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Register a new userQuantifier elimination in C*-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 $ngeq 2$.
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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 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.
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 $CL(X)$.
We present a detailed exposition (for a Dynamical System audience) of the content of the paper: R. Exel and A. Lopes, $C^*$ Algebras, approximately proper equivalence relations and Thermodynamic Formalism, {it Erg. Theo. and Dyn. Syst.}, Vol 24, pp 1051-1082 (2004). We show only the uniqueness of the beta-KMS (in a certain C*-Algebra obtained from the operators acting in $L^2$ of a Gibbs invariant probability $mu$) and its relation with the eigen-probability $ u_beta$ for the dual of a certain Ruele operator. We consider an example for a case of Hofbauer type where there exist a Phase transition for the Gibbs state. There is no Phase transition for the KMS state.
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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