Nonlocal uniform algebras on three-manifolds (Q1759643)

Let \(A\) be a uniform algebra on a compact Hausdorff space \(X\). A function \(f\in C(X)\) is said to belong locally on \(X\) to \(A\) if, for each point \(x\in X\), there are a neighborhood \(N\) of \(x\) and a function \(g\in A\) coinciding with \(f\) on \(N\). The algebra \(A\) is said to be local on \(X\) if every function that belongs locally on \(X\) to \(A\) is in \(A\). The algebra \(A\) is local if it is local on its maximal ideal space. The note under review contains a theorem which says that, on every compact \(C^\infty\)-manifold \(M\) of dimension greater than or equal to three, there exists a nonlocal uniform algebra with maximal ideal space \(M\) generated by \(C^\infty\)-smooth functions. In the proof of the result, the author uses two interesting (technical) lemmas.