What are the position matrices in noncommutative QFT?

Question

Steinacker (https://arxiv.org/pdf/1003.4134.pdf) introduces a matrix model for which there are "position" matrices $X^a$ which in the commutative limit become the traditional position variables $X^a \sim x^a$. In the article, not much information is given past that, other than $X^a\in \text{Mat}(\infty,\mathbb{C})$ and that the action is invariant under $U(\infty)$ transformations.

These objects are clearly not the same as having the usual position variables with a noncommutative star product, since such a substitution gives only what the author calls the "semiclassical" limit. If that is the case however, what mathematics has Steinacker introduced here? What are some resources which go more in depth about these objects?