If $S,T,U,V\in B(L_p(X,\mu))$, $p\in[1,\infty)$, and we regard $\begin{pmatrix} S & T \\ U & V \end{pmatrix}$ as an operator on $B(L_p\oplus_p L_p)$, then supposedly we have $\left|\left|\alpha\begin{pmatrix} S & T \\ U & V \end{pmatrix}\right|\right|\leq ||\alpha||_{B(\ell_p^2)} \left|\left|\begin{pmatrix} S & T \\ U & V \end{pmatrix}\right|\right|$ whenever $\alpha\in M_2(\mathbb{C})$.
So I was trying to verify this by computing:
For $w,x,y,z\in\mathbb{C}$ and $f,g\in L_p$, we have
$\left|\left|\begin{pmatrix} w & x \\ y & z \end{pmatrix}\begin{pmatrix} S & T \\ U & V \end{pmatrix}\begin{pmatrix} f \\ g \end{pmatrix}\right|\right| = \left|\left|\begin{pmatrix} w & x \\ y & z \end{pmatrix}\begin{pmatrix} Sf+Tg \\ Uf+Vg \end{pmatrix}\right|\right|$
but how do I extract $\left|\left|\begin{pmatrix} w & x \\ y & z \end{pmatrix}\right|\right|_{B(\ell_p^2)}$ from this?
First, you can forget $S,T,U$, and $V$. Say $A$ is that $2\times 2$ matrix with operator entries. Suppose you could prove $$||\alpha F||\le||\alpha||\,||F||\quad(i)$$for all $F\in L^p\oplus L^p$. Then it would follow that $$||(\alpha A)F||=||\alpha(AF)||\le ||\alpha||\,||AF||\le||\alpha||\,||A||\,||F||,$$which is exactly what you want. So you only need to prove (i), which is to say that the norm of $\alpha$ as an operator on $L^p\oplus L^p$ is no larger than its norm as an operator on $\mathbb C^2$.
You could get (i) from duality, i.e. Hahn-Banch. But in fact it just falls out of the definitions: $$||\alpha F||^p=\int(|wf+xg|^p+|yf+zg|^p)\le\int||\alpha||^p(|f|^p+|g|^p)=||\alpha||^p||F||^p.$$