Coronas and strongly self-absorbing C*-algebras

Abstract

Let $\mathcal{D}$ be a strongly self-absorbing ${\mathrm{C}}^{\ast }$-algebra. Given any separable ${\mathrm{C}}^{\ast }$-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 ${\mathrm{C}}^{\ast }$-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 ${\mathrm{C}}^{\ast }$-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.

29 pages. Minor corrections