Given a faithful flow ${alpha}$ on a $C^*$-algebra A, when A is ${alpha}$-simple we will show that the closed subalgebra of A consisting of elements with non-negative Arveson spectra is maximal if and only if the crossed product of A by ${alpha}$ is simple. We will also show how the general case can be reduced to the ${alpha}$-simple case, which roughly says that any flow with the above maximality is an extension of a trivial flow by a flow of the above type in the ${alpha}$-simple case. We also propose a condition of essential maximality for such closed subalgebras.