Abstract. We extend the classical Feferman-Vaught theorem to logic for metric structures. This implies that the reduced powers of elementarily equivalent structures are elementarily equivalent, and therefore they are isomorphic under the Continuum Hypothesis. We also prove the existence of two separable C*-algebras of the form i M k(i) (C) such that the assertion that their coronas are isomorphic is independent from ZFC, which gives the first example of genuinely non-commutative coronas of separable C*-algebras with this property.
We analyze the sequence obtained by consecutive applications of the Cantor-Bendixson derivative for a noncommutative scattered C * -algebra A, using the ideal I At (A) generated by the minimal projections of A. With its help, we present some fundamental results concerning scattered C * -algebras, in a manner parallel to the commutative case of scattered compact or locally compact Hausdorff spaces and superatomic Boolean algebras. It also allows us to formulate problems which have motivated the "cardinal sequences" programme in the classical topology, in the noncommutative context. This leads to some new constructions of noncommutative scattered C * -algebras and new open problems. In particular, we construct a type I C * -algebra which is the inductive limit of stable ideals Aα, along an uncountable limit ordinal λ, such that A α+1 /Aα is * -isomorphic to the algebra of all compact operators on a separable Hilbert space and A α+1 is σ-unital and stable for each α < λ, but A is not stable and where all ideals of A are of the form Aα. In particular, A is a nonseparable C * -algebra with no ideal which is maximal among the stable ideals. This answers a question of M. Rørdam in the nonseparable case. All the above C * -algebras Aαs and A satisfy the following version of the definition of an AF algebra: any finite subset can be approximated from a finite-dimensional subalgebra. Two more complex constructions based on the language developed in this paper are presented in separate papers [25,26].Proof. We prove J α = I α ⊗ K(ℓ 2 ) by induction on α ≤ ht(A). For α = 1 this follows from Lemma 5.2. At a successor ordinal by Lemma 5.2 and the inductive assumption we have J α+1 /J α =
Abstract. We consider isomorphisms between quotient algebras of ∞ n=0 M k(n) (C) associated with Borel ideals on N and prove that it is relatively consistent with ZFC that all of these isomorphisms are trivial, in the sense that they lift to a *-homomorphism from ∞ n=0 M k(n) (C) into itself. This generalizes a result of Farah-Shelah who proved this result for centers of these algebras, in its dual form.
We construct a nonhomogeneous, separably represented, type I and approximately finite dimensional C * -algebra such that its multiplier algebra is equal to its unitization. This algebra is an essential extension of the algebra K(ℓ 2 (c)) of compact operators on a nonseparable Hilbert space by the algebra K(ℓ 2 ) of compact operators on a separable Hilbert space, where c denotes the cardinality of continuum. Although both K(ℓ 2 (c)) and K(ℓ 2 ) are stable, our algebra is not. This sheds light on the permanence properties of the stability in the nonseparable setting. Namely, unlike in the separable case, an extension of a stable nonseparable C * -algebra by K(ℓ 2 ) does not have to be stable. Our construction can be considered as a noncommutative version of Mrówka's Ψ-space; a space whose one point compactification is equal to itš Cech-Stone compactification and is induced by a special uncountable family of almost disjoint subsets of N.
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