The Calkin algebra, Kazhdan's property (T), and strongly self-absorbing C*-algebras

It is well known that the relative commutant of every separable nuclear C*-subalgebra of the Calkin algebra has a unital copy of Cuntz algebra O-infinity We prove that the Calkin algebra has a separable C*-subalgebra whose relative commutant has no simple, unital, and noncom-mutative C*-subalgebra. On the other hand, the corona of every stable, separable C*-algebra that tensorially absorbs the Jiang-Su algebra Z has the property that the relative commutant of every separable C*-subalgebra contains a unital copy of Z. Analogous result holds for other strongly self-absorbing C*-algebras. As an applica-tion, the Calkin algebra is not isomorphic to the corona of the stabilization of the Cuntz algebra O-infinity, any other Kirchberg algebra, or even the corona of the stabilization of any unital, Z-stable C*-algebra.