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Let us discuss now the type II case, where the truly interesting problems are. The central result here, that we already formulated in the beginning of this chapter, is:

\begin{fact}[Reduction theory, finite case] Given a von Neumann algebra [math]A\subset B(H)[/math] coming with a trace [math]tr:A\to\mathbb C[/math], if we write its center [math]Z(A)\subset A[/math] as

[[math]] Z(A)=L^\infty(X) [[/math]]

with [math]X[/math] being a measured space, then the whole algebra and its trace decompose as

[[math]] A=\int_XA_x\,dx\quad,\quad tr=\int_Xtr_x\,dx [[/math]]

with the fibers [math]A_x[/math] being either factors of type [math]{\rm I}_N[/math], with [math]N \lt \infty[/math], or of type [math]{\rm II}_1[/math]. \end{fact} Regarding the proof, this is something quite technical, generalizing what we know, or rather what we don't, about the type I finite case, which is substantially easier. We refer here to Dixmier , and with the comment that we will see soon examples of all this.

As before in the type I case, it is possible to add a bit of infinity in the above, and we have the following result, which is a bit more general, but more technical too: \begin{fact}[Reduction theory, type II case] Given a von Neumann algebra [math]A\subset B(H)[/math] which is of type [math]{\rm II}[/math], in a suitable sense, if we write its center [math]Z(A)\subset A[/math] as

[[math]] Z(A)=L^\infty(X) [[/math]]

with [math]X[/math] being a measured space, then the whole algebra decomposes as

[[math]] A=\int_XA_x\,dx [[/math]]

with the fibers [math]A_x[/math] being von Neumann factors of type [math]{\rm I}[/math] or [math]{\rm II}[/math]. \end{fact} As before with what happened in type I, the above results are particularly interesting in the case of the von Neumann algebras of the discrete groups, [math]A=L(\Gamma)[/math], and their generalizations. In order to discuss these questions, let us recall that the center of an arbitrary group von Neumann algebra [math]A=L(\Gamma)[/math] consists, up to some standard identifications, of the functions which are constant on the finite conjugacy classes. This suggests the following definition, which is something well-known in group theory:

Definition

A discrete group [math]F[/math] is said to have the FC property if all its conjugacy classes are finite. In other words, for any [math]g\in F[/math], we must have:

[[math]] \left|\left\{hgh^{-1}\Big|h\in F\right\}\right| \lt \infty [[/math]]

If this finite conjugacy property is satisfied, we also say that [math]F[/math] is a FC group.

As basic examples of FC groups, we have the finite groups, the abelian groups, and the products of such groups. Besides being stable under taking products, the class of FC groups is stable under a number of other basic operations, such as taking subgroups, or quotients. In connection now with our reduction theory questions, we have:

Theorem

Given a group [math]F[/math] having the FC property, the center of the associated von Neumann algebra is isomorphic to the algebra of central functions on [math]F[/math],

[[math]] Z(L(F))\simeq C(F)_{central} [[/math]]

and the reduction theory applied to this algebra, which is a formula of type

[[math]] L(F)\simeq\int_{r\in X}A_r [[/math]]

appears in relation with the representation theory of [math]F[/math].

Show Proof

In what concerns the first assertion, regarding the center, this is something that we know from chapter 10. Indeed, we have the following formula for the center:

[[math]] Z(L(F))=\left\{\sum_g\lambda_gg\Big|\lambda_{gh}=\lambda_{hg},\forall h\in F\right\}'' [[/math]]

Now since on the right we have central functions on our group, [math]\lambda\in C(F)_{central}[/math], we obtain the isomorphism in the statement, namely:

[[math]] Z(L(F))\simeq C(F)_{central} [[/math]]

Regarding now the second assertion, this is something more tricky, as follows:

(1) In the finite group case, we recall from Theorem 11.8 that, by using the standard identification between representations [math]r[/math] and their characters [math]\chi_r[/math], which are central functions on [math]F[/math], the center computation that we did above reads:

