The complete classification of unital graph $C^{\ast }$-algebras: Geometric and strong

Abstract

We provide a complete classification of the class of unital graph $C^{\ast }$-algebras - prominently containing the full family of Cuntz-Krieger algebras - showing that Morita equivalence in this case is determined by ordered, filtered $K$-theory. The classification result is geometric in the sense that it establishes that any Morita equivalence between $C^{\ast }(E)$ and $C^{\ast }(F)$ in this class can be realized by a sequence of moves leading from $E$ to $F$, in a way resembling the role of Reidemeister moves on knots. As a key ingredient, we introduce a new class of such moves, establish that they leave the graph algebras invariant, and prove that after this augmentation, the list of moves becomes complete in the sense described above. Along the way, we prove that every ordered, reduced filtered $K$-theory isomorphism can be lifted to an isomorphism between the stabilized $C^{\ast }$-algebras - and, as a consequence, that every ordered, reduced filtered $K$-theory isomorphism preserving the class of the unit comes from a $\ast$-isomorphism between the unital graph $C^{\ast }$-algebras themselves. It follows that the question of Morita equivalence amongst unital graph $C^{\ast }$-algebras is a decidable one. As immediate examples of applications of our results we revisit the classification problem for quantum lens spaces and verify, in the unital case, the Abrams-Tomforde conjectures.

This article draws heavily on results and notation developed in arXiv:1602.03709, arXiv:1604.05439 and arXiv:1605.06153, and together with these papers supersedes the results of arXiv:1505.06773, which will not be published. The second version adjusts the proof of decidability in Section 14.2 to the appeared version of [BS18], corrects the statement of Corollary 3.6, and updates references

References in corpus (8)

• Adding tails to C*-correspondences
• Geometric classification of graph $C^{\ast }$-algebras over finite graphs
• Strong classification of extensions of classifiable C*-algebras
• $\ast$-isomorphism of Leavitt path algebras over $\mathbb{Z}$
• Filtered K-theory for graph algebras
• Strong classification of purely infinite Cuntz-Krieger algebras
• Invariance of the Cuntz splice
• Combinatorial classification of quantum lens spaces

Cited by in corpus (9)

• Geometric classification of graph $C^{\ast }$-algebras over finite graphs
• Flow equivalence and orbit equivalence for shifts of finite type and isomorphism of their groupoids
• Strong classification of purely infinite Cuntz-Krieger algebras
• Classification of ${\mathcal{O}}_{\infty }$-stable $C^{\ast }$-algebras
• Geometric classification of isomorphism of unital graph C*-algebras
• Graded $K$-Theory, Filtered $K$-theory and the classification of graph algebras
• Reconstructing directed graphs from generalised gauge actions on their Toeplitz algebras
• On diagonal quasi-free automorphisms of simple Cuntz-Krieger algebras
• On The Classification of Quantum Lens Spaces of Dimension at most 7