Function theory and M-ideals (Q1072679)

Let A be a uniform algebra on a compact space X, with maximal ideal space \(M_ A\) and let \(\sigma\) be a representing measure on X for an \(\phi \in M_ A\). Then \(H^{\infty}(\sigma)\) denotes the weak-star closure of A in \(L^{\infty}(\sigma)\) and \(H^{\infty}(\sigma)+C(X)\) the closed linear span of \(H^{\infty}(\sigma)\) and C(X), the algebra of continuous functions on X. The paper under review deals with the question when \([H^{\infty}(\sigma)+C(X)]/H^{\infty}(\sigma)\) is an M-ideal in \(L^{\infty}(\sigma)/H^{\infty}(\sigma)\). The authors give necessary and some sufficient conditions. In the classical case \(H^{\infty}+C\) (X is the unit circle) an affirmative answer was given earlier by \textit{D. H. Luecking} [Proc. Am. Math. Soc. 79, 222-224 (1980; Zbl 0437.46043)]. He showed by work of \textit{E. M. Alfsen} and \textit{E. G. Effros} [Ann. Math., II. Ser. 96, 98-173 (1972; Zbl 0248.46019)] that this result implies that elements of \(L^{\infty}\) have best approximants in \(H^{\infty}+C\). The authors give also an elementary proof of the Alfsen-Effros result that M-ideals have the best approximation property. The general question of best approximation from other subalgebras lying between \(H^{\infty}\) and \(L^{\infty}\) seems not well-understood.