Deformation of uniform algebras on Riemann surfaces (Q1074823)

The author continues his investiation on deformation of uniform algebras [Proc. Lond. Math. Soc., III. Ser. 39, 93-118 (1979; Zbl 0423.46042)] in studying algebras of analytic functions on Riemann surfaces. Let \({\mathcal M}\) be the space of separable uniform algebras, where two such algebras are identified if each is isomorphic to an arbitrary small deformation of the other. Equipped with the Banach-Mazur distance \({\mathcal M}\) becomes a complete metric space. Further let \({\mathcal S}\) denote the set of all connected finite bordered Riemann surfaces, and for \(S\in {\mathcal S}\) let A(S) be the Banach algebra of functions continuous on S and analytic in the interior of S. Finally \({\mathcal A}\) denotes the subset of \({\mathcal M}\) consisting of all finite direct sums of algebras in A(\({\mathcal S}):=\{A(S):\) \(S\in {\mathcal S}\}\) and their finitecodimensional subalgebras which - by technical reasons - additionally are assumed to have connected maximal ideal space, and their Shilov boundaries are to be homeomorphic to a finite number of circles. The elements of \({\mathcal A}\) are described, and as the main theme of the paper the set \({\mathcal A}\) and its subsets \({\mathcal A}_ k\) of contant defect k are considered as subsets of the metric space \({\mathcal M}\). Closeness in A(\({\mathcal S})\) can be characterised in terms of quasiconformal equivalence. The \({\mathcal A}_ k\) are shown to be open and closed in \({\mathcal M}\), and the local structure of the \({\mathcal A}_ k\) is studied. Especially a detailed description of \({\mathcal A}_ k\) is given for very small k.