Amplified graph C*-algebras II: Reconstruction

Eilers, Søren; Ruiz, Efren; Sims, Aidan

doi:10.1090/bproc/112

Amplified graph C*-algebras II: Reconstruction
HTML articles powered by AMS MathViewer

by Søren Eilers, Efren Ruiz and Aidan Sims HTML | PDF

Proc. Amer. Math. Soc. Ser. B 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