90:, gives a K-cycle which encodes index-theoretic information. The local index formula expresses the pairing of the K-group of the manifold with this K-cycle in two ways: the 'analytic/global' side involves the usual trace on the Hilbert space and commutators of functions with the phase operator (which corresponds to the 'index' part of the index theorem), while the 'geometric/local' side involves the
134:
spectral triple is a spectral triple (A, H, D) such that a.D for any a in A has a compact resolvent which belongs to the class of L-operators for a fixed p (when A contains the identity operator on H, it is enough to require D in L(H)). When this condition is satisfied, the triple (A, H, D) is said
1902:
context, an analogue of the Gromov-Hausdorff distance has been constructed on the space of metric spectral triples, allowing the discussion of the geometry of this space, and the construction of approximations of spectral triples by "simpler" (more regular, or finite dimensional) spectral triples.
1901:
These observations are the foundations of the study of noncommutative metric geometry, which deals with the geometry of the space of quantum metric spaces, many of which being constructed using spectral triples whose Connes' metric induces the weak* topology on the underlying state space. In this
101:
structure, among others. In those cases the algebraic system of the 'functions' which expresses the underlying geometric object is no longer commutative, but one may able to find the space of square integrable spinors (or, sections of a
Clifford module) on which the algebra acts, and the
82:, accompanied by the Dirac operator associated to the spin structure. From the knowledge of these objects one is able to recover the original manifold as a metric space: the manifold as a topological space is recovered as the spectrum of the algebra, while the (absolute value of)
127:-grading on H, such that the elements in A are even while D is odd with respect to this grading. One could also say that an even spectral triple is given by a quartet (A, H, D, γ) such that γ is a self adjoint unitary on H satisfying a γ = γ a for any a in A and D γ = - γ D.
1200:
1624:
Guided by this observation, it is natural to wonder what properties Connes' metric shares with
Kantorovich's distance. In general, the topology induced by Connes' distance may not be Hausdorff, or give a finite diameter to the state space of the
1672:
Rieffel worked out a necessary and sufficient condition on spectral triples (and more generally, on seminorms which play a role of analogue for
Lipschitz seminorms) for Connes' distance to indeed induce the weak* topology on the state space of
1470:
541:
188:
spectral triple is a spectral triple (A, H, D) accompanied with an anti-linear involution J on H, satisfying = 0 for a, b in A. In the even case it is usually assumed that J is even with respect to the grading on H.
114:
is a triple (A, H, D) consisting of a
Hilbert space H, an algebra A of operators on H (usually closed under taking adjoints) and a densely defined self adjoint operator D satisfying ‖‖ < ∞ for any a ∈ A. An
1892:
948:
1805:
865:
719:
1619:
410:
1766:
345:
1321:
1054:
1695:
1647:
369:
273:
755:
1576:
1284:
1047:
1021:
1733:
799:
568:
249:
995:
1229:
621:
1667:
1532:
1512:
1492:
1330:
1249:
823:
775:
665:
645:
588:
297:
97:
Extensions of the index theorem can be considered in cases, typically when one has an action of a group on the manifold, or when the manifold is endowed with a
417:
2103:
satisfying some algebraic conditions and give
Hochschild / cyclic cohomology cocycles, which describe the above maps from K-theory to the integers.
2239:
2197:
2266:
94:
and commutators with the Dirac operator (which corresponds to the 'characteristic class integration' part of the index theorem).
1810:
102:
corresponding 'Dirac' operator on it satisfying certain boundedness of commutators implied by the pseudo-differential calculus.
2190:
An invitation to noncommutative geometry. Lectures of the international workshop on noncommutative geometry, Tehran, Iran, 2005
165:(s) = Tr(b|D|) for each element b in the algebra B generated by δ(A) and δ() for positive integers n. They are related to the
2234:. University Lecture Series. Vol. 36. With a foreword by Yuri Manin. Providence, RI: American Mathematical Society.
