Nuclear dimension of graph 𝐶*-algebras with Condition (K)

Faurot, Gregory; Schafhauser, Christopher

doi:10.1090/proc/16930

Nuclear dimension of graph $C^*$-algebras with Condition (K)
HTML articles powered by AMS MathViewer

by Gregory Faurot and Christopher Schafhauser;

Proc. Amer. Math. Soc. 152 (2024), 4421-4435

DOI: https://doi.org/10.1090/proc/16930

Published electronically: August 23, 2024

HTML | PDF | Request permission

Abstract:

We prove that for any countable directed graph $E$ with Condition (K), the associated graph $C^*$-algebra $C^*(E)$ has nuclear dimension at most $2$. Furthermore, we provide a sufficient condition producing an upper bound of $1$.

References

Astrid an Huef and Iain Raeburn, The ideal structure of Cuntz-Krieger algebras, Ergodic Theory Dynam. Systems 17 (1997), no. 3, 611–624. MR 1452183, DOI 10.1017/S0143385797079200
Teresa Bates, David Pask, Iain Raeburn, and Wojciech Szymański, The $C^*$-algebras of row-finite graphs, New York J. Math. 6 (2000), 307–324. MR 1777234
Bruce Blackadar, $K$-theory for operator algebras, 2nd ed., Mathematical Sciences Research Institute Publications, vol. 5, Cambridge University Press, Cambridge, 1998. MR 1656031
Joan Bosa, Nathanial P. Brown, Yasuhiko Sato, Aaron Tikuisis, Stuart White, and Wilhelm Winter, Covering dimension of $\rm C^*$-algebras and 2-coloured classification, Mem. Amer. Math. Soc. 257 (2019), no. 1233, vii+97. MR 3908669, DOI 10.1090/memo/1233
Joan Bosa, James Gabe, Aidan Sims, and Stuart White, The nuclear dimension of $\mathcal O_\infty$-stable $C^\ast$-algebras, Adv. Math. 401 (2022), Paper No. 108250, 51. MR 4392219, DOI 10.1016/j.aim.2022.108250
Laura Brake and Wilhelm Winter, The Toeplitz algebra has nuclear dimension one, Bull. Lond. Math. Soc. 51 (2019), no. 3, 554–562. MR 3964508, DOI 10.1112/blms.12250
Lawrence G. Brown, Stable isomorphism of hereditary subalgebras of $C^*$-algebras, Pacific J. Math. 71 (1977), no. 2, 335–348. MR 454645, DOI 10.2140/pjm.1977.71.335
Nathanial P. Brown and Narutaka Ozawa, $C^*$-algebras and finite-dimensional approximations, Graduate Studies in Mathematics, vol. 88, American Mathematical Society, Providence, RI, 2008. MR 2391387, DOI 10.1090/gsm/088
Jorge Castillejos and Samuel Evington, Nuclear dimension of simple stably projectionless $\textrm {C}^*$-algebras, Anal. PDE 13 (2020), no. 7, 2205–2240. MR 4175824, DOI 10.2140/apde.2020.13.2205
Jorge Castillejos, Samuel Evington, Aaron Tikuisis, Stuart White, and Wilhelm Winter, Nuclear dimension of simple $\rm C^*$-algebras, Invent. Math. 224 (2021), no. 1, 245–290. MR 4228503, DOI 10.1007/s00222-020-01013-1
Joachim Cuntz and Wolfgang Krieger, A class of $C^{\ast }$-algebras and topological Markov chains, Invent. Math. 56 (1980), no. 3, 251–268. MR 561974, DOI 10.1007/BF01390048
D. Drinen and M. Tomforde, The $C^*$-algebras of arbitrary graphs, Rocky Mountain J. Math. 35 (2005), no. 1, 105–135. MR 2117597, DOI 10.1216/rmjm/1181069770
