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There seem to be intimate connections between the different definitions of von Neumann module. The two that I'm aware of are Hilbert von Neumann modules and correspondences (in the sense of Connes). I was wondering if there is any significant interplay between the two - for instance if any of the ideas around Jones' submodule theory (which uses correspondences of von Neumann algebras) have been extended or transfered to Hilbert von Neumann modules?

Some observations: Suppose we have a left Hilbert $N$-module (so inner-product $N$-linear in the first variable) with a faithful state $\varphi$ on $N$. Then define a complex-valued inner product on $E$ via $$\langle x,y\rangle_\varphi:=\varphi(\langle x,y\rangle).$$ If we complete then we get a vector space $\widetilde{E}$ and a von Neumann algebra $N$ acting on it - so a left von Neumann module in a sense similar to that of correspondences.

Questions:

(i) Given a faithful representation of $N$ on a Hilbert space $H$ can we always find a state $\varphi$ and left Hilbert $N$-module $E$ so that the representation of $N$ on $\widetilde{E}$ is equivalent to the one on $H$?

(ii) Can this idea be taken over to the case of correspondences and Hilbert bimodules (for instance in the case of $II_1$ factors)?

(ii) What results (if any) can be pulled up to the Hilbert bimodule level?

I'm aware that question (iii) is vague, but any input or references would be appreciated!

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Answers: (i) Yes, if we replace states by weights (not every von Neumann algebra admits a faithful state); (ii) Yes (for all von Neumann algebras); (iii) All of them.

Suppose M is an arbitrary von Neumann algebra and p≥0 is a real number. Then we define a right L_p(M)-module as a right M-module equipped with an inner product with values in L_{2p}(M), satisfying the same algebraic properties as for Hilbert W*-modules together with the appropriate completeness condition (we require completeness in the measurable topology, which coincides with the σ-weak topology for p=0 and with the norm topology for p>0). Here L_p(M)=L^{1/p} denotes the L_p-space of M, in particular, L_0(M)=L^∞(M)=M, L_1(M)=L^1(M)=M_* (the predual), L_{1/2}(M)=L^2(M)=the Hilbert space of half-densities on M. (The subscript notation is much more natural than the superscript notation because L_p-spaces form a graded algebra, p being the grading.) A morphism of right L_p(M)-modules is defined as a morphism of algebraic right M-modules that is continuous in the measurable topology. It turns out that right L_p(M)-modules form a W*-category.

We observe that the category of representations of M on Hilbert spaces is equivalent to the category of right L_{1/2}(M)-modules. If we have a right L_{1/2}(M)-module X with an inner product x,y↦(x,y)∈L_1(M), then x,y↦tr(x,y)∈C is a complex-valued inner product on X, which turns X into a Hilbert space together with an action of M. Vice versa, if X is a Hilbert space equipped with an action of M, then x,y→(w∈M↦(x,yw)∈C)∈L_1(M) is the corresponding L_1(M)-valued inner product.

Suppose 0≤p≤q are real numbers. We define a functor from the category of right L_p(M)-modules to the category of right L_q(M)-modules by sending a right L_p(M)-module X to X⊗L_{q-p}(M). Here ⊗ denotes the algebraic tensor product, without any kind of completion. Although it is non-obvious, in the end this tensor product turns out to be complete. Likewise, we define a functor from the category of right L_q(M)-modules to the category of right L_p(M)-modules by sending a right L_q(M)-module Y to Hom_M(L_{q-p}(M),Y). Here Hom_M denotes the space of algebraic homomorphisms preserving the right action of M, without any kind of continuity property. Again it is a non-obvious fact that this space is actually a right L_p(M)-module. One can prove that the two functors defined above form an adjoint unitary equivalence of the W*-categories of right L_p(M) and L_q(M) modules.

In particular, the category of Hilbert W*-modules over M and the category of representations of M on Hilbert spaces are equivalent.

The result above extends to bimodules. An M-L_p(N)-bimodule is a right L_p(N)-module X together with a morphism of von Neumann algebras A→End_N(X). (The algebra of endomorphisms of any object in a W*-category is a von Neumann algebra.) Since the above equivalence is an equivalence of W*-categories, we can immediately extend it to an equivalence of categories of M-L_p(N) and M-L_q(N) bimodules.

In particular, the category of Hilbert W*-bimodules from M to N is equivalent to the category of Connes' correspondences from M to N.

Moreover, one can observe that the bicategory of von Neumann algebras, Connes' correspondences, which compose via Connes' fusion, and their intertwiners is equivalent to the bicategory of von Neumann algebras, Hilbert W*-bimodules, which compose via the completed tensor product, and their intertwiners. This result is also valid for arbitrary p.

References:

  • The equivalence in the last paragraph of the answer was apparently first proven by Baillet, Denizeau, and Havet in their 1988 paper Indice d'une espérance conditionnelle.

  • L_p(M) modules were defined by Junge and Sherman in their 2005 paper Noncommutative L^p modules.

  • I am not aware of any paper that proves the above equivalences for arbitrary p, but I will include a proof of these statements in my thesis.

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Thanks Dmitri - the references are just what I'm after. The unifying concept of L_p module sounds fascinating.. –  Ollie Margetts Nov 14 '10 at 12:06
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