# Questions on unbounded derivations of C* algebra

In Sakai note, on the fourth part differentiation. Sakai stated the following:

It is an open question whether the result can be extended to $$n=2,3,...$$

What $$n$$ he is referring to? Also Sakai stated a conjecture:

Let $$\delta_{0}$$ be a closed *-derivation in $$U$$ and let $$\delta$$ be a *-derivation in $$U$$ with $$D(\delta_{0})=D(\delta)$$. Then $$\delta$$ is closable....

If $$\delta_{0}$$ is closed, isn't it a bounded operator? If $$D(\delta_{0})=D(\delta)$$, it means $$D(\delta)$$ is the whole space, which implies boundedness and closable. Am I missing something here?

Edit For the first question, what I really to know is which result he is referred to? Is it the unit interval one($$D(\delta)=C^{n}(I)$$) or the compact Hausdorff space one($$\cap^{\infty}_{n=1}D(\delta_{0}^{n})$$)? Are these problem solved?

• There are closed unbounded operators as, for example, all self-adjoint operators. However, if the domain of $\delta_0$ is $U$, it follows from the closed graph theorem that $\delta_0$ is bounded. – Romain Gicquaud May 4 at 9:55
• $C^n(I)$ ($n$ times continuously differentiable)'' – Nik Weaver May 4 at 11:23
• Suppose that $\delta_0, \delta$ are two $*$-derivations in $C(K)$ $\ldots$ then there is a unique continuous function $\lambda$ on $K$ such that $\delta = \lambda \delta_0$. In particular, $\delta$ is closable. It is an open question whether the result can be extended to $n = 2, 3, \ldots$.'' – Nik Weaver May 4 at 15:57
• The result stated for $C^n(I)$ is not open for $n = 2, 3, \ldots$, as it is stated, and a reference is given, for $n = \infty, 1, 2, 3, \ldots$''. – Nik Weaver May 4 at 16:01