Additive derivations of some operator algebras.

*(English)*Zbl 0705.46035Ill. J. Math. (to appear).

Summary: Let A be a standard operator algebra on an infinite dimensional Hilbert space X. We denote by $B\left(X\right)$ the algebra of all bounded linear operators on X. It is proved that every additive derivation $D:A\to B\left(X\right)$ is of the form $D\left(A\right)=AT-TA$ for some $T\in B\left(X\right)\xb7$

A complete description of all additive derivations on $B\left(X\right)$ in the case that X is finite dimensional is also given. In particular it is shown that in this case there exists an additive derivation $D:B\left(X\right)\to B\left(X\right)$ which is not inner.

Reviewer: P.Šemrl

##### MSC:

46L57 | Derivations, dissipations and positive semigroups in ${C}^{*}$-algebras |