84:
74:
53:
22:
785:
The tensor product of an infinite number of von
Neumann algebras, if done naively, is usually a ridiculously large non-separable algebra. Instead one usually chooses a state on each of the von Neumann algebras, uses this to define a state on the algebraic tensor product, which can be used to product
768:
The tensor product of two von
Neumann algebras is a von Neumann algebra. The tensor product of two finite algebras is finite, and the tensor product of an infinite algebra and a non-zero algebra is infinite. The tensor product of two von Neumann algebras of types X and Y (I II or III) has type equal
719:
topology and then with the discrete topology; the former gives rise to an injective von
Neumann algebra, the latter does not. It is nonetheless true that the majority of work nowadays is done in the case where the group is discrete, but IMHO worth knowing that a more general definition can be given.
179:
What do you mean bias? The previous version you reverted explicitly said much of the theory is still applicable in general (bicommutant theorem, Kaplansky density theorem etc.) The important fact to note is that hyperfinite von
Neumann algebras (essentially classifiiable by COnnes' classification of
1051:
very such module H can be given an M-dimension dim_M(H) (not its dimension as a complex vector space) such that modules are isomorphic if and only if they have the same M-dimension. The M-dimension is additive, and a module is isomorphic to a subspace of another module if and only if it has smaller
501:
IIRC, the algebra of locally measurable essentially bounded functions on a locally finite measure space is a von
Neumann algebra; I'm not sure whether there is any measure space not equivalent to a locally finite one, but if there is such a beast, I see no way of constructing a von Neumann algebra
841:
The term "measure space" should indeed have read "σ-finite measure space". However many of the other comments above are caused by a misunderstanding of standard von
Neumann algebra terminology. As this obviously confuses some readers I have added a section explaining some of the more confusing
718:
This comment is incorrect. The von
Neumann algebras of the real line with its usual topology, and the real line with its discrete topology, are not isomorphic, since the former has separable predual while the latter does not. Or consider the rotation group in 3 dimensions, again with its usual
898:
the infinite tensor product example needs hashing out. It's overly colloquial, and it's not terribly clear unless you know what's meant to begin with. I'm still not absolutely sure it belongs there, but no reason to throw it out as long as it's at the end and not actually
711:
Also, what's the point of considering locally compact rather than discrete groups? It makes the example much more complicated. (I believe the von
Neumann group algebra of a locally compact group is isomorphic to that of the same group considered as discrete group).
740:
functions" is also a slightly inaccurate term when locally compact groups are considered: it might not be totally obvious that the
Hilbert space is independent of the choice of Haar measure, besides the whole identification-of-equivalent-functions issue.
707:
What exactly is the point of defining a von
Neumann algebra as being "generated" by a certain set of operators? We haven't even explained how this is always possible, nevermind that it's hardly a good way to define the group von Neumann algebra.
747:
Since a left-invariant Haar measure is unique up to multiplication by a scalar factor, L is unambiguous provided one states explicitly whether one is using a left or right invariant Haar measure. Not sure what the problem is supposed to be here.
1057:
Shouldn't any initial discussion of a mathematical quantity like "M-dimension" at least state what type of quantity it is? E.g., to what set does it belong? I hope someone knowledgeable about the subject can make this a lot
798:
stands for "infinite tensor product of finite type I factors". The type of the infinite tensor product can vary dramatically as the states are changed; for example, the infinite tensor product of an infinite number of type
674:
article currently mentions only von Neumann algebras, though if memory serves a similar construction for C*-algebras exists. I am unsure whether the two coincide for von Neumann algebras considered as C*-algebras.
871:
The name of the section is Examples, R.e.b. Better not to have an example at all than having one that makes the article not self-contained. I will certainly remove such examples if there's no obvious way to fix
272:
Someone please include a definition of "locally compact measure space", does it mean a locally compact Hausdorff space with a measure that is a positive linear functional on continuous maps with compact support???
860:
Sorry, I should have made it clearer that I was deliberately approaching this subject from the point of view of a student who does not know the usual terminology about von Neumann algebras -- the article
966:
an orthogonal sum of unitarily equivalent projections need not be unitarily equivalent, i.e. suppose A ~ B and C ~ D, where ~ means unitary equivalence, it's not necessarily true that A + C ~ B + D.
140:
845:
Please do not delete paragraphs that you find too terse or difficult. Instead you can add a "too technical" tag and put a request on the talk page for someone to provide more explanation.
241:
572:
536:
496:
408:
332:
457:
372:
243:
is highly non-commutative since it doesn't seem to mean much except stating that such an algebra is non-commutative. This is not true if we make everything we can trivial.