Philip Easo, Esperanza Garijo, Sarunas Kaubrys, David Nkansah, Martin Vrabec, David Watt, Cameron Wilson, Christian Bönicke, Samuel Evington, Marzieh Forough, Sergio Girón Pacheco, Nicholas Seaton, Stuart White, Michael F. Whittaker, and Joachim Zacharias, The Cuntz-Toeplitz algebras have nuclear dimension one, J. Funct. Anal. 279 (2020), no. 7, 108690, 14. MR 4116805, DOI 10.1016/j.jfa.2020.108690
George A. Elliott, Guihua Gong, Huaxin Lin, and Zhuang Niu, On the classification of simple amenable $C^*$-algebras with finite decomposition rank, II, arXiv:1507.03437, 2015.
George A. Elliott and Dan Kucerovsky, An abstract Voiculescu-Brown-Douglas-Fillmore absorption theorem, Pacific J. Math. 198 (2001), no. 2, 385–409. MR 1835515, DOI 10.2140/pjm.2001.198.385
George A. Elliott and Andrew S. Toms, Regularity properties in the classification program for separable amenable $C^*$-algebras, Bull. Amer. Math. Soc. (N.S.) 45 (2008), no. 2, 229–245. MR 2383304, DOI 10.1090/S0273-0979-08-01199-3
Dominic Enders, On the nuclear dimension of certain UCT-Kirchberg algebras, J. Funct. Anal. 268 (2015), no. 9, 2695–2706. MR 3325534, DOI 10.1016/j.jfa.2015.01.009
Samuel Evington, Nuclear dimension of extensions of $\mathcal O_\infty$-stable algebras, J. Operator Theory 88 (2022), no. 1, 171–187. MR 4438603, DOI 10.7900/jot
James Gabe, Classification of $\mathcal O_\infty$-stable $C^*$-algebras, Mem. Amer. Math. Soc. 293 (2024), no. 1461, v+115. MR 4684339, DOI 10.1090/memo/1461
James Gabe, A new proof of Kirchberg’s $\mathcal O_2$-stable classification, J. Reine Angew. Math. 761 (2020), 247–289. MR 4080250, DOI 10.1515/crelle-2018-0010
Ruaridh Gardner and Aaron Tikuisis, The nuclear dimension of extensions of commutative $C^*$-algebras by the compact operators, to appear in J. Operator Theory, arXiv:2202.04695.
Guihua Gong, Huaxin Lin, and Zhuang Niu, A classification of finite simple amenable $\mathcal Z$-stable $C^\ast$-algebras, I: $C^\ast$-algebras with generalized tracial rank one, C. R. Math. Acad. Sci. Soc. R. Can. 42 (2020), no. 3, 63–450 (English, with English and French summaries). MR 4215379
Guihua Gong, Huaxin Lin, and Zhuang Niu, A classification of finite simple amenable $\mathcal Z$-stable $\textrm {C}^\ast$-algebras, II: $\textrm {C}^\ast$-algebras with rational generalized tracial rank one, C. R. Math. Acad. Sci. Soc. R. Can. 42 (2020), no. 4, 451–539 (English, with English and French summaries). MR 4215380
Ja A. Jeong and Gi Hyun Park, Graph $C^*$-algebras with real rank zero, J. Funct. Anal. 188 (2002), no. 1, 216–226. MR 1878636, DOI 10.1006/jfan.2001.3830
Xinhui Jiang and Hongbing Su, On a simple unital projectionless $C^*$-algebra, Amer. J. Math. 121 (1999), no. 2, 359–413. MR 1680321, DOI 10.1353/ajm.1999.0012
Eberhard Kirchberg and N. Christopher Phillips, Embedding of exact $C^*$-algebras in the Cuntz algebra $\scr O_2$, J. Reine Angew. Math. 525 (2000), 17–53. MR 1780426, DOI 10.1515/crll.2000.065
Eberhard Kirchberg and Wilhelm Winter, Covering dimension and quasidiagonality, Internat. J. Math. 15 (2004), no. 1, 63–85. MR 2039212, DOI 10.1142/S0129167X04002119
Dan Kucerovsky and P. W. Ng, The corona factorization property and approximate unitary equivalence, Houston J. Math. 32 (2006), no. 2, 531–550. MR 2219330
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
