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Decompositions of the maximal ideal space of \(L^{\infty}\). (English) Zbl 0545.30040

There are two well-known decompositions of the maximal ideal space of \(L^{\infty}\) (denoted \(M(L^{\infty})\), Shilov’s decomposition and Bishop’s decompositions for \(H^{\infty}+C\). To obtain a level set in Shilov’s decomposition, one considers QC, the largest C-* algebra contained in \(H^{\infty}+C\). The level set corresponding to an element x of \(M(L^{\infty})\) is the set of all elements in \(M(L^{\infty})\) which agree with x on all QC functions. In this context, Shilov’s theorem specializes to the following: Let f be a function in \(L^{\infty}\). If the restriction of f to each QC level set is an element of \(H^{\infty}+C\) restricted to the level set, then f lies in \(H^{\infty}+C\). The sets in Bishop’s decomposition are the maximal antisymmetric sets for \(H^{\infty}+C\), that is, the maximal sets S such that whenever f is an \(H^{\infty}+C\) function and \(f| S\) is real valued, then \(f| S\) is constant. In this case, Bishop’s theorem shows that one may replace level sets by antisymmetric sets in Shilov’s theorem above. Examples of QC level sets that are not antisymmetric were given by Sarason.
Each point x of the maximal ideal space of \(H^{\infty}+C\) has a unique representing measure \(m_ x\). The closed support of \(m_ x\) is an antisymmetric set for \(H^{\infty}+C\). By replacing level sets by support sets, Sarason further refined Shilov’s theorem. Sarason also asked the following question: Is every maximal antisymmetric set for \(H^{\infty}+C\) a support set? This question remains open. This paper gives the first examples of maximal antisymmetric sets which are also support sets. A class of examples of level sets which are not maximal antisymmetric sets are also given.

MSC:

30H05 Spaces of bounded analytic functions of one complex variable
46J15 Banach algebras of differentiable or analytic functions, \(H^p\)-spaces
46J20 Ideals, maximal ideals, boundaries
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