## The complete classification of unital graph C^{∗}-Algebras: Geometric and strong

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**The complete classification of unital graph C ^{∗}-Algebras : Geometric and strong.** / Eilers, SØren; Restorff, Gunnar; Ruiz, Efren; SØrensen, Adam P.W.

Research output: Contribution to journal › Journal article › Research › peer-review

#### Harvard

^{∗}-Algebras: Geometric and strong',

*Duke Mathematical Journal*, vol. 170, no. 11, pp. 2421-2517. https://doi.org/10.1215/00127094-2021-0060

#### APA

^{∗}-Algebras: Geometric and strong.

*Duke Mathematical Journal*,

*170*(11), 2421-2517. https://doi.org/10.1215/00127094-2021-0060

#### Vancouver

^{∗}-Algebras: Geometric and strong. Duke Mathematical Journal. 2021;170(11):2421-2517. https://doi.org/10.1215/00127094-2021-0060

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#### Bibtex

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#### RIS

TY - JOUR

T1 - The complete classification of unital graph C∗-Algebras

T2 - Geometric and strong

AU - Eilers, SØren

AU - Restorff, Gunnar

AU - Ruiz, Efren

AU - SØrensen, Adam P.W.

N1 - Publisher Copyright: © 2021 Duke University Press. All rights reserved.

PY - 2021

Y1 - 2021

N2 - 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 determined by ordered, filtered K-theory. The classification result is geometric in the sense that it establishes that any Morita equivalence between C∗(E) and C∗(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, we establish that they leave the graph algebras invariant, and we prove that after this augmentation, the list of moves becomes complete in the sense described above. Along the way, we prove that every (reduced, filtered) K-theory order isomorphism can be lifted to an isomorphism between the stabilized C∗-algebras-and, as a consequence, that every such order isomorphism preserving the class of the unit comes from a ∗-isomorphism between the unital graph C∗-algebras themselves. It follows that the question of Morita equivalence and ∗-isomorphism among unital graph C∗-algebras is a decidable one. As immediate examples of applications of our results, we revisit the classification problem for quantum lens spaces and we verify, in the unital case, the Abrams-Tomforde conjectures.

AB - 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 determined by ordered, filtered K-theory. The classification result is geometric in the sense that it establishes that any Morita equivalence between C∗(E) and C∗(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, we establish that they leave the graph algebras invariant, and we prove that after this augmentation, the list of moves becomes complete in the sense described above. Along the way, we prove that every (reduced, filtered) K-theory order isomorphism can be lifted to an isomorphism between the stabilized C∗-algebras-and, as a consequence, that every such order isomorphism preserving the class of the unit comes from a ∗-isomorphism between the unital graph C∗-algebras themselves. It follows that the question of Morita equivalence and ∗-isomorphism among unital graph C∗-algebras is a decidable one. As immediate examples of applications of our results, we revisit the classification problem for quantum lens spaces and we verify, in the unital case, the Abrams-Tomforde conjectures.

UR - http://www.scopus.com/inward/record.url?scp=85114092902&partnerID=8YFLogxK

U2 - 10.1215/00127094-2021-0060

DO - 10.1215/00127094-2021-0060

M3 - Journal article

AN - SCOPUS:85114092902

VL - 170

SP - 2421

EP - 2517

JO - Duke Mathematical Journal

JF - Duke Mathematical Journal

SN - 0012-7094

IS - 11

ER -

ID: 284199392