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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 is strong enough to affect properties 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 groups, you can consider the same question.

Good luck with your study of von Neumann algebras!

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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 is strong enough to affect properties 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 a the construction of a $II_{1}$ factor that uses 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 groups, you can consider the same question.

Good luck with your study of von Neumann algebras!

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Roughly, the idea is

As for a book on group theory that the classification of factors now reduces may be useful or interesting to the type $II_{1}$ case, as any type $III$ factor decomposes as a (continuous) crossed product of a $II_{\infty}$, which itself decomposes as a tensor product read for further study of a $II_{1}$ by $B(H)$.

As you mentionfactors, many of 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 constructions "large scale" structure of groups, and concerns ways that lead to $II_{1}$ factors closely mimic groups can be equivalent in ways that are weaker than group constructionsisomorphism. Ergodic measure preserving actions A lot of countable discrete groups on standard Borel spaces give rise to crossed product contemporary $II_{1}$ factorsfactor theory is also concerned with weak equivalence of groups and their measure-preserving actions. I'll say a bit more below, the 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 algebra construction is another one algebras, 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 original ways subject right now, and says everything I'll say here and more.

One classical construction of cooking up a $II_{1}$ factors. There are free product constructions and amalgamated free product constructions for factor using a group $II_{1}$ factors that (under G$is the (left) group von Neumann algebraconstruction) respect , i.e. the corresponding operations for commutant of the underlying groups. The right perspective for all regular represention of this is that we an i.c.c. (all nontrivial conjugacy classes are looking for properties of groups that "survive" passage to the von Neumann algebra level, infinite) group G in various ways. This suggests ways to get invariants for$II_{1}$factorsB(\ell^2 G)$. Roughly, we want to think of $II_{1}$ factors as analogous to If two i.c.c. groups in the sense that certain are isomorphic, then certainly their group properties have purely von Neumann algebraic analogues.

The famous starting point of this is amenabilityalgebras are too. The group von Neumann algebra of a countable discrete group On the other hand, it is amenable (hyperfinite) if and only very difficult in general to tell if the group is amenable. The concept of amenability of a group has a purely von Neumann algebra analogue, due construction "remembers" the group used to Connesconstruct it. First of allFor example, there is a fruitful analogue of unitary representation for any two i.c.c. amenable groups have isomorphic group von Neumann algebras, that of a correspondence, or binormal bimoduleso if you begin with an i.c.c. Leptin's condition on the space of unitary representations of a group says that the group is amenable if group and only if the trivial representation is weakly contained in the left regular representation. In whip up the group von Neumann algebrapicture, a $II_1$ factor is amenable if and it won't remember which group you used, but only if the trivial correspondence is weakly contained in the so-called coarse correspondence. So will remember the analogy with groups is fruitful for amenability.

Similarly, other properties of groups seem to extend It turns out that this construction is also sensitive to properties of von Neumann algebras. For instance, Kazhdan's property Property (T) and T), the Haagerup property (a-T-menability). Alsoa weak amenability that is strongly non-(T)), many interesting in the sense that these properties are reflected in the structure of the von Neumann algebras are 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" structure property is strong enough to affect properties of groups, for example the group von Neumann algebraof an i.c.c. Gromov-hyperbolic group always , this construction for Gromov hyperbolic groups give rise to a solid factor (not that factors. These things all seem to be "global" group properties, and so this is the only way to get a solid factor)why I'm suggesting geometric group theory.Also, as Gabriel already mentioned, freeness has a useful formulation at

We're sort of listening for echos of the group in the von Neumann algebra level and has provided an entire subject to keep us busy for a long time, Voiculescu's free probabilitybuilt from it...

As for

The broad question is: What properties of a book on group theory that may be useful or interesting to read for further study survive a construction of a $II_{1}$ factors, I think factor that de la Harpe's book Topics in Geometric Group Theory is good for this. The reason I say this is uses that geometric grouptheory ?

Another classical construction of a $II_{1}$ factor the group-measure space construction, which in modern terms is concerned with the "large scale" structure of groups. Topics such as quasi-isometry and measure equivalence yield many interesting things to think about on way we build the crossed product von Neumann algebra sidefrom a discrete group and an ergodic measure-preserving action of that group on a standard Borel space.

You should check Check out Sorin Popa's above-mentioned ICM talk , Deformation and rigidity for group actions and von Neumann algebras, on his website, and read all of itmore weak equivalences for groups surrounding this construction.This gives a really good intuition about a big part

If you look at other ways of what's going on in constructing groups, you can consider the subject right now. It's a wonderful place to startsame question.

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