Does tensor product with $L_p$ operator algebra preserve exact sequences?

Question

By $L_p$ operator algebra I mean a closed subalgebra of the algebra of bounded linear operators on some $L_p$ space where $p\in(1,\infty)$. There is a notion of tensor product of $L_p$ spaces (as described, for instance, in Defant & Floret's Tensor Norms and Operator Ideals), which then allows one to talk about tensor products of $L_p$ operator algebras. Given an exact sequence $0\rightarrow I\rightarrow A\stackrel{\pi}{\rightarrow} A/I\rightarrow 0$ of $L_p$ operator algebras, under what condition can I take the tensor product of each term with another $L_p$ operator algebra $B$ and still have an exact sequence? In particular, if I have a linear section $s:A/I\rightarrow A$, i.e., $\pi\circ s=id$ (and assumed to be $p$-completely contractive if necessary), does taking tensor products preserve exactness?