261:) of its factors, then the central projections are the projections that are direct sums (direct integrals) of identity operators of (measurable sets of) the factors. If
871:. Using this fact and a maximality argument, it can be deduced that the Murray-von Neumann partial order « on the family of projections in
998:
is partially ordered by inclusion and the above corollary shows it is non-empty. Zorn's lemma ensures the existence of a maximal element { (
1396:
402:) follows from the definition of commutant. On the other hand, is invariant under every unitary
234:), being the intersection of two von Neumann algebras, is also a von Neumann algebra. Therefore,
8:
711:
17:
1356:
1127:
be the same as in the previous proposition and again consider a maximal element { (
85:
1376:
258:
28:
is the smallest central projection, in the von
Neumann algebra, that dominates
926:
Let ~ denote the Murray-von
Neumann equivalence relation. Consider the family
648:
contain two nonzero sub-projections that are Murray-von
Neumann equivalent if
1390:
466:
One can deduce some simple consequences from the above description. Suppose
273:) is the identity operator in that factor. Informally, one would expect
573:, by the discussion in the preceding section, where 1 is the unit in
77:
356:
is precisely the projection onto the closed subspace generated by
187:
The symbol ∧ denotes the lattice operation on the projections in
309:) can be described more explicitly. It can be shown that Ran
1078:
are projections, then there exists a central projection
847:is a factor, then there exists a partial isometry
56:) denote the bounded operators on a Hilbert space
336:a projection that does not necessarily belong to
1388:
437:, applying the above to the von Neumann algebra
209:is the projection onto the closed subspace Ran(
281:) to be the direct sum of identity operators
410:. Therefore the projection onto lies in (
296:
619:) is a central projection that dominates
474:are projections in a von Neumann algebra
1060:Proposition (Generalized Comparability)
257:as a direct sum (or more accurately, a
1389:
1034:. The countable additivity of ~ means
348:). The smallest central projection in
317:) is the closed subspace generated by
265:is confined to a single factor, then
1012:) }. Maximality ensures that either
1179:. By maximality and the corollary,
13:
930:whose typical element is a set { (
461:
14:
1408:
815:⇒ The two equivalent projections
630:In turn, the following is true:
72:) be a von Neumann algebra, and
944:) } where the orthogonal sets {
1066:is a von Neumann algebra, and
332:is a von Neumann algebra, and
1:
1363:
1048:. Thus the proposition holds.
901:are projections, then either
43:
1359:with the desired properties.
1231:) = 0. So multiplication by
1052:Without the assumption that
7:
1381:C*-Algebras and W*-Algebras
883:Proposition (Comparability)
10:
1413:
717:for some partial isometry
1149:denote the "remainders":
875:becomes a total order if
644:in a von Neumann algebra
1239:) removes the remainder
32:. It is also called the
511:) are orthogonal, i.e.
297:An explicit description
148:is defined as follows:
132:}. The central carrier
1056:is a factor, we have:
721:and positive operator
626:This proves the claim.
1207:) = 0. In particular
1397:Von Neumann algebras
843:In particular, when
382:is a projection and
226:The abelian algebra
175:is a projection and
18:von Neumann algebras
1347:). This shows that
1090:) such that either
712:polar decomposition
433:is a projection in
289:is in a factor and
1357:central projection
1255:. More precisely,
398:) = . That ⊂ Ran(
140:) of a projection
16:In the context of
1383:, Springer, 1998.
1373:, Springer, 2006.
