In this paper we study expansions of infinite dimensional Hilbert spaces with a unitary representation of a discrete countable group. When the group is finite, we prove the theory of the corresponding expansion, regardless if it is existentially closed, has quantifier elimination, is $\aleph _0$-categorical, $\aleph _0$-stable and SFB. On the other hand, when the group involved is countably infinite, the theory of the Hilbert space expanded by the representation of this group is $\aleph _0$-categorical up to perturbations. Additionally, when the expansion is model complete, we prove that it is $\aleph _0$-stable up to perturbations.
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Keywords: Hilbert spaces, representation theory, $C^*$-algebras, stability theory, belle paires, classification theory, perturbations
Alexander Berenstein 1 ; Juan Perez 2
CC-BY-NC-ND 4.0
@article{CML_2025__17__33_0,
author = {Alexander Berenstein and Juan Perez},
title = {Model theory of {Hilbert} spaces with a discrete group action},
journal = {Confluentes Mathematici},
pages = {33--54},
year = {2025},
publisher = {Institut Camille Jordan},
volume = {17},
doi = {10.5802/cml.99},
language = {en},
url = {https://cml.centre-mersenne.org/articles/10.5802/cml.99/}
}
TY - JOUR AU - Alexander Berenstein AU - Juan Perez TI - Model theory of Hilbert spaces with a discrete group action JO - Confluentes Mathematici PY - 2025 SP - 33 EP - 54 VL - 17 PB - Institut Camille Jordan UR - https://cml.centre-mersenne.org/articles/10.5802/cml.99/ DO - 10.5802/cml.99 LA - en ID - CML_2025__17__33_0 ER -
%0 Journal Article %A Alexander Berenstein %A Juan Perez %T Model theory of Hilbert spaces with a discrete group action %J Confluentes Mathematici %D 2025 %P 33-54 %V 17 %I Institut Camille Jordan %U https://cml.centre-mersenne.org/articles/10.5802/cml.99/ %R 10.5802/cml.99 %G en %F CML_2025__17__33_0
Alexander Berenstein; Juan Perez. Model theory of Hilbert spaces with a discrete group action. Confluentes Mathematici, Tome 17 (2025), pp. 33-54. doi: 10.5802/cml.99
[1] Hilbert spaces expanded with a unitary operator, Math. Log. Q., Volume 55 (2009) no. 1, pp. 37-50 | DOI | Zbl
[2] On perturbations of continuous structures, J. Math. Log., Volume 8 (2008) no. 2, pp. 225-249 | DOI | Zbl
[3] Modular functionals and perturbations of Nakano spaces, J. Log. Anal., Volume 1 (2009), 1, 42 pages | DOI | Zbl
[4] On perturbations of Hilbert spaces and probability algebras with a generic automorphism, J. Log. Anal., Volume 1 (2009), 7, 18 pages | DOI | Zbl
[5] Almost indiscernible sequences and convergence of canonical bases, J. Symb. Log., Volume 79 (2014) no. 2, pp. 460-484 | DOI | Zbl
[6] Model theory for metric structures, Model theory with applications to algebra and analysis. Vol. 2 (London Mathematical Society Lecture Note Series), Volume 350, Cambridge University Press, 2008, pp. 315-427 | Zbl
[7] Continuous first order logic and local stability, Trans. Am. Math. Soc., Volume 362 (2010) no. 10, pp. 5213-5259 | DOI | Zbl | MR
[8] Generic automorphism of a Hilbert space (2008)
[9] Hilbert spaces with generic groups of automorphisms, Arch. Math. Logic, Volume 46 (2007) no. 3-4, pp. 289-299 | DOI | Zbl | MR
[10] Hilbert spaces with generic predicates, Rev. Colomb. Mat., Volume 52 (2018) no. 1, pp. 107-130 | Zbl | DOI | MR
[11] -algebras by example, Fields Institute Monographs, 6, American Mathematical Society, 1996 | Zbl | MR
[12] Linear operator theory in engineering and science, Applied Mathematical Sciences, 40, Springer, 1982 (Reprint of the 1971 original, publ. by Holt, Rinehart & Winston, Inc.) | Zbl | DOI
[13] On perturbations of continuous structures [Paires de structures stables], J. Symb. Log., Volume 48 (1983), pp. 239-249 | DOI | Zbl | MR
[14] Linear representations of finite groups, Graduate Texts in Mathematics, 42, Springer, 1977 (Translated from the French by Leonard L. Scott) | Zbl | DOI | MR
[15] Unitary representations of locally compact groups as metric structures, Notre Dame J. Formal Logic, Volume 64 (2023) no. 2, pp. 159-172 | DOI | Zbl | MR
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