Throughout, $\otimes$ denotes the minimal tensor product of $\newcommand{\Cst}{{\rm C}^{\ast}}\Cst$-algebras. and $\odot$ denotes the tensor product of (underlying) vector spaces.
Given two $\Cst$-algebras $A$ and $B$, with respective $\Cst$-subalgebras $C$ and $D$, the Fubini tensor product of $C$ and $D$ (relative to $A$ and $B$) is defined to be the following subset of $A\otimes B$:
$$C\otimes_{\mathcal F} D = \{ w \in A \otimes B \colon (\phi\otimes\iota)(w) \in C, (\iota\otimes\psi)(w)\in D \;\hbox{for all}\; \phi\in A^*\;\hbox{and all}\;\psi\in B^*\}.$$
This always contains $C\otimes D$ but example are known where it is strictly bigger; it does coincide with $C\otimes D$ if both $C$ and $D$ are nuclear, for instance.
I have a very hazy recollection of seeing some papers, possibly survey articles, where one is dealing with a Hopf $\Cst$-algebra A and wants to relax the usual definition of comodule $D$ so that the coaction takes values in $A\otimes_{\mathcal F} D$ — possibly Kirchberg's name came up, either for the technical prerequisites or as someone who had proposed a similar construction. Can anyone confirm if such a use of the Fubini tensor product has been tried before, and if so, whether it has gone anywhere? Mainly I want to quickly check if some ideas I am playing with are rediscovering old things or known not to work.
