[Paper Review] A non-stable C*-algebra with an elementary essential composition series

Saeed Ghasemi, Piotr Koszmider|arXiv (Cornell University)|Dec 6, 2017

Advanced Operator Algebra Research21 references3 citations

TL;DR

This paper constructs a non-stable AF $C^*$-algebra that is an uncountable inductive limit of separable stable ideals, each quotient isomorphic to $\mathcal{K}(\ell_2)$, demonstrating a failure of stability permanence in the nonseparable setting. The key contribution is a GCR, scattered $C^*$-algebra with an elementary essential composition series of length $\omega_1$ that is not isomorphic to its tensor product with $\mathcal{K}(\ell_2)$, despite all initial ideals being stable.

ABSTRACT

A C*-algebra $A$ is said to be stable if it is isomorphic to $A \otimes K(\ell_2)$. Hjelmborg and Rørdam have shown that countable inductive limits of separable stable 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 $A$ of $B(\ell_2)$, which is the inductive limit of length $ω_1$ of its separable stable ideals $I_α$ ($α

Motivation & Objective

• To demonstrate that stability is not preserved under uncountable inductive limits of separable stable $C^*$-algebras, even in the AF and GCR context.
• To construct a $C^*$-algebra with an elementary essential composition series of length $\omega_1$ where all successive quotients are $\mathcal{K}(\ell_2)$, yet the algebra is not stable.
• To show that the family of stable ideals in such an algebra has no maximal element, implying a structural obstruction to stability.
• To establish a counterexample to the extension of Hjelmborg and Rørdam's stability result to the nonseparable case.

Proposed method

• Construct a $C^*$-subalgebra $\mathcal{A} \subseteq \mathcal{B}(\ell_2)$ as a transfinite inductive limit of length $\omega_1$ of separable stable AF-ideals $\mathcal{I}_\alpha$.
• Ensure each quotient $\mathcal{I}_{\alpha+1}/\mathcal{I}_\alpha \cong \mathcal{K}(\ell_2)$, forming an elementary essential composition series.
• Use a representing sequence of operators in $B(\ell_2)$ that dominates a Luzin blockwise system of almost matrix units.
• Apply a contradiction argument via projection height analysis and norm estimates involving compact perturbations and almost matrix units.
• Leverage the Luzin property of the almost matrix unit system to derive a norm lower bound contradicting the zero norm of $R_\alpha Q_\beta$.
• Use the isomorphism $\mathcal{A} \cong \mathcal{A} \otimes \mathcal{K}(\ell_2)$ in the stable case to derive a sequence of projections with orthogonal supports and controlled heights.

Experimental results

Research questions

• RQ1 Does the stability of separable $C^*$-algebras persist under uncountable inductive limits?
• RQ2 Can a $C^*$-algebra with an elementary essential composition series of length $\omega_1$ and quotients isomorphic to $\mathcal{K}(\ell_2)$ fail to be stable?
• RQ3 Is there a non-stable $C^*$-algebra whose proper ideals are all stable and form a chain of length $\omega_1$?
• RQ4 Can two non-isomorphic scattered $C^*$-algebras have isomorphic composition series up to $\omega_1$, differing only at the final stage?

Key findings

• The constructed $C^*$-algebra $\mathcal{A}$ is a GCR, AF, and scattered $C^*$-algebra that is not stable, despite being an inductive limit of separable stable ideals.
• The algebra $\mathcal{A}$ has an elementary essential composition series $(\mathcal{I}_\alpha)_{\alpha \leq \omega_1}$ with $\mathcal{I}_{\alpha+1}/\mathcal{I}_\alpha \cong \mathcal{K}(\ell_2)$ for all $\alpha < \omega_1$.
• All proper two-sided ideals of $\mathcal{A}$ are of the form $\mathcal{I}_\alpha$ for some $\alpha < \omega_1$, so the family of stable ideals has no maximal element.
• The algebra $\mathcal{A}$ is not isomorphic to $\mathcal{A} \otimes \mathcal{K}(\ell_2)$, proving it is not stable.
• A stable $C^*$-algebra $\mathcal{A}' = \mathcal{A} \otimes \mathcal{K}(\ell_2)$ has a composition series with ideals isomorphic to $\mathcal{I}_\alpha$, showing non-isomorphic algebras can have isomorphic composition series up to $\omega_1$.
• The contradiction in the proof arises from norm estimates: the Luzin property forces $\|(A_\alpha + F)(B_\beta + G)\| > 3\varepsilon$, contradicting $\|R_\alpha Q_\beta\| = 0$.