# Hopf C-star algebra/comodules using a Fubini tensor product rather than the minimal tensor product?

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.