Crossed products of dual operator spaces by locally compact groups
Volume 258 / 2021
Abstract
If $ \alpha $ is an action of a locally compact group $ G $ on a dual operator space $ X $, then two generally different notions of crossed products are defined, namely the Fubini crossed product $ X\mathbin {\rtimes^{\mathcal {F}} _{\alpha }} G $ and the spatial crossed product $ X\mathbin {\overline{\rtimes} _{\alpha }} G $. It is shown that $ X\mathbin {\rtimes^{\mathcal {F}} _{\alpha }}G=X\mathbin {\overline{\rtimes} _{\alpha }}G $ if and only if the dual comodule action $ \widehat {\alpha } $ of the group von Neumann algebra $ L(G) $ on $ X\mathbin {\rtimes^{\mathcal {F}} _{\alpha }}G $ is non-degenerate. As an application, this yields an alternative proof of the result of Crann and Neufang (2019) 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 $ \mathop{\rm Bim}(J^{\perp}) $ and $ (\mathop{\rm Ran}J)^{\perp} $ defined by Anoussis, Katavolos and Todorov (2019) for a left ideal $ J $ of $ L^{1}(G) $ are respectively isomorphic to $ J^{\perp }\mathbin {\overline{\rtimes} } G $ and $ J^{\perp }\mathbin {\rtimes^{\mathcal {F}} } G $. Therefore a necessary and sufficient condition for $ \mathop{\rm Bim}(J^{\perp}) =(\mathop{\rm Ran}J)^{\perp} $ is deduced from the main result.
