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.

Kaplansky's conjectures on groups rings

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. No counterexamples have been found and the conjecture has been proved for wide classes of groups. Two related conjectures are:

K[G] does not contain any non-trivial units – if ab = 1 in K[G], then a = k.g for some k in K and g in G.
K[G] does not contain any non-trivial idempotents – if a² = a, then a = 1 or a = 0.

The zero-divisor conjecture implies the idempotent conjecture and is implied by the units conjecture. As of 2014 all three are open. Another related conjecture is the Kadison idempotent conjecture, also known as the Kadison–Kaplansky conjecture, which generalises Kaplansky's idempotent conjecture to the reduced group C*-algebra. The idempotent and zero-divisor conjectures have implications for geometric group theory. If the Farrell–Jones conjecture holds for K[G] then so does the idempotent conjecture.

Kaplansky's conjecture on Banach algebras

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.

References

H. G. Dales, Automatic continuity: a survey. Bull. London Math. Soc. 10 (1978), no. 2, 129–183.
W. Lück, L²-Invariants: Theory and Applications to Geometry and K-Theory. Berlin:Springer 2002 ISBN 3-540-43566-2
D.S. Passman, The Algebraic Structure of Group Rings, Pure and Applied Mathematics, Wiley-Interscience, New York, 1977. ISBN 0-471-02272-1
M. Puschnigg, The Kadison–Kaplansky conjecture for word-hyperbolic groups. Invent. Math. 149 (2002), no. 1, 153–194.
H. G. Dales and W. H. Woodin, An introduction to independence for analysts, Cambridge 1987

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