Alex Kumjian, David Pask, and Iain Raeburn, Cuntz-Krieger algebras of directed graphs, Pacific J. Math. 184 (1998), no. 1, 161–174. MR 1626528, DOI 10.2140/pjm.1998.184.161
Alex Kumjian, David Pask, Iain Raeburn, and Jean Renault, Graphs, groupoids, and Cuntz-Krieger algebras, J. Funct. Anal. 144 (1997), no. 2, 505–541. MR 1432596, DOI 10.1006/jfan.1996.3001
Hiroki Matui and Yasuhiko Sato, Strict comparison and $\scr {Z}$-absorption of nuclear $C^*$-algebras, Acta Math. 209 (2012), no. 1, 179–196. MR 2979512, DOI 10.1007/s11511-012-0084-4
Hiroki Matui and Yasuhiko Sato, Decomposition rank of UHF-absorbing $\mathrm {C}^*$-algebras, Duke Math. J. 163 (2014), no. 14, 2687–2708. MR 3273581, DOI 10.1215/00127094-2826908
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
Iain Raeburn and Wojciech Szymański, Cuntz-Krieger algebras of infinite graphs and matrices, Trans. Amer. Math. Soc. 356 (2004), no. 1, 39–59. MR 2020023, DOI 10.1090/S0002-9947-03-03341-5
Mikael Rørdam, A simple $C^*$-algebra with a finite and an infinite projection, Acta Math. 191 (2003), no. 1, 109–142. MR 2020420, DOI 10.1007/BF02392697
Efren Ruiz, Aidan Sims, and Adam P. W. Sørensen, UCT-Kirchberg algebras have nuclear dimension one, Adv. Math. 279 (2015), 1–28. MR 3345177, DOI 10.1016/j.aim.2014.12.042
Efren Ruiz, Aidan Sims, and Mark Tomforde, The nuclear dimension of graph $C^*$-algebras, Adv. Math. 272 (2015), 96–123. MR 3303230, DOI 10.1016/j.aim.2014.11.015
Gábor Szabó, On the nuclear dimension of strongly purely infinite $\textrm {C}^*$-algebras, Adv. Math. 306 (2017), 1262–1268. MR 3581330, DOI 10.1016/j.aim.2016.11.009
Aaron Tikuisis, Stuart White, and Wilhelm Winter, Quasidiagonality of nuclear $C^\ast$-algebras, Ann. of Math. (2) 185 (2017), no. 1, 229–284. MR 3583354, DOI 10.4007/annals.2017.185.1.4
Aaron Tikuisis and Wilhelm Winter, Decomposition rank of ${\scr Z}$-stable $\rm C^*$-algebras, Anal. PDE 7 (2014), no. 3, 673–700. MR 3227429, DOI 10.2140/apde.2014.7.673
Andrew S. Toms, On the classification problem for nuclear $C^\ast$-algebras, Ann. of Math. (2) 167 (2008), no. 3, 1029–1044. MR 2415391, DOI 10.4007/annals.2008.167.1029
Andrew S. Toms and Wilhelm Winter, Strongly self-absorbing $C^*$-algebras, Trans. Amer. Math. Soc. 359 (2007), no. 8, 3999–4029. MR 2302521, DOI 10.1090/S0002-9947-07-04173-6
Jesper Villadsen, Simple $C^*$-algebras with perforation, J. Funct. Anal. 154 (1998), no. 1, 110–116. MR 1616504, DOI 10.1006/jfan.1997.3168
Jesper Villadsen, On the stable rank of simple $C^\ast$-algebras, J. Amer. Math. Soc. 12 (1999), no. 4, 1091–1102. MR 1691013, DOI 10.1090/S0894-0347-99-00314-8
Wilhelm Winter, Decomposition rank and $\scr Z$-stability, Invent. Math. 179 (2010), no. 2, 229–301. MR 2570118, DOI 10.1007/s00222-009-0216-4
Wilhelm Winter and Joachim Zacharias, The nuclear dimension of $C^\ast$-algebras, Adv. Math. 224 (2010), no. 2, 461–498. MR 2609012, DOI 10.1016/j.aim.2009.12.005

AMS Website Down

Proceedings of the American Mathematical Society

Nuclear dimension of graph $C^*$-algebras with Condition (K)
HTML articles powered by AMS MathViewer

Abstract: