Maximal subalgebras of C⁎-algebras associated with periodic flows

We find necessary and sufficient conditions for the subalgebra of analytic elements associated with a periodic ${\mathrm{C}}^{?}$-dynamical system to be a maximal norm-closed subalgebra. Our conditions are in terms of the Arveson spectrum of the action. We also describe equivalent properties of the system in terms of the strong Connes spectrum and the simplicity of the crossed product.