In this work, we study the geometry of the unit ball of the space of operators \({\mathcal {L}}(X,Y^*)\) , by considering the projective tensor product \(X\hat{\otimes }_{\pi } Y\) as a predual. We prove that if an elementary tensor (rank one operator) of the form \(x_0^*\otimes y_0^* \) in the unit sphere \( S_{{\mathcal {L}}(X,Y^*)}\) is a \(\hbox {weak}^*\) -strongly extreme point of the unit ball, then \(x_0^*\) is \(\hbox {weak}^*\) -strongly extreme point of unit ball of \(X^*\) and \(y_0^*\) is \(\hbox {weak}^*\) -strongly extreme point of the unit ball of \(Y^*\) . We show that a similar conclusion holds if the rank one operator is a Namioka point (equivalently, a point of \(\hbox {weak}^*\) -weak continuity for the identity mapping) on the unit sphere of \({\mathcal {L}}(X,Y^*)\) . We also study extremal phenomena in the unit ball of \({\mathcal {L}}(X,Y^*)^*\) . We partly solve the open problem: when does an elementary tensor, whose components are Namioka points, become a Namioka point again? We show that if a point \(z\in S_{{\mathcal {L}}(X,Y^*)^*}\) is a \(\hbox {weak}^*\) -strongly extreme point of the unit ball, then \(z=x\otimes y\) for some \(\hbox {weak}^*\) -strongly extreme points \(x\in S_X\) and \(y\in S_Y\) , provided the space of compact operators, \(\mathcal {K}(X,Y^*)\) is separating for \(X\hat{\otimes }_{\pi } Y\) .
