Fraïssé theory for Cuntz semigroups (Q6597504)

This article is in Fraïssé theory, which is the study of generic `special' objects from a model-theoretic perspective. Fraïssé theory, after [\textit{R. Fraïssé}, C. R. Acad. Sci., Paris 237, 508--510 (1953; Zbl 0053.02904)], is an area of mathematics at the crossroads of combinatorics and model theory. Fraïssé theory studies countable homogeneous structures and their relations with properties of their finitely-generated substructures. Given a class of finitely-generated objects \(\mathcal C\), once \(\mathcal C\) satisfies a few combinatorial properties, notably amalgamation, one shows that there exists a \emph{generic}, special, object \(M\), called the Fraïssé limits of \(\mathcal C\), which is universal for countable objects constructed from \(\mathcal C\) and it has the property that all objects in \(\mathcal C\) sit inside \(M\) in a unique way (up to automorphisms of \(M\)). The perk of this approach is that often one is capable to study a quite complicated object (the limit \(M\)) by looking at `easier', `simpler' building blocks (the objects of \(\mathcal C\)) and how these interact. Fraïssé theory found profound applications in many areas of combinatorics [\textit{A. S. Kechris} et al., Geom. Funct. Anal. 15, No. 1, 106--189 (2005; Zbl 1084.54014)] and algebra, and it has recently been exploited in the context of continuous model theory, notably for functional analytic and operator algebraic considerations (see e.g. [\textit{I. Ben Yaacov}, J. Symb. Log. 80, No. 1, 100--115 (2015; Zbl 1372.03070); \textit{A. Vignati}, De Gruyter Ser. Log. Appl. 11, 453--478 (2023; Zbl 07756065); \textit{M. Lupini}, Adv. Math. 338, 93--174 (2018; Zbl 1405.46041)]).\N\NIn this article, the authors applied this way of thinking to Cuntz semigroups, in the context of metric enriched categories as developed by \textit{W. Kubiś} [``Metric-enriched categories and approximate Fraïssé limits'', Preprint, \url{arXiv:1210.6506}]. These are algebraic objects tidily connected with the study of \(\mathrm{C}^*\)-algebras. Originally introduced by Cuntz, the Cuntz semigroup of a \(\mathrm{C}^*\)-algebra offers a `refined/continuous \(K\)-theory', measuring the dimension of, and comparing, positive elements (instead that just projections). An abstract category \(\mathrm{Cu}\) of Cuntz semigroups was defined (e.g., [\textit{K. T. Coward} et al., J. Reine Angew. Math. 623, 161--193 (2008; Zbl 1161.46029)]). This category was extensively studied on its own and for its applications to \(\mathrm{C}^*\)-algebras theory, as it comes with a natural functor from the category of \(\mathrm{C}^*\)-algebras. Cuntz semigroups are, nowadays, one of the main source of new research ideas in \(\mathrm{C}^*\)-algebras, especially stably finite ones (e.g., [\textit{N. P. Brown} et al., J. Reine Angew. Math. 621, 191--211 (2008; Zbl 1158.46040); \textit{R. Antoine} et al., Duke Math. J. 171, No. 1, 33--99 (2022; Zbl 1490.19005)]).\N\NThe main results of the article focus on developing the Fraïssé theory of Cuntz semigroup, to provide a sound environment for FraÍssé theoretic considerations in this setting. They analyze the basic properties of this theory, giving the definition of a (Cuntz) FraÍssé categories and showing the existence of a generic (Fraïssé) limit for this classes satisfying of the genericity regularity conditions, such as universality and homogeneity. They also provide example of classes of Cuntz semigroups to which this approach can be applied.