The $C^*$-Algebras of Arbitrary Graphs

By Douglas Drinen, M. Tomforde. 2000

153 citations recorded by the source corpus.

To an arbitrary directed graph we associate a row-finite directed graph whose C*-algebra contains the C*-algebra of the original graph as a full corner. This allows us to generalize results for C*-algebras of row-finite graphs to C*-algebras of arbitrary graphs: the uniqueness theorem, simplicity criteria, descriptions of the ideals and primitive ideal space, and conditions under which a graph algebra is AF and purely infinite. Our proofs require only standard Cuntz-Krieger techniques and do not rely on powerful constructs such as groupoids, Exel-Laca algebras, or Cuntz-Pimsner algebras.

Subfields: Algebra and Number Theory, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/932b0acbf6e51a978fdb0ffe52d178930020255b

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Algebra and Number Theory

1. Aidan Sims
2. Trent Yeend
3. M. Tomforde
4. Roozbeh Hazrat
5. Iain Raeburn
6. P. Ara
7. David Pask
8. P. Muhly

In Mathematical Physics

1. Aidan Sims
2. M. Tomforde
3. Trent Yeend
4. Roozbeh Hazrat
5. Iain Raeburn
6. Takeshi Katsura
7. J. Hong
8. P. Ara

Citations in the corpus

• The ideal structure of the $C\sp *$-algebras of infinite graphs (2001, methodology): “s. However, we have received many enquires about this work, and in response have decided to publish it in stages. There is some overlap between the present article and the work of Drinen and Tomforde [4], who describe the primitive ideal space of the C∗-algebras of graphs satisfying Condition (K) by”
• Ideal structure of C∗-algebras associated with C∗-correspondences (2003, result): “Similar facts were observed in many papers ([Bates et al. 2002; Drinen and Tomforde 2005; Katsura 2003b; Katsura 2006a] to name a few) for C-algebras arising from graphs or topological graphs.”
• Leavitt path algebras and direct limits (2007, background): “While some of the proofs work equally well for uncountable graphs, the main results rely on a process called desingularization, introduced in [16], which only applies to graphs in which each vertex emits at most countably many edges.”
• A class of C*-algebras generalizing both graph algebras and homeomorphism C*-algebras I, fundamental results (2004, unknown): “Both of these theorems were known for graph algebras [BPRS, RaSz, DT1 , BHRS] and for homeomorphism C ∗-algebras [T1, T2, T4] (see also [ELQ]).”
• Graded Steinberg algebras and their representations (2017, background): “To see this, let E be any graph, and let E d be its Drinen–Tomforde desingularisation [Drinen and Tomforde 2005], which is row-finite.”
• Groupoid models for the C*-algebras of topological higher-rank graphs (2006, methodology): “Note that this example includes the Toeplitz and Cuntz-Krieger algebras of arbitrary directed graphs as studied in [8, 3, 7, 17, 27, 5, 23] (among others).”
• Leavitt path algebras with coefficients in a commutative ring (2009, methodology): “r algebra correspond to admissible pairs of ideals in the coefficient algebra of the Hilbert bimodule [18, Theorem 8.6]. In the past five years, methods similar to those in [8], [7], [15], [5], [6], and [9] have been used to analyze the ideal structure of Leavitt path algebras over fields. It has been s”
• The complete classification of unital graph C∗-algebras: Geometric and strong (2016, methodology): “, so Iγ(C∗(E)) is a sublattice. We collect some standard facts about graph C∗-algebras below. Remark 2.12. Every graph C∗-algebra (of a countable graph) is separable, nuclear in the UCT class ([KP99],[DT05]). A graph C∗-algebra is unital if and only if the corresponding graph has finitely many vertic”
• A class of C*-algebras generalizing both graph algebras and homeomorphism C*-algebras IV, pure infiniteness (2008, unknown): “E which is not generated by a loop, there exists a beautiful dichotomy (see [Sz1, Theorem 18], [ DT , Remark 2.16]);”
• Flow equivalence of graph algebras (2002, background): “(ii) The Drinen-Tomforde desingularisation of a graph described in [ DT , Definition 2.2] is an example of a out-delay with a proper Drinen source-vector: If v has infinite valency then the edges with source v may be written as fei : i 2 Ng; we set ds(v) = 1 and ds(ei) = i for i 2 N. If v has finite”