2-local derivations and automorphisms on von Neumann algebras and $AW^\ast$-algebras

Given an algebra $A$, a linear mapping $T: A \to A$ is called a homomorphism (respectively, a derivation) if $T(ab)=T(a)T(b)$ (respectively, $T(ab)=T(a)b+aT(b)$) for all $a, b$ in $A$. A one-to-one homomorphism is called an automorphism.
A mapping $\Delta: A \to A$ (not linear in general) is called a 2-local automorphism (respectively, a 2-local derivation) on $A$, if for every $x,y$ in $A$ there exists an automorphism $\alpha_{x,y}$ (respectively, a derivation $d_{x,y}$) on $A$ depending on $x$ and $y$, such that $\Delta(x) = \alpha_{x,y}(x)$, $\Delta(y) = \alpha_{x,y}(y)$ (respectively, $\Delta(x) = d_{x,y}(x)$ and $\Delta(y) = d_{x,y}(y)$).
The main problem concerning the above notions are to find conditions under which every 2-local automorphism or derivation automatically becomes an automorphism (respectively, a derivation).
In the present talk we give a solution of this problem in the framework of von Neumann algebras and their abstract generalization — $AW^\ast$-algebras (i.e. Kaplansky algebras).