A non-closable unbounded ∗-derivation.

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Let X be the Cantor set of [0, 1]. It is well-known that X is a perfect compact subset of [0, 1]. By Tietze’s theorem, C(X) = {f|_{X}; f ∈ C([0, 1])}. Define δ on D(δ) = {f|_{X}; f ∈ C^{1}([0, 1])} by δ(f|_{X}) = f′|_{X}. δ is a well-defined derivation (if f|_{X} = 0, then for each x_{0} ∈ X there exists a sequence {x_{n}} in X − \{x_{0}\} converging to x_{0}. So f′(x_{0}) = lim_{n} \frac{f(x_{n}) − f(x_{0})}{x_{n} − x_{0}} = 0, therefore f′|_{X} = 0). But δ is not identically zero so by [Sak2, Proposition 3.2.1] δ cannot be extended to a closed derivation in C(X). So that δ isn’t closable. This example is due to O.Bratteli and D.W. Robinson ([cf. Sak2, P.59]). Ref.

[Sak2] S. Sakai, Operator algebras in dynamical systems, Cambridge Univ. Press, 1991.