For each derivation δ of a
C∗-algebra A with δ(x∗) = −δ(x)∗ there exists a minimal positive element h in the
enveloping von Neumann algebra A′′ such that δ(x) = hx−xh. It is shown that the
generator h belongs to the class of lower semi-continuous elements in A′′. From this it
follows that if the function π →∥π ∘ δ∥ is continuous on the spectrum of A then h
multiplies A. This immediately implies that each derivation of a simple C∗-algebra is
given by a multiplier of the algebra. Another application shows that each
derivation of a countably generated monotone sequentially closed C∗. algebra is
inner.
