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تسجيل مستخدم جديدTriviality of Equivariant Maps in Crossed Products and Matrix Algebras
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تأليف
Benjamin Passer
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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 $k$.
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