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arXiv · 1307.2210

Derivations preserving quasinilpotent elements

J. Alaminos·M. Brešar·J. Extremera·Š. Špenko·A. R. Villena
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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 $A$.

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J. Alaminos, M. Brešar, J. Extremera, Š. Špenko, A. R. Villena. 2013-07-08. Derivations preserving quasinilpotent elements. https://arxiv.org/abs/1307.2210

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