Let $E$ be a reflexive Banach space and let $B(E)$ be the space of bounded operators on $E$ endowed with the weak operator topology. In particular, the unit ball of $B(E)$ is then WOT-compact. $(B(E), {\rm WOT})$ is a fairly nice locally convex space. I am interested in the weak topology of this space.

1. Does $\sigma\big( (B(E), {\rm WOT} ), (B(E), {\rm WOT} )^*)={\rm WOT}$?

2. Is $(B(E), {\rm WOT} )$ a barelled space?