# ROOM ONE: Characterising the approximation property of Haagerup-Kraus via crossed products of dual operator spaces

Let $\alpha$ be an action of a locally compact group $G$ on a dual operator space $X$ by w*-continuous completely isometric automorphisms. There are two natural ways to define the crossed product of $X$ by $\alpha$; the spatial crossed product $X\overline{\rtimes}_{\alpha}G$, which is the w*-closed linear span of "natural" generators, and the Fubini crossed product $X\rtimes^{F}_{\alpha}G$, which is the fixed point space of an appropriate action. In several special cases these two notions coincide; for example, Digernes-Takesaki established equality for the case of von Neumann algebras and normal *-automorphic actions, and Salmi-Skalski extended this to W*-TRO's and normal W*-TRO-automorphic actions. However, for general dual operator spaces the two crossed products may differ, even for trivial actions. Recently, Crann and Neufang proved that if the locally compact group $G$ has the approximation property of Haagerup-Kraus, then $X\rtimes^{F}_{\alpha}G=X\overline{\rtimes}_{\alpha}G $ for any action $\alpha$ on some dual operator space $X$.

We will discuss a duality theory for these crossed products. As an application, we give an alternative approach and an extension of the aforementioned result of Crann-Neufang: a locally compact group $G$ has the approximation property if and only if $X\rtimes^{F}_{\alpha}G=X\overline{\rtimes}_{\alpha}G $ for any action $\alpha$ on some dual operator space $X$.