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

Let C(X) be the algebra of all real-valued continuous functions on a Tychonoff space X, and \(C^*(X)\) $C^{\ast }(X)$ the subalgebra of bounded functions. We prove that if B is any subalgebra of C(X) containing \(C^*(X)\) $C^{\ast }(X)$ , then no maximal solid subspace of B contains \(C^*(X)\) $C^{\ast }(X)$ , and we derive from this that the maximal solid subspaces of B are exactly the real maximal ideals of B. Then we extend the above to the case of intermediate algebras in A, where A is a \(\varPhi \) $\Phi$ -algebra with bounded inversion.