768:
967:
837:
1069:
1077:
681:
998:
52:
67:
1138:
1126:
1148:
1143:
921:
912:
977:
48:
1040:. For example, the corona algebra of the algebra of compact operators on a Hilbert space is the
417:. The above lemma, together with the universal property of the multiplier algebra, yields that
1164:
1114:
8:
44:
1118:
1102:
883:
161:
1094:
139:
1122:
1086:
87:
1110:
864:
861:
1041:
1158:
1098:
39:
is a unital C*-algebra that is the largest unital C*-algebra that contains
372:) can be obtained via representations. The following fact will be needed:
204:
17:
1106:
1048:
33:
325:
as an essential ideal and can be identified as the multiplier algebra
426:
1090:
648:
816:, and the strict topology coincides with the norm topology. For
321:
can be given a C*-algebra structure. This C*-algebra contains
615:
is isomorphic to the compact operators on the
Hilbert module
882:), the commutative C*-algebra of continuous functions that
1047:
The corona algebra is a noncommutative analogue of the
924:
684:
172:
as an ideal, there exists a unique *-homomorphism φ:
1070:"Double centralizers and extensions of C*-algebras"
488:)) be the adjointable (resp. compact) operators on
405:Now take any faithful nondegenerate representation
223:. It also follows from the definition that for any
961:
763:{\displaystyle l_{a}(x)=\|ax\|,\;r_{a}(x)=\|xa\|.}
762:
361:by simply multiplication from the left and right.
1078:Transactions of the American Mathematical Society
525:, then any faithful nondegenerate *-homomorphism
511:). Something similar to the above lemma is true:
386:, then any faithful nondegenerate representation
1156:
556:is a faithful nondegenerate *-homomorphism of
231:as an essential ideal, the multiplier algebra
68:compact operators on a separable Hilbert space
207:is specified by the universal property. When
160:) is any C*-algebra satisfying the following
754:
745:
716:
707:
499:) can be identified via a *-homomorphism of
722:
345:) on a separable Hilbert space, then each
915:, one has the isomorphism of C*-algebras
848:) is complete in the σ-strong* topology.
55:. Multiplier algebras were introduced by
1136:
152:be a C*-algebra. Its multiplier algebra
907:), the continuous bounded functions on
1157:
317:||. The set of double centralizers of
1067:
773:The resulting topology is called the
188:extends the identity homomorphism on
56:
47:in a "non-degenerate" way. It is the
576:) is isomorphic to the idealizer of
962:{\displaystyle C_{b}(X)\simeq C(Y)}
851:
13:
634:
627:) is the adjointable operators on
357:) defines a double centralizer of
14:
1176:
1011:
134:, the "orthogonal complement" of
254:) can be shown in several ways.
997:is in fact homeomorphic to the
836:)), the strict topology is the
122:is non-trivial for every ideal
1062:K-Theory for Operator Algebras
956:
950:
941:
935:
739:
733:
701:
695:
1:
1054:
840:. It follows from above that
130:is essential if and only if
97:
1139:"Multipliers of C*-algebras"
521:is an ideal in a C*-algebra
382:is an ideal in a C*-algebra
273:) of bounded linear maps on
7:
1144:Encyclopedia of Mathematics
1137:Pedersen, Gert K. (2001) ,
999:Stone–Čech compactification
472:be a Hilbert C*-module and
53:Stone–Čech compactification
10:
1181:
1064:, MSRI Publications, 1986.
1068:Busby, Robert C. (1968),
639:Consider the topology on
604:) for any Hilbert module
337:is the compact operators
86:), the C*-algebra of all
1051:of a topological space.
445:). It is immediate that
913:Gelfand–Naimark theorem
838:σ-strong* topology
425:) is isomorphic to the
963:
764:
309:. This implies that ||
964:
789:is strictly dense in
765:
164:: for all C*-algebra
66:is the C*-algebra of
922:
682:
333:). For instance, if
243:as a C*-subalgebra.
