Derivation on operator algebras

Given an algebra $A$, a linear operator $D:A\to A$ is called a derivation, if $D(xy)=D(x)y+xD(y)$ for all $x,y\in A$. Each element $a\in A$ implements a derivation $D$ a on $A$ as ${{D}_{a}}(x)=ax-xa$, $x\in A$. Such derivations are said to be inner derivations. If the element implementing the derivation ${{D}_{a}}$ belongs to a larger algebra $B$ containing $A$ then ${{D}_{a}}$ is called a spatial derivation on $A$. In this talk we discuss derivations on algebras of operators on a Hilbert space, emphasizing their properties such as innerness and spatiality. These notions are very important in the structure theory and cohomology of abstract rings and algebras and at the same time they have deep applications in mathematical physics, in particular in the problem of constructing the dynamics in quantum statistical mechanics.
Therefore we also discuss a physical background of derivations on operator algebras. After expositions of some well-known results on derivation on ${{C}^{*}}$-algebras and von Neumann algebras we consider open problems concerning derivations on algebras of measurable operators affiliated with von Neumann algebras, which are partially solved.