The Calkin algebra, Kazhdan's property (T), and strongly self‐absorbing C∗$\mathrm{C}^*$‐algebras

Abstract

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