2-local derivations on C$^{\ast}$-algebras
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In this paper, we prove that every 2-local derivation on several classes of C$^{\ast}$-algebras, such as unital properly infinite, type $\mathrm{I}$ or residually finite-dimensional C$^{\ast}$-algebras, is a derivation. We show that the following statements are equivalent: (1) every 2-local derivation on a C$^{\ast}$-algebra is a derivation, (2) every 2-local derivation on a unital primitive antiliminal and no properly infinite C$^{\ast}$-algebra is a derivation. We also show that every 2-local derivation on a group C$^{\ast}$-algebra $C^{\ast}(\mathbb{F})$ or a unital simple infinite-dimensional quasidiagonal C$^{\ast}$-algebra, which is stable finite antiliminal C$^{\ast}$-algebra, is a derivation.
Abstract
In this paper, we prove that every 2-local derivation on several classes of C$^{\ast}$-algebras, such as unital properly infinite, type $\mathrm{I}$ or residually finite-dimensional C$^{\ast}$-algebras, is a derivation. We show that the following statements are equivalent: (1) every 2-local derivation on a C$^{\ast}$-algebra is a derivation, (2) every 2-local derivation on a unital primitive antiliminal and no properly infinite C$^{\ast}$-algebra is a derivation. We also show that every 2-local derivation on a group C$^{\ast}$-algebra $C^{\ast}(\mathbb{F})$ or a unital simple infinite-dimensional quasidiagonal C$^{\ast}$-algebra, which is stable finite antiliminal C$^{\ast}$-algebra, is a derivation.
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