See Tomiyama, Jun, On the difference of n-positivity and complete positivity in C*-algebras, J. Funct. Anal. 49, 1-9 (1982). ZBL0497.46039.


Although i do not know much more to say, i recall i have seen this variant of the definition of the comodule you are describing, used in the context of Hopf-von Neumann algebras. See for example: Crossed products of dual operator spaces by locally compact groups, Dimitrios Andreou, arXiv:1910.00433 [math.OA] (see the def in p.3) and also: Masamichi ...


Case $G=\mathbb Z/2$, generated by $g\in G$: In that case, the action of $G$ on $M$ induces a grading $M=M_0 \oplus M_1$ where $M_0=M^G=\{m\in M: gm=m\}$ and $M_1=\{m\in M: gm=-m\}$. Similarly, the action of $G$ on $L^2M$ induces a grading $L^2M=(L^2M)_0\oplus (L^2M)_1$. The subspace $(L^2M)_0$ is isomorphic to $L^2(M^G)$. And both $(L^2M)_0$ and $(L^2M)_1$...

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