We use Krivine's form of the Grothendieck inequality to renorm the space of bounded linear maps acting between Banach lattices. We construct preduals and describe the nuclear operators associated with these preduals for this renormed space of bounded operators as well as for the spaces of $p$-convex, $p$-concave and positive $p$-summing operators acting between Banach lattices and Banach spaces. The nuclear operators obtained are described in terms of factorizations through classical Banach spaces via positive operators.

Let $l^{\Phi}$ and $L^\Phi (\Omega)$ be the Orlicz sequence space and function space generated by $N$-function $\Phi(u)$ with Orlicz norm. We give equivalent expressions for the nonsquare constants $C_J (l^\Phi)$, $C_J \bigl( L^\Phi (\Omega) \bigr)$ in sense of James and $C_S (l^\Phi)$, $C_S \bigl( L^\Phi(\Omega) \bigr)$ in sense of Sch\"affer. We are devoted to get practical computational formulas giving estimates of these constants and to obtain their exact value in a class of spaces $l^{\Phi}$ and $L^\Phi (\Omega)$.