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.

Kaplansky's conjecture on group rings

Kaplansky's conjecture on group rings states that the complex group ring CG of a torsion-free group G has no nontrivial idempotent elements. It is related to the Kadison idempotent conjecture, also known as the Kadison–Kaplansky conjecture.

Kaplansky's conjecture on Banach algebras

This conjecture states that every algebra homomorphism from the Banach algebra C(X) (where X is a compact Hausdorff topological 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 proved (building on work of H. Woodin) that there is at least one model of ZFC (Zermelo–Fraenkel set theory + axiom of choice) in which Kaplansky's conjecture is true. Necessarily, in such a model the continuum hypothesis is false. Combined with the results of Dales and Esterle, this shows that the conjecture is independent of the axioms of ZFC.

See also

• List of statements 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.