Splittings of extensions of the algebra of bounded operators on a Banach space

Niels Jakob Laustsen, Richard Skillicorn

Published 2014-09-29Version 1

We show that there exist a Banach space E, a unital Banach algebra A with Jacobson radical rad A, and a continuous, surjective algebra homomorphism f from A onto the Banach algebra B(E) of bounded operators on E such that ker f = rad A and the corresponding extension {0} -> rad A -> A -> B(E) -> {0} is singular (in the sense that rad A has trivial multiplication) and splits algebraically, but it does not split strongly. This conclusion complements the work of Bade, Dales, and Lykova (Mem. Amer. Math. Soc. 1999). The Banach space E that we use is a quotient of the l_2-direct sum of an infinite sequence of James-type quasi-reflexive Banach spaces; it was originally introduced by Read (J. London Math. Soc. 1989).