  For a compact subset $K\subset \mathbb C$ and a positive finite Borel measure
$\mu$ supported on $K,$ let $\text{Rat}(K)$ denote the space of rational
functions with poles off $K,$ let $R^\infty (K,\mu)$ be the weak-star closure
of $\text{Rat}(K)$ in $L^\infty (\mu),$ and let $R^2 (K,\mu)$ be the closure of
$\text{Rat}(K)$ in $L^2(\mu).$ We show that there exists a compact subset
$K\subset \mathbb C,$ a positive finite Borel measure $\mu$ supported on $K,$
and a function $f\in R^\infty (K,\mu)$ such that $R^\infty (K,\mu)$ has no
non-trivial direct $L^\infty$ summands, $f$ is invertible in $R^2 (K,\mu)\cap
L^\infty(\mu),$ and $f$ is not invertible in $R^\infty (K,\mu).$ The result
answers an open question concerning spectral mapping theorems for subnormal
operators raised by J. Dudziak in 1984.
