Question

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 $x,y,z$.

Answer 1

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.