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We investigate the property of commutator-simplicity in algebras from both algebraic and analytic perspectives. We demonstrate that a large class of algebras possess this property. As an analytic analog, we introduce the concept of topological commutator-simplicity for Banach algebras and establish that a σ -unital C ∗ -algebra is topological commutator-simple if and only if its multiplier algebra is. Furthermore, we explore the applications of commutator-simplicity to certain equations involving commutators, emphasizing its relevance in the study of derivations. Specifically, we obtain that every continuous local derivation on L 1 (G , ω) is a derivation when G is a unimodular locally compact group with a diagonal bounded weight ω . [ABSTRACT FROM AUTHOR]

We define and study the weak Haagerup property for C*-algebras in this article. A C*-algebra with the Haagerup property always has the weak Haagerup property. We prove that a discrete group has the weak Haagerup property if and only if its reduced group C*-algebra also has that property. Moreover, we consider the permanence of the weak Haagerup property under a few canonical constructions of C*-algebras. [ABSTRACT FROM AUTHOR]