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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.

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  • $\begingroup$ 2 years later, I have the same question. Did you perhaps find other references than the ones in the answer below? $\endgroup$
    – J. De Ro
    Commented Aug 10, 2022 at 21:27
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    $\begingroup$ @QuantumSpace I'm afraid that I became busy with other projects and haven't given this further thought. I would still be interested to know of any other references in the literature $\endgroup$
    – Yemon Choi
    Commented Aug 11, 2022 at 13:42
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    $\begingroup$ Alright, I'll let know if i come across anything useful! $\endgroup$
    – J. De Ro
    Commented Aug 11, 2022 at 16:57

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Although i do not know much more to say, i recall i have seen this variant of the definition of the comodule you are describing, used in the context of Hopf-von Neumann algebras. See for example:

Crossed products of dual operator spaces by locally compact groups, Dimitrios Andreou, arXiv:1910.00433 [math.OA] (see the def in p.3)

and also:

Masamichi Hamana: Injective envelopes of dynamical systems, Toyama Math. J., Vol. 34(2011), 23-86 https://toyama.repo.nii.ac.jp/?action=repository_action_common_download&item_id=3070&item_no=1&attribute_id=18&file_no=1 (see def 2.1, p.31)

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    $\begingroup$ Thanks for these references. A quick look suggests that both of these are considering a variant of the Fubini tensor product involving embedding into B(H) and using normal slices - possibly this is the same as what I have defined above, using embedding of a Cstar algebra into its bidual, but it's not quite clear to me. But thank you for these references, which I was not aware of or had forgotten, they may be helpful for me $\endgroup$
    – Yemon Choi
    Commented Feb 14, 2020 at 13:45
  • $\begingroup$ Also worth noting that both of these seem to be using the Hopf-von Neumann perspective, whereas the work which motivated my question is really about the Hopf-Cstar perspective (for better or for worse) $\endgroup$
    – Yemon Choi
    Commented Feb 14, 2020 at 13:47

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