On maximal solid subspaces of intermediate algebras in \(C(X)\) (Q6594795)

Let \(E\) be a real vector lattice. A subset \(S\) of \(E\) is said to be \textit{solid} whenever \(|f|\leq |g|\), with \(f\in E\) and \(g\in S\), then \(f\in S\). A solid linear subspace of \(E\) is called \textit{maximal}, when it is not contained in any other proper solid linear subspace of \(E\).\N\NIn the paper under review, it is proved that, for a Tychonoff space \(X\), the maximal solid linear subspaces of an intermediate algebra \(B\) in \(C(X)\) are exactly the real maximal ideals of \(B\) (Theorem 3). Recall that an intermediate algebra in \(C(X)\) is any subalgebra of \(C(X)\) containing \(C^*(X)\), where \(C(X)\) denotes the algebra of all the real-valued continuous functions on \(X\) and \(C^*(X)\) the subalgebra of the bounded functions in \(C(X)\).\N\NAfter that, this result is extended to the general case of the so-called \(\Phi\)-algebras with bounded inversion (Theorem 7). Finally, it is proved by means of Example 8, that last result is not longer true if the given \(\Phi\)-algebra is not closed under bounded inversion.