Topological spaces associated to higher-rank graphs

  title={Topological spaces associated to higher-rank graphs},
  author={Alex Kumjian and David Pask and Aidan Sims and Michael F. Whittaker},
  journal={J. Comb. Theory, Ser. A},


Abstract We define branching systems for finitely aligned higher-rank graphs. From these, we construct concrete representations of higher-rank graph C*-algebras on Hilbert spaces. We prove a

Relative graphs and pullbacks of relative Toeplitz graph algebras

. In this note we generalize a result from a recent paper of Hajac, Reznikoff and Tobolski ([5]). In that paper they give conditions they call admissibility on a pushout diagram in the category of

Non‐surjective pullbacks of graph C*‐algebras from non‐injective pushouts of graphs

We find a substantial class of pairs of ∗ ‐homomorphisms between graph C*‐algebras of the form C∗(E)↪C∗(G)↞C∗(F) whose pullback C*‐algebra is an AF graph C*‐algebra. Our result can be interpreted as

The contravariant functoriality of graph algebras

A BSTRACT . We study pushouts in the category of directed graphs. Our first result is that the subcategory given by strongly admissible monomorphisms is closed with respect to pushouts. Next, we

Pullbacks of graph C*-algebras from admissible pushouts of graphs

We define an admissible decomposition of a graph $E$ into subgraphs $F_1$ and $F_2$, and consider the intersection graph $F_1\cap F_2$ as a subgraph of both $F_1$ and $F_2$. We prove that, if the



Topological realizations and fundamental groups of higher-rank graphs

Abstract We investigate topological realizations of higher-rank graphs. We show that the fundamental group of a higher-rank graph coincides with the fundamental group of its topological realization.

Higher Rank Graph C-Algebras

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

Remarks on some fundamental results about higher-rank graphs and their C*-algebras

Abstract Results of Fowler and Sims show that every k-graph is completely determined by its k-coloured skeleton and collection of commuting squares. Here we give an explicit description of the

Fundamental groupoids of k-graphs

k-graphs are higher-rank analogues of directed graphs which were first developed to provide combinatorial models for operator algebras of Cuntz- Krieger type. Here we develop a theory of the

On higher rank graph C ∗ -algebras

Given a row-finite k-graph Λ with no sources we investigate the K-theory of the higher rank graph C *-algebra, C * (Λ). When k = 2 we are able to give explicit formulae to calculate the K-groups of C

Skew-products of higher-rank graphs and crossed products by semigroups

We consider a free action of an Ore semigroup on a higher-rank graph, and the induced action by endomorphisms of the C∗-algebra of the graph. We show that the crossed product by this action is stably

Directed combinatorial homology and noncommutative tori (The breaking of symmetries in algebraic topology)

  • M. Grandis
  • Mathematics
    Mathematical Proceedings of the Cambridge Philosophical Society
  • 2005
This is a brief study of the homology of cubical sets, with two main purposes. First, this combinatorial structure is viewed as representing directed spaces, breaking the intrinsic symmetries of