Higher-Rank Graph C *-Algebras: An Inverse Semigroup and Groupoid Approach

By Cynthia Farthing, P. Muhly, Trent Yeend. 2004

88 citations recorded by the source corpus.

AbstractWe provide inverse semigroup and groupoid models for the Toeplitz and Cuntz-Krieger algebras of finitely aligned higher-rank graphs. Using these models, we prove a uniqueness theorem for the Cuntz-Krieger algebra.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/1e3453fa04de428b181bf0accfdfb533c2a13768

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. L. O. Clark
4. R. Exel
5. P. Ara
6. Iain Raeburn
7. M. Tomforde
8. David Pask

In Geometry and Topology

1. Aidan Sims
2. Trent Yeend
3. L. O. Clark
4. Iain Raeburn
5. P. Ara
6. Samuel B. G. Webster
7. David Pask
8. M. Rørdam

In Mathematical Physics

1. Aidan Sims
2. Trent Yeend
3. L. O. Clark
4. P. Ara
5. R. Exel
6. Iain Raeburn
7. M. Tomforde
8. David Pask

Citations in the corpus

• Inverse semigroups and combinatorial C*-algebras (2007, unknown): “*Partially supported by CNPq.”
• Simplicity of algebras associated to étale groupoids (2012, unknown): “he literature, we have taken pains to make our results self-contained and to take the most elementary path possible. There are many classes of C∗-algebras with ´etale groupoid models (see for example [8, 11, 13, 14, 16, 19, 26, 28, 31, 36]), so we expect that our results will find numerous applicatio”
• A groupoid generalisation of Leavitt path algebras (2014, background): “Versions of these theorems have been established for many generalisations of Cuntz-Krieger algebras [6, 9, 11, 12, 13, 19, 20].”
• The path space of a directed graph (2011, background): “2] and has since appeared in more general settings, for example [7,8,11].”
• Rank-two graphs whose C∗-algebras are direct limits of circle algebras (2005, methodology): “Higher-rank analogues of Cuntz-Krieger algebras and graph algebras have been introduced and studied by Robertson-Steger [29] and Kumjian-Pask [19], and are currently attracting a good deal of attention (see [14, 17, 18, 26, 33], for example).”
• Dynamical systems associated to separated graphs, graph algebras, and paradoxical decompositions☆ (2012, background): “Since then, there has been a long and fruitful interaction between the theory of combinatorially defined C*-algebras and dynamics, see [21], [40], [39], [26], [38], [12], [29], [11], [46], [55], [50], [49], [48], [28], [25].”
• Groupoid models for the C*-algebras of topological higher-rank graphs (2006, methodology): “In this section we consider an analogue of the Aperiodicity Condition used in [12, 6].”
• Remarks on some fundamental results about higher-rank graphs and their C*-algebras (2011, unknown): “Then the unit space of ∂Λ of the groupoid GΛ constructed in [8] is the closure of Λ≤∞ in WΛ; this is also homeomorphic to the spectrum of span{sλsλ : λ ∈ Λ} [26].”