647:) specified by the
413:on a Hilbert space
959:
884:vanish at infinity
760:
259:double centralizer
162:universal property
51:generalization of
22:multiplier algebra
584:). For instance,
552:Consequently, if
541:)can be extended
246:The existence of
203:Uniqueness up to
140:Hilbert C*-module
88:bounded operators
1172:
1151:
1133:
1131:
1125:, archived from
1074:
1028:is the quotient
968:
966:
965:
960:
934:
933:
852:Commutative case
769:
767:
766:
761:
732:
731:
694:
693:
394:can be extended
261:of a C*-algebra
106:in a C*-algebra
62:For example, if
1180:
1179:
1175:
1174:
1173:
1171:
1170:
1169:
1155:
1154:
1129:
1091:10.2307/1994883
1072:
1057:
1014:
988:
929:
925:
923:
920:
919:
902:
877:
865:Hausdorff space
862:locally compact
854:
775:strict topology
727:
723:
689:
685:
683:
680:
679:
674:
663:
656:
637:
635:Strict topology
611:The C*-algebra
364:Alternatively,
100:
12:
11:
5:
1178:
1168:
1167:
1153:
1152:
1134:
1065:
1060:B. Blackadar,
1056:
1053:
1042:Calkin algebra
1022:corona algebra
1013:
1012:Corona algebra
1010:
984:
970:
969:
958:
955:
952:
949:
946:
943:
940:
937:
932:
928:
898:
875:
853:
850:
771:
770:
759:
756:
753:
750:
747:
744:
741:
738:
735:
730:
726:
721:
718:
715:
712:
709:
706:
703:
700:
697:
692:
688:
666:
661:
654:
636:
633:
110:is said to be
99:
96:
49:noncommutative
9:
6:
4:
3:
2:
1177:
1166:
1163:
1162:
1160:
1150:
1146:
1145:
1140:
1135:
1132:on 2020-02-20
1128:
1124:
1120:
1116:
1112:
1108:
1104:
1100:
1096:
1092:
1088:
1084:
1080:
1079:
1071:
1066:
1063:
1059:
1058:
1052:
1050:
1045:
1043:
1039:
1035:
1031:
1027:
1023:
1019:
1009:
1007:
1003:
1000:
996:
992:
987:
983:
979:
975:
953:
947:
944:
938:
930:
926:
918:
917:
916:
914:
910:
906:
901:
897:
893:
889:
885:
881:
874:
870:
866:
863:
859:
849:
847:
843:
839:
835:
831:
827:
823:
819:
815:
811:
807:
803:
798:
796:
792:
788:
784:
780:
776:
757:
751:
748:
742:
736:
728:
724:
719:
713:
710:
704:
698:
690:
686:
678:
677:
676:
673:
669:
664:
657:
650:
646:
642:
632:
630:
626:
622:
619:. Therefore,
618:
614:
609:
607:
603:
599:
595:
591:
587:
583:
579:
575:
571:
567:
563:
559:
555:
550:
548:
544:
540:
536:
532:
528:
524:
520:
516:
512:
510:
506:
502:
498:
494:
491:
487:
483:
479:
475:
471:
466:
464:
460:
456:
452:
448:
444:
440:
436:
432:
428:
424:
420:
416:
412:
408:
403:
401:
397:
393:
389:
385:
381:
377:
373:
371:
367:
362:
360:
356:
352:
348:
344:
340:
336:
332:
328:
324:
320:
316:
312:
308:
304:
300:
296:
292:
288:
284:
280:
276:
272:
268:
264:
260:
255:
253:
249:
244:
242:
238:
234:
230:
226:
222:
218:
214:
210:
206:
201:
199:
195:
191:
187:
183:
179:
175:
171:
167:
163:
159:
155:
151:
146:
144:
141:
137:
133:
129:
125:
121:
117:
113:
109:
105:
95:
93:
89:
85:
81:
77:
73:
69:
65:
60:
58:
54:
50:
46:
42:
38:
35:
31:
27:
24:, denoted by
23:
19:
1142:
1127:the original
1085:(1): 79–99,
1082:
1076:
1061:
1046:
1037:
1033:
1029:
1025:
1021:
1017:
1015:
1005:
1001:
994:
990:
985:
981:
973:
971:
908:
904:
899:
895:
891:
887:
879:
872:
868:
857:
855:
845:
841:
833:
829:
825:
821:
817:
813:
809:
805:
801:
799:
794:
790:
786:
782:
778:
774:
772:
671:
667:
659:
652:
644:
640:
638:
628:
624:
620:
616:
612:
610:
605:
601:
597:
593:
589:
585:
581:
577:
573:
569:
565:
561:
557:
553:
551:
546:
542:
538:
534:
530:
526:
522:
518:
514:
513:
508:
504:
500:
496:
492:
489:
485:
481:
477:
473:
469:
468:Lastly, let
467:
462:
458:
454:
450:
446:
442:
438:
434:
430:
422:
418:
414:
410:
406:
404:
399:
395:
391:
387:
383:
379:
375:
374:
369:
365:
363:
358:
354:
350:
346:
342:
338:
334:
330:
326:
322:
318:
314:
310:
306:
302:
298:
294:
290:
286:
282:
278:
274:
270:
266:
262:
258:
256:
251:
247:
245:
240:
236:
232:
228:
224:
220:
216:
212:
208:
202:
197:
193:
189:
185:
184:) such that
181:
177:
173:
169:
165:
157:
153:
149:
147:
142:
135:
131:
127:
123:
119:
115:
111:
107:
103:
101:
91:
83:
79:
75:
71:
63:
61:
57:Busby (1968)
40:
36:
29:
25:
21:
15:
1165:C*-algebras
804:is unital,
265:is a pair (
239:) contains
227:containing
211:is unital,
205:isomorphism
168:containing
126:. An ideal
18:mathematics
1055:References
1049:corona set
277:such that
98:Definition
34:C*-algebra
1149:EMS Press
1099:0002-9947
945:≃
911:. By the
755:‖
746:‖
717:‖
708:‖
649:seminorms
480:) (resp.
