Triviality of Equivariant Maps in Crossed Products and Matrix Algebras

TL;DR

This paper explores a twisted noncommutative join construction for $C^*$-algebras with abelian group actions, establishing conditions for the non-existence of equivariant maps akin to Borsuk-Ulam theorems, and examining their implications for twisted spheres.

Contribution

It introduces a twisted join procedure for $C^*$-algebras with abelian group actions and identifies conditions under which equivariant maps cannot exist, extending noncommutative Borsuk-Ulam theory.

Findings

Multiple sufficient conditions for twisted Borsuk-Ulam theorems.

Existence of equivariant maps under restrictive assumptions.

Extension of unital contractibility modulo $k$.

Abstract

We consider a "twisted" noncommutative join procedure for unital $C^*$-algebras which admit actions by a compact abelian group $G$ and its discrete abelian dual $\Gamma$, so that we may investigate an analogue of Baum-Dabrowski-Hajac noncommutative Borsuk-Ulam theory in the twisted setting. Namely, under what conditions is it guaranteed that an equivariant map $\phi$ from a unital $C^*$-algebra $A$ to the twisted join of $A$ and $C^*(\Gamma)$ cannot exist? This pursuit is motivated by the twisted analogues of even spheres, which admit the same $K_0$ groups as even spheres and have an analogous Borsuk-Ulam theorem that is detected by $K_0$, 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 $\phi$ that corresponds to the choice of twist. However, we also find multiple examples of equivariant maps $\phi$ that exist even under fairly restrictive assumptions. Finally, we consider an extension of unital contractibility (in the sense of Dabrowski-Hajac-Neshveyev) "modulo $k$."