Maximality of the analytic subalgebras of $C^*$-algebras with flows

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.

Keywords: Arveson spectrum, maximal subalgebra, crossed product

MSC numbers: 46L55