Kaplansky's conjectures


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.

Group rings

Let be a field, and a torsion-free group. Kaplansky's zero divisor conjecture states:
Two related conjectures are known as, respectively, Kaplansky's idempotent conjecture:
  • does not contain any non-trivial idempotents, i.e., if, then or.
and Kaplansky's unit conjecture :
  • does not contain any non-trivial units, i.e., if in, then for some in and in.
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 crystallographic group, namely the fundamental group of the Hantzsche–Wendt manifold; see also Fibonacci group. A later preprint by Gardam claims that essentially the same element also gives a counter-example in characteristic 0.
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 elementary amenable groups, since their group algebras are known to be Ore domains. It follows that the conjecture holds more generally for all residually torsion-free elementary amenable groups. Note that when is a field of characteristic zero, then the zero-divisor conjecture is implied by the Atiyah conjecture, which has also been established for large classes of groups.
The idempotent conjecture has a generalisation, the Kadison idempotent conjecture, also known as the Kadison-Kaplansky conjecture, for elements in the Group algebra of a [locally compact group#The reduced group C*-algebra Cr*(G)|reduced group C*-algebra]. In this setting, it is known that if the Farrell–Jones conjecture holds for, 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.
The unit conjecture is also known to hold in many groups, but its partial solutions are much less robust than the other two. This conjecture is not known to follow from any analytic statement like the other two, and so the cases where it is 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.

Banach algebras

This conjecture states that every algebra homomorphism from the Banach algebra C into any other Banach algebra, is necessarily continuous. The conjecture is equivalent to the statement that every algebra norm on C is equivalent to the usual uniform norm.
In the mid-1970s, H. Garth Dales and J. Esterle independently proved that, if one furthermore assumes the validity of the continuum hypothesis, there exist compact Hausdorff spaces X and discontinuous homomorphisms from C to some Banach algebra, giving counterexamples to the conjecture.
In 1976, R. M. Solovay exhibited a model of ZFC in which Kaplansky's conjecture is true. Kaplansky's conjecture is thus an example of a List of statements [undecidable in ZFC|statement undecidable in ZFC].

Quadratic forms

In 1953, Kaplansky proposed the conjecture that finite values of u-invariants can only be powers of 2.
In 1989, the conjecture was refuted by Alexander Merkurjev who demonstrated fields with u-invariants of any even m. In 1999, Oleg Izhboldin built a field with u-invariant m = 9 that was the first example of an odd u-invariant. In 2006, Alexander Vishik demonstrated fields with u-invariant for any integer k starting from 3.