So I was having a discussion with a friend as follows:
Let $\mathcal H$ be a Hilbert space and $\mathcal H^{\otimes n} = \mathcal H \otimes \mathcal H \otimes \cdots \otimes \mathcal H$ be a Hilbert space with inner product $$\left\langle \bigotimes_{i=1}^{n} u_i, \bigotimes_{i=1}^{n} v_i \right\rangle = \prod_{i=1}^{n} \langle u_i, v_i \rangle.$$ We define a Matrix or linear transformation over $\mathcal H$ as an array of the form
$$ U = \begin{bmatrix} u_{11} & u_{12} & \ldots & u_{1n}\\ u_{21} & u_{22} & \ldots & u_{2n}\\ \vdots & \vdots & \ddots & \vdots\\ u_{n1} & u_{n2} &\ldots & u_{nn} \end{bmatrix} $$
where $u_{ij} \in \mathcal H$.
Define the product of two matrices in $\mathcal H$ as $$ UV = \begin{bmatrix} \prod \limits_{i=1}^{n} \langle u_{1i},v_{i1} \rangle & \prod \limits_{i=1}^{n} \langle u_{1i},v_{i2} \rangle & \ldots & \prod \limits_{i=1}^{n} \langle u_{1i},v_{in} \rangle \\ \prod \limits_{i=1}^{n} \langle u_{2i},v_{i1} \rangle & \prod \limits_{i=1}^{n} \langle u_{2i},v_{i2} \rangle & \ldots \prod \limits_{i=1}^{n} \langle u_{2i},v_{in} \rangle \\ \vdots & \vdots & \ddots & \vdots\\ \prod \limits_{i=1}^{n} \langle u_{ni},v_{i1} \rangle & \prod \limits_{i=1}^{n} \langle u_{ni},v_{i2} \rangle &\ldots & \prod \limits_{i=1}^{n} \langle u_{ni},v_{in} \rangle \end{bmatrix} $$
Just as in normal martrix multiplication, where we identify a row of the left matrix and a column of the right matrix with elements $\mathbb{R}^n$, and then take the inner (dot) products, what we're doing here is identifying a row of the left matrix with $\mathcal H^{\otimes n}$ and a column with the right matrix with $\mathcal H^{\otimes n}$, and returning an array of complex numbers.
So, for example, we could interpret normal matrix multiplication (say, of an $n \times 2$ and $2 \times n$ matrix) in $\mathbb{R}^n$ as a Hilbert Space product of the form
\begin{align} \begin{bmatrix} x_1 & x_2 \end{bmatrix} \begin{bmatrix} y_{1} \\ y_2 \end{bmatrix} \end{align}
where $x_i,y_i \in \mathbb{R}^n$
or perhaps we could take elements in $L^2(\mathbb{R},m)$ and take a product
\begin{align} \begin{bmatrix} f_1 & f_2 \end{bmatrix} \begin{bmatrix} g_{1} \\ g_2 \end{bmatrix} \end{align}
which would just give $$\left(\int_\mathbb{R} f_1 \overline{g_1} \, dm\right)\left(\int_\mathbb{R} f_2 \overline{g_2} \, dm\right)$$
So what we've really done is given a ($n$-linear?) map from (n times, cartesian product) $$\mathcal H^{\otimes n} \times \mathcal H^{\otimes n} \times \cdots \times \mathcal H^{\otimes n} \to M_n(\mathbb{C})$$
I suppose my question would be if (i) this is a reasonable thing to construct and (ii) if this construction could potentially have any uses. Again, this was just a friend and I throwing out ideas. Is there anything, for example, that could be pulled back from $M_n(\mathbb{C})$ to be studied over Hilbert Spaces?