2071:
When the spectral triple is finitely summable, one may write the above indexes using the (super) trace, and a product of
197:
Given a spectral triple (A, H, D), one can apply several important operations to it. The most fundamental one is the
49:
870:
1649:, whereas Kantorovich's metric always induces the weak* topology on the space of Radon probability measures over
1771:
831:
670:
32:
is a set of data which encodes a geometric phenomenon in an analytic way. The definition typically involves a
1581:
374:
17:
150:
when the elements in A and the operators of the form for a in A are in the domain of the iterates δ of δ.
962:
1195:{\displaystyle k(\mu ,\nu )=\sup\{|\int _{X}fd\mu -\int _{X}fd\nu |:f\in C(X),\mathrm {Lip} (f)\leq 1\},}
1895:
1738:
317:
86:
retains the metric. On the other hand, the phase part of the Dirac operator, in conjunction with the
37:
1289:
594:, that the restriction of this pseudo-metric to the pure states, i.e. the characters of the C*-algebra
202:
1676:
1628:
1324:
350:
254:
25:
724:
958:
591:
198:
954:
74:
A motivating example of spectral triple is given by the algebra of smooth functions on a compact
1537:
146:
Let δ(T) denote the commutator of |D| with an operator T on H. A spectral triple is said to be
1254:
1026:
1000:
2235:
2193:
828:
Moreover, Connes observed that this distance is bounded if, and only if, there exists a state
201:
D = F|D| of D into a self adjoint unitary operator F (the 'phase' of D) and a densely defined
1700:
784:
553:
216:
2245:
2227:
2203:
2185:
2161:
170:
87:
968:
52:
and sought its extension to 'noncommutative' spaces. Some authors refer to this notion as
2249:
2207:
1465:{\displaystyle \mathrm {Lip} (f)=\sup\{{\frac {|f(x)-f(y)|}{d(x,y)}}:x,y\in X,x\not =y\}.}
1205:
597:
62:
1652:
1517:
1497:
1477:
1234:
808:
778:
760:
650:
630:
573:
282:
83:
1918:
into integers by taking
Fredholm index as follows. In the even case, each projection
2260:
2215:
802:
624:
300:
91:
75:
33:
2174:
2157:
536:{\displaystyle d(\varphi ,\psi )=\sup\{|\varphi (a)-\psi (a)|:a\in A,\|\|\leq 1\}.}
308:
304:
45:
41:
2218:. "Cyclic Theory, Bivariant K-Theory and the Bivariant Chern-Connes Character".
2112:
166:
21:
570:, and it may be zero between different states. Connes originally observed, for
961:
over a compact metric space, as introduced by
Kantorovich during his study of
312:
276:
55:
98:
2143:
A. Connes, H. Moscovici; The Local Index
Formula in Noncommutative Geometry
1023:
are two such probability measures, then
Kantorovich's distance between
1474:
This analogy is more than formal: in the case described above, where
79:
44:
operator, endowed with supplemental structures. It was conceived by
1049:, as was observed by Kantorovich and Rubinstein, can be defined by
153:
When a spectral triple (A, H, D) is p-summable, one may define its
173:. The collection of the poles of the analytic continuation of ζ
623:, whose space is naturally homeomorphic (when endowed with the
1231:
is the C*-algebra of complex valued continuous functions over
667:
induced by the
Riemannian metric, when the spectral triple is
1887:{\displaystyle \left\{a\in A:\|\|\leq 1,\mu (a)=0\right\}}
2134:
A. Connes, Noncommutative Geometry, Academic Press, 1994
953:
This construction is reminiscent of the construction by
647:, recovers the path metric for a Riemannian metric over
161:(s) = Tr(|D|); more generally there are zeta functions ζ
2053: − 1)/4 gives an additive mapping from
1697:, namely: Connes' metric induced by a spectral triple
