# Hereditary C*-subalgebras of graph C*-algebras

@article{Arklint2020HereditaryCO, title={Hereditary C*-subalgebras of graph C*-algebras}, author={Sara E. Arklint and James Gabe and Efren Ruiz}, journal={arXiv: Operator Algebras}, year={2020}, pages={99} }

We show that a $C^*$-algebra $\mathfrak{A}$ which is stably isomorphic to a unital graph $C^*$-algebra, is isomorphic to a graph $C^*$-algebra if and only if it admits an approximate unit of projections. As a consequence, a hereditary $C^*$-subalgebra of a unital real rank zero graph $C^*$-algebra is isomorphic to a graph $C^*$-algebra. Furthermore, if a $C^*$-algebra $\mathfrak{A}$ admits an approximate unit of projections, then its minimal unitization is isomorphic to a graph $C^*$-algebra if…

## 2 Citations

Corners of Leavitt path algebras of finite graphs are Leavitt path algebras

- MathematicsJournal of Algebra
- 2020

Exchange rings and real rank zero C*-algebras associated with finitely separated graphs

- Mathematics
- 2017

We introduce a generalisation of Condition (K) to finitely separated graphs and show that it is equivalent to essential freeness of the associated partial action as well as the exchange property of…

## References

SHOWING 1-10 OF 19 REFERENCES

The complete classification of unital graph C∗-algebras: Geometric and strong

- MathematicsDuke Mathematical Journal
- 2021

We provide a complete classification of the class of unital graph $C^*$-algebras - prominently containing the full family of Cuntz-Krieger algebras - showing that Morita equivalence in this case is…

Nonstable K-theory for Graph Algebras

- Mathematics
- 2004

We compute the monoid V(LK(E)) of isomorphism classes of finitely generated projective modules over certain graph algebras LK(E), and we show that this monoid satisfies the refinement property and…

Corners of Cuntz-Krieger algebras

- Mathematics
- 2012

We show that if $A$ is a unital $C^*$-algebra and $B$ is a Cuntz-Krieger algebra for which $A\otimes\mathbb{K} \cong B\otimes\mathbb{K}$, then $A$ is a Cuntz-Krieger algebra. Consequently, corners of…

Ideals in Graph Algebras

- Mathematics
- 2012

We show that the graph construction used to prove that a gauge-invariant ideal of a graph C ∗ -algebra is isomorphic to a graph C ∗ -algebra, and also used to prove that a graded ideal of a Leavitt…

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…

The isomorphism problem for semigroup C*-algebras of right-angled Artin monoids

- Mathematics
- 2016

Semigroup C*-algebras for right-angled Artin monoids were introduced and studied by Crisp and Laca. In the paper at hand, we are able to present the complete answer to their question of when such…

On Stability of C-algebras

- Mathematics
- 1997

LetA be a -unitalC -algebra, i.e.A admits a countable approximate unit. It is proved thatA is stable, i.e.A is isomorphic toA K whereK is the algebra of compact operators on a separable Hilbert…

On Stability ofC*-Algebras

- Mathematics
- 1998

LetAbe aσ-unitalC*-algebra, i.e.,Aadmits a countable approximate unit. It is proved thatAis stable, i.e.,Ais isomorphic toA⊗K where K is the algebra of compact operators on a separable Hilbert space,…

Invariance of the Cuntz splice

- Mathematics
- 2016

We show that the Cuntz splice induces stably isomorphic graph $$C^*$$C∗-algebras. This result is a key step towards the recent complete classification of unital graph $$C^*$$C∗-algebras both with…

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).