Timeline for A left inverse for the comultiplication on a Hopf von Neumann algebra
Current License: CC BY-SA 3.0
9 events
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| Dec 20, 2023 at 23:09 | comment | added | J. De Ro | @YemonChoi The comment I mention holds more generally for locally compact quantum groups btw. More concretely, if $\mathbb{G}$ is a lcqg with associated Hopf-von Neumann algebra $(L^\infty(\mathbb{G}), \Delta)$, then the action $\Delta: L^\infty(\mathbb{G})\curvearrowleft \mathbb{G}$ is amenable if and only if $\hat{\mathbb{G}}$ is inner amenable. | |
| Dec 20, 2023 at 21:15 | comment | added | Yemon Choi | @J.DeRo Thanks - I think I did once know something similar (using the language of QSIN groups, I forget the exact connection with inner amenablity in the Paterson-Lau sense). I must confess that the original motivation for my question was a project that got put on hiatus and is now gathering dust, but your comment might prompt me to try and revive it. | |
| Dec 19, 2023 at 8:40 | comment | added | J. De Ro | You might know this, but if $G$ is an inner amenable locally compact group and you consider the associated Hopf-von Neumann algebra $(\mathscr{L}(G), \Delta)$, then there exists an ucp map $E: \mathscr{L}(G)\bar{\otimes}\mathscr{L}(G) \to \mathscr{L}(G)$ such that $E\circ \Delta= \iota$ and $(E\otimes \iota)\circ(\iota \otimes \Delta)= \Delta\circ E$. In other words, there exists an equivariant version of your desired map. | |
| Nov 16, 2011 at 12:19 | answer | added | Matthew Daws | timeline score: 2 | |
| Nov 14, 2011 at 22:05 | history | edited | Yemon Choi | CC BY-SA 3.0 |
corrected erroneous claim (h/t Matt Daws)
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| Nov 14, 2011 at 21:59 | comment | added | Yemon Choi | Argh, you're right - I think I had examples where $\mathcal M$ is injective or $(\mathcal M)_*$ is operator biflat in mind. Will edit accordingly | |
| Nov 14, 2011 at 9:38 | comment | added | Matthew Daws | Erm... How do you use the fundamental unitary to slice? You have that $\Delta(x) = W^*(1\otimes x)W$ and so $W\Delta(x)W^* = 1\otimes x$. So it would be tempting to use the map $M\overline\otimes M \rightarrow B(H), z\mapsto (\omega\otimes\iota)(WzW^*)$ but I don't see why this lands in $M$?? Or did you have another argument in mind (in which case, I'd love to see it!) | |
| Nov 14, 2011 at 4:13 | history | edited | Mariano Suárez-Álvarez | CC BY-SA 3.0 |
Silly edit to avoid empty space on top of the post
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| Nov 14, 2011 at 3:53 | history | asked | Yemon Choi | CC BY-SA 3.0 |