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comment Ultraweak topology on B(X): Is the map X\otimes X* -> B(X)* isometric?
@YemonChoi: I would look at $B(X,X^{**})$ instead of $B(X^*)$, then we're asking if the ball of $B(X,X)$ is weak$^*$-dense in the ball of $B(X,X^{**})$. The PLR shows that this is true for finite-rank operators; and the metric approximation property allows you to reduce from all operators to just the finite-rank ones. If $X$ is reflexive, then there is nothing to prove. What I don't see is how to combine these two rather different viewpoints...!
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