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.

- Does this norm change if x,y,z can be taken to arbitrary
*complex*unit vectors? - 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 $x,y,z$.