If $A$ is an element of $\mathbb{R}^n \otimes\mathbb{R}^n \otimes\mathbb{R}^n$, then define its [injective tensor norm](http://en.wikipedia.org/wiki/Topological_tensor_product) 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 $x,y,z$.


edit: My motivation for this question is related to the paper
<https://arxiv.org/abs/1205.4484>



injective tensor norms for real tensors
I just found [this paper](http://dmle.cindoc.csic.es/pdf/RRACEFN_2000_94_04_05.pdf), 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.


