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Discontinuous Derivations on the Algebra of Bounded Operators on a Banach Space
Author(s) -
Read C. J.
Publication year - 1989
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms/s2-40.2.305
Subject(s) - bounded function , banach algebra , mathematics , banach space , operator space , algebra over a field , finite rank operator , approximation property , pure mathematics , space (punctuation) , infinite dimensional vector function , banach manifold , lp space , mathematical analysis , computer science , operating system
It is a question of general interest whether, on a given Banach algebra A , there is any discontinuous derivation from A into M , where M is a Banach A ‐bimodule. In the case when A is B( E ), the algebra of all bounded operators on a Banach space E , there are a number of known conditions on E which ensure that derivations from B( E ) are automatically continuous (for example, if E is isomorphic to its square). Until now, there have been no examples of Banach spaces E such that discontinuous derivations from B( E ) are known to exist. In this paper we exhibit a class of Banach spaces E such that discontinuous derivations exist on B(E) ; in the process, we discover various new Banach spaces which are not isomorphic to their squares.