  A Banach space X has the SHAI (surjective homomorphisms are injective)
property provided that for every Banach space Y, every continuous surjective
algebra homomorphism from the bounded linear operators on X onto the bounded
linear operators on Y is injective. The main result gives a sufficient
condition for X to have the SHAI property. The condition is satisfied for L^p
(0, 1) for 1 < p < \infty, spaces with symmetric bases that have finite cotype,
and the Schatten p-spaces for 1 < p < \infty.
