Derivations preserving quasinilpotent elements

TL;DR

This paper investigates Banach algebras with a property related to irreducible representations and quasinilpotent elements, showing that derivations preserving quasinilpotent elements map into the algebra's radical.

Contribution

It introduces a broad class of Banach algebras with this property and proves that derivations preserving quasinilpotent elements have ranges in the radical.

Findings

Includes $C^*$-algebras, group algebras, commutative algebras, and $L(X)$.

Derivations preserving quasinilpotent elements map into the radical.

The class of algebras with the property is quite large.

Abstract

We consider a Banach algebra $A$ with the property that, roughly speaking, sufficiently many irreducible representations of $A$ on nontrivial Banach spaces do not vanish on all square zero elements. The class of Banach algebras with this property turns out to be quite large -- it includes $C^*$-algebras, group algebras on arbitrary locally compact groups, commutative algebras, $L(X)$ for any Banach space $X$, and various other examples. Our main result states that every derivation of $A$ that preserves the set of quasinilpotent elements has its range in the radical of $A$.