Let $A$ be an algebra of operators on a Hilbert space $H$. Let $A^m$ and $A^n$ be free $A$modules with bases $e_1, \ldots, e_m$ and $f_1, \ldots, f_n$. Define a norm on $A^m \otimes A^n$ as follows: $$\left \sum a_i e_i \otimes b_j f_j \right = \sup_{X_i, Y_j} \left\sum a_i X_i b_j Y_j\right.$$ The sup is taken over all collections of operators $X_i$, $Y_j$ on $H$ of norm no more than 1. The norm on the right is the operator norm. Has anyone considered these? Is there any chance that the norms defined for $A^m \otimes A^n$ and for $A^n \otimes A^m$ be compared?
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$\begingroup$ to get some comparison inequalities, you might benefit from searching for Pisier's survey on Grothendieck inequalities... $\endgroup$ – Suvrit May 1 '13 at 22:47

1$\begingroup$ Should't it be $\sum_{i,j,k} a_{k,i}e_i\otimes b_{k,j}f_j$, since it's a tensor product? $\endgroup$ – Narutaka OZAWA May 1 '13 at 23:23

$\begingroup$ Exactly, sorry, I suppressed the index k. $\endgroup$ – Boris Tsygan May 4 '13 at 0:53