323:
610:
189:
431:
396:
360:
492:
457:
119:
162:
552:
235:
212:
139:
89:
247:
751:
710:
497:
The
Koecher–Vinberg theorem now states that these properties precisely characterize the positive cones of Jordan algebras.
756:
557:
42:
399:
727:
167:
404:
662:
706:
369:
332:
654:
462:
436:
50:
22:
54:
98:
144:
537:
530:
504:
220:
197:
124:
74:
46:
38:
30:
745:
238:
215:
57:
700:
69:
34:
666:
318:{\displaystyle C^{*}=\{a\in A:\forall b\in C\langle a,b\rangle >0\}}
658:
534:. The domain of positivity associated with a real Jordan algebra
702:
The
Minnesota Notes on Jordan Algebras and Their Applications
524:
Convex cones satisfying these four properties are called
560:
540:
465:
439:
407:
372:
335:
250:
223:
200:
170:
147:
127:
101:
77:
49:and so-called domains of positivity. Thus it links
645:Koecher, Max (1957). "Positivitatsbereiche im R".
604:
546:
486:
451:
425:
390:
354:
317:
229:
206:
183:
156:
133:
113:
83:
503:: There is a one-to-one correspondence between
625:
8:
680:Vinberg, E. B. (1961). "Homogeneous Cones".
605:{\displaystyle A_{+}=\{a^{2}\colon a\in A\}}
599:
574:
312:
303:
291:
264:
60:views on state spaces of physical systems.
581:
565:
559:
539:
464:
438:
406:
371:
346:
334:
255:
249:
222:
199:
171:
169:
146:
126:
100:
76:
637:
621:
554:is the interior of the 'positive' cone
29:is a reconstruction theorem for real
7:
279:
14:
33:. It was proved independently by
647:American Journal of Mathematics
184:{\displaystyle {\overline {C}}}
475:
469:
443:
433:that restricts to a bijection
426:{\displaystyle T\colon A\to A}
417:
1:
505:formally real Jordan algebras
47:formally real Jordan algebras
176:
732:Analysis on Symmetric Cones
626:Faraut & Koranyi (1994)
507:and convex cones that are:
18:Theorem of operator algebra
773:
734:. Oxford University Press.
43:one-to-one correspondence
752:Non-associative algebras
391:{\displaystyle a,b\in C}
366:when to any two points
355:{\displaystyle C=C^{*}}
41:in 1961. It provides a
27:Koecher–Vinberg theorem
606:
548:
488:
487:{\displaystyle T(a)=b}
453:
452:{\displaystyle C\to C}
427:
392:
356:
319:
231:
208:
185:
158:
135:
115:
85:
699:Koecher, Max (1999).
607:
549:
526:domains of positivity
489:
454:
428:
400:linear transformation
393:
357:
325:. The cone is called
320:
232:
209:
186:
159:
136:
116:
86:
558:
538:
463:
437:
405:
370:
333:
248:
221:
198:
168:
145:
125:
99:
75:
757:Theorems in algebra
164:are in the closure
114:{\displaystyle a=0}
602:
544:
484:
449:
423:
388:
352:
315:
227:
204:
181:
157:{\displaystyle -a}
154:
131:
111:
81:
51:operator algebraic
682:Soviet Math. Dokl
620:For a proof, see
547:{\displaystyle A}
230:{\displaystyle A}
207:{\displaystyle C}
179:
134:{\displaystyle a}
84:{\displaystyle C}
764:
736:
735:
723:
717:
716:
696:
690:
689:
677:
671:
670:
642:
611:
609:
608:
603:
586:
585:
570:
569:
553:
551:
550:
545:
493:
491:
490:
485:
458:
456:
455:
450:
432:
430:
429:
424:
398:there is a real
397:
395:
394:
389:
361:
359:
358:
353:
351:
350:
324:
322:
321:
316:
260:
259:
236:
234:
233:
228:
213:
211:
210:
205:
190:
188:
187:
182:
180:
172:
163:
161:
160:
155:
140:
138:
137:
132:
120:
118:
117:
112:
90:
88:
87:
82:
23:operator algebra
772:
771:
767:
766:
765:
763:
762:
761:
742:
741:
740:
739:
725:
724:
720:
713:
698:
697:
693:
679:
678:
674:
659:10.2307/2372563
644:
643:
639:
634:
618:
577:
561:
556:
555:
536:
535:
531:symmetric cones
461:
460:
435:
434:
403:
402:
368:
367:
362:. It is called
342:
331:
330:
251:
246:
245:
219:
218:
196:
195:
166:
165:
143:
142:
123:
122:
97:
96:
73:
72:
66:
58:order theoretic
31:Jordan algebras
19:
12:
11:
5:
770:
768:
760:
759:
754:
744:
743:
738:
737:
718:
711:
691:
672:
653:(3): 575–596.
