double NCG seminar

[Emanuele Tuillier Illingworth]

Mariusz Tobolski (IMPAN/Warsaw University)

Title: A Noncommutative Borsuk-Ulam theorem for free actions of compact 
groups with torsion

Abstract:
In recent years, a collection of results that generalize the celebrated 
Borsuk-Ulam theorem to the C*-algebraic setting have appeared. Motivated 
by the conjecture of Baum, Dąbrowski and Hajac, Ben Passer proved that 
for a compact Hausdorff group G with torsion there is no equivariant map 
from a unital C*-algebra A to its equivariant join with the unital 
C*-algebra C(G) of complex valued continuous functions on G. First part 
of the talk will be devoted to the classical Borsuk-Ulam theorem and its 
reformulation in terms of joins that can be generalized to 
noncommutative geometry. In the second part, I will present the 
mentioned result of Passer and discuss its proof.

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Piotr M. Hajac (IMPAN)

NONCOMMUTATIVE BORSUK-ULAM-TYPE CONJECTURES REVISITED

Abstract:
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.)

Friday 3 February 2017, 16:00
SISSA, room 136