What is the story behind this Hilbert space in the definition of Hilbert Modules

Question

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?

Answer 1

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").