269:
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
193:
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
206:. 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).
270:
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.
209:
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
473:
235:
964:
249:
262:, then so does the idempotent conjecture. The latter has been positively solved for an extremely large class of groups, including for example all
935:
869:
850:
800:
375:
566:
217:. It follows that the conjecture holds more generally for all residually torsion-free elementary amenable groups. Note that when
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253:
199:
210:
541:
98:
545:
917:
195:
429:
959:
954:
888:
344:
279:
77:
663:
401:
309:
757:
683:
642:
593:
523:
505:
248:
idempotent conjecture, also known as the
Kadison–Kaplansky conjecture, for elements in the
147:
52:
879:
Puschnigg, Michael (July 2002). "The
Kadison-Kaplansky conjecture for word-hyperbolic groups".
439:
931:
904:
865:
846:
796:
634:
363:
238:
213:(a class including all virtually solvable groups), since their group algebras are known to be
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765:
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717:
701:
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626:
515:
263:
20:
769:
571:
374:) in which Kaplansky's conjecture is true. Kaplansky's conjecture is thus an example of a
371:
305:
245:
203:
892:
737:
496:
Gardam, Giles (2021-02-23). "A counterexample to the unit conjecture for group rings".
413:
283:
220:
845:. Ergebnisse der Mathematik und ihrer Grenzgebiete. Berlin ; New York: Springer.
237:
is a field of characteristic zero, then the zero-divisor conjecture is implied by the
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687:
527:
302:
214:
134:
321:
73:
28:
792:
519:
394:
387:
592:
Gardam, Giles (December 11, 2023). "Non-trivial units of complex group rings".
64:
24:
927:
908:
638:
833:
722:
705:
339:
In the mid-1970s, H. Garth Dales and J. Esterle independently proved that,
312:. The conjecture is equivalent to the statement that every algebra norm on
27:
in several branches of mathematics, including a list of ten conjectures on
900:
761:
679:
646:
614:
785:
Algebra, Arithmetic, and
Geometry, Volume II: In Honor of Yu. I. Manin
615:"Applications of a New $ K$ -Theoretic Theorem to Soluble Group Rings"
753:
630:
359:) to some Banach algebra, giving counterexamples to the conjecture.
598:
510:
666:(1991). "Kaplansky conjecture in the theory of quadratic forms".
386:
In 1953, Kaplansky proposed the conjecture that finite values of
84:
Two related conjectures are known as, respectively, Kaplansky's
843:
L²-invariants: theory and applications to geometry and K-theory
241:, which has also been established for large classes of groups.
820:
Dales, H. G. (July 1978). "Automatic
Continuity: a Survey".
783:
Vishik, Alexander (2009). "Fields of u-Invariant 2 r + 1".
366:(building on work of H. Woodin) exhibited a model of ZFC (
613:
Kropholler, P. H.; Linnell, P. A.; Moody, J. A. (1988).
567:"Mathematician Disproves 80-Year-Old Algebra Conjecture"
787:. Progress in Mathematics. Vol. 270. p. 661.
16:
Numerous conjectures by mathematician Irving
Kaplansky
442:
223:
244:
The idempotent conjecture has a generalisation, the
467:
324:. (Kaplansky himself had earlier shown that every
229:
864:. Pure and applied mathematics. New York: Wiley.
619:Proceedings of the American Mathematical Society
308:) into any other Banach algebra, is necessarily
8:
919:An Introduction to Independence for Analysts
822:Bulletin of the London Mathematical Society
658:
656:
252:. In this setting, it is known that if the
922:(1 ed.). Cambridge University Press.
293:) (continuous complex-valued functions on
721:
597:
509:
453:
441:
424:= 9 that was the first example of an odd
222:
488:
400:In 1989, the conjecture was refuted by
347:, there exist compact Hausdorff spaces
862:The algebraic structure of group rings
336:) is equivalent to the uniform norm.)
198:, namely the fundamental group of the
351:and discontinuous homomorphisms from
7:
916:Dales, H. G.; Woodin, W. H. (1987).
