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| Paper IPM / M / 18013 |
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| Abstract: | |
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or a discrete group $\Gamma$, a Hopf von Neumann algebra $(\mathfrak{M},\Delta)$
and a $W^*$-dynamical system $(\mathfrak{M},\Gamma,\alpha)$
such that
$(\alpha_s\otimes\alpha_s)\circ\Delta=\Delta\circ\alpha_s$, we show that the crossed product
$\mathfrak{M}\rtimes_\alpha\Gamma$ with a co-multiplication is a Hopf von Neumann algebra.
Furthermore, we prove
that the inner amenability of the predual $\mathfrak{M}_*$ is equivalent to the inner amenability of
$(\mathfrak{M}\rtimes_\alpha\Gamma)_*$. Finally, we conclude that if the action
$\alpha:\Gamma\rightarrow\mathrm{Aut}(\ell^\infty(\Gamma))$ is defined by
$\alpha_s(f)(t)=f(s^{-1}ts)$, then the inner amenability of discrete group $\Gamma$
is equivalent to the inner amenability of $(\ell^\infty(\Gamma)\rtimes_\alpha\Gamma)_*$.
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