Local automorphisms and derivations on $\mathcal {B}(H)$

Abstract:

Let ${\mathcal {A}}$ be an algebra. A mapping $\theta :{\mathcal {A}}\longrightarrow {\mathcal {A}}$ is called a $2$-local automorphism if for every $a,b\in {\mathcal {A}}$ there is an automorphism $\theta _{a,b}:{\mathcal {A}}\longrightarrow {\mathcal {A}}$, depending on $a$ and $b$, such that $\theta _{a,b}(a)=\theta (a)$ and $\theta _{a,b}(b)=\theta (b)$ (no linearity, surjectivity or continuity of $\theta$ is assumed). Let $H$ be an infinite-dimensional separable Hilbert space, and let ${\mathcal {B}}(H)$ be the algebra of all linear bounded operators on $H$. Then every $2$-local automorphism $\theta :{\mathcal {B}}(H)\longrightarrow {\mathcal {B}}(H)$ is an automorphism. An analogous result is obtained for derivations.