A Dichotomy for Derivations on ${\mathcal{O}}_n$

Abstract

Let ${\mathcal{O}}_n$ be the Cuntz algebra generated by $s_1,\cdots ,s_n$, and let $\mathcal{P}({\mathcal{O}}_n)$ be the ${}^{\ast }$-subalgebra of ${}^{\ast }$-polynomials in the generators. We show that if $\delta$ is a gauge-invariant derivation mapping $\mathcal{P}({\mathcal{O}}_n)$ into $\mathcal{P}({\mathcal{O}}_n)$, and $\delta$ is approximately inner, then $\delta$ is inner.