This paper focuses on the applications of continuous model theory to the study of operator algebras. In particular, it explores (weak) Fraïssé theory, with the scope of showing, using only model theoretic tools, that the Jiang-Su algebra \(\mathcal Z\) is strongly self-absorbing [X.-H. Jiang and H.-B. Su, Am. J. Math. 121, No. 2, 359–413 (1999; Zbl 0923.46069)]. The author gives a self-contained and rather elementary proof of this fact.
The \(\mathrm{C}^*\)-algebra \(\mathcal Z\) can be viewed as the (stably finite) infinite dimensional version of the complex numbers \(\mathbb C\); \(\mathcal Z\) plays a pivotal role in the classification of infinite-dimensional simple separable nuclear \(\mathrm{C}^*\)-algebras, where tensorially absorbing \(\mathcal Z\) is proved to be equivalent to a finite-dimensionality condition [J. Castillejos et al., Invent. Math. 224, No. 1, 245–290 (2021; Zbl 1467.46055)]. \(\mathcal Z\) was approached with Fraïssé theoretic methods in [C. J. Eagle et al., J. Symb. Log. 81, No. 2, 755–773 (2016; Zbl 1383.03046)] and [S. Masumoto, J. Symb. Log. 82, No. 4, 1541–1559 (2017; Zbl 1470.03019)].
The notions of Fraïssé classes and Fraïssé limits were originally introduced by R. Fraïssé [Ann. Sci. Éc. Norm. Supér. (3) 71, 363–388 (1954; Zbl 0057.04206)] as a method to construct countable homogeneous structures. Since then, Fraïssé theory has become an influential area of mathematics at the crossroads of combinatorics and model theory. Broadly speaking, Fraïssé theory studies the correspondence between homogeneous structures and properties of the classes of their finitely generated substructures. In the setting of continuous model theory, Fraïssé theory was developed in [I. Ben Yaacov, J. Symb. Log. 80, No. 1, 100–115 (2015; Zbl 1372.03070)] and further in [C. J. Eagle et al., J. Symb. Log. 81, No. 2, 755–773 (2016; Zbl 1383.03046)] where it was brought specifically to the setting of \(\mathrm{C}^*\)-algebras.
In both [C. J. Eagle et al., J. Symb. Log. 81, No. 2, 755–773 (2016; Zbl 1383.03046)] and [S. Masumoto, J. Symb. Log. 82, No. 4, 1541–1559 (2017; Zbl 1470.03019)], it was shown that \(\mathcal Z\) is a Fraïssé limit of the class of dimension drop algebras together with a specified faithful diffuse trace. (A dimension drop algebra is one of the form \[ \mathcal Z_{p,q}=\{f\in C([0,1],M_{p}\otimes M_q)\mid f(0)\in 1\otimes M_q\wedge f(1)\in M_p\otimes 1\} \] where \(p\) and \(q\) are coprime.) Here the author considers the class of all dimension drop algebras and their tensor products \(Z_{p,q}\otimes Z_{p',q'}\) with distinguished faithful diffuse traces. Although this is not a Fraïssé class if all trace preserving \(^*\)-homomorphisms are considered, it is a weak Fraïssé class, in that one only asks about the weak near amalgamation property: for all \(A\) in the weak Fraïssé class \(\mathcal K\), \(\epsilon>0\) and a finite \(F\subseteq A\), there is an object \(A'\in\mathcal K\) and a \(\mathcal K\)-morphism \(A\to A'\) such that for all objects \(B\) and \(C\) in \(\mathcal K\) and \(\mathcal K\)-morphisms \(A'\to B\) and \(A'\to C\), one can amalgamate over \(A\). Weak Fraïssé classes were introduced in [A. A. Ivanov, J. Symb. Log. 64, No. 2, 775–789 (1999; Zbl 0930.03034)] and [A. S. Kechris and C. Rosendal, Proc. Lond. Math. Soc. (3) 94, No. 2, 302–350 (2007; Zbl 1118.03042)]; later their study was boosted by W. Kubiś [Ann. Pure Appl. Logic 165, No. 11, 1755–1811 (2014; Zbl 1329.18002)] and Kubiś’s unpublished work on weak Fraïssé categories.
The author shows that the class whose objects are \(Z_{p,q}\) and \(Z_{p,q}\otimes Z_{p',q'}\) (with specified traces) and whose morphisms are certain trace preserving \(^*\)-homomorphisms is a weak Fraïssé class, and that the Jiang-Su sequence and its tensor product are weak generic sequences, showing therefore that \(\mathcal Z\) and \(\mathcal Z\otimes\mathcal Z\) are isomorphic in a very strong sense (i.e., \(\mathcal Z\) is strongly self-absorbing).