A twisted action of a group \(G\) on a unital \(C^*\)-algebra \(A\) is given by automorphisms \(\alpha_g\) and unitaries \(u_{g,h}\) for \(g,h\in G\) such that \(u_{g,h} \alpha_g (\alpha_h(a)) u_{g,h}^* = \alpha_{g h}(a)\) for all \(g,h\in G\), \(a\in A\), and \(u\) satisfies the cocycle condition \(u_{gh,k}^* u_{g,hk} \alpha_g(u_{hk}) u_{g,h}^* = 1\) for all \(g,h,k\in G\); the latter product of unitaries is always central in \(A\). More generally, one may ask the product above to be equal to a given \(3\)-cocycle \(\omega(g,h,k)\) on \(G\) with values in \(U(1)\) or the centre of \(A\). This is called an anomalous action of \(G\) on \(A\). Anomalous actions appear naturally in the study of \(C^*\)-tensor categories and their actions on \(C^*\)-algebras. They may also be rewritten as actions of \(2\)-groups or crossed modules.
This article studies the existence of such anomalous actions on some classes of \(C^*\)-algebras, namely, commutative \(C^*\)-algebras and their stabilisations, Roe \(C^*\)-algebras, and the quotients of Roe \(C^*\)-algebras by the ideal of compact operators. On the one hand, some \(C^*\)-algebras admit no anomalous actions at all. Namely, this is true for the Roe \(C^*\)-algebra of any discrete metric space of bounded geometry; for \(C(X)\) for a connected, locally path connected compact space \(X\) provided \(H^1(X,\mathbb Z)\) vanishes; and it remains true for the \(C^*\)-stabilisation \(C(X) \otimes \mathbb K\) if, in addition, to the assumptions above, all complex line bundles over \(X\) are trivial. On the other hand, for any \(n\ge 2\), group \(G\) and 3-cocycle \(\omega\), there is a connected closed \(n\)-manifold \(M\) and an \(\omega\)-anomalous action on \(C(M)\otimes \mathbb K\). In addition, there is a discrete metric space \(X\) with bounded geometry and property A and an \(\omega\)-anomalous action on the quotient of the Roe \(C^*\)-algebra of \(X\) by the ideal of compact operators.
Another interesting result in the article says that if a \(C^*\)-algebra admits an anomalous action with a cocycle that is not a coboundary, then the induced action cannot fix any pure state. For instance, if the \(C^*\)-algebra admits a unique trace, then any automorphism must fix it, and so the \(C^*\)-algebra cannot admit any anomalous action.
A key step to prove existence of anomalous actions is a construction of anomalous actions on certain twisted crossed products. This is reminiscent of a construction of crossed module actions on untwisted crossed products in Example 8 in [A. Buss et al., Math. Ann. 352, No. 1, 73–97 (2012; Zbl 1242.46075)].