Consider the following fragment from Effros and Ruan's book "Operator spaces"
Why is a decomposition as in the red box possible? In fact, it is not even clear to me that any such decomposition is possible! But I don't have a lot of intuition about the Kronecker product of matrices. I am guessing some explicit construction here exists.
Any help or hints is highly appreciated!
Ok, here is how it works for $u=v_1\otimes w_1+v_2\otimes w_1$ with $v_1,v_2\in V$, $w_1,w_2\in W$. In this case let $$ v=\begin{pmatrix}v_1&v_2\\0&0\end{pmatrix},\;w=\begin{pmatrix}w_1&w_2\\0&0\end{pmatrix}. $$ We have (well, there is some choice in the isomorphism $M_2\otimes M_2\cong M_4$) $$ v\otimes w=\begin{pmatrix}v_1\otimes w_1&v_2\otimes w_1&v_1\otimes w_2&v_2\otimes w_2\\0\otimes w_1&0\otimes w_1&0\otimes w_2&0\otimes w_2\\v_1\otimes 0&v_2\otimes 0&v_1\otimes 0&v_2\otimes 0\\0\otimes 0&0\otimes 0&0\otimes 0&0\otimes 0\end{pmatrix}=\begin{pmatrix}v_1\otimes w_1&v_2\otimes w_1&v_1\otimes w_2&v_2\otimes w_2\\0&0&0&0\\0&0&0&0\\0&0&0&0\end{pmatrix}. $$ With $$ \alpha=\begin{pmatrix}1&0&0&0\end{pmatrix},\,\beta=\begin{pmatrix}1\\0\\0\\1\end{pmatrix} $$ we get $$ \alpha(v\otimes w)\beta=\begin{pmatrix}1&0&0&0\end{pmatrix}\begin{pmatrix}v_1\otimes w_1+v_2\otimes w_2\\0\\0\\0\end{pmatrix}=v_1\otimes w_1+v_2\otimes w_2. $$ For sums of more than two elementary tensors and elements of $M_n(V\otimes W)$ the idea is the same, just that things get real ugly to write down.
To get elements $\alpha,\beta,v,w$ with the desired norm, first note that by definition of the norm they can be chosen such that $\lVert \alpha\rVert\lVert v\rVert\lVert w\rVert\lVert\beta\rVert\leq \lVert u\rVert_\wedge+\epsilon$ for every $\epsilon>0$. Then one can use the multilinearity of the expression to rescale it.