Metric groups, unitary representations and continuous logic

Aleksander Ivanov

Communications in Mathematics (2021)

  • Issue: 1, page 35-48
  • ISSN: 1804-1388

Abstract

top
We describe how properties of metric groups and of unitary representations of metric groups can be presented in continuous logic. In particular we find L ω 1 ω -axiomatization of amenability. We also show that in the case of locally compact groups some uniform version of the negation of Kazhdan’s property (T) can be viewed as a union of first-order axiomatizable classes. We will see when these properties are preserved under taking elementary substructures.

How to cite

top

Ivanov, Aleksander. "Metric groups, unitary representations and continuous logic." Communications in Mathematics (2021): 35-48. <http://eudml.org/doc/297841>.

@article{Ivanov2021,
abstract = {We describe how properties of metric groups and of unitary representations of metric groups can be presented in continuous logic. In particular we find $L_\{\omega _1 \omega \}$-axiomatization of amenability. We also show that in the case of locally compact groups some uniform version of the negation of Kazhdan’s property (T) can be viewed as a union of first-order axiomatizable classes. We will see when these properties are preserved under taking elementary substructures.},
author = {Ivanov, Aleksander},
journal = {Communications in Mathematics},
keywords = {Continuous logic; metric groups; unitary representations; amenable groups},
language = {eng},
number = {1},
pages = {35-48},
publisher = {University of Ostrava},
title = {Metric groups, unitary representations and continuous logic},
url = {http://eudml.org/doc/297841},
year = {2021},
}

TY - JOUR
AU - Ivanov, Aleksander
TI - Metric groups, unitary representations and continuous logic
JO - Communications in Mathematics
PY - 2021
PB - University of Ostrava
IS - 1
SP - 35
EP - 48
AB - We describe how properties of metric groups and of unitary representations of metric groups can be presented in continuous logic. In particular we find $L_{\omega _1 \omega }$-axiomatization of amenability. We also show that in the case of locally compact groups some uniform version of the negation of Kazhdan’s property (T) can be viewed as a union of first-order axiomatizable classes. We will see when these properties are preserved under taking elementary substructures.
LA - eng
KW - Continuous logic; metric groups; unitary representations; amenable groups
UR - http://eudml.org/doc/297841
ER -

References

top
  1. Bekka, B., Harpe, P. de la, Valette, P., Kazhdan's Property (T), New Mathematical Monographs, 11, Cambridge University Press, Cambridge, 2008, (2008) MR2415834
  2. Yaacov, I. Ben, Berenstein, A., Henson, W., Usvyatsov, A., Model theory for metric structures, in: Model theory with Applications to Algebra and Analysis, v.2, Z. Chatzidakis, H.D. Macpherson, A. Pillay, A. Wilkie (eds.), London Math. Soc. Lecture Notes, v. 350, Cambridge University Press, 2008, 315 - 427, (2008) MR2436146
  3. Yaacov, I. Ben, Iovino, J., 10.1016/j.apal.2007.10.011, Annals of Pure and Appl. Logic, 158, 2009, 163 - 174, (2009) MR2500090DOI10.1016/j.apal.2007.10.011
  4. Bunina, E.I., Mikhalev, A.V., 10.1023/B:JOTH.0000036655.73484.a8, J. Math. Sci. (N. Y.), 123, 2004, 3921 - 3985, (2004) MR2096270DOI10.1023/B:JOTH.0000036655.73484.a8
  5. Doucha, M., 10.1007/s00605-019-01277-7, Monatsh. Mathematik, 189, 2020, 51 - 74, (2020) MR3948286DOI10.1007/s00605-019-01277-7
  6. Ershov, M., Jaikin-Zapirain, A., 10.1007/s00222-009-0218-2, Invent. Math., 179, 2010, 303 - 347, (2010) MR2570119DOI10.1007/s00222-009-0218-2
  7. Goldbring, I., Enforceable operator algebras, Journal of the Institute of Mathematics of Jussieu, 2019, 1 - 33, (2019) MR4205777
  8. Goldbring, I., On the nonexistence of Folner sets, arXiv:1901.02445, 2019, (2019) 
  9. Grigorchuk, R., Harpe, P. de la, Amenability and ergodic properties of topological groups: from Bogolyubov onwards, in: Groups, graphs and random walks, London Math. Soc. Lecture Note Ser., 436, Cambridge Univ. Press, Cambridge, 2017, 215 - 249, (2017) MR3644011
  10. Hernandes, S., Hofmann, K.H., Morris, S.A., The weights of closed subgroups of a locally compact subgroup, J. Group Theory, 15, 2012, 613 - 630, (2012) MR2982605
  11. Ivanov, A., 10.1007/s00153-016-0516-5, Arch. Math. Logic, 56, 2017, 67 - 78, (2017) MR3598797DOI10.1007/s00153-016-0516-5
  12. Ivanov, A., Actions of metric groups and continuous logic, arXiv:1706.04157, 2017, (2017) MR4251310
  13. Pestov, V., 10.1090/tran/7256, Trans. Amer. Math. Soc., 370, 2018, 7417 - 7436, (2018) MR3841853DOI10.1090/tran/7256
  14. Schneider, F.M., Thom, A., 10.1112/S0010437X1800708X, Compositio Mathematica, 154, 2018, 1333 - 1361, (2018) MR3809992DOI10.1112/S0010437X1800708X

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.