On the Takai duality for LP operator crossed products

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((G) over cap, F-p(G, A, alpha), (alpha) over cap) to K(l(p)(G)) circle times(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 (alpha) over cap of G on F-p((G) over cap, F-p(G, A, alpha), (alpha) over cap) and the action Ad rho circle times alpha of G on K(l(p)(G)) circle times(p) A.