I have a problem with understanding the notion of complete boundedness in tensor notation.

One possible way of saying a mapping $\phi\colon \mathcal{A}\to \mathcal{B}$ is completely bounded is to require that norms of $\phi_n\colon M_n(\mathcal{A})\to M_n(\mathcal{B})$ are uniformly bounded, where $\phi_n$ denotes an "entrywise" application of $\phi$ to every entry of the relevant matrix. In tensor notation, this means that we require that $\phi\otimes \mathcal{id}_n$ has uniformly bounded norm.

On the other hand, when one defines tensor products of Banach spaces, one usually immediately requires that $\|A\otimes B\|=\|A\|\|B\|$. Thus, treating $B(\mathcal{A},\mathcal{B})$ and $B(M_n)$ as Banach spaces, it seems obvious that all $\phi_n$ are then uniformly bounded with $\|\phi_n\|=\|\phi\|$.

I am aware of the fact that there is a flaw in this reasoning, but ever since I learned the definition of an operator space I haven't been able to quite understand where the flaw lies, and curiously enough none of the operator space books mention this problem in understanding the tensor interpretation of complete boundedness.

I would be very grateful for explaining where am I going wrong in that reasoning. I was trying to find a similar question on StackExchange but failed to do so.


Sir, actually in general the norm on each matirx level $M_n(\mathcal{A})$ is not equal to tensor product norms of Banach spaces $\mathcal{A}$ and $M_n$.


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