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)C(X), where X is 0-dimensional, to topological properties of X  , particularly the saturation of CL(X)CL(X). We also characterize elementary equivalence of the algebras C(X)C(X) in terms of CL(X)CL(X) when X   is 0-dimensional, and show that elementary equivalence of the generalized Calkin algebras of densities ℵαℵα and ℵβℵβ implies elementary equivalence of the ordinals α and β.