Yaacov, Itaï Ben; Goldbring, Isaac Unitary representations of locally compact groups as metric structures. (English) Zbl 1537.03043 Notre Dame J. Formal Logic 64, No. 2, 159-172 (2023). For a locally compact group \(G\), the authors show that the class of continuous unitary representations of \(G\) is elementary (that is: axiomatizable) in the sense of continuous logic. They also relate the notion of ultraproduct in the sense of (continuous) logic with other notions of ultraproduct of representations appearing in the literature. The authors also obtain an interesting result (Theorem 4.2) about a model-theoretic characterization of Kazhdan’s property (T) in a locally compact group \(G\) in terms of definability of certain sets of fixed points in the associated structure. Reviewer: Piotr Kowalski (Wrocław) MSC: 03C60 Model-theoretic algebra 03C20 Ultraproducts and related constructions 22D10 Unitary representations of locally compact groups 22D20 Representations of group algebras 22D55 Kazhdan’s property (T), the Haagerup property, and generalizations Keywords:continuous logic; continuous unitary representation; locally compact group; metric structure; nondegenerate \(\ast\)-representation; property (T); ultrapower; ultraproduct × Cite Format Result Cite Review PDF Full Text: DOI arXiv HAL References: [1] Bekka, B., P. de la Harpe, and A. Valette, Kazhdan’s Property (T), vol. 11 of New Mathematical Monographs, Cambridge University Press, Cambridge, 2008. · Zbl 1146.22009 · doi:10.1017/CBO9780511542749 [2] Ben Yaacov, I., “Modular functionals and perturbations of Nakano spaces,” Journal of Logic and Analysis, vol. 1 (2009), no. 1. · Zbl 1163.03021 · doi:10.4115/jla.2009.1.1 [3] Berenstein, A., “Hilbert spaces with generic groups of automorphisms,” Archive for Mathematical Logic, vol. 46 (2007), pp. 289-99. · Zbl 1115.03028 · doi:10.1007/s00153-007-0044-4 [4] Cherix, P.-A., M. Cowling, and B. Straub, “Filter products of \(C_0\)-semigroups and ultraproduct representations for Lie groups,” Journal of Functional Analysis vol. 208 (2004), pp. 31-63. · Zbl 1060.47043 · doi:10.1016/j.jfa.2003.06.007 [5] Dacunha-Castelle, D., and J.-L. Krivine, “Applications des ultraproduits à l’étude des espaces et des algèbres de Banach,” Studia Mathematica, vol. 41 (1972), pp. 315-34. · Zbl 0275.46023 · doi:10.4064/sm-41-3-315-334 [6] Folland, G B., A Course in Abstract Harmonic Analysis, CRC Press, Boca Raton, 1995. · Zbl 0857.43001 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.