For a compact set K⊂C, a finite positive Borel measure μ
on K, and 1 \le t < \i, let Rat(K) be the set of rational
functions with poles off K and let Rt(K,μ) be the closure of
Rat(K) in Lt(μ). For a bounded Borel subset D⊂C, let \area_{\mathcal D} denote the area (Lebesgue) measure
restricted to D and let H^\i (\mathcal D) be the weak-star closed
sub-algebra of L^\i(\area_{\mathcal D}) spanned by f, bounded and analytic
on C∖Ef for some compact subset Ef⊂C∖D. We show that if Rt(K,μ) contains no non-trivial
direct Lt summands, then there exists a Borel subset R⊂K
whose closure contains the support of μ and there exists an isometric
isomorphism and a weak-star homeomorphism ρ from Rt(K,μ)∩L∞(μ) onto H∞(R) such that ρ(r)=r for all
r∈Rat(K). Consequently, we obtain some structural decomposition
theorems for \rtkmu.Comment: arXiv admin note: text overlap with arXiv:2212.1081