Positive and negative results on the numerical index of Banach spaces and duality

Martín, Miguel

doi:10.1090/S0002-9939-09-09837-2

Positive and negative results on the numerical index of Banach spaces and duality
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Proc. Amer. Math. Soc. 137 (2009), 3067-3075 Request permission

Abstract:

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 $\ell _\infty$ 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, $1/\mathrm {e}$ in the complex case).

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