1813:
1774:
1741:
1703:
1679:
1655:
1631:
1584:
1540:
1520:
1500:
1494:
is a connected compact spin Riemannian manifold, and
1480:
1333:
1292:
1257:
1237:
1208:
1057:
1029:
1003:
971:
873:
834:
811:
787:
763:
757:
is the algebra of smooth functions over the manifold
727:
673:
653:
633:
600:
576:
556:
420:
377:
353:
320:
285:
257:
219:
2166:
Noncommutative geometry, quantum fields and motives
2005:. In the odd case the eigenspace decomposition of
1886:
1799:
1760:
1727:
1689:
1661:
1641:
1613:
1570:
1526:
1506:
1486:
1464:
1315:
1278:
1243:
1223:
1194:
1041:
1015:
989:
942:
859:
817:
793:
769:
749:
713:
659:
639:
615:
582:
562:
535:
404:
363:
339:
291:
267:
243:
1735:topolgizes the weak* topology on the state space
1357:
1079:
781:acting on a dense subspace of the Hilbert space
442:
943:{\displaystyle \{a\in A:\|\|\leq 1,\mu (a)=0\}}
8:
1849:
1831:
1559:
1541:
1456:
1360:
1186:
1082:
937:
907:
889:
874:
592:connected, compact, spin Riemannian manifold
527:
518:
500:
445:
2220:Cyclic homology in non-commutative geometry
143:when e is of trace class for any t > 0.
2176:An introduction to noncommutative geometry
2130:
2128:
1800:{\displaystyle \mu \in S({\mathfrak {A}})}
860:{\displaystyle \mu \in S({\mathfrak {A}})}
714:{\displaystyle (C^{\infty }(X),\Gamma ,D)}
1812:
1788:
1787:
1773:
1749:
1748:
1740:
1702:
1681:
1680:
1678:
1654:
1633:
1632:
1630:
1585:
1583:
1539:
1519:
1499:
1479:
1398:
1366:
1363:
1334:
1332:
1293:
1291:
1256:
1236:
1207:
1160:
1131:
1116:
1094:
1085:
1056:
1028:
1002:
970:
872:
848:
847:
833:
810:
786:
762:
732:
726:
681:
672:
652:
632:
599:
575:
555:
480:
448:
419:
376:
355:
354:
352:
328:
327:
319:
284:
259:
258:
256:
218:
208:
2124:
1614:{\displaystyle \mathrm {Lip} (f)\leq 1}
1768:if, and only if, there exists a state
405:{\displaystyle \varphi ,\psi \in S(A)}
801:of square integrable sections of the
7:
2192:. Hackensack, NJ: World Scientific.
777:, and D is the closure of the usual
40:of operators on it and an unbounded
1789:
1750:
1682:
1634:
849:
356:
329:
260:
139:. A spectral triple is said to be
78:, acting on the Hilbert space of L-
2232:Arithmetic Noncommutative Geometry
1761:{\displaystyle S({\mathfrak {A}})}
1592:
1589:
1586:
1341:
1338:
1335:
1300:
1297:
1294:
1167:
1164:
1161:
997:is a compact metric space, and if
788:
733:
699:
682:
557:
371:, by setting, for any two states
340:{\displaystyle S({\mathfrak {A}})}
14:
2013:, and each invertible element in
1953:becomes a Fredholm operator from
1514:is the associated path metric on
1316:{\displaystyle \mathrm {Lip} (f)}
209:Connes' Metric on the state space
119:is an odd spectral triple with a
1990:defines an additive mapping of
1914:gives a map of the K-theory of
1690:{\displaystyle {\mathfrak {A}}}
1642:{\displaystyle {\mathfrak {A}}}
364:{\displaystyle {\mathfrak {A}}}
268:{\displaystyle {\mathfrak {A}}}
2099: + 1)-functional on
2095:). This can be encoded as a (
2025: − 1)/4 from (
1870:
1864:
1846:
1834:
1794:
1784:
1755:
1745:
1722:
1704:
1602:
1596:
1556:
1544:
1420:
1408:
1399:
1395:
1389:
1380:
1374:
1367:
1351:
1345:
1310:
1304:
1273:
1267:
1218:
1212:
1177:
1171:
1154:
1148:
1132:
1086:
1073:
1061:
984:
972:
963:Monge's transportation problem
957:of a distance on the space of
928:
922:
904:
892:
854:
844:
750:{\displaystyle C^{\infty }(X)}
744:
738:
708:
693:
687:
674:
610:
604:
515:
503:
481:
477:
471:
462:
456:
449:
436:
424:
399:
393:
334:
324:
238:
220:
1:
2049: + 1) u (
2021: + 1) u (
1669:--- which is weak* compact.
