On the extension of inner derivations from dense ideals in Banach algebras

TL;DR

This paper investigates whether inner derivations on dense ideals in Banach algebras imply all derivations are inner, providing counterexamples in operator algebras and verifying results with formal proof tools.

Contribution

It provides the first rigorous counterexamples showing that inner derivations on dense ideals do not guarantee all derivations are inner, using operator algebra examples and formal verification.

Findings

Counterexamples in $K(H)$ and $F(H)$ show outer derivations exist.

Inner derivations on dense ideals do not imply all derivations are inner.

Formal verification of results using the Lean theorem prover.

Abstract

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.