I will describe joint work with Baruch Solel in which we investigate the following type of operator algebra that arises naturally in the noncommutative function theory that is based on free algebras. Let $X$ denote a compact subset of the $d$-tuples of $n\times n$ complex matrices $M_{n}(\mathbb{C})^{d}$; and let $\mathfrak{A}(X)$ denote the closure of the space of $d$-generic $n\times n$ matrices in $C(X,M_{n}(\mathbb{C}))$. I will discuss auspicious circumstances under which we can identify the $C^{*}$-algebra generated by $\mathfrak{A}(X)$ and Arveson's $C^{*}$-envelope of $\mathfrak{A}(X)$. It turns out that under these circumstances, functions in $\mathfrak{A}(X)$ don't really ``live'' on $X$. Rather, their natural habitat is a certain quotient of a subset of $X$.