The \(C^*\)-algebras of row-finite graphs
Let \(E\) be a row-finite directed graph, i.e., each vertex emits at most finitely many edges. Denote the vertices by \(v\)'s, the edges by \(e\)'s and let \(s(e)\) and \(r(e)\) be the source and range, respectively, of the edge \(e\). A Cuntz-Krieger E-family in a C\(^{*}\)-algebra consists of partial isometries \(s_{e}\) and mutually orthogonal projections \(p_{v}\) are such that \(s^{*}_{e}s_{e} = p_{r(e)}\), and for a \(v\) which is not a sink, \( p_{v} = \sum_{e:s(e)=v}{s_{e}s^{*}_{e}}\). Let \(C^{*}(E)\) denote the universal C\(^{*}\)-algebra generated by these relations [cf. \textit{A. Kumjian, D. Pask} and \textit{I. Raeburn}, Pac. J. Math. 184, No. 1, 161-174 (1998; Zbl 0717.46056)]. If \(\{ s_{e},p_{v}\}\) is an E-family generating \(C^{*}(E)\) then, for \(z \in {\mathbf{T}}\), also \(\{ zs_{e},p_{v} \}\) is an E-family generating \(C^{*}(E)\). The universal property of \(C^{*}(E)\) gives an endomomorphism \(\gamma_{z}\) of \(C^{*}(E)\) such that \(\gamma_{z}\{ s_{e},p_{v}\} = \{ zs_{e}, p_{v} \}\), and \(\gamma\) is a strongly continuous action of the ``gauge group'' \({\mathbf{T}}\) on \(C^{*}(E)\). The authors prove the following gauge-invariant uniqueness theorem: suppose \(\{ S_{e},P_{v}\}\) is a concrete E-family, the \(P_{v}\) assumed all non-zero, and that \(\pi\) is the representation \(\{ s_{e},p_{v}\} \mapsto \{ S_{e},P_{v}\}\); if there is a strongly continuous action \(\beta\) of \({\mathbf{T}}\) on \(C^{*}(S_{e},P_{v})\) such that \(\pi.\gamma_{z} =\beta_{z}.\pi\) for \(z \in {\mathbf{T}}\) then \(\pi\) is faithful. This gauge-invariance eliminates the necessity, for existence of the Cuntz algebras, of the condition I set in [\textit{J. Cuntz} and \textit{W. Krieger}, Invent. Math. 56, 251-268 (1980; Zbl 0434.46045)]. The authors effectively eliminate any sinks which occur by adding an infinite tail to each sink. A finite graph with a sink is converted to an infinite graph without the sink. The graph \(E\) extends to a larger graph \(F\) and \(C^{*}(E)\) can be embedded in \(C^{*}(F)\). Most of the theorems proved are already essentially known, but the methods of proof here are more basic. A transitive ``pre-order'' is imposed on the vertex set by setting \( v \geq w\) if there is a possibly infinite path with source \(v\) and range \(w\). A set \(H\) of vertices is said to be hereditary and saturated if \(v \in H\) implies \(w \in H\) whenever \(v \geq w\) and if every vertex which feeds into \(H\) and only into \(H\) is again in \(H\). Letting \(I_{H}\) denote the ideal of \(C^{*}(E)\) generated by \(\{ p_{v}: v \in H\}\), as in \textit{A. Kumjian, D. Pask, I. Raeburn} and \textit{J. Renault} [J. Funct. Anal. 144, No. 2, 505-541 (1997; Zbl 0929.46055)] the mapping \(H \to I_{H}\) is an isomorphism of saturated hereditary sets of vertices onto the lattice of closed guage-invariant ideals. The authors then characterise the primitive ideals when \(E\) is such that every vertex lies either on no loops or at least two loops [cf. \textit{A. an Huef} and \textit{I. Raeburn}, Ergod. Theory Dyn. Syst. 17, No. 3, 611-624 (1997; Zbl 0886.46061) and condition II of \textit{J. Cuntz}, Invent. Math 63, 25-40 (1981; Zbl 0461.46047)]. ``Maximal tails'' can be constructed by taking the vertices on any infinite path and including the vertices which connect to the path. A topology on the set of maximal tails is so constructed that it is homeomorphic to the primitive ideal space.
scientific article; zbMATH DE number 1531353
scientific article