636:
635:
633:
630:
622:Koecher (1999)
617:
614:
601:
598:
595:
592:
589:
584:
580:
576:
573:
568:
564:
543:
522:
521:
518:
515:
512:
483:
480:
477:
474:
471:
468:
459:and satisfies
448:
445:
442:
422:
419:
416:
413:
410:
387:
384:
381:
378:
375:
349:
345:
341:
338:
314:
311:
308:
305:
302:
299:
296:
293:
290:
287:
284:
281:
278:
275:
272:
269:
266:
263:
258:
254:
226:
203:
194:A convex cone
178:
175:
153:
150:
130:
121:whenever both
110:
107:
104:
80:
65:
62:
39:Ernest Vinberg
17:
13:
10:
9:
6:
4:
3:
2:
769:
758:
755:
753:
750:
749:
747:
733:
729:
722:
719:
714:
712:3-540-66360-6
708:
704:
703:
695:
692:
687:
683:
676:
673:
668:
664:
660:
656:
652:
648:
641:
638:
631:
629:
627:
623:
615:
613:
596:
593:
590:
587:
582:
578:
571:
566:
562:
541:
533:
532:
527:
519:
516:
513:
510:
509:
508:
506:
502:
498:
495:
481:
478:
472:
466:
446:
440:
420:
414:
411:
408:
401:
385:
382:
379:
376:
373:
365:
347:
343:
339:
336:
328:
309:
306:
300:
297:
294:
288:
285:
282:
276:
273:
270:
267:
261:
256:
252:
244:
240:
239:inner product
224:
217:
201:
192:
173:
151:
148:
128:
108:
105:
102:
94:
78:
71:
63:
61:
59:
56:
52:
48:
44:
40:
36:
32:
28:
24:
16:
731:
726:Faraut, J.;
721:
705:. Springer.
701:
694:
685:
681:
675:
650:
646:
640:
619:
529:
525:
523:
517:homogeneous;
500:
499:
496:
363:
326:
242:
216:vector space
193:
92:
67:
37:in 1957 and
26:
20:
15:
728:Koranyi, A.
364:homogeneous
70:convex cone
35:Max Koecher
746:Categories
688:: 787–790.
632:References
520:self-dual.
91:is called
594:∈
588::
444:→
418:→
412::
383:∈
348:∗
327:self-dual
304:⟩
292:⟨
286:∈
280:∀
271:∈
257:∗
243:dual cone
177:¯
149:−
64:Statement
730:(1994).
514:regular;
237:with an
45:between
667:2372563
501:Theorem
93:regular
709:
665:
241:has a
55:convex
25:, the
663:JSTOR
616:Proof
511:open;
329:when
214:in a
707:ISBN
307:>
141:and
53:and
655:doi
624:or
528:or
95:if
21:In
748::
684:.
661:.
651:97
649:.
628:.
612:.
494:.
191:.
68:A
715:.
686:1
669:.
657::
600:}
597:A
591:a
583:2
579:a
575:{
572:=
567:+
563:A
542:A
482:b
479:=
476:)
473:a
470:(
467:T
447:C
441:C
421:A
415:A
409:T
386:C
380:b
377:,
374:a
344:C
340:=
337:C
313:}
310:0
301:b
298:,
295:a
289:C
283:b
277::
274:A
268:a
265:{
262:=
253:C
225:A
202:C
174:C
152:a
129:a
109:0
106:=
103:a
79:C
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.