I am having to deal with the problem of passing between $M_{kn}(A)$ and $M_k(M_n(A))$ where $A$ is a Banach algebra, removing parentheses in one direction, and adding parentheses in the opposite direction. When I equip the matrix algebras with the maximum row sum norm (i.e. $||[a_{ij}]||=\max_i\sum_j||a_{ij}||$), I get that removing parentheses is a contraction but adding parentheses increases norm with the increment depending on matrix size.
For example, $\begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix}\in M_4(\mathbb{C})$ will have norm 1 while $\begin{pmatrix} \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix} & \begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix} \\ \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix} & \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix} \end{pmatrix}\in M_2(M_2(\mathbb{C}))$ will have norm 2.
Is there a choice of norm such that adding parentheses is also a contraction, or such that the increment does not depend on the matrix size? If $A$ is a $C^*$-algebra, then the $C^*$ norm would do but I'm wondering about a general Banach algebra.
I was also trying out operator norms, which means that in the case of $M_k(M_n(A))$ I will have to multiply by a $k$-tuple of matrices in $M_n(A)$, get another $k$-tuple of matrices in $M_n(A)$ and then consider each of their norms, which caused me some confusion.
If you use the operatornorm with respect to the Euclidean norm on $\mathbb{C}^m$, you will end up with isoemtrically isomorphic spaces: Both $M_{kn}(\mathbb{C})$ and $M_n(M_k(\mathbb{C}))$ are obviously complete in this norm and the norms satisfy the $C^\ast$-property: $\|x^\ast x\|=\|x\|^2$ in both cases. But such a norm is uniquely determined.