Unbounded derivations of commutativeC*-algebras

It is shown that an unbounded *-derivation of a unital commutativeC*-algebraA is quasi well-behaved if and only if there is a dense open subsetU of the spectrum ofA such that, for anyf in the domain of , (f) vanishes at any point ofU wheref attains its norm. An example is given to show that even if is closed it need not be quasi well-behaved. This answers negatively a question posed by Sakai for arbitraryC*-algebras.It is also shown that there are no-zero closed derivations onA if the spectrum ofA contains a dense open totally disconnected subset.