Identifying derivations through the spectra of their values

TL;DR

This paper explores how the spectra of derivations in Banach and $C^*$-algebras can identify their relationships, revealing that in certain cases derivations are uniquely determined or related by simple transformations.

Contribution

It characterizes derivations in Banach and $C^*$-algebras based on spectral inclusion, providing new insights especially for von Neumann and $C^*$-algebras with inner derivations.

Findings

In $B(X)$, derivations are either equal, zero, or negatives of each other.

For von Neumann algebras, spectral conditions strongly restrict derivations.

In $C^*$-algebras, inner derivations by selfadjoint elements are characterized by spectral properties.

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 $C^*$-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 $\|[b,x]\|\leq M\|[a,x]\|$ for all selfadjoint elements $x$ from a $C^*$-algebra $B$, where $a,b\in B$ and $a$ is normal.