple
1762:James Glimm
1742:Borel space
1266:defined by
1028:) with the
854:)) = {
142:theory for
94:dimensional
66:irreducible
43:irreducible
3122:Categories
2715:Continuous
2530:C*-algebra
2525:B*-algebra
2279:C*-algebra
2094:Properties
1967:J. Glimm,
1944:References
1851:See also:
1810:Lie groups
1768:Definition
1012:where min(
903:Thus Prim(
891:} is Prim(
539:singletons
476:Definition
400:surjective
323:idempotent
132:unimodular
33:, denoted
23:C*-algebra
2501:-algebras
2253:Unbounded
2248:Transpose
2206:Operators
2135:Separable
2130:Reflexive
2115:Algebraic
2101:Barrelled
1807:nilpotent
1803:connected
1750:separable
1746:G. Mackey
1641:∈
1622:∈
1619:ξ
1616:∀
1605:ξ
1593:π
1590:→
1587:ξ
1569:π
1520:∈
1501:∈
1498:η
1492:ξ
1489:∀
1485:⟩
1482:η
1479:∣
1476:ξ
1464:π
1461:⟨
1458:→
1455:⟩
1452:η
1449:∣
1446:ξ
1428:π
1424:⟨
1326:^
1317:→
1305:
1302:PureState
1296:κ
1199:⟩
1196:ξ
1184:π
1181:∣
1178:ξ
1175:⟨
1158:ξ
1061:
1055:≅
1049:^
977:
971:∈
964:⨁
960:≅
779:
773:≅
767:^
691:
685:∈
664:
620:
611:
605:≅
437:
431:→
425:^
359:¯
346:¯
342:¯
298:π
290:∈
287:π
283:⋂
279:⊇
276:ρ
261:
255:∈
252:ρ
239:¯
207:with the
135:separable
3102:Weyl law
3047:Lax pair
2994:Examples
2828:With an
2747:Discrete
2725:Residual
2661:Spectrum
2646:operator
2638:operator
2630:operator
2545:Spectrum
2455:Category
2267:Algebras
2149:Theorems
2106:Complete
2075:Schwartz
2021:glossary
1873:, where
938:Suppose
524:Examples
494:of Prim(
174:topology
90:non-null
2643:Unitary
2258:Unitary
2238:Nuclear
2223:Compact
2218:Bounded
2213:Adjoint
2187:Minâmax
2080:Sobolev
2065:Nuclear
2055:Hilbert
2050:Fréchet
2015: (
1900:. Let
1898:Theorem
1825:Theorem
1805:(real)
1664:Theorem
1284:Theorem
1116:Theorem
865:}. Now
861:,
581:natural
574:compact
152:abelian
80:) with
2627:Normal
2233:Normal
2070:Orlicz
2060:Hölder
2040:Banach
2029:Spaces
2017:topics
1758:smooth
1720:Remark
1666:. Let
1635:
1514:
1118:. Let
514:states
126:and a
97:spaces
2720:Point
2045:Besov
1928:) of
1881:of a
1861:is a
1827:. If
1774:is a
572:is a
199:) of
195:Prim(
57:on a
53:Ï of
2651:Unit
2499:and
2393:(or
2111:Dual
1863:ring
1691:) â
1262:) â
1250:pure
1232:The
1058:Prim
811:Let
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