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

@article{Sims2003GaugeInvariantII, title={Gauge-Invariant Ideals in the C*-Algebras of Finitely Aligned Higher-Rank Graphs}, author={Aidan Sims}, journal={Canadian Journal of Mathematics}, year={2003}, volume={58}, pages={1268 - 1290} }

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 which every ideal is gauge-invariant. We give conditions on $\Lambda $ under which ${{C}^{*}}(\Lambda )$ satisfies the hypotheses of the Kirchberg–Phillips classification theorem.

## 153 Citations

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## References

SHOWING 1-10 OF 22 REFERENCES

### The ideal structure of the $C\sp *$-algebras of infinite graphs

- Mathematics
- 2001

We classify the gauge-invariant ideals in the C*-algebras of infinite directed graphs, and describe the quotients as graph algebras. We then use these results to identify the gauge-invariant…

### 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…

### The primitive ideal space of the $C^{*}$-algebras of infinite graphs

- Mathematics
- 2002

For any countable directed graph E we describe the primitive ideal space of the corresponding generalized Cuntz-Krieger algebra C*(E).

### 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)…

### The Toeplitz algebra of a Hilbert bimodule

- Mathematics
- 1998

Suppose a C*-algebra A acts by adjointable operators on a Hilbert A-module X. Pimsner constructed a C*-algebra O_X which includes, for particular choices of X, crossed products of A by Z, the Cuntz…

### On higher rank graph C ∗ -algebras

- Mathematics
- 2000

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…

### 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…

### CROSSED PRODUCTS BY SEMIGROUPS OF ENDOMORPHISMS AND THE TOEPLITZ ALGEBRAS OF ORDERED GROUPS

- Mathematics
- 1994

Let r+ be the positive cone in a totally ordered abelian group F. We construct crossed products by actions of r1" as endomorphisms of C- algebras, and give criteria which ensure a given…

### A FUNCTORIAL APPROACH TO THE C*-ALGEBRAS OF A GRAPH

- Mathematics
- 2001

A functor from the category of directed trees with inclusions to the category of commutative C*-algebras with injective *-homomorphisms is constructed. This is used to define a functor from the…