\(C^*\)-algebras associated with branched coverings (Q2701584)

Let \(X\) be a l.c. second countable Hausdorff space and \(\sigma:X\to X\) its branched covering, with a singular set \(S\), whence \(\sigma\) is an open surjective map, \(S\) is closed, \(X\setminus S\) is dense and possibly disconnected and \(T=\sigma|_{X\setminus S}\) is a local homeomorphism. Assume that \((X,T)\) is essentially free, i.e., for all \(m,n\in{\mathbb N}\) there is no nonempty open set on which \(T^m\) and \(T^n\) agree. The associated \(C^*\)-algebra \(C^*(X,\sigma)\) is defined as the \(C^*\)-algebra \(C^*(\Gamma(X,T))\) of the l.c. \(r\)-discrete groupoid \(\Gamma(X,T)=\{x,m-n,y\mid m,n\in{\mathbb N}\), \(x\in \text{dom} (T^m)\), \(y\in \text{dom} (T^n)\), \(T^mx=T^ny\}\) associated to \(T\) by \textit{J. Renault} [``A groupoid approach to \(C^*\)-algebras'', Lect. Notes Math. 793 (1980; Zbl 0433.46049)]. Theorem 3.2 establishes isomorphism of \(C^*(\Gamma(X,T))\) and the augmented Cuntz-Pimsner algebra \(\widetilde {\mathcal O}_E\) of the specified \(C^*\)-correspondence \((A,E)\) [see \textit{M. V. Pimsner} in: Fields Inst. Commun. 12, 189-212 (1997; Zbl 0871.46028)]; where \(A=C_0(X)\), and \(E=\overline{C_c(X\setminus S)}\) is given the structure of \(A\)-Hilbert bimodule. Namely, \(C^*(\Gamma(X,T))\) and \(\widetilde {\mathcal O}_E\) are shown to be isomorphic to the \(C^*\)-algebras associated to isomorphic Fell bundles over \(\mathbb Z\) [c.f. \textit{B. Abadie, S. Eilers} and \textit{R. Exel}, Trans. Am. Math. Soc. 350, No. 8, 3043-3054 (1998; Zbl 0899.46053)].NEWLINENEWLINENEWLINEThe conditions on \(T\) implying that \(C^*(\Gamma(X,\sigma))\) is simple or purely infinite follow from the results by \textit{J. Renault} [J. Op. Theory 18, 67-97 (1987; Zbl 0659.46058)]. The six-term exact sequence obtained for \(\widetilde {\mathcal O}_E\) by Pimsner (Theorem 4.9., loc. cit.) is used to compute the \(K\)-theory of \(C^*(\Gamma(X,\sigma))\). Examples include the folding of \([0,1]\) and rational maps on the Riemann sphere.