Geodesic
Approximation of Piecewise Continuous Functions by Quotients of Bounded Analytic Functions
Canadian journal of mathematics, Tome 24 (1972) no. 4, pp. 642-657

Voir la notice de l'article provenant de la source Cambridge University Press

This paper concerns a certain subalgebra of the Banach algebra of complex valued, essentially bounded, Lebesgue measurable functions on the unit circle in the complex plane (denoted here by L∞). My interest in this subalgebra was prompted by a question of R. G. Douglas. Let H∞ denote the space of functions in L∞ whose Fourier coefficients with negative indices vanish (equivalently, the space of boundary functions for bounded analytic functions in the unit disk). Douglas [5] has asked whether every closed subalgebra of L∞ containing H∞ is determined by the functions in H∞ that it makes invertible. More precisely, is such an algebra generated by H∞ and the inverses of the functions in H∞ that are invertible in the algebra? An affirmative answer is known for L∞ itself and for certain subalgebras of L∞ recently studied by Davie, Gamelin, and Garnett [3]. At the time of this writing, no algebra is known for which the above question can be answered negatively. 
Sarason, Donald. Approximation of Piecewise Continuous Functions by Quotients of Bounded Analytic Functions. Canadian journal of mathematics, Tome 24 (1972) no. 4, pp. 642-657. doi: 10.4153/CJM-1972-060-9
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