2-Local Derivations on Some $$ C^* $$ C ∗ -Algebras

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.