arXiv subjects
Publications and source records attributed to Mohsen Niazi.
We prove that, for every separable complex Hilbert space $H$, every weak-2-local $^*$-derivation on $B(H)$ is a linear $^*$-derivation. We also establish that every (non-necessarily linear nor continuous) weak-2-local derivation on a finite dimensional C$^*$-algebra is a linear derivation.
We introduce the notion of weak-2-local derivation (respectively, $^*$-derivation) on a C$^*$-algebra $A$ as a (non-necessarily linear) map $Δ: A\to A$ satisfying that for every $a,b\in A$ and $ϕ\in A^*$ there exists a derivation (respectively, a $^*$-derivation) $D_{a,b,ϕ}: A\to A$, depending on $a$, $b$ and $ϕ$, such that $ϕΔ(a) = ϕD_{a,b,ϕ} (a)$ and $ϕΔ(b) = ϕD_{a,b,ϕ} (b)$. We prove that every weak-2-local $^*$-derivation on $M_n$ is a linear derivation. We also show that the same conclusion remains true for weak-2-local $^*$-derivations on finite dimensional C$^*$-algebras.
We prove that, for a complex Hilbert space $H$ with dimension bigger or equal than three, every linear mapping $T: B(H)\to B(H)$ satisfying the 3-local property is a $^*$-monomorphism, that is, every linear mapping $T: B(H) \to B(H)$ satisfying that for every $a$ in $B(H)$ and every $ξ,η$ in $H$, there exists a $^*$-automorphism $π_{a,ξ,η}: B(H)\to B(H)$, depending on $a$, $ξ$, and $η$, such that $$T(a) (ξ) = π_{a,ξ,η} (a) (ξ), \hbox{ and } T(a) (η) = π_{a,ξ,η} (a) (η),$$ is a $^*$-monomorphism. This solves a question posed by L. Molnár in [\emph{Arch. Math.} \textbf{102}, 83-89 (2014)].