**Edit: incorrect claim at end of earlier version; thanks to Matthew Daws for pointing this out in comments.**

$\newcommand{\cM}{{\mathcal M}}\newcommand{\stp}{{\overline{\otimes}}}$The following technical question arose in some work I'm doing, which concerns traces on Banach algebras, but which has wandered into territory that I don't know well.

Let $(\cM,\Delta)$ be a Hopf von Neumann algebra: that is,
 $\cM$ is a von Neumann algebra and $\Delta: \cM\to \cM\stp\cM$ is a coassociative, injective, normal $*$-homomorphism.

> Does there always exist a completely bounded, linear map $T:\cM\stp\cM\to \cM$ such that $T\Delta$ is the identity? If so, can we always choose $T$ to be normal?

This works for <strike>many</strike> several  examples, for instance when $\cM$ is injective as a von Neumann algebra [the image of $\Delta$ is then complemented in $B(H\otimes H)$ by a norm-one projection], <strike>or $\cM$ is a locally compact quantum group in the sense of Kustermans-Vaes [use the fundamental unitary and then slice].</strike> or if the predual $\cM_*$ has a bounded approximate identity for the natural product induced by $\Delta_*$.