Higher rank graph C*-algebras

Abstract

Building on recent work of Robertson and Steger, we associate a C*-algebra to a combinatorial object which may be thought of as a higher rank graph. This C*-algebra is shown to be isomorphic to that of the associated path groupoid. Sufficient conditions on the higher rank graph are found for the associated C*-algebra to be simple, purely infinite and AF. Results concerning the structure of crossed products by certain natural actions of discrete groups are obtained; a technique for constructing rank 2 graphs from ``commuting'' rank 1 graphs is given.

14 pages, see also http://nyjm.albany.edu:8000/j/2000/6-1nf.htm

References in corpus (2)

• Skew products and crossed products by coactions
• A Classification Theorem for Nuclear Purely Infinite Simple C*-Algebras

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• The primitive ideals of the Cuntz-Krieger algebra of a row-finite higher-rank graph with no sources
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• Representations of Higher Rank Graph Algebras
• Crossed products of k-graph C*-algebras by Z^l
• Type III von Neumann Algebras associated with ${Ø}_{\theta }$
• Purely infinite $C^{\ast }$-algebras associated to étale groupoids
• von Neuman algebras of strongly connected higher-rank graphs
• Classifying Higher Rank Analytic Toeplitz Algebras
• Fundamental groupoids of k-graphs
• Topological higher-rank graphs and the $C^{\ast }$-algebras of topological 1-graphs
• Removing sources from higher-rank graphs
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• Poisson transform for higher-rank graph algebras and its applications
• A Going-Down principle for ample groupoids and the Baum-Connes conjecture
• Cartan subalgebras for non-principal twisted groupoid $C^{\ast }$-algebras
• Reconstruction of twisted Steinberg algebras
• $K$-theory and homotopies of 2-cocycles on higher-rank graphs
• Iterating the Pimsner construction
• Simplicity of finitely-aligned k-graph C*-algebras
• Unbounded quasitraces, stable finiteness and pure infiniteness
• Operator algebras associated with unitary commutation relations
• Semiprojectivity for Kirchberg algebras
• Examples of *-commuting maps
• Aperiodicity and the primitive ideal space of a row-finite $k$-graph $C^{\ast }$-algebra
• Subproduct systems over N$\times$N
• Topological spaces associated to higher-rank graphs
• Finitely Correlated Representations of Product Systems of $C^{\ast }$-Correspondences over ${\mathbb{N}}^k$
• A unified framework for graph algebras and quantum causal histories
• Product-system models for twisted $C^{\ast }$-algebras of topological higher-rank graphs
• Simplicity of twisted C*-algebras of Deaconu--Renault groupoids
• The ideal structures of self-similar $k$-graph C*-algebras
• KMS states on the Toeplitz algebras of higher-rank graphs
• Topological realizations and fundamental groups of higher-rank graphs
• Cuntz's $ax+b$-semigroup C*-algebra over $\mathbb{N}$ and product system C*-algebras
• C*-algebras of commuting endomorphisms
• Cuntz-Krieger type uniqueness theorem for topological higher-rank graph C*-algebras
• The ideal intersection property for essential groupoid C*-algebras
• Nonself-adjoint 2-graph algebras
• Cohomology for small categories: $k$-graphs and groupoids
• Purely atomic representations of higher-rank graph C*-algebras
• The primitive ideals of some étale groupoid C*-algebras
• Interpolation in Semigroupoid Algebras
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• $\mathrm{K}$-theory and homotopies of twists on ample groupoids
• Groupoid algebras as covariance algebras
• Separable representations of higher-rank graphs
• Crossed products and twisted $k$-graph algebras
• Homotopy of product systems and K-theory of Cuntz-Nica-Pimsner algebras
• KMS states on the C*-algebras of Fell bundles over groupoids
• Amenability for Fell bundles over groupoids
• $K$-theory for real $k$-graph $C^{\ast }$-algebras
• Realising the Toeplitz algebra of a higher-rank graph as a Cuntz-Krieger algebra
• Orbit equivalence of higher-rank graphs
• Multiply generated dynamical systems and the duality of higher rank graph algebras
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• Dirichlet forms and ultrametric Cantor sets associated to higher-rank graphs
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• Remarks On the K-theory of C*-Algebras of Products of Odometers
• The K-theory of the C*-algebras of 2-rank graphs associated to complete bipartite graphs
• Cocycles on Certain Groupoids Associated to ${\mathbb{N}}^k$-Actions
• Computing the fundamental group of a higher-rank graph
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• The simplicity of the C*-algebras associated to arbitrary labeled spaces
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• The bicategory of groupoid correspondences
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• The Jacobson topology of the primitive ideal space of self-similar k-graph C*-algebras
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• Dense subalgebras of purely infinite simple groupoid C*-algebras