After over half a year's study on operator algebra (especially on von Neumann algebra) by doing exercises in Fundamentals of the theory of operator algebras 1, 2 –Kadison, I was told that the current research focus is on the Ⅱ1 factor, and certain background on group theory is necessary, such as studying the free product, specific group construction and the ergodic action. Then, I want to know are there any good books on the group theory that might be necessary for further study on von Neumann theory?
[Math] Group theory required for further study in von Neumann algebra
oa.operator-algebrassoft-question
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You definitely need some extra structure on your von Neumann algebra, but I'm not quite sure what you're asking for. Intuitively I would think that just as different topological spaces share the same measure space structure, trying to extract NC-topological information out of a von Neumann algebra is going to need extra structure. (For instance, no one does topological K-theory of von Neumann algebras as far as I know.)
I see that on page 7 of that Connes paper, he shows that the WOT-closure A'' does remember the original algebra A if extra data are given (the Dirac operator and its interaction with A).
Although it's probably not what you want: if you're looking at group von Neumann algebras and looking at the "geometry of the dual group", then the original group can be recovered from a suitable coproduct on the original (Hopf) von Neumann algebra. This is vaguely on the lines of Weil's theorem that "essentially" recovers a locally compact group and its Haar measure from a measurable group.
By the way, here's the "correct" functorial property. If G and H are abelian, and $f:G\rightarrow H$ is a continuous group homomorphism, then we get a continuous group homomorphism $\hat f:\hat H\rightarrow \hat G$ between the dual groups. By the pull-back, we get a *-homomorphism $\hat f_*:C_0(\hat G) \rightarrow C^b(\hat H)$. We should think of $C^b(\hat H)$ as the multiplier algebra of $C_0(\hat G)$. Then $C_0(\hat G) \cong C^*_r(G)$, and so we do get a *-homomorphism $C^*_r(G) \rightarrow M(C^*_r(H))$; the strict-continuity extension of this is a *-homomorphism $M(C^*_r(G)) \rightarrow M(C^*_r(H))$ which does indeed send $\lambda(s)$ to $\lambda(f(s))$.
For non-abelian group (in fact, non-amenable groups) it's necessary to work with $C^*(G)$ instead.
We cannot ensure a map to $C^*_r(H)$ itself, as we cannot ensure a map from $C_0(\hat G)$ to $C_0(\hat H)$; indeed, this would only happen when $\hat f$ were a proper map.
Similarly, we don't get maps at the von Neumann algebra level, as we don't get a map $L^\infty(\hat G) \rightarrow L^\infty(\hat H)$: we would need that $\hat f$ pulled-back null sets in $\hat G$ to null sets in $\hat H$.
Best Answer
As for a book on group theory that may be useful or interesting to read for further study of $II_{1}$ factors, I think that de la Harpe's book Topics in Geometric Group Theory is good for this. The reason I say this is that geometric group theory is concerned with the "large scale" structure of groups, and concerns ways that groups can be equivalent in ways that are weaker than group isomorphism. A lot of contemporary $II_{1}$ factor theory is also concerned with weak equivalence of groups and their measure-preserving actions. I'll say a bit more below, for context.
Before I do, though, let me mention that you should check out Sorin Popa's ICM talk, Deformation and rigidity for group actions and von Neumann algebras, a preprint listed on his website, and read all of it. This gives a really good intuition about a big part of what's going on in the subject right now, and says everything I'll say here and more.
One classical construction of a $II_{1}$ factor using a group $G$ is the (left) group von Neumann algebra, i.e. the commutant of the right regular represention of an i.c.c. (all nontrivial conjugacy classes are infinite) group G in $B(\ell^2 G)$. If two i.c.c. groups are isomorphic, then certainly their group von Neumann algebras are too. On the other hand, it is very difficult in general to tell if the group von Neumann algebra construction "remembers" the group used to construct it. For example, any two i.c.c. amenable groups have isomorphic group von Neumann algebras, so if you begin with an i.c.c. amenable group and whip up the group von Neumann algebra, it won't remember which group you used, but only will remember the amenability. It turns out that this construction is also sensitive to Kazhdan's Property (T), the Haagerup property (a weak amenability that is strongly non-(T)), in the sense that these properties are reflected in the structure of the von Neumann algebra. This construction is also sensitive to freeness, as in the freeness of the generators of a group. (See Gabriel's answer above.) Gromov's hyperbolicity is also reflected in the structure of the group von Neumann algebra, in that this "large scale" property severely governs the structure of the von Neumann algebra: this construction for Gromov hyperbolic groups give rise to solid factors. These things all seem to be "global" group properties, and so this is why I'm suggesting geometric group theory.
We're sort of listening for echos of the group in the von Neumann algebra built from it...
The broad question is: What properties of a group survive the construction of a $II_{1}$ factor using that group?
Another classical construction of a $II_{1}$ factor the group-measure space construction, which in modern terms is the way we build the crossed product von Neumann algebra from a discrete group and an ergodic measure-preserving action of that group on a standard Borel space. Check out Popa's above-mentioned ICM talk for more weak equivalences for groups surrounding this construction.
If you look at other ways of constructing $II_{1}$ factors, you can consider the same question.
Good luck with your study of von Neumann algebras!