2017:gives a Fredholm operator (
965:. Indeed, in that case, if
50:Atiyah-Singer index theorem
2283:
1571:{\displaystyle \|\|\leq 1}
959:Radon probability measures
550:can indeed take the value
251:is a spectral triple, and
1910:The self adjoint unitary
1279:{\displaystyle f\in C(X)}
1042:{\displaystyle \mu ,\nu }
1016:{\displaystyle \mu ,\nu }
205:|D| (the 'metric' part).
177:for b in B is called the
48:who was motivated by the
20:and related branches of
2267:Noncommutative geometry
2045: → Ind (
1728:{\displaystyle (A,H,D)}
1251:, and for any function
794:{\displaystyle \Gamma }
563:{\displaystyle \infty }
244:{\displaystyle (A,H,D)}
18:noncommutative geometry
1977: → Ind
1940:under the grading and
1888:
1801:
1762:
1729:
1691:
1663:
1643:
1615:
1572:
1528:
1508:
1488:
1466:
1317:
1280:
1245:
1225:
1196:
1043:
1017:
991:
944:
861:
819:
795:
771:
751:
715:
661:
641:
617:
584:
564:
537:
406:
365:
341:
309:extended pseudo-metric
293:
269:
245:
2029: − 1)
1906:Pairing with K-theory
1889:
1802:
1763:
1730:
1692:
1664:
1644:
1616:
1573:
1529:
1509:
1489:
1467:
1318:
1281:
1246:
1226:
1197:
1044:
1018:
992:
990:{\displaystyle (X,d)}
945:
862:
820:
796:
772:
752:
716:
662:
642:
618:
585:
565:
538:
407:
366:
342:
294:
270:
246:
2083:) and commutator of
1811:
1772:
1739:
1701:
1677:
1653:
1629:
1582:
1538:
1518:
1498:
1478:
1331:
1290:
1255:
1235:
1224:{\displaystyle C(X)}
1206:
1055:
1027:
1001:
969:
871:
832:
809:
785:
761:
725:
671:
651:
631:
616:{\displaystyle C(X)}
598:
574:
554:
418:
375:
351:
318:
283:
255:
217:
117:even spectral triple
88:algebra of functions
26:mathematical physics
2184:Khalkhali, Masoud;
2173:Várilly, Joseph C.
