UNBOUNDED DERIVATIONS OF COMMUTATIVE CSTAR-ALGEBRAS - Batty, C (Springer-Verlag, 1978)
It is shown that an unbounded *-derivation δ of a unital commutative C*-algebra A is quasi well-behaved if and only if there is a dense open subset U of the spectrum of A such that, for any f in the domain of δ, δ(f) vanishes at any point of U where f attains its norm. An example is given to show that even if δ is closed it need not be quasi well-behaved. This answers negatively a question posed by Sakai for arbitrary C*-algebras. It is also shown that there are no-zero closed derivations on A if the spectrum of A contains a dense open totally disconnected subset. © 1978 Springer-Verlag.