Let $A$ be a normed $\mathbb{C}$-algebra. Let $n \in \mathbb{N}^*$. Consider the right-$A$-module $A^n$, endowed with the $2$-norm.
Consider the usual isomorphism $M_n(A) \rightarrow End_A(A^n)$ and endow $M_n(A)$ with the operator $\Vert \cdot \Vert_2\to \Vert \cdot \Vert_2$ norm.
Now consider the isomorphism $A \otimes_{\otimes} M_n(\mathbb{C}) \rightarrow M_n(A)$ sending $\sum_{i,j} a_{ij} \otimes E_{ij}$ to $(a_{ij})_{i,j}$ (where $E_{ij}$ denotes the usual matrix with only a $1$ at position $(i,j)$ and $0$ everywhere else).
Questions:
- I am aware that in general, there are several norms on tensor products of algebras. Is there some canonical norm on tensor products that gives the norm on $A\otimes M_n(\mathbb{C})$ that is transported from $M_n(A)$ by the previous isomorphism?
- I would like to see if I can make computations with this $\Vert \cdot \Vert_2 \to \Vert \cdot \Vert_2$ norm. Can it be described in a more comfortable way?
More generally, can you provide me with a reference where this kind of things is treated in some detail? The references I have seen go very fast on this (for example, because they want to explain K-theory and skip directly to it)!