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Please help me with these questions or give some references.
How this can be restated in the case of a non-commutative Banach algebra?


Does there exist a similar result for a non-commutative Banach algebra?



The image of a derivation on a Banach algebra is contained in the kernel of a character
I shall assume that in your question, you are asking about *continuous* derivations on a Banach algebra $A$. Results for everywhere-defined-but-not-continuous derivations were studied intensively 30-40 years ago but my understanding is that the remaining open problems are thought to be inaccessible.


So, let $A$ be a Banach algebra and let $D:A\to A$ be a continuous derivation. Without loss of generality, we can assume that $A$ is unital — if not, then we simply adjoin an identity element and extend $D$ by making it $0$ on the adjoined identity.


We then have the following old result of A. M. Sinclair [(Proc. AMS 20 (1969), 166–170)](http://www.ams.org/journals/proc/1969-020-01/S0002-9939-1969-0233207-X/home.html):



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> Let $P$ be a primitive ideal of $A$. Then $D(P)\subseteq P$.
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(This can also be found as Proposition 2.7.22(ii) of Dales's behemoth *Banach Algebras and Automatic Continuity*, but the original paper of Sinclair is reasonably self-contained and easy to follow.)


**If $A$ is commutative**, then primitive ideals are the same thing as maximal ideals, and so Sinclair's theorem implies that $D(M)\subseteq M$ for every maximal ideal $M$. But since $A={\mathbb C}1\_A+M$ and $D(1\_A)=0$, we get $D(A)\subseteq M$ for all maximal ideals $M$, which is to say $D(A)\subseteq {\rm rad}(A)$ — and that is the result which you quoted in your question.


