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Benjamin W Passer, Triviality of equivariant maps in crossed products and matrix algebras, The Quarterly Journal of Mathematics, Volume 70, Issue 1, March 2019, Pages 289–303, https://doi.org/10.1093/qmath/hay044
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Abstract
We consider a twisted non-commutative join procedure for unital C*-algebras which admit actions by a compact abelian group G and its discrete abelian dual , so that we may investigate an analogue of Baum–Dąbrowski–Hajac non-commutative Borsuk–Ulam theory. Namely, under what conditions is it guaranteed that an equivariant map from a unital C*-algebra A to the twisted join of A and cannot exist? This pursuit is motivated by the twisted analogues of even spheres, which admit the same groups as even spheres and have an analogous Borsuk–Ulam theorem that is detected by , despite the fact that the objects are not themselves deformations of a sphere. We find multiple sufficient conditions for twisted Borsuk–Ulam theorems to hold, one of which is the addition of another equivariance condition on that corresponds to the choice of twist. However, we also find multiple examples of equivariant maps that exist even under fairly restrictive assumptions. Finally, we consider an extension of unital contractibility (in the sense of Dąbrowski–Hajac–Neshveyev) modulo k.
