Gauge-Invariant Ideals in the C*-Algebras of Finitely Aligned Higher-Rank Graphs

By Aidan Sims. Canadian Journal of Mathematics - Journal Canadien de Mathematiques, 2004

159 citations recorded by the source corpus.

Abstract We produce a complete description of the lattice of gauge-invariant ideals in ${{C}^{*}}(\Lambda )$ for a finitely aligned $k$ -graph $\Lambda $ . We provide a condition on $\Lambda $ under which every ideal is gauge-invariant. We give conditions on $\Lambda $ under which ${{C}^{*}}(\Lambda )$ satisfies the hypotheses of the Kirchberg–Phillips classification theorem.

Subfields: Algebra and Number Theory, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/ea941f36d98e726779bd9db10d4fcbf3b8c6d7a5

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. David Pask
4. Iain Raeburn
5. P. Ara
6. R. Exel
7. K. Davidson
8. G. Pino

In Mathematical Physics

1. Aidan Sims
2. Trent Yeend
3. P. Ara
4. Iain Raeburn
5. David Pask
6. R. Exel
7. K. Davidson
8. G. Pino

Citations in the corpus

• The Noncommutative Geometry of k-graph C*-Algebras (2005, unknown): “Recall from [ Si ] that a loop with an entrance is a path λ ∈ � with r(λ) = s(λ) such that d(λ) ≥ ei for some 1 ≤ i ≤ k, together with an e ∈ � ei with r(e) = r(λ) but λ(0, ei) 6 e.”
• 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).”
• Groupoid models for the C*-algebras of topological higher-rank graphs (2006, background): “13] that C(GΛ) ∼= T C(Λ) and C(GΛ) ∼= C(Λ), where T C(Λ) and C(Λ) are defined and studied in [24] and [26], respectively.”
• Inverse semigroups and combinatorial C*-algebras (2007, unknown): “Mathematical subject classification: Primary: 46L05; Secondary: 18B40.”
• C*-algebras associated to product systems of hilbert bimodules (2007, methodology): “We will achieve this indirectly by combining Theorem 4.1 with the uniqueness theorems of [7, 18, 30 ].”
• On the K-theory of higher rank graph C ∗ -algebras (2004, background): “For further examples of k-graphs see, for example, [13, 20, 21, 29].”
• Periodicity in Rank 2 Graph Algebras (2007, background): “Our 2-graphs have a single vertex and are particularly amenable to analysis while exhibiting a wealth of interesting phenomena.”
• 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].”
• Kumjian-Pask algebras of higher-rank graphs (2011, background): “There are satisfactory C∗-algebraic uniqueness theorems for larger families of k-graphs, but they can be very complicated to work with (look at the proof of Cuntz-Krieger uniqueness in [31], for example).”