[Paper Review] Corners of Cuntz-Krieger algebras

Sara E. Arklint, Efren Ruiz|arXiv (Cornell University)|Sep 19, 2012

Advanced Operator Algebra Research8 references4 citations

TL;DR

This paper establishes that unital C*-algebras stably isomorphic to Cuntz-Krieger algebras are themselves Cuntz-Krieger algebras, proving that corners of Cuntz-Krieger algebras are also Cuntz-Krieger algebras. The key result relies on a characterization of Cuntz-Krieger algebras as unital graph C*-algebras with finite graphs lacking sinks or sources, and uses K-theory rank equality to identify them via stable isomorphism.

ABSTRACT

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 Cuntz-Krieger algebras are Cuntz-Krieger algebras.

Motivation & Objective

• To characterize Cuntz-Krieger algebras via outer properties, specifically through K-theory ranks and unitality.
• To resolve the open question of whether corners of Cuntz-Krieger algebras inherit the Cuntz-Krieger property.
• To establish that stable isomorphism to a Cuntz-Krieger algebra implies isomorphism, thereby proving permanence under corners.
• To provide a combinatorial model using finite graphs without sinks or sources to represent Cuntz-Krieger algebras.
• To support the conjecture that full corners of semiprojective C*-algebras are semiprojective, in the special case of Cuntz-Krieger algebras.

Proposed method

• Use graph C*-algebra theory to model Cuntz-Krieger algebras as universal C*-algebras generated by partial isometries satisfying Cuntz-Krieger relations.
• Characterize Cuntz-Krieger algebras as unital graph C*-algebras arising from finite directed graphs with no sinks and no sources.
• Apply K-theory to show that a unital graph C*-algebra is a Cuntz-Krieger algebra if and only if the rank of its $K_0$-group equals the rank of its $K_1$-group.
• Prove that if $A times ext{K} o B times ext{K}$ is an isomorphism with $B$ a Cuntz-Krieger algebra, then $A$ is isomorphic to a Cuntz-Krieger algebra.
• Use gauge-invariant ideals and hereditary saturated subsets of graph vertices to analyze ideals generated by projections in $C^*(E)$.
• Leverage stable isomorphism and Morita equivalence to show that corners of $A$ or $A times ext{K}$ are stably isomorphic to Cuntz-Krieger algebras, hence isomorphic to them via the main theorem.

Experimental results

Research questions

• RQ1 Can Cuntz-Krieger algebras be characterized by outer invariants such as K-theory ranks and unitality?
• RQ2 Do corners of Cuntz-Krieger algebras remain Cuntz-Krieger algebras?
• RQ3 Is stable isomorphism to a Cuntz-Krieger algebra sufficient for a unital C*-algebra to be isomorphic to a Cuntz-Krieger algebra?
• RQ4 Do Cuntz-Krieger algebras satisfy the semiprojectivity property under cornering?
• RQ5 Does the permanence of the Cuntz-Krieger class under corners hold despite the failure of this property in the broader class of graph C*-algebras?

Key findings

• A unital graph C*-algebra is a Cuntz-Krieger algebra if and only if the rank of its $K_0$-group equals the rank of its $K_1$-group.
• If a unital C*-algebra $A$ satisfies $A times ext{K} o B times ext{K}$ for a Cuntz-Krieger algebra $B$, then $A$ is isomorphic to a Cuntz-Krieger algebra.
• Corners of Cuntz-Krieger algebras are isomorphic to Cuntz-Krieger algebras, as shown in Corollary 4.10.
• Stabilized Cuntz-Krieger algebras are semiprojective, and so are their unital corners, as established in Corollary 4.11.
• The class of Cuntz-Krieger algebras is closed under taking unital corners, even though the broader class of graph C*-algebras is not.
• The result confirms a special case of Blackadar’s conjecture on semiprojectivity of full corners of semiprojective C*-algebras.