291:
The unit conjecture is also known to hold in many groups, but its partial solutions are much less robust than the other two (as witnessed by the earlier-mentioned counter-example). This conjecture is not known to follow from any analytic statement like the other two, and so the cases where it is
211:
The zero-divisor conjecture implies the idempotent conjecture and is implied by the unit conjecture. As of 2021, the zero divisor and idempotent conjectures are open. The unit conjecture, however, was disproved in characteristic 2 by Giles Gardam by exhibiting an explicit counterexample in a
228:. A later preprint by Gardam claims that essentially the same element also gives a counter-example in characteristic 0 (finding an inverse is computationally much more involved in this setting, hence the delay between the first result and the second one).
292:
known to hold have all been established via a direct combinatorial approach involving the so-called unique products property. By Gardam's work mentioned above, it is now known to not be true in general.
231:
There are proofs of both the idempotent and zero-divisor conjectures for large classes of groups. For example, the zero-divisor conjecture is known for all torsion-free
495:
257:
986:
271:
284:, then so does the idempotent conjecture. The latter has been positively solved for an extremely large class of groups, including for example all
957:
891:
872:
822:
397:
588:
239:. It follows that the conjecture holds more generally for all residually torsion-free elementary amenable groups. Note that when
389:
275:
221:
232:
563:
116:
567:
939:
213:
451:
981:
976:
910:
366:
301:
95:
685:
423:
331:
70:
59:
779:
705:
664:
615:
545:
527:
270:
idempotent conjecture, also known as the
Kadison–Kaplansky conjecture, for elements in the
165:
67:
901:
Puschnigg, Michael (July 2002). "The
Kadison-Kaplansky conjecture for word-hyperbolic groups".
461:
953:
926:
887:
868:
818:
656:
385:
260:
235:(a class including all virtually solvable groups), since their group algebras are known to be
217:
945:
918:
851:
810:
787:
771:
739:
723:
697:
648:
537:
285:
31:
17:
791:
593:
396:) in which Kaplansky's conjecture is true. Kaplansky's conjecture is thus an example of a
393:
327:
267:
225:
914:
759:
518:
Gardam, Giles (2021-02-23). "A counterexample to the unit conjecture for group rings".
435:
305:
242:
867:. Ergebnisse der Mathematik und ihrer Grenzgebiete. Berlin ; New York: Springer.
259:
is a field of characteristic zero, then the zero-divisor conjecture is implied by the
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709:
549:
324:
236:
152:
343:
91:
39:
814:
541:
416:
409:
614:
Gardam, Giles (December 11, 2023). "Non-trivial units of complex group rings".
82:
35:
949:
930:
660:
855:
744:
727:
361:
In the mid-1970s, H. Garth Dales and J. Esterle independently proved that,
334:. The conjecture is equivalent to the statement that every algebra norm on
38:
in several branches of mathematics, including a list of ten conjectures on
922:
783:
701:
668:
636:
807:
Algebra, Arithmetic, and
Geometry, Volume II: In Honor of Yu. I. Manin
637:"Applications of a New $ K$ -Theoretic Theorem to Soluble Group Rings"
775:
652:
381:) to some Banach algebra, giving counterexamples to the conjecture.
620:
532:
688:(1991). "Kaplansky conjecture in the theory of quadratic forms".
408:
In 1953, Kaplansky proposed the conjecture that finite values of
102:
Two related conjectures are known as, respectively, Kaplansky's
865:
L²-invariants: theory and applications to geometry and K-theory
263:, which has also been established for large classes of groups.
842:
Dales, H. G. (July 1978). "Automatic
Continuity: a Survey".
805:
Vishik, Alexander (2009). "Fields of u-Invariant 2 r + 1".
388:(building on work of H. Woodin) exhibited a model of ZFC (
635:
Kropholler, P. H.; Linnell, P. A.; Moody, J. A. (1988).
589:"Mathematician Disproves 80-Year-Old Algebra Conjecture"
809:. Progress in Mathematics. Vol. 270. p. 661.
27:
Numerous conjectures by mathematician Irving
Kaplansky
464:
245:
266:
The idempotent conjecture has a generalisation, the
489:
346:. (Kaplansky himself had earlier shown that every
251:
886:. Pure and applied mathematics. New York: Wiley.
641:Proceedings of the American Mathematical Society
330:) into any other Banach algebra, is necessarily
8:
941:An Introduction to Independence for Analysts
844:Bulletin of the London Mathematical Society
680:
678:
274:. In this setting, it is known that if the
944:(1 ed.). Cambridge University Press.
315:) (continuous complex-valued functions on
743:
619:
531:
475:
463:
446:= 9 that was the first example of an odd
244:
510:
422:In 1989, the conjecture was refuted by
369:, there exist compact Hausdorff spaces
884:The algebraic structure of group rings
358:) is equivalent to the uniform norm.)
373:and discontinuous homomorphisms from
7:
938:Dales, H. G.; Woodin, W. H. (1987).
