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If $A$ is an element of $\mathbb{R}^n \otimes\mathbb{R}^n \otimes\mathbb{R}^n$, then define its injective tensor norm to be $$\|A\|_{\rm inj} := \max_{x,y,z\in \mathbb{R}^n, \|x\|=\|y\|=\|z\|=1} |\langle A, x\otimes y\otimes z\rangle|.$$ Here the norm on vectors is the usual Euclidean norm.

I have two questions.

  1. Does this norm change if x,y,z can be taken to arbitrary complex unit vectors?
  2. What if $A$ is symmetric under exchange of the first two positions? That is,
    $\langle A, x\otimes y\otimes z\rangle = \langle A, y\otimes x\otimes z\rangle$ for all $x,y,z$.
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I'm having trouble getting the $$ equation to work. – aram Sep 30 '11 at 23:01
Fixed your LaTeX and added the "banach-spaces" tag – Yemon Choi Sep 30 '11 at 23:08
As clarification: what exactly is the second question? Is it whether the norm changes if $A$ is symmetric in two factors and the vectors are allowed to be complex (as in the first question)? – ARupinski Sep 30 '11 at 23:22
Yes. Sorry for being unclear. – aram Oct 1 '11 at 0:58

I just found this paper, which gives an example in which the real and complex version of the norm are different. The tensor in this example is also symmetric, which provides an example for part 2 as well.

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