A note on the centralizer of a subalgebra of the Steinberg algebra
HTML articles powered by AMS MathViewer
- by R. Hazrat and Huanhuan Li
- St. Petersburg Math. J. 33 (2022), 179-184
- DOI: https://doi.org/10.1090/spmj/1695
- Published electronically: December 28, 2021
- PDF | Request permission
Abstract:
For an ample Hausdorff groupoid $\mathcal {G}$, and the Steinberg algebra $A_R(\mathcal {G})$ with coefficients in the commutative ring $R$ with unit, the centralizer is described for the subalgebra $A_R(U)$ with $U$ an open closed invariant subset of the unit space of $\mathcal {G}$. In particular, it is shown that the algebra of the interior of the isotropy is indeed the centralizer of the diagonal subalgebra of the Steinberg algebra. This will unify several results in the literature, and the corresponding results for Leavitt path algebras follow.References
- Gene Abrams, Pere Ara, and Mercedes Siles Molina, Leavitt path algebras, Lecture Notes in Mathematics, vol. 2191, Springer, London, 2017. MR 3729290, DOI 10.1007/978-1-4471-7344-1
- Pere Ara, Joan Bosa, Roozbeh Hazrat, and Aidan Sims, Reconstruction of graded groupoids from graded Steinberg algebras, Forum Math. 29 (2017), no. 5, 1023–1037. MR 3692025, DOI 10.1515/forum-2016-0072
- Gonzalo Aranda Pino, John Clark, Astrid an Huef, and Iain Raeburn, Kumjian-Pask algebras of higher-rank graphs, Trans. Amer. Math. Soc. 365 (2013), no. 7, 3613–3641. MR 3042597, DOI 10.1090/S0002-9947-2013-05717-0
- Gonzalo Aranda Pino and Kathi Crow, The center of a Leavitt path algebra, Rev. Mat. Iberoam. 27 (2011), no. 2, 621–644. MR 2848533, DOI 10.4171/RMI/649
- Jonathan H. Brown and Astrid an Huef, Centers of algebras associated to higher-rank graphs, Rev. Mat. Iberoam. 30 (2014), no. 4, 1387–1396. MR 3293437, DOI 10.4171/RMI/818
- Jonathan H. Brown, Gabriel Nagy, Sarah Reznikoff, Aidan Sims, and Dana P. Williams, Cartan subalgebras in $C^*$-algebras of Hausdorff étale groupoids, Integral Equations Operator Theory 85 (2016), no. 1, 109–126. MR 3503181, DOI 10.1007/s00020-016-2285-2
- Toke Meier Carlsen, Efren Ruiz, Aidan Sims, and Mark Tomforde, Reconstruction of groupoids and $C^\ast$-rigidity of dynamical systems, Adv. Math. 390 (2021), Paper No. 107923, 55. MR 4292963, DOI 10.1016/j.aim.2021.107923
- Toke Meier Carlsen, $\ast$-isomorphism of Leavitt path algebras over $\Bbb Z$, Adv. Math. 324 (2018), 326–335. MR 3733888, DOI 10.1016/j.aim.2017.11.018
- Lisa Orloff Clark, Cristóbal Gil Canto, and Alireza Nasr-Isfahani, The cycline subalgebra of a Kumjian-Pask algebra, Proc. Amer. Math. Soc. 145 (2017), no. 5, 1969–1980. MR 3611313, DOI 10.1090/proc/13439
- L. O. Clark and R. Hazrat, Étale groupoid and Steinberg algebras, a concise introduction, Indian Stat. Inst. Leavitt Path Algebras and Classical K-Theory Ser., Springer, Singapore, 2020.
- Lisa Orloff Clark, Ruy Exel, and Enrique Pardo, A generalized uniqueness theorem and the graded ideal structure of Steinberg algebras, Forum Math. 30 (2018), no. 3, 533–552. MR 3794898, DOI 10.1515/forum-2016-0197
- Cristóbal Gil Canto and Alireza Nasr-Isfahani, The commutative core of a Leavitt path algebra, J. Algebra 511 (2018), 227–248. MR 3834772, DOI 10.1016/j.jalgebra.2018.06.016
- Gabriel Nagy and Sarah Reznikoff, Abelian core of graph algebras, J. Lond. Math. Soc. (2) 85 (2012), no. 3, 889–908. MR 2927813, DOI 10.1112/jlms/jdr073
- Jean Renault, Cuntz-like algebras, Operator theoretical methods (Timişoara, 1998) Theta Found., Bucharest, 2000, pp. 371–386. MR 1770333
- Jean Renault, A groupoid approach to $C^{\ast }$-algebras, Lecture Notes in Mathematics, vol. 793, Springer, Berlin, 1980. MR 584266, DOI 10.1007/BFb0091072
- Benjamin Steinberg, A groupoid approach to discrete inverse semigroup algebras, Adv. Math. 223 (2010), no. 2, 689–727. MR 2565546, DOI 10.1016/j.aim.2009.09.001
- Benjamin Steinberg, Diagonal-preserving isomorphisms of étale groupoid algebras, J. Algebra 518 (2019), 412–439. MR 3873946, DOI 10.1016/j.jalgebra.2018.10.024
- Samuel B. G. Webster, The path space of a directed graph, Proc. Amer. Math. Soc. 142 (2014), no. 1, 213–225. MR 3119197, DOI 10.1090/S0002-9939-2013-11755-7
Bibliographic Information
- R. Hazrat
- Affiliation: Centre for Research in Mathematics and Data Sceince, Western Sydney University, Australia
- MR Author ID: 654632
- Email: r.hazrat@westernsydney.edu.au
- Huanhuan Li
- Affiliation: School of Mathematical Sciences, Anhui University, Hefei 230601, Anhui, PR China
- Email: lihuanhuan2005@163.com
- Received by editor(s): March 23, 2020
- Published electronically: December 28, 2021
- Additional Notes: The authors would like to acknowledge Australian Research Council grant DP160101481
- © Copyright 2021 American Mathematical Society
- Journal: St. Petersburg Math. J. 33 (2022), 179-184
- MSC (2020): Primary 22A22, 18B40, 19D55
- DOI: https://doi.org/10.1090/spmj/1695
- MathSciNet review: 4219511