Sufficient conditions for a linear functional to be multiplicative (Q2719014)

The authors prove the following theorems:NEWLINENEWLINENEWLINE(1) Let \(R(X)\) be the uniform closure of the algebra of rational functions \(\text{Rat}(X)\) with poles off a compact nowhere dense subset \(X\) of the plane. If \(M\) is a finite-codimensional subspace of \(R(X)\) such that every element of it has at least \(k\) zeros in \(X\), then there are \(k\) common zeros for \(M\) in \(X\). That is, the \(P(k,n)\) property holds for \(R(X)\).NEWLINENEWLINENEWLINE(2) If \(X\) is a totally disconnected compact Hausdorff space, then \(\text{Re }C(X)\) has the \(P(1,n)\) property for all \(n\in\mathbb N\).NEWLINENEWLINENEWLINE(3) If \(X\) is a totally disconnected compact Hausdorff space such that each point of \(X\) is a \(G_\delta\), then the \(P(k,n)\) property holds for \(\text{Re }C(X)\) for all \(k, n\in\mathbb{N}\).NEWLINENEWLINENEWLINE(4) Let \(A\) be a unital real commutative Banach algebra with a totally disconnected maximal ideal space \(X\). If \(M\) is a finite-codimensional subspace of \(A\) such that the Gelfand transform of each element of \(M\) has a zero in \(X\), then \(M\) is contained in a maximal ideal.