Real rank and topological dimension of higher rank graph algebras
@article{Pask2015RealRA, title={Real rank and topological dimension of higher rank graph algebras}, author={David Pask and Adam Sierakowski and Aidan Sims}, journal={arXiv: Operator Algebras}, year={2015} }
We study dimension theory for the $C^*$-algebras of row-finite $k$-graphs with no sources. We establish that strong aperiodicity - the higher-rank analogue of condition (K) - for a $k$-graph is necessary and sufficient for the associated $C^*$-algebra to have topological dimension zero. We prove that a purely infinite $2$-graph algebra has real-rank zero if and only if it has topological dimension zero and satisfies a homological condition that can be characterised in terms of the adjacency…
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References
SHOWING 1-10 OF 47 REFERENCES
Graph C*-Algebras with Real Rank Zero
- Mathematics
- 2002
Given a row-finite directed graph E, a universal C*-algebra C*(E) generated by a family of partial isometries and projections subject to the relations determined by E is associated to the graph E.…
Stable rank and real rank of graph C*-algebras
- Mathematics
- 2001
For a row finite directed graph E, Kumjian, Pask, and Raeburn proved that there exists a universal C*-algebra C* (E) generated by a Cuntz-Krieger E-family. In this paper we consider two density…
Remarks on some fundamental results about higher-rank graphs and their C*-algebras
- MathematicsProceedings of the Edinburgh Mathematical Society
- 2013
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…
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…
$C^*$-algebras associated to coverings of $k$-graphs
- Mathematics, Computer ScienceDocumenta Mathematica
- 2008
This work shows how to realise a direct limit of k-graph algebras under embeddings induced from coverings as the universal algebra of a (k+1-graph) whose universal algebra encodes this embedding.
Nuclear dimension and Z-stability of non-simple C*-algebras
- Mathematics
- 2013
We investigate the interplay of the following regularity properties for non-simple C*-algebras: finite nuclear dimension, Z-stability, and algebraic regularity in the Cuntz semigroup. We show that…
Purely infinite C*-algebras of real rank zero
- Mathematics
- 2006
Abstract We show that a separable purely infinite C*-algebra is of real rank zero if and only if its primitive ideal space has a basis consisting of compact-open sets and the natural map K 0(I) → K…
Permanence properties for crossed products and fixed point algebras of finite groups
- Mathematics
- 2012
For an action of a finite group on a C*-algebra, we present some conditions under which properties of the C*-algebra pass to the crossed product or the fixed point algebra. We mostly consider the…