Say you have Euclidean vectors $\mathbf{a}=a_i \mathbf{p}_i$ and $\mathbf{b}=b_j \mathbf{q}_j$ in $\mathbb{R}^3$, with bases $\mathbf{p}_i$ and $\mathbf{q}_j$. Then you could use a typical inner product to find $\mathbf{a}\cdot\mathbf{b}=a_i b_j (\mathbf{p}_i \cdot \mathbf{q}_j)=a_i b_j g_{ij}^{pq}$, where $g_{ij}^{pq}$ is a metric between the two bases. You can also perform a tensor product $\mathbf{a}\otimes\mathbf{b}=a_i b_j (\mathbf{p}_i \otimes \mathbf{q}_j)$, where for $\mathbf{c}=c_k \mathbf{r}_k$, we have products of vectors with tensors $(\mathbf{a}\otimes\mathbf{b})\cdot\mathbf{c}=\mathbf{a}(\mathbf{b}\cdot\mathbf{c})$ and $\mathbf{c}\cdot(\mathbf{a}\otimes\mathbf{b})=(\mathbf{c}\cdot\mathbf{a})\mathbf{b}$.
Now say $\mathbf{p}_i=\mathbf{q}_i=\mathbf{r}_i=\hat{\mathbf{e}}_i$, where $\hat{\mathbf{e}}_i$ are the standard Cartesian unit vectors. Everything is unchanged, although $g_{ij}^{pq}$ becomes $\delta_{ij}$. For the moment, I’m not going to worry about defining co- or contra-variant bases and the details of their inner products.
That’s great. But what happens when we look now at $\mathbb{C}^3$, in the general case that both the scalar components and the bases can be complex. I’m fine with defining the inner product to include a complex conjugate (in either the first or second term), e.g. $\mathbf{a}\cdot\mathbf{b}=\overline{a}_i b_j (\mathbf{p}_i \cdot \mathbf{q}_j)=\overline{a}_i b_j g_{ij}^{pq}$. This means that now $\mathbf{a}\cdot\mathbf{b}=\overline{\mathbf{b}\cdot\mathbf{a}}$.
But here I start to get myself confused. Texts often define the tensor (outer) product to include a conjugate on the side opposing the inner product, e.g. $\mathbf{a}\otimes\mathbf{b}=a_i \overline{b}_j (\mathbf{p}_i \otimes \mathbf{q}_j)$ to match the inner product defined above. Which leads me to several interrelated questions:
Is this conjugate in the outer product necessary, and how does this affect $(\mathbf{a}\otimes\mathbf{b})\cdot\mathbf{c}=[a_i \overline{b}_j (\mathbf{p}_i \otimes \mathbf{q}_j)]\cdot(c_k \mathbf{r}_k)$ and similar vector/tensor products?
How does this affect tensors of higher order, such as $\mathbf{a}\otimes\mathbf{b}\otimes\mathbf{c}$? Does $(\mathbf{a}\otimes\mathbf{b})\otimes\mathbf{c}=\mathbf{a}\otimes(\mathbf{b}\otimes\mathbf{c})$?
Is there a need to use complex conjugate basis vectors (i.e. dual spaces) such as $\overline{\mathbf{p}}_i$? If the bases were all Cartesian $\hat{\mathbf{e}}_i$ bases, would there be a need to keep track of something like $\overline{\hat{\mathbf{e}}}_i$, or would this be identical to $\hat{\mathbf{e}}_i$?
I’ve been pondering this for a while, but haven’t found quite the right references to figure it out. Any pointers would be great!
Edit: As commented below, this might be a bit of an overly broad question. Really, the question about whether a complex conjugate would be required for such outer products (#1) is the most critical.