  This paper aims to study the $L^p$ Takai duality problem raised by N. C.
Phillips. Let $G$ be a countable discrete Abelian group, $A$ be a separable
unital $L^p$ operator algebra with $p\in [1,\infty)$, and $\alpha$ be an
isometric action of $G$ on $A$. When $A$ is $p$-incompressible and has unique
$L^p$ operator matrix norms, it is proved in this paper that the iterated $L^p$
operator crossed product $F^{p}(\hat{G},F^p(G,A,\alpha),\hat{\alpha})$ is
isometrically isomorphic to $\overline{M}_{G}^{p}\otimes_{p}A$ if and only if
$p=2$.
