237:
674:
showed, these spaces can be realised as unbounded operators only by using a certain positive power of the reference modular operator. Instead of being characterised by the simple affiliation relation
64:
could naturally be phrased in terms of unbounded operators affiliated with the von
Neumann algebra of the group. Algebraic properties of affiliated operators have proved important in
363:
465:
need not necessarily be well-behaved under either addition or composition. However in the presence of a faithful semi-finite normal trace τ and the standard
434:
778:
771:
764:
728:
714:
700:
153:
466:
798:
553:
682:, there is a more complicated bimodule relation involving the analytic continuation of the modular automorphism group.
509:
sufficiently large. This algebra of unbounded operators is complete for a natural topology, generalising the notion of
49:
663:
395:, a positive self-adjoint operator commutes with a unitary operator if and only if each of its spectral projections
793:
604:
510:
100:
41:
596:
25:
591:
factor, then every affiliated operator is automatically measurable, so the affiliated operators form a
281:
33:
304:
89:
37:
325:
774:
767:
760:
724:
710:
696:
61:
53:
573:
392:
369:
120:
29:
569:
57:
398:
670:
spaces are no longer realised by operators affiliated with the von
Neumann algebra. As
45:
787:
486:
93:
65:
671:
380:
17:
662:, the theory takes a quite different form. Indeed in this case, thanks to the
592:
494:
127:
517:
spaces defined by the trace and was introduced to facilitate their study.
73:
69:
315:
The last condition follows by uniqueness of the polar decomposition. If
461:
In general the operators affiliated with a von
Neumann algebra
721:
L-Invariants: Theory and
Applications to Geometry and K-Theory
647:), we simply recover the *-algebra of measurable functions on
232:{\displaystyle G(A)=\{(x,Ax):x\in D(A)\}\subseteq H\oplus H}
497:
with nice properties: these are operators such that τ(
444:| and the partial isometry in the polar decomposition of
520:
This theory can be applied when the von
Neumann algebra
401:
328:
156:
40:
to study modules generated by a single vector. Later
556:, it gives the well-known theory of non-commutative
76:
that evolved from the study of such index theorems.
723:, (Chapter 8: the algebra of affiliated operators)
428:
357:
231:
595:, as originally observed in the first paper of
436:does. This gives another equivalent condition:
8:
214:
172:
587:von Neumann algebra, for example a type II
303:should commute with both operators in the
400:
354:
346:
338:
327:
155:
666:, it is known that the non-commutative
631:. Of course in the classical case when
757:Theory of Operator Algebras I, II, III
513:. It contains all the non-commutative
7:
746:Notes on non-commutative integration
741:(1936), 116–229 (Chapter XVI).
146:should leave invariant the graph of
623:defines a measurable operator with
607:: for on the closure of its image
134:. Equivalent conditions are that:
14:
733:F. J. Murray and J. von Neumann,
467:Gelfand–Naimark–Segal
505:()) < ∞ for
493:affiliated operators do form a
599:and von Neumann. In this case
544:) acting on the Hilbert space
423:
420:
408:
405:
347:
339:
211:
205:
190:
175:
166:
160:
1:
440:each spectral projection of |
103:and densely defined operator
635:is a probability space and
319:has a polar decomposition
288:, onto itself and satisfy
815:
387:should be affiliated with
611:has a measurable inverse
554:Hilbert–Schmidt operators
358:{\displaystyle A=V|A|,\,}
36:as a technique for using
737:, Annals of Mathematics
693:Non-commutative geometry
605:von Neumann regular ring
752:(1974), 103–116.
664:Tomita–Takesaki theory
511:convergence in measure
430:
379:and that the positive
359:
233:
431:
360:
234:
799:Von Neumann algebras
707:Von Neumann algebras
457:Measurable operators
399:
326:
242:the projection onto
154:
119:commutes with every
34:von Neumann algebras
22:affiliated operators
501: −
429:{\displaystyle E()}
305:polar decomposition
90:von Neumann algebra
38:unbounded operators
24:were introduced by
748:, J. Funct. Anal.