740:(2001). "Fields of u-Invariant 9".
565:Erica Klarreich (April 12, 2021).
278:This conjecture states that every
23:is notable for proposing numerous
14:
368:Zermelo–Fraenkel set theory
146:does not contain any non-trivial
97:does not contain any non-trivial
965:Unsolved problems in mathematics
137:and popularized by Kaplansky):
133:(which was originally made by
1:
542:"Interview with Giles Gardam"
404:who demonstrated fields with
320:) is equivalent to the usual
793:10.1007/978-0-8176-4747-6_22
376:statement undecidable in ZFC
72:does not contain nontrivial
31:. They are usually known as
860:Passman, Donald S. (1977).
520:10.4007/annals.2021.194.3.9
981:
341:if one furthermore assumes
211:elementary amenable groups
468:{\displaystyle m=2^{k}+1}
432:demonstrated fields with
928:10.1017/cbo9780511662256
881:Inventiones Mathematicae
408:-invariants of any even
254:Farrell–Jones conjecture
250:reduced group C*-algebra
200:Hantzsche–Wendt manifold
841:LĂĽck, Wolfgang (2002).
544:. Mathematics MĂĽnster,
57:zero divisor conjecture
33:Kaplansky's conjectures
469:
231:
196:crystallographic group
901:10.1007/s002220200216
834:10.1112/blms/10.2.129
742:Annals of Mathematics
723:10.2969/jmsj/00520200
546:University of MĂĽnster
498:Annals of Mathematics
470:
428:-invariant. In 2006,
232:
86:idempotent conjecture
440:
345:continuum hypothesis
343:the validity of the
280:algebra homomorphism
221:
893:2002InMat.149..153P
416:built a field with
402:Alexander Merkurjev
76:, that is, it is a
738:Izhboldin, Oleg T.
680:10.1007/BF01100118
465:
227:
53:torsion-free group
19:The mathematician
937:978-0-521-33996-4
871:978-0-471-02272-5
852:978-3-540-43566-2
802:978-0-8176-4746-9
744:. Second Series.
710:J. Math. Soc. Jpn
706:"Quadratic forms"
479:starting from 3.
264:hyperbolic groups
239:Atiyah conjecture
230:{\displaystyle K}
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328:algebra norm on
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129:and Kaplansky's
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21:Irving Kaplansky
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382:Quadratic forms
372:axiom of choice
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274:Banach algebras
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204:Fibonacci group
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131:unit conjecture
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887:(1): 153–194.
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748:(3): 529–587.
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504:(3): 967–979.
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414:Oleg Izhboldin
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284:Banach algebra
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55:. Kaplansky's
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576:. Retrieved
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393:can only be
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960:Conjectures
955:Ring theory
674:(6): 3489.
436:-invariant
420:-invariant
412:. In 1999,
395:powers of 2
391:-invariants
215:Ore domains
202:; see also
150:, i.e., if
101:, i.e., if
99:idempotents
39:Group rings
25:conjectures
949:Categories
770:0998.11015
668:J Math Sci
599:2312.05240
578:2021-04-13
551:2021-03-10
511:2102.11818
483:References
310:continuous
256:holds for
65:group ring
909:0020-9910
688:122865942
639:0002-9939
528:232013430
362:In 1976,
282:from the
173:for some
704:(1951).
326:complete
297:, where
59:states:
889:Bibcode
762:3062141
647:2046771
303:compact
246:Kadison
163:, then
111:, then
934:
907:
868:
849:
799:
768:
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686:
645:
637:
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78:domain
758:JSTOR
684:S2CID
643:JSTOR
594:arXiv
524:S2CID
506:arXiv
301:is a
148:units
932:ISBN
905:ISSN
866:ISBN
847:ISBN
797:ISBN
635:ISSN
181:and
63:The
43:Let
924:doi
897:doi
885:149
830:doi
789:doi
766:Zbl
750:doi
746:154
718:doi
676:doi
627:doi
623:104
516:doi
502:194
185:in
177:in
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