## Amplified graph C*-algebras II: Reconstruction

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Søren Eilers, Efren Ruiz and Aidan Sims
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**9**(2022), 297-310

## Abstract:

Let $E$ be a countable directed graph that is amplified in the sense that whenever there is an edge from $v$ to $w$, there are infinitely many edges from $v$ to $w$. We show that $E$ can be recovered from $C^*(E)$ together with its canonical gauge-action, and also from $L_\mathbb {K}(E)$ together with its canonical grading.## References

- G. Abrams and G. Aranda Pino,
*The Leavitt path algebras of arbitrary graphs*, Houston J. Math.**34**(2008), no. 2, 423–442. MR**2417402** - Pere Ara and Kenneth R. Goodearl,
*Leavitt path algebras of separated graphs*, J. Reine Angew. Math.**669**(2012), 165–224. MR**2980456**, DOI 10.1515/crelle.2011.146 - Pere Ara, Roozbeh Hazrat, Huanhuan Li, and Aidan Sims,
*Graded Steinberg algebras and their representations*, Algebra Number Theory**12**(2018), no. 1, 131–172. MR**3781435**, DOI 10.2140/ant.2018.12.131 - Sara E. Arklint, Søren Eilers, and Efren Ruiz,
*Geometric classification of isomorphism of unital graph $C^*$-algebras*, New York J. Math., to appear, arXiv:1910.11514. - Kevin Aguyar Brix,
*Balanced strong shift equivalence, balanced in-splits, and eventual conjugacy*, Ergodic Theory Dynam. Systems**42**(2022), no. 1, 19–39. MR**4348408**, DOI 10.1017/etds.2020.126 - Nathan Brownlowe, Toke Meier Carlsen, and Michael F. Whittaker,
*Graph algebras and orbit equivalence*, Ergodic Theory Dynam. Systems**37**(2017), no. 2, 389–417. MR**3614030**, DOI 10.1017/etds.2015.52 - Nathan Brownlowe, Marcelo Laca, Dave Robertson, and Aidan Sims,
*Reconstructing directed graphs from generalized gauge actions on their Toeplitz algebras*, Proc. Roy. Soc. Edinburgh Sect. A**150**(2020), no. 5, 2632–2641. MR**4153628**, DOI 10.1017/prm.2019.36 - Toke Meier Carlsen, Søren Eilers, Eduard Ortega, and Gunnar Restorff,
*Flow equivalence and orbit equivalence for shifts of finite type and isomorphism of their groupoids*, J. Math. Anal. Appl.**469**(2019), no. 2, 1088–1110. MR**3860463**, DOI 10.1016/j.jmaa.2018.09.056 - Toke Meier Carlsen, Efren Ruiz, and Aidan Sims,
*Equivalence and stable isomorphism of groupoids, and diagonal-preserving stable isomorphisms of graph $C^*$-algebras and Leavitt path algebras*, Proc. Amer. Math. Soc.**145**(2017), no. 4, 1581–1592. MR**3601549**, DOI 10.1090/proc/13321 - Adam Dor-On, Søren Eilers, and Shirly Geffen,
*Classification of irreversible and reversible Pimsner operator algebras*, Compos. Math.**156**(2020), no. 12, 2510–2535. MR**4205042**, DOI 10.1112/s0010437x2000754x - Søren Eilers and Efren Ruiz,
*Refined moves for structure-preserving isomorphism of graph $C^*$-algebras*, arXiv:1908.03714, 2019. - Søren Eilers, Efren Ruiz, and Adam P. W. Sørensen,
*Amplified graph $C^*$-algebras*, Münster J. Math.**5**(2012), 121–150. MR**3047630** - Søren Eilers, Gunnar Restorff, Efren Ruiz, and Adam P. W. Sørensen,
*The complete classification of unital graph $C^*$-algebras: geometric and strong*, Duke Math. J.**170**(2021), no. 11, 2421–2517. MR**4302548**, DOI 10.1215/00127094-2021-0060 - Neal J. Fowler, Marcelo Laca, and Iain Raeburn,
*The $C^*$-algebras of infinite graphs*, Proc. Amer. Math. Soc.**128**(2000), no. 8, 2319–2327. MR**1670363**, DOI 10.1090/S0002-9939-99-05378-2 - Roozbeh Hazrat,
*Graded rings and graded Grothendieck groups*, London Mathematical Society Lecture Note Series, vol. 435, Cambridge University Press, Cambridge, 2016. MR**3523984**, DOI 10.1017/CBO9781316717134 - Damon Hay, Marissa Loving, Martin Montgomery, Efren Ruiz, and Katherine Todd,
*Non-stable $K$-theory for Leavitt path algebras*, Rocky Mountain J. Math.**44**(2014), no. 6, 1817–1850. MR**3310950**, DOI 10.1216/RMJ-2014-44-6-1817 - Pierre Julg,
*$K$-théorie équivariante et produits croisés*, C. R. Acad. Sci. Paris Sér. I Math.**292**(1981), no. 13, 629–632 (French, with English summary). MR**625361** - Alex Kumjian and David Pask,
*$C^*$-algebras of directed graphs and group actions*, Ergodic Theory Dynam. Systems**19**(1999), no. 6, 1503–1519. MR**1738948**, DOI 10.1017/S0143385799151940 - Kengo Matsumoto and Hiroki Matui,
*Continuous orbit equivalence of topological Markov shifts and Cuntz-Krieger algebras*, Kyoto J. Math.**54**(2014), no. 4, 863–877. MR**3276420**, DOI 10.1215/21562261-2801849 - N. Christopher Phillips,
*Equivariant $K$-theory and freeness of group actions on $C^*$-algebras*, Lecture Notes in Mathematics, vol. 1274, Springer-Verlag, Berlin, 1987. MR**911880**, DOI 10.1007/BFb0078657 - Iain Raeburn,
*Graph algebras*, CBMS Regional Conference Series in Mathematics, vol. 103, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2005. MR**2135030**, DOI 10.1090/cbms/103 - Mark Tomforde,
*Uniqueness theorems and ideal structure for Leavitt path algebras*, J. Algebra**318**(2007), no. 1, 270–299. MR**2363133**, DOI 10.1016/j.jalgebra.2007.01.031 - Mark Tomforde,
*Stability of $C^\ast$-algebras associated to graphs*, Proc. Amer. Math. Soc.**132**(2004), no. 6, 1787–1795. MR**2051143**, DOI 10.1090/S0002-9939-04-07411-8

