# Twisted $C^*$-algebras associated to finitely aligned higher-rank graphs

@article{Sims2013TwistedA, title={Twisted \$C^*\$-algebras associated to finitely aligned higher-rank graphs}, author={Aidan Sims and Benjamin Whitehead and Michael F. Whittaker}, journal={Documenta Mathematica}, year={2013} }

We introduce twisted relative Cuntz-Krieger algebras associated to finitely aligned higher-rank graphs and give a comprehensive treatment of their fundamental structural properties. We establish versions of the usual uniqueness theorems and the classification of gauge-invariant ideals. We show that all twisted relative Cuntz-Krieger algebras associated to finitely aligned higher-rank graphs are nuclear and satisfy the UCT, and that for twists that lift to real-valued cocycles, the $K$-theory of…

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## References

SHOWING 1-10 OF 32 REFERENCES

### Twisted relative Cuntz-Krieger algebras associated to finitely aligned higher-rank graphs

- Mathematics
- 2013

To each finitely aligned higher-rank graph $\Lambda$ and each $\mathbb{T}$-valued 2-cocycle on $\Lambda$, we associate a family of twisted relative Cuntz-Krieger algebras. We show that each of these…

### On twisted higher-rank graph C*-algebras

- Mathematics
- 2011

We define the categorical cohomology of a k-graphand show that the first three terms in this cohomology are isomorphic to the corresponding terms in the cohomology defined in our previous paper. This…

### HOMOLOGY FOR HIGHER-RANK GRAPHS AND TWISTED

- Mathematics
- 2011

We introduce a homology theory for k-graphs and explore its fundamental properties. We establish connections with algebraic topology by showing that the homol- ogy of a k-graph coincides with the…

### Co-universal C*-algebras associated to generalised graphs

- Mathematics
- 2010

We introduce P-graphs, which are generalisations of directed graphs in which paths have a degree in a semigroup P rather than a length in ℕ. We focus on semigroups P arising as part of a…

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

- MathematicsCanadian Journal of Mathematics
- 2006

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…

### Product systems of graphs and the Toeplitz algebras of higher-rank graphs | NOVA. The University of Newcastle's Digital Repository

- Mathematics
- 2003

There has recently been much interest in the C � -algebras of directed graphs. Here we consider product systems E of directed graphs over semigroups and associated C � -algebras C � (E) and T C � (E)…

### The primitive ideals of the Cuntz–Krieger algebra of a row-finite higher-rank graph with no sources ☆

- Mathematics
- 2014

### 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…