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Noncommutative Borsuk-Ulam-type conjectures revisited

Piotr M. Hajac
Friday, February 3, 2017 - 16:00

Let H be the C*-algebra of a non-trivial compact quantum group acting freely on a unital C*-algebra A. Baum, Dabrowski and Hajac conjectured that there does not exist an equivariant *-homomorphism from A to the equivariant noncommutative join C*-algebra A*H. When A is the C*-algebra of functions on a sphere, and H is the C*-algebra of functions on Z/2Z acting antipodally on the sphere, then the conjecture becomes the celebrated Borsuk-Ulam theorem. Recently, Passer proved the conjecture when H is the commutative C*-algebra of functions on a non-trivial compact group with a torsion element. The first goal of this talk is to show how to extend this result to the quantum setting. Next, with a stronger assumption that our compact quantum group is a q-deformation of a compact connected semisimple Lie group, we prove a stronger result that there exists a finite-dimensional representation of the compact quantum group such that, for any C*-algebra A admitting a character, the finitely generated projective module associated with A*H via this representation is not stably free. (Based on joint work with L. Dabrowski and S. Neshveyev.)

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