METRIC VERSIONS OF POSNER’S THEOREMS

Let $S$ and $T$ be continuous linear operators on an ultraprime Banach algebra $A$. We show that if $S$, $T$, and $ST$ are close to satisfy the derivation identity on $A$, then either $S$ or $T$ approaches to zero. If $T$ is close to satisfy the derivation identity and $[T(a),a]$ is near the centre of $A$ for each $a \in A$, then either $T$ approaches to zero or $A$ is nearly commutative. Further, we give quantitative estimates of these phenomena.