937:
At some point we should say that von Neumann algebras can be characterized abstractly as a C*-algebra with a predual. We need to decide where to put this in.--
868:
The tensor product case should certainly go out. The definition isn't trivial, and it should certainly not be dispensed with as a mere question of terminology!
334:
is not always a von Neumann algebra. Sure, the counterexamples might be pathological and uninteresting, but that doesn't mean incorrect statements are okay.
1086:
130:
779:
to be a von Neumann algebra, and the interesting statement is that the result of this completion is independent of certain choices made to define it.
283:
I couldn't find the term "locally compact measure space" anywhere in the article. Are you suggesting we include a definition of it in any case? --
809:
Von Neumann algebras are virtually always non-separable, in the norm topology. This most definitely no longer belongs in the examples section.
106:
1081:
1059:
775:
The tensor product of two von Neumann algebras is most certainly not a von Neumann algebra (except in some very simple cases). It can be
298:
I'm not sure a footnote is the best way to do it, but in the interest of mathematical correctness, it is important to point out that
277:
1029:
wow, that's terrible. pasted it from a prev paragraph, my bad. i suggest that the sentence be added back, with obvious corrections.
97:
58:
538:
short of considering its closure (but, of course, the resulting von Neumann algebra would be commutative, and thus isomorphic to
345:
33:
1063:
287:
274:
753:
724:
664:
of a von Neumann algebra by a discrete (or more generally locally compact) group is a von Neumann algebra.
895:
I'd like a real section about the tensor products of von Neumann algebras; it is more than terminology.
39:
83:
195:
21:
862:
541:
505:
465:
377:
301:
105:
on Knowledge. If you would like to participate, please visit the project page, where you can join
163:
89:
417:
264:
244:
73:
52:
351:
749:
720:
974:
590:
The examples section was rather error-ridden: here are the changes, and reasons for them:
889:
Oops, sorry. Just saw you didn't revert to the old wording of the tensor product example.
671:
661:
923:
Why is their definition in the von Neumann-article? I'm moving them to the C*-algebras.
1075:
192:
I would like to delete the comment that the space of bounded linear transforms on a
1030:
971:
924:
904:
875:
832:
577:
498:
isn't order complete, but the lattice of projections in a von Neumann algebra is).
1017:
970:
was removed with the edit summary claiming it's incorrect. why is it incorrect?
849:
102:
596:
The essentially bounded functions on a measure space form a commutative (type I
1067:
1033:
1020:
951:
948:
941:
938:
927:
907:
878:
852:
835:
757:
728:
580:
284:
267:
257:
254:
247:
181:
166:
79:
786:
a Hilbert space and a (reasonably small) von Neumann algebra. The factors of
374:
uncountable with uncountable complement, and consider a net of operators in
654:
I fail to see why this is discussed before von Neumann group algebras are.
410:
that converge in the weak or strong topology towards the projection onto
610:
Fails in the case of certain pathological measure spaces, see above.
632:′, whose projections correspond exactly to the closed subspaces of
689:
is the von Neumann algebra generated by the left translations of
15:
829:
If the removed examples can be salvaged, feel free to do so.
803:
factors can have any type depending on the choice of states.
1046:
The section "Modules over a factor" contains this passage:
825:
What are constructed from what? This isn't even a stub.
462:(Alternatively, note that the lattice of projections in
544:
508:
468:
420:
380:
354:
304:
198:
574:
for some nice (i.e. locally finite) measure space).
101:, a collaborative effort to improve the coverage of
566:
530:
490:
451:
402:
366:
326:
235:
180:hyperfinite factors) are all separably realizable.
616:If we have any unitary representation of a group
818:s factors are constructed from ergodic actions.
348:-- the counting measure will do fine. Choose
344:be an uncountable set and μ a measure on the
8:
624:then the bounded operators commuting with
162:How 'bout giving some concrete examples?
47:
549:
543:
513:
507:
473:
467:
459:: but that projection isn't measurable.
431:
419:
385:
379:
353:
309:
303:
209:
197:
980:Your definition of unitary equivalence
49:
19:
865:should certainly not presuppose that.
7:
600:) von Neumann algebra acting on the
95:This article is within the scope of
294:The pathological measure space case
38:It is of interest to the following
1087:High-priority mathematics articles
962:a comment that says, essentially:
550:
514:
474:
386:
310:
14:
115:Knowledge:WikiProject Mathematics
236:{\displaystyle B(L^{2}(X,\mu ))}
118:Template:WikiProject Mathematics
82:
72:
51:
20:
794:were found like this. The term
135:This article has been rated as
1068:02:24, 26 September 2018 (UTC)
928:01:47, 16 September 2006 (UTC)
648:is also a von Neumann algebra.
