$C^*$-algebras associated to coverings of $k$-graphs
@article{Kumjian2006CalgebrasAT, title={\$C^*\$-algebras associated to coverings of \$k\$-graphs}, author={Alex Kumjian and David Pask and Aidan Sims}, journal={Documenta Mathematica}, year={2006} }
A covering of k-graphs (in the sense of Pask-Quigg-Raeburn) induces an embedding of universal C*-algebras. We show how to build a (k+1)-graph whose universal algebra encodes this embedding. More generally we show how to realise a direct limit of k-graph algebras under embeddings induced from coverings as the universal algebra of a (k+1)-graph. Our main focus is on computing the K-theory of the (k+1)-graph algebra from that of the component k-graph algebras.
Examples of our construction include…
18 Citations
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References
SHOWING 1-10 OF 62 REFERENCES
THE C -ALGEBRAS OF ROW-FINITE GRAPHS
- Mathematics
- 2000
NSKI Abstract. We prove versions of the fundamental theorems about Cuntz-Krieger algebras for the C -algebras of row-finite graphs: directed graphs in which each vertex emits at most finitely many…
The $C^*$-Algebras of Arbitrary Graphs
- Mathematics
- 2000
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…
HIGHER-RANK GRAPHS AND THEIR $C^*$-ALGEBRAS
- MathematicsProceedings of the Edinburgh Mathematical Society
- 2003
Abstract We consider the higher-rank graphs introduced by Kumjian and Pask as models for higher-rank Cuntz–Krieger algebras. We describe a variant of the Cuntz–Krieger relations which applies to…
SimpleC*-algebra generated by isometries
- Mathematics
- 1977
AbstractWe consider theC*-algebra
$$\mathcal{O}_n $$
generated byn≧2 isometriesS1,...,Sn on an infinite-dimensional Hilbert space, with the property thatS1S*1+...+SnS*n=1. It turns out that…
The range of K-invariants for C*-algebras of infinite graphs
- Mathematics
- 2002
It is shown that for any pair (K 0 ,K 1 ) of countable abelian groups, with K 1 free abelian, and any element Ξ ∈ K 0 there exists a purely infinite and simple, stable C * -algebra C * (E)…
On the K-theory of higher rank graph C ∗ -algebras
- Mathematics
- 2004
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…
CUNTZ-KRIEGER ALGEBRAS OF DIRECTED GRAPHS
- Mathematics
- 1998
We associate to each row-nite directed graph E a universal Cuntz-Krieger C-algebra C(E), and study how the distribution of loops in E aects the structure of C(E) .W e prove that C(E) is AF if and…
$C^*$-algebras of directed graphs and group actions
- MathematicsErgodic Theory and Dynamical Systems
- 1999
Given a free action of a group $G$ on a directed graph $E$ we show that the crossed product of $C^* (E)$, the universal $C^*$-algebra of $E$, by the induced action is strongly Morita equivalent to…
A class ofC*-algebras and topological Markov chains
- Mathematics
- 1980
In this paper we present a class of C*-algebras and point out its close relationship to topological Markov chains, whose theory is part of symbolic dynamics. The C*-algebra construction starts from a…
Actions of $\mathbb{Z}^k$ associated to higher rank graphs
- MathematicsErgodic Theory and Dynamical Systems
- 2003
An action of $\mathbb{Z}^k$ is associated to a higher rank graph $\Lambda$ satisfying a mild assumption. This generalizes the construction of a topological Markov shift arising from a non-negative…