On the numerical index of real $L_p(\mu )$-spaces

We give a lower bound for the numerical index of the real space $L_p(\mu )$ showing, in particular, that it is non-zero for $p\mathrm{\ne }2$. In other words, it is shown that for every bounded linear operator $T$ on the real space $L_p(\mu )$, one has $$\sup \mid \int \mathrm{\mid }x{\mathrm{\mid }}^{p-1}\text{sign}(x)Txd\mu \mid :x\in L_p(\mu ),\mathrm{\parallel }x\mathrm{\parallel }=1\ge \frac{M_p}{12\text{e}}\mathrm{\parallel }T\mathrm{\parallel }$$ where $M_p={\max }_{t\in [0,1]}\frac{\mathrm{\mid }{t}^{p-1}-t\mathrm{\mid }}{1+t^p}>0$ for every $p\mathrm{\ne }2$. It is also shown that for every bounded linear operator $T$ on the real space $L_p(\mu )$, one has $$\sup \int \mathrm{\mid }x{\mathrm{\mid }}^{p-1}\mathrm{\mid }Tx\mathrm{\mid }d\mu :x\in L_p(\mu ),\mathrm{\parallel }x\mathrm{\parallel }=1\ge \frac{1}{2\text{e}}\mathrm{\parallel }T\mathrm{\parallel }\mathrm{.}$$