In this paper, the gauge-invariant primitive ideals in the graph \(C^{*}\)-algebra \(C^{*}(E)\) associated to an arbitrary directed graph \(E\) are identified in terms of the structural properties of the graph \(E.\)
A countable directed graph is a quadruple \(E=(E^0,E^1,r,s)\) consisting of a set \(E^0\) of vertices, a set \(E^1\) of edges, and range and source maps \(r,s:E^1\rightarrow E^0\). The graph \(C^{*}\)-algebra \(C^{*}(E)\) is generated by a set of mutually orthogonal projections \(\{p_v;v\in E^0\}\) and a set of partial isometries \(\{s_e;e\in E^1\}\) that satisfy the equality \(s_e^{*}s_e=p_{r(e)},\) the inequality \(s_es_e^{*}\leq p_{s(e)},\) and the equality \(p_v=\sum_{\{e;s(e)=v\}}s_es_e^{*}\) for every vertex \(v\in E^0\) which emits a finite number of edges. The sets \(\{p_v,s_e;v\in E^0,e\in E^1\} \) are called Cuntz-Krieger \(E\)-families. For a saturated hereditary subset \(H\) of \(E^0\) and a subset \(B\) of \(H_\infty ^{\text{fin}}=\{v\in E^0\backslash H;| s^{-1}(v)| =\infty \) and \(0<| s^{-1}(v)\cap r^{-1}(E^0\backslash H)| <\infty \}\), the ideal of the graph \(C^{*}\)-algebra \(C^{*}(E)\) generated by \(\{p_v;v\in H\}\cup \{p_v-\sum_{\{e;s(e)=v,r(e)\notin H\}}s_es_e^{*};v\in B\}\) is denoted by \(J_{H,B}\).
First, the authors define the notion of quotient graph and show that the graph \(C^{*}\)-algebra \(C^{*}(E/H)\) associated to the quotient graph \(E/H\), where \(H\) is a hereditary subset of \(E^0\), is isomorphic to the \(C^{*}\)-algebra \(C^{*}(E)/I_H\), where \(I_H\) is the ideal of \(C^{*}(E)\) generated by \(\{p_v;v\in H\}\). Then, using this isomorphism, they extend to the case of directed graphs, a result from [T. Bates, D. Pask, I. Raeburn and W. Szymanski, New York J. Math. 6, 307–324 (2000; Zbl 0976.46041)] about gauge-invariant ideals in the graph \(C^{*}\)-algebra \(C^{*}(E)\) associated to a row-finite graph \(E\). Thus they show that any gauge-invariant ideal in the graph \(C^{*}\)-algebra \(C^{*}(E)\) associated to a directed graph \(E\) is of the form \(J_{H,B}\), where \(H\) is a saturated hereditary subset of \(E^0\) and \(B\) is a subset of \(H_\infty ^{\text{fin}}\). Moreover, if the graph \(E\) satisfies Condition \((K)\), then any ideal in \(C^{*}(E)\;\)is of the form \(J_{H,B}.\)
If \(X\) is a non-empty subset of \(E^0\), the set of all vertices \(w\in E^0\backslash X\) such that there is no path from \(w\) to any vertex in \(X\) is denoted by \(\Omega (X).\) Bates, Pask, Raeburn and Szymanski [loc. cit.] proved that there is a bijective correspondence between the primitive ideals in the graph \(C^{*}\)-algebra associated to a row-finite graph and certain subsets of the vertex set, called maximal tails. The authors extend the notion of maximal tail to the case of directed graphs, and prove that any gauge-invariant primitive ideal of the graph \(C^{*}\)-algebra \(C^{*}(E)\) is of the form \(J_{\Omega (M),\Omega (M)_\infty ^{\text{fin}}}\), where \(M\) is a maximal tail in which all loops have exits or \(J_{\Omega (v),\Omega (v)_\infty ^{\text{fin}}\backslash \{v\}}\;\) where \(v\) is a vertex such that \(| s^{-1}(v)| =\infty \) and \(0<| s^{-1}(v)\backslash r^{-1}(\Omega (v))| <\infty .\) Moreover, if the graph \(E\) satisfies Condition \((K)\), then any primitive ideal in the graph \(C^{*}\)-algebra \(C^{*}(E)\) is of the form \(J_{\Omega (M),\Omega (M)_\infty ^{\text{fin}}}\) or \(J_{\Omega (v),\Omega (v)_\infty ^{\text{fin}}\backslash \{v\}}.\)
As an application, the authors describe the \(K\)-theory of the graph \(C^{*}\)-algebras associated to arbitrary infinite graphs.