Takesaki duality for weak* closed $L^p$-operator crossed products

The aim of this paper is to study Takesaki duality for weak* closed $L^p$-operator crossed product $W^*_p(G,A,\alpha)$, where $G$ is a countable discrete Abelian group, $A$ is a unital separable weak* closed $L^p$-operator algebra ($p>1$), and $\alpha$ is a weak* continuous $p$-completely isometric action of $G$ on $A$. In this paper, we construct a weak* continuous homomorphism $\Phi$ from $W^*_p(\hat{G},W^*_p(G,A,\alpha),\hat{\alpha})$ to $\mathcal{B}(l^{p}(G))\bar{\otimes}A$. We show that $\Phi$ is an isomorphism if and only if either $p=2$ or $G$ is finite, and $\Phi$ is an isometric isomorphism if either $p=2$ or $G$ is trivial. It is also proved that $\Phi$ is equivariant for the double dual action $\hat{\hat{\alpha}}$ of $G$ on $W^*_p(\hat{G},W^*_p(G,A,\alpha),\hat{\alpha})$ and the action $\mathrm{Ad}\rho_p\otimes\alpha$ of $G$ on $\mathcal{B}(l^p(G))\bar{\otimes} A$. Furthermore, we prove that $W^*_p(\hat{G},W^*_p(G,A,\alpha),\hat{\alpha})$ is weak* continuous isometrically isomorphic to $\mathcal{B}(l^{p}(G))\bar{\otimes}A$ if and only if either $p=2$ or $G$ is trivial, and $W^*_p(\hat{G},W^*_p(G,A,\alpha),\hat{\alpha})$ is weak* continuous isomorphic to $\mathcal{B}(l^{p}(G))\bar{\otimes}A$ if and only if either $p=2$ or $G$ is finite when $A=M_n^p$. This shows that Takesaki duality theorem of von Neumann algebras can be generalized to weak* closed $L^2$-operator algebras, and this theorem can not be generalized to weak* closed $L^p$-operator algebras when $p\in (1,\infty)\setminus\{2\}$.