427:idealizer
200:) = {0}.
112:essential
102:An ideal
1159:Category
1123:54047557
978:spectrum
675:, where
568:), then
543:uniquely
396:uniquely
297:for all
145:is {0}.
32:), of a
1115:0225175
1107:1994883
976:is the
886:. Then
313:|| = ||
138:in the
1121:
1113:
1105:
1097:
1018:corona
972:where
533:into
515:Lemma.
376:Lemma.
43:as an
20:, the
1130:(PDF)
1119:S2CID
1103:JSTOR
1073:(PDF)
894:) is
860:be a
800:When
596:)) =
560:into
503:into
457:)) =
437:) in
78:) is
45:ideal
1095:ISSN
1016:The
856:Let
824:) =
812:) =
797:) .
301:and
285:) =
219:) =
192:and
148:Let
1087:doi
1083:132
1024:of
1020:or
1004:of
993:).
980:of
785:).
777:on
545:to
529:of
517:If
465:).
429:of
409:of
398:to
390:of
378:If
305:in
114:if
90:on
16:In
1161::
1147:,
1141:,
1117:,
1111:MR
1109:,
1101:,
1093:,
1081:,
1075:,
1044:.
1036:)/
1008:.
1002:βX
871:=
867:,
670:∈
658:,
631:.
608:.
549:.
402:.
349:∈
279:aL
269:,
257:A
176:→
118:∩
94:.
70:,
59:.
1089::
1038:A
1034:A
1032:(
1030:M
1026:A
1006:X
995:Y
991:X
989:(
986:b
982:C
974:Y
957:)
954:Y
951:(
948:C
942:)
939:X
936:(
931:b
927:C
909:X
905:X
903:(
900:b
896:C
892:A
890:(
888:M
880:X
878:(
876:0
873:C
869:A
858:X
846:H
844:(
842:B
834:H
832:(
830:K
828:(
826:M
822:H
820:(
818:B
814:A
810:A
808:(
806:M
802:A
795:A
793:(
791:M
787:A
783:A
781:(
779:M
758:.
752:a
749:x
743:=
740:)
737:x
734:(
729:a
725:r
720:,
714:x
711:a
705:=
702:)
699:x
696:(
691:a
687:l
672:A
668:a
665:}
662:a
660:r
655:a
653:l
651:{
645:A
643:(
641:M
629:A
625:A
623:(
621:M
617:A
613:A
606:E
602:E
600:(
598:B
594:E
592:(
590:K
588:(
586:M
582:A
580:(
578:π
574:A
572:(
570:M
566:E
564:(
562:B
558:A
554:π
547:B
539:E
537:(
535:B
531:I
527:π
523:B
519:I
509:E
507:(
505:B
501:A
497:A
495:(
493:M
490:E
486:E
484:(
482:K
478:E
476:(
474:B
470:E
463:H
461:(
459:B
455:H
453:(
451:K
449:(
447:M
443:H
441:(
439:B
435:A
433:(
431:π
423:A
421:(
419:M
415:H
411:A
407:π
400:B
392:I
388:π
384:B
380:I
370:A
368:(
366:M
359:A
355:H
353:(
351:B
347:x
343:H
341:(
339:K
335:A
331:A
329:(
327:M
323:A
319:A
315:R
311:L
307:A
303:b
299:a
295:b
293:)
291:a
289:(
287:R
283:b
281:(
275:A
271:R
267:L
263:A
252:A
250:(
248:M
241:D
237:A
235:(
233:M
229:A
225:D
221:A
217:A
215:(
213:M
209:A
198:A
196:(
194:φ
190:A
186:φ
182:A
180:(
178:M
174:D
170:A
166:D
158:A
156:(
154:M
150:A
143:B
136:I
132:I
128:I
124:J
120:J
116:I
108:B
104:I
92:H
84:H
82:(
80:B
76:A
74:(
72:M
64:A
41:A
37:A
30:A
28:(
26:M
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