Douglas algebras which admit codimension 1 linear isometries (Q2718973)

\(L^\infty\) denote the space of essentially bounded Lebesgue measurable functions on the unit circle and \(H^\infty\) the closed subalgebra associated with the bounded analytic functions on the disc. A closed subalgebra \(B\) satisfying \(H^\infty\subseteq B \subseteq L^\infty\) is called a Douglas algebra, and a famous theorem of Chang and Marshall asserts that every Douglas algebra is generated by \(H^\infty\) together with the complex conjugates of some interpolating Blaschke products. Associated to a Douglas algebra \(B\) is another Douglas algebra \(B_b\), called the Bourgain algebra of \(B\), which satisfies \(B\subseteq B_b\). NEWLINENEWLINENEWLINEIn the paper under review, the author establishes the interesting result that a Douglas algebra \(B\) admits a linear isometry of codimension 1 if, and only if, \(B\neq B_b\). Moreover, a precise form is given for such an isometry. An example is presented of a Douglas algebra \(B\), other than \(H^\infty\), for which \(B\neq B_b\), thus refuting a conjecture of Araujo and Font. The paper assumes a reasonable familiarity with the structure of Banach spaces of analytic functions.