\(H^{\infty}+BUC\) does not have the best approximation property (Q1070494)

Any closed algebra between \(H^{\infty}\) and \(L^{\infty}\) is called a Douglas algebra. The question had been raised of whether all Douglas algebras had the best approximation property. That is, given any Douglas algebra \({\mathcal D}\) and \(f\in L^{\infty}\) does there exist a \(g\in {\mathcal D}\) such that \(\| f-g\|_{\infty}=\inf \{\| f- g\|_{\infty}:g\in {\mathcal D}\}\). An example is given of such an algebra that does not have this property.