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

Abstract

It is well known that the relative commutant of every separable nuclear (Formula presented.) -subalgebra of the Calkin algebra has a unital copy of Cuntz algebra (Formula presented.). We prove that the Calkin algebra has a separable (Formula presented.) -subalgebra whose relative commutant has no simple, unital, and noncommutative (Formula presented.) -subalgebra. On the other hand, the corona of every stable, separable (Formula presented.) -algebra that tensorially absorbs the Jiang–Su algebra (Formula presented.) has the property that the relative commutant of every separable (Formula presented.) -subalgebra contains a unital copy of (Formula presented.). Analogous result holds for other strongly self-absorbing (Formula presented.) -algebras. As an application, the Calkin algebra is not isomorphic to the corona of the stabilization of the Cuntz algebra (Formula presented.), any other Kirchberg algebra, or even the corona of the stabilization of any unital, (Formula presented.) -stable (Formula presented.) -algebra.