  Nonreflexive Banach spaces that are complemented in their bidual by an
L-projection - like preduals of von Neumann algebras or the Hardy space $H^1$ -
contain, roughly speaking, many copies of $l^1$ which are very close to
isometric copies. Such $l^1$-copies are known to fail the fixed point property.
Similar dual results hold for $c_0$.
