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We realize the Jiang-Su algebra, all UHF algebras, and the hyperfinite II$_{1}$ factor as Fraisse limits of suitable classes of structures. Moreover by means of Fraisse theory we provide new examples of AF algebras with strong homogeneity properties. As a consequence of our analysis we deduce Ramsey-theoretic results about the class of full-matrix algebras.
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The only C*-algebras that admit elimination of quantifiers in continuous logic are $mathbb{C}, mathbb{C}^2$, $C($Cantor space$)$ and $M_2(mathbb{C})$. We also prove that the theory of C*-algebras does not have model companion and show that the theory of $M_n(mathcal {O_{n+1}})$ is not $forallexists$-axiomatizable for any $ngeq 2$.
We study the saturation properties of several classes of $C^*$-algebras. Saturation has been shown by Farah and Hart to unify the proofs of several properties of coronas of $sigma$-unital $C^*$-algebras; we extend their results by showing that some coronas of non-$sigma$-unital $C^*$-algebras are countably degree-$1$ saturated. We then relate saturation of the abelian $C^*$-algebra $C(X)$, where $X$ is $0$-dimensional, to topological properties of $X$, particularly the saturation of $CL(X)$.
The general theory developed by Ben Yaacov for metric structures provides Fraisse limits which are approximately ultrahomogeneous. We show here that this result can be strengthened in the case of relational metric structures. We give an extra condition that guarantees exact ultrahomogenous limits. The condition is quite general. We apply it to stochastic processes, the class of diversities, and its subclass of $L_1$ diversities.
Work of Eagle, Farah, Goldbring, Kirchberg, and Vignati shows that the only separable C*-algebras that admit quantifier elimination in continuous logic are $mathbb{C},$ $mathbb{C}^2,$ $M_2(mathbb{C}),$ and the continuous functions on the Cantor set. We show that, among finite dimensional C*-algebras, quantifier elimination does hold if the language is expanded to include two new predicate symbols: One for minimal projections, and one for pairs of unitarily conjugate projections. Both of these predicates are definable, but not quantifier-free definable, in the usual language of C*-algebras. We also show that adding just the predicate for minimal projections is sufficient in the case of full matrix algebras, but that in general both new predicate symbols are required.
We study the $C^*$-algebra $mathbb{E}_{mathscr{M}}$ of a smooth 4-dimensional manifold $mathscr{M}$ introduced by Gabor Etesi. It is proved that the $mathbb{E}_{mathscr{M}}$ is a stationary AF-algebra. We calculate the topological and smooth invariants of $mathscr{M}$ in terms of the K-theory of the $C^*$-algebra $mathbb{E}_{mathscr{M}}$. Using Gompfs Stable Diffeomorphism Theorem, it is shown that all smoothings of $mathscr{M}$ form a torsion abelian group. The latter is isomorphic to the Brauer group of a number field associated to the K-theory of $mathbb{E}_{mathscr{M}}$.
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