  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 A\rtimes_{\alpha} \Sigma $ be the associated semicrossed product.
Using Takai duality we establish a stable isomorphism \[ \mathcal
A\rtimes_{\alpha} \Sigma \sim_{s} \big(\mathcal A \otimes \mathcal K(\mathcal
G, \Sigma, \mu)\big)\rtimes_{\alpha\otimes {\rm Ad}\: \rho} \mathcal G, \]
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 $\mathcal A\rtimes_{\alpha} \Sigma$ and the
$(\alpha\otimes {\rm Ad}\: \rho)$-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.
