Triviality of Equivariant Maps in Crossed Products and Matrix Algebras

We show that for certain unital C*-algebras A with free actions of a finite cyclic group G, equivariant self-maps of A are path connected to finite-dimensional representations, including one-dimensional representations, within the homomorphisms from A to certain crossed products of A with G. Therefore, our previously proposed extensions of Baum-Dabrowski-Hajac noncommutative Borsuk-Ulam theory, in which certain crossed products replace tensor products in their definitions, do not apply universally. We examine the consequences of this claim for extending contractibility (in the sense of Dabrowski-Hajac-Neshveyev) "modulo torsion" by embedding A into matrix algebras over A.