Kernel Stabilization of Unbounded Derivations on C*-algebras

A derivation $\delta$ on a $C^{\ast }$-algebra has kernel stabilization if for all $n\in \mathbb{N}$, $\ker {\delta }^n=\ker \delta \mathrm{.}$ Our main result shows that a weakly-defined derivation studied recently by E. Christensen has kernel stabilization. As corollaries, we (1) show that a family of $\ast$-derivations on $C^{\ast }$-algebras studied by Bratteli and Robinson has kernel stabilization and (2) provide sufficient conditions for when operators satisfying the Heisenberg Commutation Relation must both be unbounded.