Isolated point theorems for uniform algebras on two- and three-manifolds (Q2814412)

The author establishes that the Gleason conjecture holds for two important classes of uniform algebras considered by John Andersen, Alexander Izzo and John Wermer in connection with the peak point conjecture. The author obtains the following theorems.NEWLINENEWLINETheorem 2.1 (two-dimensional isolated point theorem). Suppose \(M\) is a compact two-dimensional real manifold with boundary of class \(C^1\). Let \(A\) be a uniform algebra on \(M\) generated by a collection of \(C^1\) functions. If (i) the maximal ideal space of \(A\) is \(M\), and (ii) every point of \(M\) is isolated in the Gleason metric for \(A\), then \(A = C(M)\).NEWLINENEWLINETheorem 2.2. Suppose \(X\) is a compact subset of \(M\), a two-dimensional real manifold with boundary of class \(C^1\), and \(A\) is a uniform algebra on \(X\) generated by the continuous functions that extend to \(C^1\) functions on a neighborhood of \(X\). If (i) the maximal ideal space of \(A\) is \(X\), and (ii) each point of \(X\) is isolated in the Gleason metric for \(A\), then \(A = C(X)\).NEWLINENEWLINETheorem 2.5 Suppose \(A\) is a uniform algebra on \(X\), and \(L\) is a closed subset of \(X\) containing the essential set \(E\) for \(A\). Then \(A|L\) is uniformly closed in \(C(L)\). Moreover, the maximal ideal space of \(A|L\) is \(L\) if and only if the maximal ideal space of \(A\) is \(X\).NEWLINENEWLINETheorem 3.1 (embedded three-dimensional isolated point theorem). Suppose \(M\) is a real-analytic three-dimensional submanifold of \(\mathbb C^n\). Assume that \(X\) is a compact subset of \(M\) such that the boundary \(\partial X\) of \(X\) relative to \(M\) is a two-dimensional submanifold of class \(C^1\). If (i) \(X\) is polynomially convex, and (ii) every point of \(X\) is an isolated point in the Gleason metric for \(P(X)\), then \(P(X) = C(X)\).