2009:gives a grading on
867:such that the set:
199:polar decomposition
112:odd spectral triple
1884:
1807:such that the set
1797:
1758:
1725:
1687:
1659:
1639:
1611:
1568:
1524:
1504:
1484:
1462:
1325:Lipschitz seminorm
1313:
1276:
1241:
1221:
1192:
1039:
1013:
987:
940:
857:
815:
791:
767:
747:
711:
657:
637:
613:
580:
560:
533:
402:
361:
337:
289:
265:
241:
193:Important concepts
179:dimension spectrum
2241:978-0-8218-3833-4
2228:Marcolli, Matilde
2199:978-981-270-616-4
2186:Marcolli, Matilde
2162:Marcolli, Matilde
1662:{\displaystyle X}
1578:if, and only if,
1527:{\displaystyle X}
1507:{\displaystyle d}
1487:{\displaystyle X}
1424:
1244:{\displaystyle X}
818:{\displaystyle X}
770:{\displaystyle X}
660:{\displaystyle X}
640:{\displaystyle X}
583:{\displaystyle X}
292:{\displaystyle A}
203:positive operator
132:finitely summable
2274:
2253:
2223:
2211:
2180:
2169:
2144:
2141:
2135:
2132:
1893:
1891:
1890:
1885:
1883:
1879:
1806:
1804:
1803:
1798:
1793:
1792:
1767:
1765:
1764:
1759:
1754:
1753:
1734:
1732:
1731:
1726:
1696:
1694:
1693:
1688:
1686:
1685:
1668:
1666:
1665:
1660:
1648:
1646:
1645:
1640:
1638:
1637:
1620:
1618:
1617:
1612:
1595:
1577:
1575:
1574:
1569:
1533:
1531:
1530:
1525:
1513:
1511:
1510:
1505:
1493:
1491:
1490:
1485:
1471:
1469:
1468:
1463:
1425:
1423:
1403:
1402:
1370:
1364:
1344:
1322:
1320:
1319:
1314:
1303:
1285:
1283:
1282:
1277:
1250:
1248:
1247:
1242:
1230:
1228:
1227:
1222:
1201:
1199:
1198:
1193:
1170:
1135:
1121:
1120:
1099:
1098:
1089:
1048:
1046:
1045:
1040:
1022:
1020:
1019:
1014:
996:
994:
993:
988:
949:
947:
946:
941:
866:
864:
863:
858:
853:
852:
824:
822:
821:
816:
800:
798:
797:
792:
776:
774:
773:
768:
756:
754:
753:
748:
737:
736:
720:
718:
717:
712:
686:
685:
666:
664:
663:
658:
646:
644:
643:
638:
622:
620:
619:
614:
589:
587:
586:
581:
569:
567:
566:
561:
546:In general, the
542:
540:
539:
534:
484:
452:
411:
409:
408:
403:
370:
368:
367:
362:
360:
359:
346:
344:
343:
338:
333:
332:
298:
296:
295:
290:
274:
272:
271:
266:
264:
263:
250:
248:
247:
242:
171:Mellin transform
169:exp(-t|D|) by a
63:Fredholm modules
2282:
2281:
2277:
2276:
2275:
2273:
2272:
2271:
2257:
2256:
2242:
2226:
2214:
2200:
2183:
2172:
2156:
2153:
2148:
2147:
2142:
2138:
2133:
2126:
2121:
2109:
2059:
2037: + 1)
1996:
1989:
1983:
1969:
1959:
1952:
1946:
1939:
1932:
1908:
1896:totally bounded
1818:
1814:
1809:
1808:
1770:
1769:
1737:
1736:
1699:
1698:
1675:
1674:
1651:
1650:
1627:
1626:
1580:
1579:
1536:
1535:
1516:
1515:
1496:
1495:
1476:
1475:
1404:
1365:
1329:
1328:
1288:
1287:
1286:, we denote by
1253:
1252:
1233:
1232:
1204:
1203:
1112:
1090:
1053:
1052:
1025:
1024:
999:
998:
967:
966:
869:
868:
830:
829:
807:
