Kaplansky's conjecture

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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.

[edit] 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 idempotents. It is related to the Kadison idempotent conjecture, also known as the Kadison–Kaplansky conjecture.

[edit] Kaplansky's conjecture on Banach algebras

This conjecture states that there exists a discontinuous homomorphism from the Banach algebra C(X) (where X is an infinite compact Hausdorff topological space) into any other Banach algebra.

In 1976, Garth Dales and Robert M. Solovay proved that this conjecture is independent of the axioms of Zermelo–Fraenkel set theory and the axiom of choice, but is implied by the continuum hypothesis.

[edit] See also

• List of statements undecidable in ZFC

[edit] References

• Lück, W., 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

• Puschnigg, Michael, The Kadison-Kaplansky conjecture for word-hyperbolic groups. Invent. Math. 149 (2002), no. 1, 153--194.