This is a bit of a dumb question I know, but I was reading "Morita equivalence for C*-algebras and W*-algebras" by Rieffel, in this section about Morita equivalences and how they relate to forming tensor products of $W^*$-algebras (page 90) I came across this proof that I just can't seem to grasp and was seeking clarification about a few things.

On page 90 there's this proposition:

8.5. Proposition.Let $M$, $M_1$, $N$, $N_1$ be $W^*$-algebras. Let $X$ be a normal $N$-rigged $M$-module and $Y$ a normal $N_1$-rigged $M_1$-module. Then the algebraic tensor product $X \otimes Y$ over the complex numbers , with $N \otimes N_1$-valued inner product defined by$\langle x \otimes y,x' \otimes y' \rangle_{N \otimes N_1} = \langle x,x' \rangle_{N} \otimes \langle y,y' \rangle_{N_1}$

and completed for the

usual norm, is a normal $N \otimes N_1$-rigged $M \otimes M_1$-bimodule. If $X$ and $Y$ are in fact equivalence bimodules and if an $M \otimes M_1$-valued inned product is defined by$ \langle x \otimes y , x' \otimes y' \rangle_{M \otimes M_1} = \langle x,x' \rangle_{M} \otimes \langle y,y' \rangle_{M_1}$,

then $X \otimes Y$ (completed) becomes and equivalence module.

The proof focuses on proving that the inner products are positive and the "rest of the proof of this proposition is carried out by routine calculations", which, as it were, I'm stuck with, pardon my lack of familiarity with the whole subject but I just have to ask:

What is this "usual norm" they're referring to? I've seen a tensor product of $W^*$-algebras be referred to as the completion of the algebraic tensor product $X \otimes Y$ with respect to certain norm (don't know if that's Sakai's definition they refer to in the paper).

How does the range of $\langle - , - \rangle_{M \otimes M_1}$ and $\langle - , - \rangle_{N \otimes N_1}$ span weakly dense subsets on $M \otimes M_1$ and $N \otimes N_1$ respectively?