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.


1 Answer 1


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 $M_n$.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.