1371:Operator Algebras
889:is a factor, and
363:. In symbols, if
253:If one thinks of
1404:
636:Two projections
422:then yields Ran(
418:. Minimality of
24:of a projection
1412:
1411:
1407:
1406:
1405:
1403:
1402:
1401:
1387:
1386:
1366:
1321:
1306:
1291:
1272:
1177:
1162:
1139:
1132:
1046:
1039:
1032:
1021:
1010:
1003:
992:
985:
974:
963:
956:
949:
942:
935:
495:if and only if
464:
462:Related results
352:that dominates
301:The projection
299:
259:direct integral
222:
215:
208:
201:
46:
34:central support
22:central carrier
12:
11:
5:
1410:
1400:
1399:
1385:
1384:
1374:
1369:B. Blackadar,
1365:
1362:
1361:
1360:
1319:
1304:
1289:
1270:
1247:while leaving
1175:
1160:
1137:
1130:
1050:
1049:
1044:
1037:
1030:
1019:
1008:
1001:
990:
983:
972:
961:
954:
947:
940:
933:
841:
840:
813:
730:
704:
689:
628:
627:
624:
593:
578:
559:
548:
463:
460:
459:
458:
392:
391:
340:and has range
298:
295:
220:
213:
206:
199:
185:
184:
45:
42:
9:
6:
4:
3:
2:
1409:
1398:
1395:
1394:
1392:
1382:
1378:
1375:
1372:
1368:
1367:
1358:
1354:
1350:
1346:
1342:
1338:
1334:
1330:
1326:
1322:
1315:
1311:
1307:
1300:
1296:
1292:
1285:
1281:
1277:
1273:
1266:
1262:
1258:
1254:
1250:
1246:
1242:
1238:
1234:
1230:
1226:
1222:
1218:
1214:
1210:
1206:
1202:
1198:
1194:
1190:
1186:
1182:
1178:
1171:
1167:
1163:
1156:
1152:
1148:
1144:
1140:
1133:
1126:
1122:
1121:
1120:
1119:
1115:
1113:
1109:
1105:
1101:
1097:
1093:
1089:
1085:
1081:
1077:
1073:
1069:
1065:
1061:
1057:
1055:
1047:
1040:
1033:
1026:
1022:
1015:
1011:
1004:
997:
994:. The family
993:
986:
979:
975:
968:
964:
957:
950:
943:
936:
929:
925:
924:
923:
922:
918:
916:
912:
908:
904:
900:
896:
892:
888:
884:
880:
879:is a factor.
878:
874:
870:
866:
862:
858:
854:
850:
846:
838:
834:
830:
826:
822:
818:
814:
811:
807:
803:
799:
796:); therefore
795:
791:
787:
783:
779:
775:
771:
767:
763:
759:
755:
751:
747:
743:
739:
735:
731:
728:
724:
720:
716:
713:
709:
705:
702:
698:
695:≠ 0 for some
694:
690:
687:
683:
679:
675:
672:
671:
670:
669:
665:
663:
659:
655:
651:
647:
643:
639:
635:
631:
625:
622:
618:
614:
611:), since 1 -
610:
606:
602:
598:
594:
591:
587:
583:
579:
576:
572:
568:
564:
560:
557:
553:
549:
546:
542:
538:
535:
534:
533:
532:
528:
526:
522:
518:
514:
510:
506:
502:
498:
494:
490:
486:
483:
479:
477:
473:
469:
456:
452:
448:
447:
446:
444:
440:
436:
432:
427:
425:
421:
417:
413:
409:
405:
401:
397:
389:
385:
381:
377:
373:
369:
366:
365:
364:
362:
359:
355:
351:
347:
343:
339:
335:
331:
326:
324:
320:
316:
312:
308:
304:
294:
292:
288:
284:
280:
276:
272:
268:
264:
260:
256:
251:
249:
245:
241:
237:
233:
229:
224:
219:
212:
205:
198:
194:
190:
182:
178:
174:
170:
166:
162:
159:) = ∧ {
158:
154:
151:
150:
149:
147:
143:
139:
135:
131:
127:
123:
119:
115:
111:
107:
103:
99:
95:
91:
87:
83:
79:
75:
71:
67:
63:
59:
55:
51:
41:
39:
38:central cover
35:
31:
27:
23:
19:
1380:
1370:
1352:
1348:
1344:
1340:
1336:
1332:
1328:
1324:
1317:
1313:
1309:
1302:
1298:
1294:
1287:
1283:
1279:
1275:
1268:
1264:
1260:
1256:
1252:
1248:
1244:
1240:
1236:
1232:
1228:
1224:
1220:
1216:
1212:
1208:
1204:
1200:
1196:
1192:
1188:
1184:
1183:= 0 for all
1180:
1173:
1169:
1165:
1158:
1154:
1150:
1146:
1142:
1135:
1128:
1124:
1117:
1116:
1111:
1107:
1103:
1099:
1095:
1091:
1087:
1083:
1079:
1075:
1071:
1067:
1063:
1059:
1058:
1053:
1051:
1042:
1035:
1028:
1024:
1017:
1013:
1006:
999:
995:
988:
981:
977:
970:
966:
959:
952:
945:
938:
931:
927:
920:
919:
914:
910:
906:
902:
898:
894:
890:
886:
882:
881:
876:
872:
868:
864:
860:
856:
852:
848:
844:
842:
836:
832:
828:
824:
820:
816:
809:
805:
801:
797:
793:
789:
785:
781:
777:
773:
769:
765:
761:
757:
753:
749:
745:
741:
737:
733:
726:
722:
718:
714:
707:
700:
696:
692:
685:
681:
677:
673:
667:
666:
661:
657:
653:
649:
645:
641:
637:
633:
632:
629:
620:
616:
612:
608:
604:
600:
596:
589:
585:
581:
574:
570:
566:
562:
555:
551:
544:
540:
539:= 0 for all
536:
530:
529:
524:
520:
516:
512:
508:
504:
500:
496:
492:
488:
487:= 0 for all
484:
481:
480:
475:
471:
467:
465:
457:) = = = .
