Гипотезы Ирвинга Капланского и их независимость от ZFC

Numerous conjectures by mathematician Irving Kaplansky
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.

This conjecture states that every algebra homomorphism from the Banach algebra C(X) (continuous complex valued functions on X, where X is a compact Hausdorff space) into any other Banach algebra, is necessarily continuous. The conjecture is equivalent to the statement that every algebra norm on C(X) is equivalent to the usual uniform norm. (Kaplansky himself had earlier shown that every complete algebra norm on C(X) is equivalent to the 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(X) to some Banach algebra, giving counterexamples to the conjecture. In 1976, R. M. Solovay (building on work of H. Woodin) exhibited a model of ZFC (Zermelo–Fraenkel set theory + axiom of choice) in which Kaplansky's conjecture is true. Kaplansky's conjecture is thus an example of a statement undecidable in ZFC.

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 2006, Alexander Vishik demonstrated fields with u invariant for any integer k starting from 3.