On the Takai duality for $L^{p}$ operator crossed products

Abstract

The aim of this paper is to study a problem raised by Phillips concerning the existence of Takai duality for $L^p$ operator crossed products $F^{p}(G,A,\alpha )$, where G is a locally compact Abelian group, A is an $L^{p}$ operator algebra and $\alpha$ is an isometric action of G on A. Inspired by Williams’ proof for the Takai duality theorem for crossed products of $C^*$-algebras, we construct a homomorphism $\Phi$ from $F^{p}({\hat{G}},F^p(G,A,\alpha ),{\hat{\alpha }})$ to $\mathcal {K}(l^{p}(G))\otimes _{p}A$ which is a natural $L^p$-analog of Williams’ map. For countable discrete Abelian groups G and separable unital $L^p$ operator algebras A which have unique $L^p$ operator matrix norms, we show that $\Phi$ is an isomorphism if and only if either G is finite or $p=2$; in particular, $\Phi$ is an isometric isomorphism in the case that $p=2$. Moreover, it is proved that $\Phi$ is equivariant for the double dual action $\hat{{\hat{\alpha }}}$ of G on $F^p({\hat{G}},F^p(G,A,\alpha ),{\hat{\alpha }})$ and the action $\textrm{Ad}\rho \otimes \alpha$ of G on $\mathcal {K}(l^p(G))\otimes _p A$.