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I am interested in operators on non-reflexive Banach space. Let $X$ be a Banach space and let $L(X)$ be the algebra of operators acting on $X$. We may embed $L(X)$ into $L(X^{**})$ by $\Phi(T)=T^{**}$; this is an algebra homomorphism. Is the image $\Phi(L(X))$ dense in $L(X^{**})$ in the sense of the classical operator topologies like WOT, SOT etc when $X$ lacks the approximation property?

EDIT: Now, cross-posted at MO.

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