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Let $V$ and $W$ be Banach spaces.

$V^* \otimes W^*$ embeds into $(V \otimes W)^*$ (projective tensor product). I am looking for criteria for it to be an isomorphism.

If $V$ and $W$ are $C^*$-algebras, is this an isomorphism?

If $V$ and $W$ are reflexive, is this an isomorphism?

This property is one which commonly is useful, especially when I want to take the dual of an algebra (which makes a coalgebra if this holds).

Thanks very much!

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    $\begingroup$ If your tensor denotes the projective tensor product then the map you have described below is usually not bounded below when both V and W are infinite dimensional. Much as one might like this property to hold it will fail if both V and W are infinite-dimensional Cstar algebras or infinite-dimensional Hilbert spaces, and indeed probably for "most" infinite-dimensional Banach spaces. $\endgroup$
    – Yemon Choi
    Commented Mar 9, 2021 at 3:33
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    $\begingroup$ Loosely speaking, the "dual" tensor norm to the projective tensor norm is the injective tensor norm. You can see this if you take $V=W=c_0$, and then $V^* \widehat{\otimes} W^*$ is (isometrically) isomorphic to $(V \otimes_\varepsilon W)^*$. You might find some of the discussion on this old MO question helpful: mathoverflow.net/questions/105502/… $\endgroup$
    – Yemon Choi
    Commented Mar 9, 2021 at 3:36
  • $\begingroup$ Thanks Yemon! If you write this as an answer, I will accept and upvote. $\endgroup$
    – user30211
    Commented Mar 9, 2021 at 6:10
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    $\begingroup$ Returning to your question concerning reflexive spaces: The dual of $V\otimes_\varepsilon W$ is then $V^* \otimes _\pi W^*$ provided one of these spaces has Grothendieck's approximation property. (There's a more general such theorem involving spaces with the Radon-Nikodym property.) $\endgroup$
    – Dirk Werner
    Commented Mar 9, 2021 at 20:36
  • $\begingroup$ Good point @DirkWerner - also Dean, you might want to keep in mind that in general $\ell_\infty({\bf N}) \otimes_\varepsilon \ell_\infty({\bf N})$ is a closed subspace of $\ell_\infty({\bf N}\times {\bf N})$ but it is much smaller, and so if one wants to do Hopf-von Neumann algebras one can't base it on these "small" tensor products, one needs some kind of "extended" or "weak" version $\endgroup$
    – Yemon Choi
    Commented Mar 11, 2021 at 11:59

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