Let $X$ be a compact subset of $\mathbb{C}$ such that:
$\mathbb{C} \setminus X$ is not connected
$0\notin X$.
The prototypical example of such an $X$ is an annulus centered at the origin. Let $\mathscr{B}$ denote the uniform algebra on $X$ generated by the functions $f_1(z) = z$ and $f_2(z) = \frac{1}{z}$. That is, $\mathscr{B}$ is the closure in the uniform norm of the Laurent polynomials.
Question: Can one describe the functions in $\mathscr{B}$ for "nice" $X$ (e.g. connected, smooth boundary, etc.)?
As a trivial example, if $X = \{\lambda: |\lambda| = 1\}$, then $\mathscr{B} = C(X)$. I have found results dating back to Wermer dealing with similar questions for $X = \{\lambda: |\lambda \leq 1\}$, but nothing particularly enlightening for when $X$ is not simply connected. Thank you in advance.