Let $(mathcal G, Sigma)$ be an ordered abelian group with Haar measure $mu$, let $(mathcal A, mathcal G, alpha)$ be a dynamical system and let $mathcal Artimes_{alpha} Sigma $ be the associated semicrossed product. Using Takai duality we establish a stable isomorphism [ mathcal Artimes_{alpha} Sigma sim_{s} big(mathcal A otimes mathcal K(mathcal G, Sigma, mu)big)rtimes_{alphaotimes {rm Ad}: rho} mathcal G, ] where $mathcal K(mathcal G, Sigma, mu)$ denotes the compact operators in the CSL algebra ${rm Alg}:mathcal L(mathcal G, Sigma, mu)$ and $rho$ denotes the right regular representation of $mathcal G$. We also show that there exists a complete lattice isomorphism between the $hat{alpha}$-invariant ideals of $mathcal Artimes_{alpha} Sigma$ and the $(alphaotimes {rm Ad}: rho)$-invariant ideals of $mathcal A otimes mathcal K(mathcal G, Sigma, mu)$. Using Takai duality we also continue our study of the Radical for the crossed product of an operator algebra and we solve open problems stemming from the earlier work of the authors. Among others we show that the crossed product of a radical operator algebra by a compact abelian group is a radical operator algebra. We also show that the permanence of semisimplicity fails for crossed products by $mathbb R$. A final section of the paper is devoted to the study of radically tight dynamical systems, i.e., dynamical systems $(mathcal A, mathcal G, alpha)$ for which the identity ${rm Rad}(mathcal A rtimes_alpha mathcal G)=({rm Rad}:mathcal A) rtimes_alpha mathcal G$ persists. A broad class of such dynamical systems is identified.
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