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

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?

• Warning: the phrase "Hilbert module" gets used in two different contexts to mean two completely different things – Yemon Choi Jan 27 '19 at 12:34
• Modules of the form $H\otimes \ell^2(G)$ are "free" in some sense -- as ${\mathcal N}(G)$-modules they correspond to amplifying the canonical ${\mathcal N}(G)$-module $\ell^2(G)$, as you can check by fixing an orthonormal basis of $H$. So Lueck's definition seems to be that he wants to consider the closed submodules of these "free modules" – Yemon Choi Jan 27 '19 at 12:37

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").