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

TL;DR

This paper investigates the structural differences of the Calkin algebra compared to other well-known C*-algebras, using properties of relative commutants, and highlights its unique non-isomorphism characteristics.

Contribution

It demonstrates that the Calkin algebra is not isomorphic to various corona algebras associated with Kirchberg and ${ m C}^*$-algebras, using properties of relative commutants.

Findings

Calkin algebra is not isomorphic to the corona of the stabilization of ${ m O}_infty$

Calkin algebra is not isomorphic to the corona of any unital, ${ m Z}$-stable ${ m C}^*$-algebra

Proof relies on properties of relative commutants of separable ${ m C}^*$-subalgebras

Abstract

The Calkin algebra is not isomorphic to the corona of the stabilization of the Cuntz algebra~${\mathcal O}_\infty$, any other Kirchberg algebra, or even the corona of the stabilization of any unital, ${\mathcal Z}$-stable ${\mathrm C}^*$-algebra. The proof relies on properties of relative commutants of separable ${\mathrm C}^*$-subalgebras.