  In this paper we prove that a positive commutator between a positive compact
operator $A$ and a positive operator $B$ is in the radical of the Banach
algebra generated by $A$ and $B$. Furthermore, on every at least
three-dimensional Banach lattice we construct finite rank operators $A$ and $B$
satisfying $AB\geq BA\geq 0$ such that the commutator $AB-BA$ is not contained
in the radical of the Banach algebra generated by $A$ and $B$. These two
results now completely answer to two open questions published in [4]. We also
obtain relevant results in the case of the Volterra and the Donoghue operator.