762:(2001). "Fields of u-Invariant 9".
587:Erica Klarreich (April 12, 2021).
300:This conjecture states that every
34:is notable for proposing numerous
25:
390:Zermelo–Fraenkel set theory
164:does not contain any non-trivial
115:does not contain any non-trivial
987:Unsolved problems in mathematics
155:and popularized by Kaplansky):
151:(which was originally made by
1:
564:"Interview with Giles Gardam"
426:who demonstrated fields with
342:) is equivalent to the usual
815:10.1007/978-0-8176-4747-6_22
398:statement undecidable in ZFC
90:does not contain nontrivial
42:. They are usually known as
882:Passman, Donald S. (1977).
542:10.4007/annals.2021.194.3.9
18:Kaplansky's conjecture
1003:
363:if one furthermore assumes
233:elementary amenable groups
490:{\displaystyle m=2^{k}+1}
454:demonstrated fields with
950:10.1017/cbo9780511662256
903:Inventiones Mathematicae
430:-invariants of any even
276:Farrell–Jones conjecture
272:reduced group C*-algebra
222:Hantzsche–Wendt manifold
863:LĂĽck, Wolfgang (2002).
566:. Mathematics MĂĽnster,
75:zero divisor conjecture
44:Kaplansky's conjectures
491:
253:
214:crystallographic group
923:10.1007/s002220200216
856:10.1112/blms/10.2.129
764:Annals of Mathematics
745:10.2969/jmsj/00520200
568:University of MĂĽnster
520:Annals of Mathematics
492:
450:-invariant. In 2006,
254:
104:idempotent conjecture
462:
367:continuum hypothesis
365:the validity of the
302:algebra homomorphism
243:
915:2002InMat.149..153P
438:built a field with
424:Alexander Merkurjev
94:, that is, it is a
760:Izhboldin, Oleg T.
702:10.1007/BF01100118
487:
249:
30:The mathematician
959:978-0-521-33996-4
893:978-0-471-02272-5
874:978-3-540-43566-2
824:978-0-8176-4746-9
766:. Second Series.
732:J. Math. Soc. Jpn
728:"Quadratic forms"
501:starting from 3.
286:hyperbolic groups
261:Atiyah conjecture
252:{\displaystyle K}
218:fundamental group
16:(Redirected from
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963:
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859:
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686:Merkur'ev, A. S.
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497:for any integer
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494:
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488:
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452:Alexander Vishik
350:algebra norm on
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147:and Kaplansky's
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32:Irving Kaplansky
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836:Further reading
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776:10.2307/3062141
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404:Quadratic forms
394:axiom of choice
328:Hausdorff space
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296:Banach algebras
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241:
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226:Fibonacci group
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149:unit conjecture
137:
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55:
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5:
1000:
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909:(1): 153–194.
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850:(2): 129–183.
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797:
770:(3): 529–587.
751:
738:(2): 200–207.
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647:(3): 675–684.
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579:
555:
526:(3): 967–979.
509:
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436:Oleg Izhboldin
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306:Banach algebra
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100:
99:
73:. Kaplansky's
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724:Kaplansky, I.
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216:, namely the
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153:Graham Higman
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92:zero divisors
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40:Hopf algebras
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609:
598:. Retrieved
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582:
571:. Retrieved
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415:can only be
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344:uniform norm
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101:
86:
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68:torsion-free
53:
43:
29:
982:Conjectures
977:Ring theory
696:(6): 3489.
458:-invariant
442:-invariant
434:. In 1999,
417:powers of 2
413:-invariants
237:Ore domains
224:; see also
168:, i.e., if
119:, i.e., if
117:idempotents
50:Group rings
36:conjectures
971:Categories
792:0998.11015
690:J Math Sci
621:2312.05240
600:2021-04-13
573:2021-03-10
533:2102.11818
505:References
332:continuous
278:holds for
83:group ring
931:0020-9910
710:122865942
661:0002-9939
550:232013430
384:In 1976,
304:from the
191:for some
726:(1951).
348:complete
319:, where
77:states:
911:Bibcode
784:3062141
669:2046771
325:compact
268:Kadison
220:of the
181:, then
129:, then
956:
929:
890:
871:
821:
790:
782:
708:
667:
659:
548:
96:domain
62:, and
780:JSTOR
706:S2CID
665:JSTOR
616:arXiv
546:S2CID
528:arXiv
323:is a
166:units
71:group
60:field
58:be a
954:ISBN
927:ISSN
888:ISBN
869:ISBN
819:ISBN
657:ISSN
199:and
81:The
54:Let
946:doi
919:doi
907:149
852:doi
811:doi
788:Zbl
772:doi
768:154
740:doi
698:doi
649:doi
645:104
538:doi
524:194
203:in
195:in
175:in
173:= 1
141:= 0
136:or
134:= 1
973::
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925:.
917:.
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848:10
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786:.
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827:.
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552:.
540::
530::
499:k
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482:+
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466:m
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205:G
201:g
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178:K
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