Isomorphisms of quotients of FDD-algebras

Abstract

We consider isomorphisms between quotient algebras of ${\prod }_{n=0}^{\infty }{\mathbb{M}}_{k(n)}(\mathbb{C})$ associated with Borel ideals on $\mathbb{N}$ and prove that it is relatively consistent with \textbf{ZFC} that all of these isomorphisms are trivial, in the sense that they lift to a *-homomorphism from ${\prod }_{n=0}^{\infty }{\mathbb{M}}_{k(n)}(\mathbb{C})$ into itself. This generalizes a result of Farah-Shelah who proved this result for centers of these algebras (in its dual form). We also use a simpler forcing notion and completely remove the large cardinal assumption used by Farah-Shelah.

References in corpus (2)

• Automorphisms of corona algebras, and group cohomology
• Reduced products of UHF algebras under forcing axioms