Coronas and strongly self-absorbing C*-algebras
Ilijas Farah, Gábor Szabó
TL;DR
The paper develops a general framework linking strongly self-absorbing $\\mathcal{D}$-algebras with the structure of corona algebras. It proves that for a $\\sigma$-unital, separably $\\mathcal{D}$-stable $A$, the corona $\\mathcal{Q}(A)$ is $\\mathcal{D}$-saturated, and establishes a partial converse showing that separable $\\mathcal{D}$-stability of $A$ is equivalent to separable $\\mathcal{D}$-stability of both $\\mathcal{Q}(\\mathcal{K}\\otimes A)$ and $\\mathcal{M}(A)$ under suitable hypotheses. A technical backbone comprises a refinement of Schur’s Lemma for property (T) groups and a transfer mechanism that moves approximately central sequences from stable coronas to multiplier algebras. The work yields notable applications, including model-theoretic and classification consequences and partial answers to rigidity questions about corona isomorphisms, while also clarifying the non-equivalence of the Calkin algebra with nuclear algebras in first-order theory.
Abstract
Let $\mathcal D$ be a strongly self-absorbing $\mathrm{C}^*$-algebra. Given any separable $\mathrm{C}^*$-algebra $A$, our two main results assert the following. If $A$ is $\mathcal D$-stable, then the corona algebra of $A$ is $\mathcal D$-saturated, i.e., $\mathcal D$ embeds unitally into the relative commutant of every separable $\mathrm{C}^*$-subalgebra. Conversely, assuming that the stable corona of $A$ is separably $\mathcal D$-stable, we prove that $A$ is $\mathcal D$-stable. This generalizes recent work by the first-named author on the structure of the Calkin algebra. As an immediate corollary, it follows that the multiplier algebra of a separable $\mathcal D$-stable $\mathrm{C}^*$-algebra is separably $\mathcal D$-stable. Appropriate versions of the aforementioned results are also obtained when $A$ is not necessarily separable. The article ends with some non-trivial applications.