On the Takai duality of $L^p$ operator crossed products, II

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 $p\in [1,\mathrm{\infty })$, 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 $F^p(\widehat{G},F^p(G,A,\alpha ),\widehat{\alpha })$ is isometrically isomorphic to ${\bar{M}}_G^p{\otimes }_{p}A$ if and only if $p=2$.