Here is Deflnition 1.5 of Hilbert module in "L^2-invariants: theory and applications to geometry and K-theory", Springer-Verlag, 2002, by W. Lück: 



> 
> A Hilbert $\mathcal N(G)$-module $V$ is a Hilbert space $V$ together
>  with a linear isometric $G$-action such that there exists a Hilbert
>  space $H$ and an isometric linear $G$-embedding of $V$ into the tensor
>  product of Hilbert spaces $H\bar\otimes\ell^2(G)$ with the obvious
>  $G$-action.
> 
> 
> 


($\mathcal N(G)$ is the group von Neumann algebra of a group $G$).


My question is about the Hilbert space $H$ in the definition. What is important about inserting this Hilbert space in this definition?



What is the story behind this Hilbert space in the definition of Hilbert Modules
Consider the structure theory for normal $\*$-homomorphisms of a von Neumann algebra $M$. Namely, if $M\subseteq B(H)$, and $M\rightarrow B(K)$ is a normal $\*$-homomorphism then, up to unitary conjugation, we may suppose that there is another Hilbert space $H'$ so that $K$ is an invariant (for the $M$ action) subspace of $H\otimes H'$ with $M$ acting as $M\otimes 1$.


So applying to $\mathcal N(G)\subseteq B(\ell^2(G))$ we obtain subspaces of $H \otimes \ell^2(G)$, with $\mathcal N(G)$ acting as $1\otimes\mathcal N(G)$. As $\mathcal N(G)$ is generated by $\{\lambda(g) : g\in G\}$ the translation operators on $\ell^2(G)$, it is not hard to see that $V\subseteq H\otimes \ell^2(G)$ is $\mathcal N(G)$-invariant if and only if it is $G$-invariant, for the translation action on $\ell^2(G)$.


Thus, your definition seems to capture exactly the spaces $V$ with $\mathcal N(G)\rightarrow B(V)$ a normal unital $\*$-homomorphism. Again, given such a map, we need only know the image of $\lambda(g)$, as $g$ varies, to reconstruct the whole homomorphism; that is, we only need the $g$ action on $V$. The definition nicely captures exactly which $G$-spaces which can occur, without having to talk about the von Neumann algebra theory of $\mathcal N(G)$. Finally, this would indeed seem to be a reasonable notion of what an "$\mathcal N(G)$-module" should be (which, as Yemon suggests, is different from the notion of a "Hilbert $C^\*$-module").


