Crossed products of dual operator spaces by locally compact groups

TL;DR

This paper investigates the conditions under which two different notions of crossed products of dual operator spaces by locally compact groups coincide, providing new insights and proofs related to the structure of these crossed products.

Contribution

It establishes a criterion for the equality of Fubini and spatial crossed products in dual operator spaces and links these notions to group properties like the approximation property.

Findings

Fubini and spatial crossed products coincide if the dual comodule action is non-degenerate.

Provides an alternative proof that the two notions coincide when the group has the approximation property.

Identifies bimodules associated with ideals in $L^1(G)$ as crossed products, giving a necessary and sufficient condition for their equality.

Abstract

For an action $\alpha$ of a locally compact group $G$ on a dual operator space $X$ by w*-continuous completely isometric isomorphisms one can define two generally different notions of crossed products, namely the Fubini crossed product $X\rtimes_{\alpha}^{F}G$ and the spatial crossed product $X\overline{\rtimes}_{\alpha} G$. It is shown that $X\rtimes_{\alpha}^{F}G=X\overline{\rtimes}_{\alpha} G$ if and only if the dual comodule action $\widehat{\alpha}$ of the group von Neumann algebra $L(G)$ on the Fubini crossed product of $X\rtimes_{\alpha}^{F}G$ is non-degenerate. As an application, this yields an alternative proof of the result of Crann and Neufang that the two notions coincide when G satisfies the approximation property (AP) of Haagerup and Kraus. Also, it is proved that the $L(G)$-bimodules $Bim(J^{\perp})$ and $Ran(J)^{\perp}$ defined by Anoussis, Katavolos and Todorov for a left closed ideal J of $L^{1}(G)$ can be identified respectively with a spatial crossed product and a Fubini crossed product of the annihilator of $J$ by $G$. Therefore a necessary and sufficient condition so that $Bim(J^{\perp})=Ran(J)^{\perp}$ is obtained by the main result.