567:{\displaystyle L^{\infty }(X)}
561:
555:
531:{\displaystyle L^{\infty }(X)}
525:
519:
491:{\displaystyle L^{\infty }(X)}
485:
479:
446:
443:
437:
424:
403:{\displaystyle L^{\infty }(X)}
397:
391:
327:{\displaystyle L^{\infty }(X)}
321:
315:
230:
227:
215:
202:
1:
933:Abstract von Neumann algebras
758:11:13, 24 November 2008 (UTC)
729:11:13, 24 November 2008 (UTC)
346:countable-cocountable algebra
109:and see a list of open tasks.
1082:B-Class mathematics articles
1034:15:54, 26 January 2007 (UTC)
1021:15:31, 26 January 2007 (UTC)
975:05:18, 26 January 2007 (UTC)
957:
952:17:12, 9 October 2006 (UTC)
942:17:05, 9 October 2006 (UTC)
892:Anyway, about the article:
685:of a locally compact group
628:form a von Neumann algebra
586:Fixing the examples section
452:{\displaystyle B(L^{2}(X))}
1103:
769:to the maximum of X and Y.
367:{\displaystyle A\subset X}
268:22:51, 22 April 2006 (UTC)
258:14:40, 22 April 2006 (UTC)
248:13:42, 22 April 2006 (UTC)
947:Oops its already there!--
683:von Neumann group algebra
288:19:38, 17 July 2006 (UTC)
278:13:39, 16 July 2006 (UTC)
134:
67:
46:
908:22:12, 16 May 2006 (UTC)
879:22:03, 16 May 2006 (UTC)
853:21:40, 16 May 2006 (UTC)
836:01:44, 13 May 2006 (UTC)
640:. The double commutator
581:01:11, 13 May 2006 (UTC)
184:18:30, 3 Jan 2005 (UTC)
167:02:07, 31 May 2004 (UTC)
141:project's priority scale
337:For the non-believers:
263:O boy o boy, I did it!
98:WikiProject Mathematics
568:
532:
492:
453:
404:
368:
328:
237:
188:High non-commutativity
28:This article is rated
1052:or equal M-dimension.
569:
533:
493:
454:
405:
369:
329:
238:
542:
506:
466:
418:
378:
352:
302:
196:
121:mathematics articles
1042:Unclear M-dimension
863:von Neumann algebra
620:on a Hilbert space
253:Go ahead, delete.--
958:what's the dealio?
919:Type I-C*-algebras
564:
528:
488:
449:
400:
364:
324:
275:Kuratowski's Ghost
233:
90:Mathematics portal
34:content assessment
999:for some unitary
987:is equivalent to
175:Separability bias
155:
154:
151:
150:
147:
146:
1094:
636:invariant under
573:
571:
570:
565:
554:
553:
537:
535:
534:
529:
518:
517:
497:
495:
494:
489:
478:
477:
458:
456:
455:
450:
436:
435:
409:
407:
406:
401:
390:
389:
373:
371:
370:
365:
333:
331:
330:
325:
314:
313:
242:
240:
239:
234:
214:
213:
123:
122:
119:
116:
113:
92:
87:
86:
76:
69:
68:
63:
55:
48:
31:
25:
24:
16:
1102:
1101:
1097:
1096:
1095:
1093:
1092:
1091:
1072:
1071:
1044:
960:
935:
921:
802:
672:crossed product
662:crossed product
599:
588:
545:
540:
539:
509:
504:
503:
469:
464:
463:
427:
416:
415:
414:(an element of
381:
376:
375:
350:
349:
305:
300:
299:
296:
205:
194:
193:
190:
177:
160:
120:
117:
114:
111:
110:
88:
81:
61:
32:on Knowledge's
29:
12:
11:
5:
1100:
1098:
1090:
1089:
1084:
1074:
1073:
1060:108.245.209.39
1043:
1040:
1039:
1038:
1037:
1036:
1024:
1023:
1006:
1005:
1004:
968:
967:
959:
956:
955:
954:
934:
931:
920:
917:
915:
913:
912:
911:
910:
902:
901:
900:
896:
890:
884:
883:
882:
881:
873:
869:
866:
847:
846:
843:
828:
824:
822:
821:
820:
819:
807:
806:
805:
804:
800:
773:
772:
771:
770:
763:
762:
761:
760:
734:
733:
732:
731:
705:
704:
703:
702:
693:acting on the
668:
667:
666:
665:
652:
651:
650:
649:
608:
607:
606:
605:
597:
587:
584:
563:
560:
557:
552:
548:
527:
524:
521:
516:
512:
