Christopher J. Eagle

Saturation and elementary equivalence of C*-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 σ-unital C*-algebras; we extend their results by showing that some coronas of non-σ-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 also characterize elementary equivalence of the algebras C(X) in terms of CL(X) when X is 0-dimensional, and show that elementary equivalence of the generalized Calkin algebras of densities ℵα and ℵβ implies elementary equivalence of the ordinals α and β.

THE PSEUDOARC IS A CO-EXISTENTIALLY CLOSED CONTINUUM

Abstract Answering a question of P. Bankston, we show that the pseudoarc is a co-existentially closed continuum. We also show that C ( X ) , for X a nondegenerate continuum, can never have quantifier elimination, answering a question of the first and third named authors and Farah and Kirchberg.

Fraïssé limits of C*-algebras

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.

Omitting types for infinitary [0,1]-valued logic

Abstract We describe an infinitary logic for metric structures which is analogous to L ω 1 , ω . We show that this logic is capable of expressing several concepts from analysis that cannot be expressed in finitary continuous logic. Using topological methods, we prove an omitting types theorem for countable fragments of our infinitary logic. We use omitting types to prove a two-cardinal theorem, which yields a strengthening of a result of Ben Yaacov and Iovino concerning separable quotients of Banach spaces.