Kaplansky's conjecture
The mathematician Irving Kaplansky is notable for proposing numerous conjectures in several branches of mathematics, including a list of ten conjectures on Hopf algebras. They are usually known as Kaplansky's conjectures.
Let K be a field, and G a torsion-free group. Kaplansky's zero divisor conjecture states that the group ring K[G] does not contain nontrivial zero divisors, that is, it is a domain. Two related conjectures are:
The zero-divisor conjecture implies the idempotent conjecture and is implied by the unit conjecture. As of 2017 all three are open, though there are positive solutions for large classes of groups for both the idempotent and zero-divisor conjectures. For example the zero-divisor conjecture is known to hold for all virtually solvable groups and more generally also for all residually torsion-free solvable groups. These solutions go through establishing first the conclusion to the Atiyah conjecture on -Betti numbers, from which the zero-divisor conjecture easily follows.
The idempotent conjecture has a generalisation, the Kadison idempotent conjecture, also known as the Kadison–Kaplansky conjecture, for elements in the reduced group C*-algebra. In this setting it is known that if the Farrell–Jones conjecture holds for K[G] then so does the idempotent conjecture. The latter has been positively solved for an extremely large class of groups, including for example all hyperbolic groups.
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