Noncommutative Borsuk–Ulam-type conjectures revisited

  • Ludwik Dąbrowski

    SISSA, Trieste, Italy
  • Piotr M. Hajac

    IMPAN, Warsaw, Poland
  • Sergey Neshveyev

    University of Oslo, Norway
Noncommutative Borsuk–Ulam-type conjectures revisited cover

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Let be the C*-algebra of a non-trivial compact quantum group acting freely on a unital C*-algebra . It was recently conjectured that there does not exist an equivariant *-homomorphism from (type-I case) or (type-II case) to the equivariant noncommutative join C*-algebra . When is the C*-algebra of functions on a sphere, and is the C*-algebra of functions on acting antipodally on the sphere, then the conjecture of type I becomes the celebrated Borsuk–Ulam theorem. Taking advantage of recent work of Passer, we prove the conjecture of type I for compact quantum groups admitting a non-trivial torsion character. Next, we prove that, if the compact quantum group admits a representation whose -class is non-trivial and admits a character, then a stronger version of the type-II conjecture holds: the finitely generated projective module associated with via this representation is not stably free. In particular, we apply this result to the -deformations of compact connected semisimple Lie groups and to the reduced group C*-algebras of free groups on generators.

Cite this article

Ludwik Dąbrowski, Piotr M. Hajac, Sergey Neshveyev, Noncommutative Borsuk–Ulam-type conjectures revisited. J. Noncommut. Geom. 14 (2020), no. 4, pp. 1305–1324

DOI 10.4171/JNCG/352