Coronas and strongly self-absorbing C*-algebras

TL;DR

This paper investigates the structure of corona and multiplier algebras of $ ext{C}^*$-algebras, establishing conditions under which $ ext{D}$-stability is preserved and providing new insights into their embeddings and stability properties.

Contribution

It generalizes recent results on $ ext{C}^*$-algebra stability, characterizing $ ext{D}$-stability of corona and multiplier algebras for strongly self-absorbing $ ext{C}^*$-algebras.

Findings

Corona algebra of a $ ext{D}$-stable algebra is $ ext{D}$-saturated.

If the stable corona is $ ext{D}$-stable, then the algebra itself is $ ext{D}$-stable.

Multiplier algebra of a separable $ ext{D}$-stable algebra is also $ ext{D}$-stable.

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.