487:
484:
481:
476:
472:
448:
445:
442:
439:
434:
430:
426:
423:
399:
396:
393:
388:
384:
363:
360:
357:
323:
320:
317:
312:
308:
295:
292:
291:
290:
261:
260:
232:
229:
226:
223:
220:
217:
212:
208:
204:
201:
189:
186:
176:
173:
171:
159:
156:
153:
152:
149:
148:
145:
144:
133:
127:
126:
124:
107:the discussion
94:
93:
77:
65:
64:
56:
44:
43:
37:
26:
13:
10:
9:
6:
4:
3:
2:
1099:
1088:
1085:
1083:
1080:
1079:
1077:
1070:
1069:
1065:
1061:
1055:
1053:
1047:
1041:
1035:
1032:
1028:
1027:
1026:
1025:
1022:
1019:
1015:
1011:
1008:implies that
1007:
1002:
998:
994:
990:
986:
982:
981:
979:
978:
977:
976:
973:
965:
964:
963:
953:
950:
946:
945:
944:
943:
940:
932:
930:
929:
926:
918:
916:
909:
906:
903:
897:
894:
893:
891:
888:
887:
886:
885:
880:
877:
874:
870:
867:
864:
859:
858:
857:
856:
855:
854:
851:
844:
840:
839:
838:
837:
834:
830:
826:
817:
814:
813:
812:
811:
810:
797:
793:
789:
784:
783:
782:
781:
780:
778:
767:
766:
765:
764:
759:
755:
751:
746:
745:
744:
743:
742:
739:
730:
726:
722:
717:
716:
715:
714:
713:
709:
700:
697:functions on
696:
692:
688:
684:
680:
679:
678:
677:
676:
673:
663:
659:
658:
657:
656:
655:
647:
643:
639:
635:
631:
627:
623:
619:
615:
614:
613:
612:
611:
603:
595:
594:
593:
592:
591:
585:
583:
582:
579:
575:
558:
546:
522:
510:
499:
482:
470:
460:
440:
432:
428:
421:
413:
394:
382:
361:
358:
355:
347:
343:
338:
335:
318:
306:
293:
289:
286:
282:
281:
280:
279:
276:
270:
269:
266:
259:
256:
252:
251:
250:
249:
246:
224:
221:
218:
210:
206:
199:
187:
185:
183:
174:
172:
169:
168:
165:
164:Michael Hardy
157:
142:
138:
137:High-priority
132:
129:
128:
125:
108:
104:
100:
99:
91:
85:
80:
78:
75:
71:
70:
66:
62:High‑priority
60:
57:
54:
50:
45:
41:
35:
27:
23:
18:
17:
1056:
1050:
1048:
1045:
1013:
1009:
1000:
996:
992:
988:
984:
969:
961:
936:
922:
914:
848:
831:
827:
823:
815:
808:
795:
791:
787:
776:
774:
750:NowhereDense
737:
735:
721:NowhereDense
710:
706:
698:
694:
690:
686:
682:
669:
653:
645:
641:
637:
633:
629:
625:
621:
617:
609:
601:
589:
576:
500:
461:
411:
341:
339:
336:
297:
271:
262:
191:
178:
170:
161:
136:
96:
40:WikiProjects
792:Araki-Woods
265:Herman Stel
112:Mathematics
103:mathematics
59:Mathematics
1076:Categories
604:functions.
777:completed
1058:clearer.
983:"we say
158:Untitled
1031:Mct mht
972:Mct mht
925:KennyDC
905:RandomP
876:RandomP
833:RandomP
816:Krieger
578:RandomP
502:out of
139:on the
30:B-class
1018:R.e.b.
899:wrong.
850:R.e.b.
842:terms.
788:Powers
644:′′ of
36:scale.
949:CSTAR
939:CSTAR
872:them.
796:ITPFI
285:CSTAR
255:CSTAR
245:HStel
182:CSTAR
1064:talk
1016:=1.
997:F=uu
995:and
993:E=uu
790:and
754:talk
725:talk
681:The
670:The
660:The
340:Let
131:High
991:if
1078::
1066:)
1054:"
1003:."
756:)
727:)
551:∞
515:∞
475:∞
387:∞
359:⊂
311:∞
225:μ
1062:(
1049:"
1014:F
1012:=
1010:E
1001:u
989:F
985:E
801:2
799:I
752:(
738:L
736:"
723:(
701:.
699:G
695:L
691:G
687:G
646:G
642:G
638:G
634:H
630:G
626:G
622:H
618:G
602:L
598:1
562:)
559:X
556:(
547:L
526:)
523:X
520:(
511:L
486:)
483:X
480:(
471:L
447:)
444:)
441:X
438:(
433:2
429:L
425:(
422:B
412:A
398:)
395:X
392:(
383:L
362:X
356:A
342:X
322:)
319:X
316:(
307:L
231:)
228:)
222:,
219:X
216:(
211:2
207:L
203:(
200:B
143:.
42::
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
