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Derivations preserving quasinilpotent elements
Author(s) -
Alaminos J.,
Brešar M.,
Extremera J.,
Špenko Š.,
Villena A. R.
Publication year - 2014
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/blms/bdt102
Subject(s) - mathematics , commutative property , banach algebra , pure mathematics , approximation property , banach space , property (philosophy) , class (philosophy) , set (abstract data type) , zero (linguistics) , space (punctuation) , square (algebra) , algebra over a field , philosophy , linguistics , epistemology , artificial intelligence , computer science , programming language , geometry
We consider a Banach algebra A with the property that, roughly speaking, sufficiently many irreducible representations of A on non‐trivial 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 .