On the extension of inner derivations from dense ideals in Banach algebras
Let $A$ be a Banach algebra and $I$ a dense ideal in $A$. A natural question in the theory of operator algebras is whether the property that all derivations $D: A \to I$ are inner (implemented by elements in $I$) implies that all derivations $D: A \to A$ are inner (implemented by elements in $A$). We present a rigorous negative answer to this question. By utilizing the algebra of compact operators $A = K(H)$ and the dense ideal of finite-rank operators $I = F(H)$ on a separable infinite-dimensional Hilbert space $H$, we demonstrate that while every derivation into $F(H)$ is inner, there exist outer derivations on $K(H)$. Furthermore, we generalize this result to Schatten $p$-classes and discuss the cohomological implications and the role of approximate identities. Moreover, the main results and counterexamples presented in this paper have been formally verified using the Lean theorem prover.