Kernel Stabilization of Unbounded Derivations on C*-algebras

TL;DR

This paper investigates kernel stabilization of derivations on C*-algebras, proving that certain weakly-defined derivations and families of *-derivations exhibit this property, with implications for operators satisfying the Heisenberg relation.

Contribution

It demonstrates kernel stabilization for specific classes of derivations on C*-algebras, including weakly-defined and *-derivations, and provides conditions for unbounded operators satisfying the Heisenberg relation.

Findings

Weakly-defined derivation studied by E. Christensen has kernel stabilization.

A family of *-derivations on C*-algebras also exhibits kernel stabilization.

Provides conditions under which operators satisfying the Heisenberg relation are unbounded.

Abstract

A derivation $\delta$ on a $C^*$-algebra has kernel stabilization if for all $n\in \mathbb{N}$, $\ker \delta^n=\ker \delta.$ 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 $*$-derivations on $C^*$-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.