806:
783:
782:
759:
758:
728:
723:
722:
677:
669:
668:
649:
648:
629:
628:
596:
595:
572:
571:
552:
551:
545:
416:
415:
373:
372:
349:
348:
316:
315:
281:
280:
253:
252:
215:
214:
211:
195:
176:
164:
160:
108:
72:
30:spectral triple
12:
11:
5:
2280:
2278:
2270:
2269:
2259:
2258:
2255:
2254:
2240:
2224:
2216:Cuntz, Joachim
2212:
2198:
2181:
2170:
2152:
2149:
2146:
2145:
2136:
2123:
2122:
2120:
2117:
2116:
2115:
2108:
2105:
2057:
1994:
1987:
1981:
1967:
1957:
1950:
1944:
1937:
1930:
1926:decomposes as
1907:
1904:
1882:
1878:
1875:
1872:
1869:
1866:
1863:
1860:
1857:
1854:
1851:
1848:
1845:
1842:
1839:
1836:
1833:
1830:
1827:
1824:
1821:
1817:
1796:
1791:
1786:
1783:
1780:
1777:
1757:
1752:
1747:
1744:
1724:
1721:
1718:
1715:
1712:
1709:
1706:
1684:
1658:
1636:
1610:
1607:
1604:
1601:
1598:
1594:
1591:
1588:
1567:
1564:
1561:
1558:
1555:
1552:
1549:
1546:
1543:
1523:
1503:
1483:
1461:
1458:
1455:
1452:
1449:
1446:
1443:
1440:
1437:
1434:
1431:
1428:
1422:
1419:
1416:
1413:
1410:
1407:
1401:
1397:
1394:
1391:
1388:
1385:
1382:
1379:
1376:
1373:
1369:
1362:
1359:
1356:
1353:
1350:
1347:
1343:
1340:
1337:
1312:
1309:
1306:
1302:
1299:
1296:
1275:
1272:
1269:
1266:
1263:
1260:
1240:
1220:
1217:
1214:
1211:
1191:
1188:
1185:
1182:
1179:
1176:
1173:
1169:
1166:
1163:
1159:
1156:
1153:
1150:
1147:
1144:
1141:
1138:
1134:
1130:
1127:
1124:
1119:
1115:
1111:
1108:
1105:
1102:
1097:
1093:
1088:
1084:
1081:
1078:
1075:
1072:
1069:
1066:
1063:
1060:
1038:
1035:
1032:
1012:
1009:
1006:
986:
983:
980:
977:
974:
939:
936:
933:
930:
927:
924:
921:
918:
915:
912:
909:
906:
903:
900:
897:
894:
891:
888:
885:
882:
879:
876:
856:
851:
846:
843:
840:
837:
814:
790:
779:Dirac operator
766:
746:
743:
740:
735:
731:
710:
707:
704:
701:
698:
695:
692:
689:
684:
680:
676:
656:
636:
625:weak* topology
612:
609:
606:
603:
579:
559:
532:
529:
526:
523:
520:
517:
514:
511:
508:
505:
502:
499:
496:
493:
490:
487:
483:
479:
476:
473:
470:
467:
464:
461:
458:
455:
451:
447:
444:
441:
438:
435:
432:
429:
426:
423:
401:
398:
395:
392:
389:
386:
383:
380:
358:
336:
331:
326:
323:
307:introduces an
288:
262:
240:
237:
234:
231:
228:
225:
222:
210:
207:
194:
191:
181:of (A, H, D).
174:
162:
158:
107:
104:
84:Dirac operator
71:
68:
13:
10:
9:
6:
4:
3:
2:
2279:
2268:
2265:
2264:
2262:
2251:
2247:
2243:
2237:
2233:
2229:
2225:
2221:
2217:
2213:
2209:
2205:
2201:
2195:
2191:
2187:
2182:
2178:
2177:
2171:
2167:
2163:
2159:
2158:Connes, Alain
2155:
2154:
2150:
2140:
2137:
2131:
2129:
2125:
2118:
2114:
2111:
2110:
2106:
2104:
2102:
2098:
2094:
2090:
2086:
2082:
2078:
2074:
2069:
2067:
2063:
2056:
2052:
2048:
2044:
2040:
2036:
2032:
2028:
2024:
2020:
2016:
2012:
2008:
2004:
