Lushness, numerical index one and duality

We construct an example of a Banach space which is not lush, but whose dual space is lush. This example shows that lushness is not equivalent to numerical index one. A characterization of lushness for some quotient spaces of L 1 ( μ ) spaces and new results on C-rich subspaces of (scalar- or vector-valued) C ( K ) spaces are also presented.