Partial actions of groups on hyperspaces

Luis Martínez; Héctor Pinedo Tapia; Edwar Ramirez

doi:10.4995/agt.2022.15745

Authors

Luis Martínez Universidad Industrial de Santander https://orcid.org/0000-0002-3957-3119
Héctor Pinedo Tapia Universidad Industrial de Santander https://orcid.org/0000-0003-4432-419X
Edwar Ramirez Universidad Industrial de Santander https://orcid.org/0000-0003-4919-9439

DOI:

https://doi.org/10.4995/agt.2022.15745

Keywords:

Partial action, globalization, hyperspace, monad

Abstract

Let X be a compact Hausdorff space. In this work we translate partial actions of X to partial actions on some hyperspaces determined by X, this gives an endofunctor 2^- in the category of partial actions on compact Hausdorff spaces which generates a monad in this category. Moreover, structural relations between partial actions θ on X and partial determined by 2^θ as well as their corresponding globalizations are established.

Downloads

Download data is not yet available.

Author Biographies

Luis Martínez, Universidad Industrial de Santander

Escuela de Matemáticas

Héctor Pinedo Tapia, Universidad Industrial de Santander

Escuela de Matemáticas

Edwar Ramirez, Universidad Industrial de Santander

Escuela de Matemáticas

References

F. Abadie, Enveloping actions and Takai duality for partial actions, J. Funct. Anal. 197 (2003), 14-67. https://doi.org/10.1016/S0022-1236(02)00032-0

J. Camargo, D. Herrera and S. Macías, Cells and $n$-fold hyperspaces, Colloq. Math. 145, no. 2 (2016), 157-166. https://doi.org/10.1016/j.topol.2015.10.001

K. Choi, Birget-Rhodes expansions of topological groups, Advanced Studies in Contemporary Mathematics 23, no. 1 (2013), 203-211.

M. Dokuchaev, Recent developments around partial actions, S~ao Paulo J. Math. Sci. 13, no. 1 (2019), 195-247. https://doi.org/10.1007/s40863-018-0087-y

M. Dokuchaev and R. Exel, Partial actions and subshifts, J. Funct. Analysis 272 (2017), 5038-5106. https://doi.org/10.1016/j.jfa.2017.02.020

R. Exel, Circle actions on $C^*$-algebras, partial automorphisms and generalized Pimsner-Voiculescu exact sequences, J. Funct. Anal. 122, no. 3 (1994), 361-401. https://doi.org/10.1006/jfan.1994.1073

R. Exel, Partial actions of group and actions of inverse semigroups, Proc. Amer. Math. Soc. 126, no. 12 (1998), 3481-3494. https://doi.org/10.1090/S0002-9939-98-04575-4

R. Exel, Partial dynamical systems, Fell bundles and applications, Mathematical surveys and monographs; volume 224, Providence, Rhode Island: American Mathematical Society, 2017. https://doi.org/10.1090/surv/224

R. Exel, T. Giordano, and D. Gonçalves, Enveloping algebras of partial actions as groupoid $C^*$-algebras, J. Operator Theory 65 (2011), 197-210.

J. Kellendonk and M. V. Lawson, Partial actions of groups, International Journal of Algebra and Computation 14 (2004), 87-114. https://doi.org/10.1142/S0218196704001657

L. Martínez, H. Pinedo and A. Villamizar, Partial actions on profinite spaces, preprint.

S. Nadler and A. Illanes, Hyperspaces: Fundamentals and Recent Advances, Pure and Applied Mathematics, Marcel Dekker, Inc., New York, 1999.

H. Pinedo and C. Uzcátegui, Polish globalization of Polish group partial actions, Math. Log. Quart. 63, no. 6 (2017), 481-490. https://doi.org/10.1002/malq.201600018

H. Pinedo and C. Uzcátegui, Borel globalization of partial actions of Polish groups, Arch. Mat. Log. 57 (2018), 617-627. https://doi.org/10.1007/s00153-017-0598-8

J. C. Quigg and I. Raeburn, Characterizations of crossed products by partial actions, J. Operator Theory 37 (1997), 311-340.

B. Steinberg, Partial actions of groups on cell complexes, Monatsh. Math. 138, no. 2 (2003), 159-170. https://doi.org/10.1007/s00605-002-0521-0