2000:
1993:
1986:
1980:
1976:
1973:. Thus
1972:
1966:
1962:
1956:
1949:
1943:
1936:
1933: ⊕
1929:
1925:
1921:
1917:
1913:
1905:
1903:
1899:
1897:
1880:
1876:
1873:
1867:
1861:
1858:
1855:
1852:
1843:
1840:
1837:
1828:
1825:
1822:
1819:
1815:
1781:
1778:
1775:
1742:
1719:
1716:
1713:
1710:
1707:
1670:
1656:
1622:
1608:
1605:
1599:
1565:
1562:
1553:
1550:
1547:
1521:
1501:
1481:
1472:
1459:
1453:
1450:
1447:
1444:
1441:
1438:
1435:
1432:
1429:
1426:
1417:
1414:
1411:
1405:
1392:
1386:
1383:
1377:
1371:
1354:
1348:
1326:
1307:
1270:
1264:
1261:
1258:
1238:
1215:
1209:
1189:
1183:
1180:
1174:
1157:
1151:
1145:
1142:
1139:
1136:
1128:
1125:
1122:
1117:
1113:
1109:
1106:
1103:
1100:
1095:
1091:
1076:
1070:
1067:
1064:
1058:
1050:
1036:
1033:
1030:
1010:
1007:
1004:
981:
978:
975:
964:
960:
956:
951:
934:
931:
925:
919:
916:
913:
910:
901:
898:
895:
886:
883:
880:
877:
841:
838:
835:
826:
812:
804:
803:spinor bundle
780:
764:
741:
729:
705:
702:
696:
690:
678:
654:
634:
626:
607:
601:
593:
577:
549:
548:Connes metric
543:
530:
524:
521:
512:
509:
506:
497:
494:
491:
488:
485:
474:
468:
465:
459:
453:
439:
433:
430:
427:
421:
413:
396:
390:
387:
384:
381:
378:
321:
314:
310:
306:
302:
301:operator norm
286:
278:
235:
232:
229:
226:
223:
206:
204:
200:
192:
190:
187:
182:
180:
172:
168:
156:
155:zeta function
151:
149:
144:
142:
138:
133:
128:
126:
122:
118:
113:
105:
103:
100:
95:
93:
92:Dixmier trace
89:
85:
81:
77:
76:spin manifold
69:
67:
65:
64:
58:
57:
51:
47:
43:
39:
35:
34:Hilbert space
31:
27:
23:
19:
2231:
2219:
2189:
2175:
2165:
2139:
2100:
2096:
2092:
2091:(resp.
2088:
2084:
2080:
2079:(resp.
2076:
2072:
2070:
2065:
2061:
2054:
2050:
2046:
2042:
2038:
2034:
2030:
2026:
2022:
2018:
2014:
2010:
2006:
2002:
1998:
1991:
1984:
1978:
1974:
1970:
1964:
1960:
1954:
1947:
1941:
1934:
1927:
1923:
1919:
1915:
1911:
1909:
1900:
1671:
1623:
1473:
1051:
952:
950:is bounded.
827:
547:
544:
414:
212:
196:
185:
183:
178:
154:
152:
147:
145:
140:
136:
131:
129:
124:
120:
116:
111:
109:
96:
73:
60:
53:
46:Alain Connes
42:self-adjoint
29:
15:
2113:JLO cocycle
955:Kantorovich
313:state space
167:heat kernel
22:mathematics
2250:1081.58005
2208:1135.14002
2151:References
2064:) to
141:θ-summable
137:p-summable
106:Definition
70:Motivation
61:unbounded
54:unbounded
1862:μ
1853:≤
1850:‖
1832:‖
1823:∈
1779:∈
1776:μ
1606:≤
1563:≤
1560:‖
1542:‖
1439:∈
1384:−
1262:∈
1181:≤
1143:∈
1129:ν
1114:∫
1110:−
1107:μ
1092:∫
1071:ν
1065:μ
1037:ν
1031:μ
1011:ν
1005:μ
920:μ
911:≤
908:‖
890:‖
881:∈
839:∈
836:μ
789:Γ
734:∞
700:Γ
683:∞
558:∞
522:≤
519:‖
501:‖
492:∈
469:ψ
466:−
454:φ
434:ψ
428:φ
388:∈
385:ψ
379:φ
99:foliation
2261:Category
2230:(2005).
2188:(2005).
