Remarks on Anomalous Symmetries of C*-Algebras

For a group $G$ and $\omega\in Z^{3}(G, \text{U}(1))$, an $\omega$-anomalous action on C*-algebra $B$ is $\text{U}(1)$-linear monoidal functor between 2-groups $\text{2-Gr}(G, \text{U}(1), \omega)\rightarrow \underline{\text{Aut}}(B)$, where the latter denotes 2-group of $*$-automorphisms $B$. The class $[\omega]\in H^{3}(G, \text{U}(1))$ called anomaly action. We show for every $n\ge 2$ finite $G$, can be realized stabilization commutative $C(M)\otimes \mathcal{K}$ some closed connected $n$-manifold $M$. also that although there are no anomalous symmetries Roe C*-algebras coarse spaces, corona $C^{*}(X)/\mathcal{K}$ bounded geometry metric space $X$ with property $A$.