Identifying derivations through the spectra of their values

Abstract

We consider the relationship between derivations $d$ and $g$ of a Banach algebra $B$ that satisfy $\s(g(x)) \subseteq \s(d(x))$ for every $x\in B$, where $\s(\, . \,)$ stands for the spectrum. It turns out that in some basic situations, say if $B=B(X)$, the only possibilities are that $g=d$, $g=0$, and, if $d$ is an inner derivation implemented by an algebraic element of degree 2, also $g=-d$. The conclusions in more complex classes of algebras are not so simple, but are of a similar spirit. A rather definitive result is obtained for von Neumann algebras. In general $C^{\ast }$-algebras we have to make some adjustments, in particular we restrict our attention to inner derivations implemented by selfadjoint elements. We also consider a related condition $\Vert [b,x]\Vert \le M\Vert [a,x]\Vert$ for all selfadjoint elements $x$ from a $C^{\ast }$-algebra $B$, where $a,b\in B$ and $a$ is normal.

12 pages