454:
450:
442:
438:
434:
430:
428:
423:
419:
415:
411:
407:
403:
399:
395:
393:
387:
383:
379:
375:
371:
367:
360:
357:
353:
349:
345:
341:
337:
333:
329:
327:
322:
318:
314:
310:
306:
302:
300:
290:
286:
282:
278:
274:
270:
266:
262:
254:
252:
247:
243:
239:
235:
231:
227:
225:
217:
210:
203:
196:
192:
188:
186:
180:
176:
172:
168:
164:
160:
156:
152:
145:
141:
137:
133:
129:
125:
121:
117:
113:
109:
105:
101:
97:
93:
89:
81:
73:
69:
65:
61:
57:
53:
49:
47:
37:
33:
29:
25:
21:
15:
804:)) ⊂
550:⇔ ⊂
482:Proposition
370:= ∧ {
1364:References
1219:) = 0 and
958:} satisfy
855:such that
748:) ⊂
242:) lies in
44:Definition
1355:) is the
1141:) }. Let
1106:) «
756:). Also,
634:Corollary
394:then Ran(
291:I · E ≠ 0
78:commutant
1391:Category
1377:S. Sakai
1286:) = (Σ
1267:) = (Σ
823:satisfy
603:) ≤ 1 -
569:) ≤ 1 -
445:) gives
216:) ∩ Ran(
124:for all
1316:) ≤ (Σ
1301:) ~ (Σ
1094:«
951:} and {
913:«
905:«
664:) ≠ 0.
527:) = 0.
429:Now if
1118:Proof:
980:, and
921:Proof:
688:) ≠ 0.
668:Proof:
584:≤ 1 -
531:Proof:
503:) and
426:) ⊂ .
344:= Ran(
285:where
86:center
84:. The
20:, the
1243:from
1191:. So
1110:(1 -
1102:(1 -
414:)' =
1335:) =
1327:) ·
1308:) ·
1293:) ·
1278:) ·
1172:- Σ
1164:and
1157:- Σ
1145:and
1123:Let
1098:and
1041:~ Σ
1027:= Σ
1016:= Σ
863:and
831:and
819:and
788:) ⊃
786:ET*F
780:) =
772:) =
764:) =
740:) =
710:has
640:and
470:and
449:Ran
321:Ran(
171:) |
100:) =
76:the
48:Let
1251:in
1187:in
1181:RTS
1114:).
1062:If
1023:or
909:or
885:If
865:U*U
857:UU*
833:U*U
825:UU*
821:U*U
817:UU*
806:Ran
798:Ker
790:Ker
782:Ker
778:ETF
774:Ran
766:Ran
758:Ker
750:Ran
746:ETF
742:Ran
734:Ran
725:in
708:ETF
699:in
693:ETF
552:Ker
543:in
537:ETF
491:in
485:ETF
424:F'
420:F'
408:N'
406:in
400:F'
396:F'
368:F'
358:N'
328:If
325:).
250:).
223:).