735:Rings of Operators
426:
391:. However, by the
355:
229:
68:, an area between
54:elliptic operators
583:is in addition a
368:it says that the
62:fundamental group
32:in the theory of
806:
489:proved that the
485:, τ),
435:
433:
432:
427:
393:spectral theorem
370:partial isometry
364:
362:
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238:
236:
235:
230:
121:unitary operator
58:closed manifolds
814:
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809:
808:
807:
805:
804:
803:
794:Operator theory
784:
783:
688:
590:
459:
397:
396:
324:
323:
256:
152:
151:
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375:should lie in
366:
365:
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189:
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180:
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107:is said to be
81:
78:
60:with infinite
50:index theorems
13:
10:
9:
6:
4:
3:
2:
811:
800:
797:
795:
792:
791:
789:
780:
779:3-540-42913-1
776:
773:
772:3-540-42914-X
769:
766:
765:3-540-42248-X
762:
758:
755:M. Takesaki,
754:
751:
747:
743:
740:
736:
732:
730:
729:3-540-43566-2
726:
722:
718:
716:
715:0-444-86308-7
712:
708:
704:
702:
701:0-12-185860-X
698:
694:
690:
689:
685:
683:
681:
678: =
677:
673:
669:
665:
661:
657:
652:
650:
646:
642:
639: =
638:
634:
630:
627: =
626:
622:
619: =
618:
614:
610:
606:
602:
598:
594:
586:
582:
577:
575:
571:
567:
563:
559:
555:
551:
547:
543:
539:
536: =
535:
531:
527:
523:
518:
516:
512:
508:
504:
500:
496:
492:
488:
487:Edward Nelson
484:
480:
477: =
476:
472:
468:
464:
456:
451:
447:
443:
439:
438:
437:
417:
414:
411:
402:
394:
390:
386:
382:
378:
374:
371:
351:
343:
335:
332:
329:
322:
321:
320:
318:
310:
306:
302:
298:
295:each unitary
294:
291:
287:
283:
279:
275:
272:should carry
271:
267:
264:each unitary
263:
260:
253:
249:
245:
241:
226:
223:
220:
217:
208:
202:
199:
196:
193:
187:
184:
181:
178:
169:
163:
157:
149:
145:
141:
138:each unitary
137:
136:
135:
133:
129:
125:
122:
118:
114:
110:
106:
102:
98:
95:
94:Hilbert space
91:
87:
79:
77:
75:
71:
67:
63:
59:
55:
51:
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43:
39:
35:
31:
27:
23:
19:
756:
749:
745:
738:
734:
720:
706:
705:J. Dixmier,
692:
679:
675:
667:
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655:
653:
648:
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640:
636:
632:
628:
624:
620:
616:
612:
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600:
584:
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549:
545:
541:
537:
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525:
521:
519:
514:
506:
502:
498:
490:
482:
478:
474:
470:
462:
460:
449:
445:
441:
388:
384:
381:self-adjoint
376:
372:
367:
316:
314:
308:
300:
296:
289:
285:
277:
273:
269:
265:
258:
251:
247:
243:
147:
143:
139:
131:
123:
116:
112:
108:
104:
96:
92:acting on a
85:
83:
66:L cohomology
48:showed that
21:
15:
744:E. Nelson,
691:A. Connes,
654:If however
574:von Neumann
150:defined by
30:von Neumann
18:mathematics
788:Categories
686:References
491:measurable
469:action of
109:affiliated
80:Definition
719:W. LĂĽck,
615:and then
593:*-algebra
568:) due to
495:*-algebra
383:operator
224:⊕
218:⊆
200:∈
128:commutant
660:type III
570:Schatten
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70:analysis
560:spaces
532:. When
530:type II
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770:
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699:
672:Connes
597:Murray
585:finite
526:type I
292:there.
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101:closed
46:Singer
42:Atiyah
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579:When
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572:and
99:. A
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625:ATA
609:|A|
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