## Additional Information

**Søren Eilers**- Affiliation: Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, DK-2100 Copenhagen, Denmark
- MR Author ID: 609837
- ORCID: 0000-0002-3009-0524
- Email: eilers@math.ku.dk
**Efren Ruiz**- Affiliation: Department of Mathematics, University of Hawaii, Hilo, 200W. Kawili St., Hilo, Hawaii 96720-4091
- MR Author ID: 817213
- ORCID: 0000-0002-3009-0524
- Email: ruize@hawaii.edu
**Aidan Sims**- Affiliation: School of Mathematics and Applied Statistics, The University of Wollongong, NSW 2522, Australia
- MR Author ID: 671497
- ORCID: 0000-0002-1965-6451
- Email: asims@uow.edu.au
- Received by editor(s): July 7, 2020
- Received by editor(s) in revised form: September 30, 2021
- Published electronically: June 28, 2022
- Additional Notes: This research was supported by Australian Research Council Discovery Project DP200100155, by DFF-Research Project 2 ‘Automorphisms and Invariants of Operator Algebras’, no. 7014-00145B, and by a Simons Foundation Collaboration Grant, #567380.
- Communicated by: Adrian Ioana
- © Copyright 2022 by the authors under Creative Commons Attribution 3.0 License (CC BY 3.0)
- Journal: Proc. Amer. Math. Soc. Ser. B
**9**(2022), 297-310 - MSC (2020): Primary 46L35
- DOI: https://doi.org/10.1090/bproc/112
- MathSciNet review: 4446255