Octahedral norms in spaces of operators

We study octahedral norms in the space of bounded linear operators between Banach spaces. In fact, we prove that L(X,Y)L(X,Y) has octahedral norm whenever X⁎X⁎ and Y   have octahedral norm. As a consequence the space of operators L(ℓ1,X)L(ℓ1,X) has octahedral norm if, and only if, X has octahedral norm. These results also allow us to get the stability of strong diameter 2 property for projective tensor products of Banach spaces, which is an improvement of the known results about the size of nonempty relatively weakly open subsets in the unit ball of the projective tensor product of Banach spaces.