The Takai duality of $L^p$-operator algebras with incompressibility

Abstract

Let $G$ be a countable discrete Abelian group, $A$ be a separable unital $L^p$-operator algebra which has unique $L^p$-operator matrix norms for $p\in [1,\infty )$, and $\alpha$ be an isometric action of $G$ on $A$. We prove in this paper that if $A$ is $p$-isometrically incompressible, then $F^p(\hat{G},F^p(G,A,\alpha ),\hat{\alpha })$, the iterated $L^p$-operator crossed product, is isometrically isomorphic to ${\overline{M}}_G^p{\otimes }_{p}A$ if and only if $p=2$. Furthermore, we prove that if $A=M_n^p$, then $F^p(\hat{G},F^p(G,A,\alpha ),\hat{\alpha })$ is isomorphic to ${\overline{M}}_G^p{\otimes }_{p}A$ if and only if either $p=2$ or $G$ is finite. This shows that the Takai duality for $C^{\ast }$-algebras can not be generalized to $L^p$-operator algebras with $p\ne 2$, and solves a problem raised by N. C. Phillips in the negative.

arXiv comment18 pages