195:):
144:in
108:= {
92:is
88:of
80:of
36:or
1393::
1379:,
1339:·
1323:+
1274:+
1259:·
1223:·
1211:·
1168:=
1153:=
1134:,
1096:FP
1092:EP
1082:∈
1074:∈
1070:,
1005:,
987:~
976:≤
969:,
965:≤
937:,
917:.
897:∈
893:,
867:≤
859:≤
851:∈
835:≤
827:≤
812:).
732:⇒
715:UH
706:⇒
691:⇒
595:⇔
592:).
580:⇔
561:⇔
558:).
478:.
412:N'
386:≥
378:|
374:∈
293:.
202:∧
183:}.
179:≥
163:∈
128:∈
122:MT
120:=
118:TM
116:|
112:∈
104:∩
102:M'
74:M'
64:⊂
60:,
40:.
1353:S
1351:(
1349:C
1345:S
1343:(
1341:C
1337:F
1333:S
1331:(
1329:C
1325:S
1320:j
1318:F
1314:S
1312:(
1310:C
1305:j
1303:F
1299:S
1297:(
1295:C
1290:j
1288:E
1284:S
1282:(
1280:C
1276:R
1271:j
1269:E
1265:S
1263:(
1261:C
1257:E
1253:F
1249:S
1245:E
1241:R
1237:S
1235:(
1233:C
1229:S
1227:(
1225:C
1221:S
1217:S
1215:(
1213:C
1209:R
1205:S
1203:(
1201:C
1199:)
1197:R
1195:(
1193:C
1189:M
1185:T
1176:j
1174:F
1170:F
1166:S
1161:j
1159:E
1155:E
1151:R
1147:S
1143:R
1138:j
1136:F
1131:j
1129:E
1125:S
1112:P
1108:E
1104:P
1100:F
1088:M
1086:(
1084:Z
1080:P
1076:M
1072:F
1068:E
1064:M
1054:M
1045:j
1043:F
1038:j
1036:E
1031:j
1029:F
1025:F
1020:j
1018:E
1014:E
1009:j
1007:F
1002:j
1000:E
996:S
991:i
989:F
984:i
982:E
978:F
973:i
971:F
967:E
962:i
960:E
955:i
953:F
948:i
946:E
941:i
939:F
934:i
932:E
928:S
915:E
911:F
907:F
903:E
899:M
895:F
891:E
887:M
877:M
873:M
869:F
861:E
853:M
849:U
845:M
839:.
837:F
829:E
810:F
808:(
802:U
800:(
794:F
792:(
784:(
776:(
770:H
768:(
762:U
760:(
754:E
752:(
744:(
738:U
736:(
729:.
727:M
723:H
719:U
703:.
701:M
697:T
686:F
684:(
682:C
680:)
678:E
676:(
674:C
662:F
660:(
658:C
656:)
654:E
652:(
650:C
646:M
642:F
638:E
623:.
621:E
617:F
615:(
613:C
609:F
607:(
605:C
601:E
599:(
597:C
590:F
588:(
586:C
582:E
577:.
575:M
571:E
567:F
565:(
563:C
556:E
554:(
547:.
545:M
541:T
525:F
523:(
521:C
519:)
517:E
515:(
513:C
509:F
507:(
505:C
501:E
499:(
497:C
493:M
489:T
476:M
472:F
468:E
455:E
453:(
451:C
443:M
441:(
439:Z
435:M
431:E
416:N
404:U
390:}
388:E
384:F
380:F
376:N
372:F
361:K
354:E
350:N
346:E
342:K
338:N
334:E
330:N
323:E
319:M
315:E
313:(
311:C
307:E
305:(
303:C
287:I
283:I
279:E
277:(
275:C
271:E
269:(
267:C
263:E
255:M
248:M
246:(
244:Z
240:E
238:(
236:C
232:M
230:(
228:Z
221:2
218:F
214:1
211:F
207:2
204:F
200:1
197:F
193:M
191:(
189:Z
181:E
177:F
173:F
169:M
167:(
165:Z
161:F
157:E
155:(
153:C
146:M
142:E
138:E
136:(
134:C
130:M
126:M
114:M
110:T
106:M
98:M
96:(
94:Z
90:M
82:M
70:H
68:(
66:L
62:M
58:H
54:H
52:(
50:L
30:E
26:E
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