Let $A$ be a $\text{ von Neumann algebra }$, $\mathcal{H}$ is a cyclic $A$ module, $G$ be a finite group acting on $A$, is $\mathcal{H}$ cyclic module over fixed point subalgebra of the action? Further for which class of $A$-s the result is positive!!

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    $\begingroup$ Did you at least consider the finite-dimensional case? Obviously not, since if we take $A = H = M_n C$ (with $n > 1$), and $G$ a finite group acting irreducibly on $A$ (lots exist), then the fixed point algebra is just $C$, so $H$ is not cyclic over the fixed point algebra. If $A$ is a type II finite factor and the fixed point algebra is itself a factor, then there is a dimension associated, and the hoped-for result is still not true. $\endgroup$ – David Handelman May 8 at 13:29
  • $\begingroup$ I am not taking ergodic action, so the fixed point algebra as huge as $M$ itself. $\endgroup$ – user136400 May 9 at 9:06

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