Consider the following fragment from the book "Operator spaces" by Effros-Ruan:
In this fragment, we encounter the operatornorm $\|\varphi_{p,r}$. This begs the question:
What is the norm on $M_p(V)\times M_r(W)$ used here? If I take a look at the proofs that follow this fragment in the book, it appears to be important that we have something like $$\|\varphi_{p,r}(v,w)\|\le \|v\|\|w\|.$$
I think it is important to note that we want to prove the identification $$CB(V\hat{\otimes}W, X) = CB(V\times W, X)$$ where $\hat{\otimes}$ is the projective tensor product norm of operator spaces.
Many thanks in advance for your help!
You don't need to specify a norm on $M_p(V)\times M_r(W)$. The norm of a bilinear map $\beta\colon X\times Y\to Z$ is defined as the infimum over all $C>0$ such that $\lVert\beta(x,y)\rVert\leq C\lVert x\rVert\lVert y\rVert$ for all $x\in X$, $y\in Y$. With this definition one trivially has $\lVert\beta(x,y)\rVert\leq \lVert \beta\rVert\lVert x\rVert\lVert y\rVert$.