In relation to various problems in understanding entanglement and non-locality, I have come across the following mathematical problem. It is most concise by far to state in its most mathematical form and not go into the background much. However, I hope people interested in entanglement theory might be able to see how the problem is interesting/useful.

Here goes. I have two finite dimension vector spaces $A$ and $B$ and each is equipped with a norm (Banach spaces) such that $||...||: A \rightarrow \mathbb{R}$ and $||...||: B \rightarrow \mathbb{R}$. Both the vector spaces and norms are isomorphic to each other. My question concerns norms on the tensor product of these spaces (for simplicity, lets say just the algebraic tensor product) $A \otimes B$ and the dual norms. First let me state something I know to be true.

Lem 1:

If a norm $||...||$ on $A \otimes B$ satisfies:

$||a \otimes b || \leq ||a|| . ||b||$ (sub-multiplicativity)

then the dual norm satisfies

$||a \otimes b ||_{D} \geq ||a||_{D} . ||b||_{D}$ (super-multiplicativity)

where we define the dual of a norm in the usual way as

$|| a ||_{D}= \mathrm{sup} \{ |b^{\dagger}a| ; ||b|| \leq 1 \}$

This lemma crops up often such as in Horn and Johnson Matrix analysis where it is used to prove the duality theorem (that in finite dimensions the bidual equals the original norm $||..||_{DD}=||...|$ ).

I wish to know the status of the converse, which I conjecture will be answered in the affirmative:

Conjecture:

If a norm $||...||$ on $A \otimes B$ satisfies:

$||a \otimes b || \geq ||a|| . ||b||$ (super-multiplicativity)

then the dual norm satisfies

$||a \otimes b ||_{D} \leq ||a||_{D} . ||b||_{D}$ (sub-multiplicativity)

My question is simply "is my conjecture true or does anyone have a counterexample?".

Although I am inclined to think the conjecture is true, it is certainly not as easy to prove as the first stated lemma (which is a 3-4 line proof). The asymmetry enters in the definition of a dual norm, which allows us to "guess" a separable answer at the cost of having underestimated the size of the norm, but we can not so easily overestimate it!

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