2107:See also
2041:. Thus
1451:≠
721:, where
299:for the
56:K-cycles
1534:, then
311:on the
303:, then
277:closure
275:is the
148:regular
80:spinors
38:algebra
2248:
2238:
2206:
2196:
1202:where
305:Connes
135:to be
59:or as
2119:Notes
2087:with
2001:) to
805:over
627:) to
36:, an
2236:ISBN
2194:ISBN
2033:to (
1323:its
186:real
28:, a
24:and
2246:Zbl
2204:Zbl
1963:to
1922:in
1894:is
1358:sup
1080:sup
443:sup
347:of
279:of
213:If
110:An
16:In
2263::
2244:.
2202:.
2164:.
2160:;
2127:^
2075:,
2068:.
1985:Fe
1948:Fe
1898:.
1621:.
1327::
825:.
590:a
412::
184:A
130:A
123:/2
66:.
2252:.
2222:.
2210:.
2179:.
2168:.
2101:A
2097:p
2093:u
2089:e
2085:F
2081:u
2077:e
2073:F
2066:Z
2062:A
2060:(
2058:1
2055:K
2051:F
2047:F
2043:u
2039:H
2035:F
2031:H
2027:F
2023:F
2019:F
2015:A
2011:H
2007:F
2003:Z
1999:A
1997:(
1995:0
1992:K
1988:0
1982:1
1979:e
1975:e
1971:H
1968:1
1965:e
1961:H
1958:0
1955:e
1951:0
1945:1
1942:e
1938:1
1935:e
1931:0
1928:e
1924:A
1920:e
1916:A
1912:F
1881:}
1877:0
1874:=
1871:)
1868:a
1865:(
1859:,
1856:1
1847:]
1844:a
1841:,
1838:D
1835:[
1829::
1826:A
1820:a
1816:{
1795:)
1790:A
1785:(
1782:S
1756:)
1751:A
1746:(
1743:S
1723:)
1720:D
1717:,
1714:H
1711:,
1708:A
1705:(
1683:A
1657:X
1635:A
1609:1
1603:)
1600:f
1597:(
1593:p
1590:i
1587:L
1566:1
1557:]
1554:f
1551:,
1548:D
1545:[
1522:X
1502:d
1482:X
1460:.
1457:}
1454:y
1448:x
1445:,
1442:X
1436:y
1433:,
1430:x
1427::
1421:)
1418:y
1415:,
1412:x
1409:(
1406:d
1400:|
1396:)
1393:y
1390:(
1387:f
1381:)
1378:x
1375:(
1372:f
1368:|
1361:{
1355:=
1352:)
1349:f
1346:(
1342:p
1339:i
1336:L
1311:)
1308:f
1305:(
1301:p
1298:i
1295:L
1274:)
1271:X
1268:(
1265:C
1259:f
1239:X
1219:)
1216:X
1213:(
1210:C
1190:,
1187:}
1184:1
1178:)
1175:f
1172:(
1168:p
1165:i
1162:L
1158:,
1155:)
1152:X
1149:(
1146:C
1140:f
1137::
1133:|
1126:d
1123:f
1118:X
1104:d
1101:f
1096:X
1087:|
1083:{
1077:=
1074:)
1068:,
1062:(
1059:k
1034:,
1008:,
985:)
982:d
979:,
976:X
973:(
938:}
935:0
932:=
929:)
926:a
923:(
917:,
914:1
905:]
902:a
899:,
896:D
893:[
887::
884:A
878:a
875:{
855:)
850:A
845:(
842:S
813:X
765:X
745:)
742:X
739:(
730:C
709:)
706:D
703:,
697:,
694:)
691:X
688:(
679:C
675:(
655:X
635:X
611:)
608:X
605:(
602:C
578:X
531:.
528:}
525:1
516:]
513:a
510:,
507:D
504:[
498:,
495:A
489:a
486::
482:|
478:)
475:a
472:(
463:)
460:a
457:(
450:|
446:{
440:=
437:)
431:,
425:(
422:d
400:)
397:A
394:(
391:S
382:,
357:A
335:)
330:A
325:(
322:S
287:A
261:A
239:)
236:D
233:,
230:H
227:,
224:A
221:(
175:b
163:b
159:D
157:ζ
125:Z
121:Z
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