Equivariant dimensions of graph \(C^\ast\)-algebras (Q2219463)

Let \(E = (E^0, E^1, r, s)\) be a graph with countable vertex set \(E^0\), countable edge set \(E^1\), the rival and source maps \(r,s : E^1 \to E^0\) and the adjacency matrix \(A_E = (A_{vw}), A_{vw} = \#\{ \mbox{edges with source \textit{v} and rival \textit{w}}\}\). One associates to \(E\) a g\textit{raph \(C^\ast\)-algebra} \(C^*(E)\) generated by the mutually orthogonal projections \(P_v, v\in E^0\) corresponding to vertices and the mutually orthogonal partial isometries \(S_e, e\in E^1\) corresponding to edges satisfying the conditions: for each \(e\in E^1\), \(S_e^*S_e= P_{r(e)}\), \(S_eS_e^* = P_{s(e)}\), and for each \(v\in E^0\), \(P_v = \sum_{e\in s^{-1}(v)} S_eS^*_e\). The gauge action \(\mathbb S^1 \curvearrowright C^*(E)\) is defined by \(S_e \mapsto \lambda E_e\) and \(P_v \mapsto P_v, \forall \lambda\in \mathbb S^1\). The restriction of the \textit{gauge action} to subgroup \(\mathbb Z/k \hookrightarrow \mathbb S^1\). For a subgroup \(G\) acting on a unital \(C^\ast\)-algebra \(A\), the \textit{local-triviality dimension} \(\dim_{LT}^G(A)\) is the smallest \(n\) for which there exist \(G\)-equivalent *-homomorphism \(\rho_0, \dots, \rho_n: C_0((0,1]) \otimes C(G) \to A\) such that \(\sum_{i=0}^n \rho_i(t\otimes 1) = 1\). The weak (resp., strong) local-triviality dimension \(\dim_{WLT}^G(A)\)(resp., \(\dim_{SLT}^G(A)\)) is the smallest \(n\) for which there exist \(G\)-equivalent *-homomorphism \(\rho_0, \dots, \rho_n: C_0((0,1]) \otimes C(G) \to A\) such that \(\sum_{i=0}^n \rho_i(t\otimes 1)\) is invertible (resp. there is a unital *-homomorphism \(C(E_nG) \to A\), \(E_nG := E_{n-a}G*G, E_0G:= G)\). It is clear that \(\dim_{WLT}^G(A) \leq \dim_{LT}^G(A) \leq \dim_{SLT}^G(A) \). For \(C^\ast\)-algebras of finite acyclic graphs and finite cycles, as the main result, the authors \textit{characterize the finiteness of these dimensions} (Theorems 3.4, 4.1, 4.4), and then study the gauge actions on various examples of graph \(C^\ast\)-algebras, including Cuntz algebras (\S5.1), the Toeplitz algebra (\S5.2), and the antipodal actions on quantum spheres (\S5.4).