The following is an abstraction of a more specific problem I've been grappling with, so I might give some extra unnecessary information. (The `bounded approximate left identities' assumption is probably not necessary, but it is definitely the case in the specific problem that this question is abstracted from).

Let $A$ and $B$ be Banach algebras, both having bounded approximate
left identities. We assume that we have non-degenerate, *isometric*
representations $\pi:A\to B(X)$ and $\rho:B\to B(Y)$ on some Banach
spaces $X$ and $Y$.

We can define the following algebra representation $$ \pi\otimes\rho:A\otimes B\to B(X\hat{\otimes}Y) $$ (where $X\hat{\otimes}Y$ denotes the projective tensor product) in the usual way, for all $a\in A$, $b\in B$, $x\in X$ and $y\in Y$, by $$ \left(\pi\otimes\rho(a\otimes b)\right)x\otimes y:=\pi(a)x\otimes\rho(b)y. $$ Now, since $\pi$ and $\rho$ were assumed to be isometric, we can prove that the map $\pi\otimes\rho$ is injective and satisfies $$ ||\pi\otimes\rho(a\otimes b)||_{op} = ||a||_A ||b||_B $$ for all elementary tensors $a\otimes b\in A\otimes B$, where the norm on the left hand side denotes the operator norm on $B(X\hat{\otimes}Y)$.

My question is the following: Does the map $$ \sum_{i}a_{i}\otimes b_{i}\mapsto\left\Vert \pi\otimes\rho\left(\sum_{i}a_{i}\otimes b_{i}\right)\right\Vert _{\mbox{op}} $$ define a reasonable cross norm on $A\otimes B$ (Ryan's book, Ch. 6)? Seeing that it is dominated by the projective tensor norm on $A\otimes B$ is easy, but is it bounded from below by the injective tensor norm on $A\otimes B$?

Note that we can extend bounded functionals on $A$ to bounded functionals on $B(X)$ by using Hahn Banach and that the representations are isometric; similarly with $B$. Still, using this, a proof still seems just out of reach for want of being able to extend functionals on $B(X)\hat{\otimes}B(Y)$ to functionals on $B(X\hat{\otimes}Y)$ because the norms on these algebras can't be favourably related (projective norm on the one, operator norm on the other).

Any ideas, helpful references or counterexamples showing how this might fail will be greatly appreciated.