Crossed products of dual operator spaces by locally compact groups

2019/10/01 by Andreou, Dimitrios
#FOS: Mathematics #Operator Algebras (math.OA)

paper · doi:10.48550/arxiv.1910.00433

Abstract

For an action α 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_α^FG and the spatial crossed product X\rtimes_α G. It is shown that X\rtimes_α^FG=X\rtimes_α G if and only if the dual comodule action \widehatα of the group von Neumann algebra L(G) on the Fubini crossed product of X\rtimes_α^FG 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^⊥) and Ran(J)^⊥ defined by Anoussis, Katavolos and Todorov for a left closed ideal J of L¹(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^⊥)=Ran(J)^⊥ is obtained by the main result.

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