A nonstable 𝐶*-algebra with an elementary essential composition series
Saeed Ghasemi, Piotr Koszmider · Proceedings of the American Mathematical Society · 2019
A C ∗ C^* -algebra A \mathcal {A} is said to be stable if it is isomorphic to A ⊗ K ( ℓ 2 ) \mathcal {A} \otimes \mathcal {K}(\ell _2) . Hjelmborg and Rørdam have shown that countable inductive limits of separable stable C ∗ C^* -algebras are stable. We show that this is no longer true in the nonseparable context even for the most natural case of an uncountable inductive limit of an increasing chain of separable stable and AF ideals: we construct a GCR, AF (in fact, scattered) subalgebra A \mathcal {A} of B ( ℓ 2 ) \mathcal {B}(\ell _2) , which is the inductive limit of length ω 1 \omega _1 of its separable stable ideals I α \mathcal {I}_\alpha ( α > ω 1 \alpha >\omega _1 ) satisfying I α + 1 / I