The ideal structure of the $C\sp *$-algebras of infinite graphs

By T. Bates, J. Hong, Iain Raeburn, W. Szymański. 2001

221 citations recorded by the source corpus.

We classify the gauge-invariant ideals in the C*-algebras of infinite directed graphs, and describe the quotients as graph algebras. We then use these results to identify the gauge-invariant primitive ideals in terms of the structural properties of the graph, and describe the K-theory of the C*-algebras of arbitrary infinite graphs.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/95f0ddb07c41770d85d7f6c8f0fc36882aa68684

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In Applied Mathematics

1. Aidan Sims
2. M. Tomforde
3. David W. Kribs
4. Takeshi Katsura
5. M. Jury
6. Trent Yeend
7. J. Hong
8. W. Szymański

In Mathematical Physics

1. Aidan Sims
2. M. Tomforde
3. Trent Yeend
4. R. Exel
5. G. Pino
6. Iain Raeburn
7. Toke Meier Carlsen
8. David W. Kribs

Citations in the corpus

• Kumjian-Pask algebras of higher-rank graphs (2011, methodology): “The parametrization of ideals in Cuntz-Krieger algebras by the saturated hereditary subsets comes from [11], and was extended to various classes of graph C∗-algebras in [23, 9, 8, 18].”
• C*-algebras associated to product systems of hilbert bimodules (2007, methodology): “results to the non-row-finite setting in [ 2 ].”
• Gauge-Invariant Ideals in the C*-Algebras of Finitely Aligned Higher-Rank Graphs (2004, methodology): “C ∗(E) of a directed graph E was given in [ 2 ], and conditions on E were described under which C ∗(E) is simple and purely infinite.”
• IDEAL STRUCTURE IN FREE SEMIGROUPOID ALGEBRAS FROM DIRECTED GRAPHS (2003, methodology): “On the other hand, the “coisometric” component of the decomposition is characterized by determining a representation of a Cuntz-Krieger directed graph C⁄-algebra [ 4 ], [21], [22].”
• Uniqueness theorems and ideal structure for Leavitt path algebras (2006, background): “This result is analogous to the Gauge-Invariant Uniqueness Theorem for graph C ∗ -algebras ([9, Theorem 2.1] and [ 10, Theorem 2.1 ]).”
• Stable rank of graph algebras. Type I graph algebras and their limits (2002, methodology): “[ 1 ]) and let B ⊆ X fin ∞ . (Recall from [1] that X fin ∞ is the subset of E 0 \ X consisting of all those vertices v such that |s −1 (v)| = ∞ and 0 1 ] that X fin ∞ is the subset of E 0 \ X consisting of all those vertices v such that |s −1 (v)| = ∞ and 0 1 ], for v ∈ X fin ∞ we denote by Pv,X t”
• A class of C*-algebras generalizing both graph algebras and homeomorphism C*-algebras I, fundamental results (2004, unknown): “Among others, investigation of graph algebras has rapidly progressed these days ([BPRS, BHRS , HS, DT1], among others), and many structures of graph algebras had been characterized in terms of graphs.”
• Free Semigroupoid Algebras (2003, methodology): “From the C-algebra perspective, the partial isometries {Le} are of Cuntz-Krieger-Toeplitz type in the sense that the C-algebra generated by {Le}e∈E(G) is generally the extension by the compact operators of a Cuntz-Krieger C-algebra [3, 26, 27].”