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

Hamid Shafieasl, Amir Mohammad Tavakkoli

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.

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

Abstract

Let be a Banach algebra and a dense ideal in . A natural question in the theory of operator algebras is whether the property that all derivations are inner (implemented by elements in ) implies that all derivations are inner (implemented by elements in ). We present a rigorous negative answer to this question. By utilizing the algebra of compact operators and the dense ideal of finite-rank operators on a separable infinite-dimensional Hilbert space , we demonstrate that while every derivation into is inner, there exist outer derivations on . Furthermore, we generalize this result to Schatten -classes and discuss the cohomological implications and the role of approximate identities.
Paper Structure (8 sections, 3 theorems, 8 equations)

This paper contains 8 sections, 3 theorems, 8 equations.

Table of Contents

  1. Introduction
  2. Definitions and Mathematical Preliminaries
  3. Discussion and Background
  4. The Main Result and Counterexample
  5. Generalization to Schatten $p$-Classes
  6. Approximation and Bounded Approximate Identities
  7. Cohomological Interpretation
  8. Conclusion

Key Result

Theorem 4.1

Every derivation $D: K(H) \to F(H)$ is inner and implemented by an element in $F(H)$, but there exist derivations $D: K(H) \to K(H)$ that are not inner in $K(H)$.

Theorems & Definitions (8)

  • Definition 2.1: Schatten $p$-Classes
  • Definition 2.2: Hochschild Cohomology
  • Definition 2.3: Approximate Identity
  • Theorem 4.1
  • proof
  • Theorem 5.1
  • proof
  • Corollary 7.1