POSITIVE AND NEGATIVE RESULTS ON THE NUMERICAL INDEX OF BANACH SPACES AND DUALITY

We show that the numerical index of an L-embedded space and that of its dual coincide. In particular, the numerical index of the predual of a real or complex von Neumann algebra or JBW*-triple coincides with the numerical index of the space. Also, we prove that when X is an M-embedded Banach space with numerical index 1, then every closed subspace of X** containing X also has numerical index 1 (in particular, X* and X** have numerical index 1). Finally, we show that any Banach space X containing a complemented copy of c(0) or a copy of l(infinity) admits an equivalent norm for which the numerical index of its dual space is strictly less than the index of the space. In the special case of a separable space X containing c(0), it is actually possible to renorm X with the maximum value of the numerical index (namely 1) while the numerical index of the dual is as small as possible (namely, 0 in the real case, 1/e in the complex case).