# Universal characterization and explicit description of elements of the group von Neumann algebra and the crossed product

Group von Neumann algebras and crossed products for a locally compact group G can be constructed in many different ways. For example, one can take the von Neumann algebra generated by certain operators on a certain Hilbert space.

However, none of these constructions give an explicit description of elements of the group algebra or the crossed product.

I am looking for such an explicit description. I suspect that distributions with bounded Fourier transform might be involved, but I am not entirely sure about this.

I am also looking for a more abstract description of these constructions. Can we characterize the group von Neumann algebra and the crossed product by some universal property?

Any references on this matter will be appreciated.

• I'm having a hard time parsing this question. What's a simple example? E.g. would you be interested in: Let M be a vn alg on H, with an action of $\mathbb R$. Then we can form the cross-product, which lives on the Hilbert space $L^2(\mathbb R,H)$. Do you want to understand what an element in the resulting algebra looks like? If M is the complex numbers, then you just get the group von Neumann algebra VN(G). The predual of this is the Fourier algebra A(G): you can indeed then think of VN(G) as being "distributions" on G, but I'd have to think to find the details... – Matthew Daws May 9 '10 at 10:21
• I can't help thinking that "an explicit description" of elements of the group von Neumann algebra is rather a lot to hope for: the case of the free group on two generators has had quite a lot of blood sweat & tears expended on it. (At least, that's my non-expert opinion.) – Yemon Choi May 9 '10 at 11:39
• Even if $G$ is discrete, so elements of VN(G) are elements of $\ell^2(G)$, it's hard to see any characterization of $VN(G)$ inside $\ell^2(G)$ except the tautological one. Though I suppose for certain groups you have Sobolev-type norm estimates: this seems to have started with Haagerup's paper proving (inter alia) that $C_r^*(F_2)$ has bounded AP & the generalization by Ramagge-Robertson-Steger. – Yemon Choi May 9 '10 at 11:44
• @Matthew: Yes, the case when G equals R is very interesting for me. Actually I am interested in the case when R acts on M by the modular automorphism group of some faithful weight; the crossed product is then the core of M, which is an extremely important object in the noncommutative integration theory. – Dmitri Pavlov May 9 '10 at 19:46

Okay, this is very far from being a complete answer. But...

Firstly, if you are interested in $\mathbb R$ acting on $M$ by the modular automorphism group (which is an incredibly interesting object!) then you're getting dangerously close to trying to understand all the Type III factors. Compare with Takesaki's "solution" to this problem (see Takesaki volume 2, or his paper in Acta, 1973.

Another place to look is two (IMHO much overlooked) papers by Haagerup, "On the dual weights for crossed products of von Neumann algebras. I." and "II." in Math. Scand. 43 (1978/79). In particular, Section 2 of the 2nd paper is an excellent resource if you are interested in the weight on the group von Neumann algebra.

But, I doubt this will answer your original question. Maybe a real expert will wonder by...

• Takesaki's construction is precisely the one that I described as “the von Neumann algebra generated by certain operators on a certain Hilbert space”. [There are two constructions of the core of M in his Volume 2, and both are of this type. One is Lemma XII.6.8, the other one is Definition X.1.6, also found in his 1973 Acta paper.] This question arose from an attempt to make rigorous my intuition for dual weights, the canonical trace on the core of M, and the canonical operator valued weight from the core of M to M (of which the Definition 2.2 of Haagerup's second paper is a particular case). – Dmitri Pavlov May 11 '10 at 7:34
• Can we at least make rigorous this Fourier algebra stuff? Is there a reference for the fact that A(G) is the predual of VN(G)? – Dmitri Pavlov May 11 '10 at 7:47
• Dmitri: If you have Takesaki Volume 2, then read Chapyer VII, Section 3 for the Fourier algebra (you could also look at Eymard's original paper, but it's in French, and I prefer the modern viewpoint). – Matthew Daws May 11 '10 at 7:57

I am not sure of what you want, but let me offer a few ideas.

First, let me give you a silly answer: by the bicommutant theorem, if $M\subset B(H)$ is a von Neumann algebra, then $M = (M') '$. Now pick some subset $F$ of $M'$ that generates it as a $W^*$-algebra. Then a description of $M$ is of course $\{T\in B(H): Tx = xT \forall x\in F\}$.

There are, however, some instances where a more "explicit" description is available. This is the case, for example, for von Neumann algebras related to lattices in $PSL_2(\mathbb{R})$ via Berezin quantization (the symbol is required to be equivariant with respect to the group action). You can find more details in some papers of Radulescu about this.

Another situation where a description is available is in the case of a II$_\infty$ (or type III) factor associated to a foliation; in this case an element of the algebra ends up being a measurable field of bounded operators on Hilbert spaces associated to the leaves of the foliation. This is described in Connes' non-commutative geometry book.