Let $X$ be a Banach space, and consider the algebra $B(X)$ of continuous linear operators mapping $X$ into itself. Then each operator $A \in B(X)$ has an adjoint which I denote by $A^*$. The operator $A^*$ is of course a member of $B(X^*)$, the operator algebra on the dual. Basic functional analysis says that the corresponding adjoint map $ad: B(X) \to B(X^*)$ is an isometric (anti)linear embedding, isometry being with respect to the norm topologies on both operator algebras.
A sloppy and brief calculation seems to tell me that the mapping is surjective if and only if $X$ is a reflexive space. My question is if there are classes of Banach spaces in which the image of this embedding is dense in other popular topologies on $B(X^*)$, like the weak and strong operator topologies, or the ultraweak topology? What about sequential convergence?