Crossed products and $\mathrm{C}^{*}$-covers of semi-Dirichlet operator algebras

In this paper, we show that the semi-Dirichlet \mathrm{C}^{*} -covers of a semi-Dirichlet operator algebra form a complete lattice, establishing that there is a maximal semi-Dirichlet \mathrm{C}^{*} -cover. Given an operator algebra dynamical system we prove a dilation theory that shows that the full crossed product is isomorphic to the relative full crossed product with respect to this maximal semi-Dirichlet cover. In this way, we can show that every semi-Dirichlet dynamical system has a semi-Dirichlet full crossed product.