Abstract
In this paper, we introduce the concept of trace-open projections in the second dual \( \mathcal {A}^{**} \), of a \( C^* \)-algebra \( \mathcal {A} \). This new concept is applied to show that if there is a faithful normal semi-finite trace \( \tau \) on \( \mathcal {A}^{**} \) such that \( 1_{ \mathcal {A}^{**} } \) is a \( \tau \)-open projection, then every 2-local derivation \( \Delta \) from \( \mathcal {A} \) to \( \mathcal {A}^{**} \) is an inner derivation. We also prove that the same conclusion holds for approximately 2-local derivations when \( \mathcal {A}^{**} \) is a finite von Neumann algebra without extra assumptions on its unit element.
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We would like to thank the two anonymous referees for their constructive comments and suggestion.
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Communicated by Hamid Reza Ebrahimi Vishki.
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Habibzadeh Fard, M., Sahleh, A. 2-Local Derivations on Some \( C^* \)-Algebras. Bull. Iran. Math. Soc. 45, 649–656 (2019). https://doi.org/10.1007/s41980-018-0156-0
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DOI: https://doi.org/10.1007/s41980-018-0156-0