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Geometric classification of unital graph C*-algebras of real rank zero
Operator Algebras · Submitted 25 May 2015
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim We generalize the classification result of Restorff on Cuntz-Krieger algebras to cover all unital graph C*-algebras with real rank zero, showing that Morita equivalence in this case is determined by ordered, filtered K-theory as conjectured by three of the authors.
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We generalize the classification result of Restorff on Cuntz-Krieger algebras to cover all unital graph C*-algebras with real rank zero, showing that Morita equivalence in this case is determined by ordered, filtered K-theory as conjectured by three of the authors. 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 technical step, we prove that the so-called Cuntz splice leaves unital graph C*-algebras invariant up to Morita equivalence. We note that we have recently found a way to generalize the results of the present paper to cover general unital graph C*-algebras. The improved methods needed render some parts of the present paper obsolete, and hence we do not intend to publish it. Instead, we will present a complete solution (drawing heavily on many of the methods presented here) in a forthcoming paper.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We generalize the classification result of Restorff on Cuntz-Krieger algebras to cover all unital graph C*-algebras with real rank zero, showing that Morita equivalence in this case is determined by ordered, filtered K-theory as conjectured by three of the authors. | result 0.58 | 0.95 | 0.92 | asserted 1.00 |
| s1 | 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. | interpretation 0.84 | 0.91 | 0.14 | asserted 1.00 |
| s2 | As a key technical step, we prove that the so-called Cuntz splice leaves unital graph C*-algebras invariant up to Morita equivalence. | result 0.42 | 0.95 | 0.05 | asserted 1.00 |
| s3 | We note that we have recently found a way to generalize the results of the present paper to cover general unital graph C*-algebras. | future work 0.78 | 0.94 | 0.03 | asserted 0.99 |
| s4 | The improved methods needed render some parts of the present paper obsolete, and hence we do not intend to publish it. | future work 0.52 | 0.71 | 0.02 | asserted 1.00 |
| s5 | Instead, we will present a complete solution (drawing heavily on many of the methods presented here) in a forthcoming paper. | future work 1.00 | 0.48 | 0.03 | asserted 0.99 |
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Original abstract
We generalize the classification result of Restorff on Cuntz-Krieger algebras to cover all unital graph C*-algebras with real rank zero, showing that Morita equivalence in this case is determined by ordered, filtered K-theory as conjectured by three of the authors. 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 technical step, we prove that the so-called Cuntz splice leaves unital graph C*-algebras invariant up to Morita equivalence. We note that we have recently found a way to generalize the results of the present paper to cover general unital graph C*-algebras. The improved methods needed render some parts of the present paper obsolete, and hence we do not intend to publish it. Instead, we will present a complete solution (drawing heavily on many of the methods presented here) in a forthcoming paper.
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This gluing operation is called “performing a Cuntz splice”, and it has played an important role in the study Cuntz–Krieger algebras, graph $C^*$-algebras, and Leavitt path algebras [ref] .
The Cuntz splice does not preserve $*$-isomorphism of Leavitt path algebras over $\mathbb{Z}$ · Introduction · 0.98
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