Can we construct an unbounded derivation on abelian C* algebra which is not closable?
One of possible construction may be found in the paper by Bratteli and Robinson(Unbounded derivations of C*-algebras). In the paper they construct $\delta_{0}$ in theorem 15. The only question is that there are no "theorem 15" in this paper. I doubt this is a typo. The only related theorem seems to be theorem 12, but $\delta_{0}$ there is differentiation, which is closable. So I don't know if that counts.