[[math]] Z(L(F))\simeq L^\infty(Irr(F)) [[/math]]

In order to discuss now the reduction theory for [math]L(F)[/math], we recall that the Peter-Weyl theory applied to [math]F[/math] gives a direct sum decomposition as follows, which is technically an isomorphism of linear spaces, which is in addition a [math]*[/math]-coalgebra isomorphism:

[[math]] L^\infty(F)\simeq\bigoplus_{r\in Irr(F)}M_{\dim(r)}(\mathbb C) [[/math]]

Thus by dualizing, we obtain a direct sum decomposition of the group von Neumann algebra as follows, which is this time a [math]*[/math]-algebra isomorphism:

[[math]] L(F)\simeq\bigoplus_{r\in Irr(F)}M_{\dim(r)}(\mathbb C) [[/math]]

But this is exactly what comes out from von Neumann's reduction theory, applied to the algebra [math]L(F)[/math], and so we are fully done with the finite group case.

(2) As a second key particular case, let us discuss now the case where [math]F[/math] is abelian. In the simplest infinite group case, where our group is [math]F=\mathbb Z[/math], the group algebra is:

[[math]] L(\mathbb Z)\simeq L^\infty(\mathbb T) [[/math]]

More generally, for the abelian groups [math]F=\mathbb Z^N[/math], which are those which are finitely generated and without torsion, we obtain the algebras of functions on various tori:

[[math]] L(\mathbb Z^N)\simeq L^\infty(\mathbb T_N) [[/math]]

In general now, assuming that [math]F[/math] is finitely generated and abelian, here we know from Pontrjagin duality that we have an isomorphism as follows:

[[math]] L(F)\simeq L^\infty(\widehat{F}) [[/math]]

More explicitly now, let us write our finitely generated abelian group [math]F[/math] as a product of cyclic groups, possibly taken infinite, as follows:

[[math]] F=\mathbb Z^N\times\left(\prod_i\mathbb Z_{n_i}\right) [[/math]]

The Pontrjagin dual of [math]F[/math] is then the following compact abelian group:

[[math]] F=\mathbb T^N\times\left(\prod_i\mathbb Z_{n_i}\right) [[/math]]

Thus, things are very explicit here, and we are done with the abelian case too.

(3) In the general case now, where our discrete group [math]F[/math] is only assumed to have the FC property, the reduction theory for the corresponding von Neumann algebra [math]L(F)[/math] appears somewhat as a mixture of what happens for the finite and for the abelian groups, discussed in (1) and (2) above. For more on all this, we refer to Dixmier .

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Regarding the corresponding problems for the discrete quantum groups, these are not solved yet. In fact, the knowledge here stops at a very basic level, with the analogue of the ICC property, leading to the factoriality of [math]L(\Gamma)[/math], not being known yet, and for more on all this, we refer to the discussion made in chapter 10.

Moving ahead from these difficulties, let us go back now to the usual group von Neumann algebras [math]L(\Gamma)[/math], and discuss what happens in general. Once again inspired by the basic computation that we have, namely that of the center of an arbitrary group algebra [math]L(\Gamma)[/math], let us formulate the following purely group-theoretical definition:

Definition

Given a discrete group [math]\Gamma[/math], its FC subgroup [math]F\subset\Gamma[/math] is the subgroup

[[math]] F=\left\{g\in\Gamma\Big|\left|\left\{hgh^{-1}\Big|h\in\Gamma\right\}\right| \lt \infty\right\} [[/math]]

consisting of the elements in the finite conjugacy classes of [math]\Gamma[/math].

Here the fact that [math]F[/math] is indeed a subgroup is clear from definitions, with the fact that [math]F[/math] is stable under multiplication coming from the following trivial observation:

[[math]] h(gk)h^{-1}=hgh^{-1}\cdot hkh^{-1} [[/math]]

Observe that [math]\Gamma[/math] has the FC property, in the sense of Definition 11.16, precisely when the inclusion [math]F\subset\Gamma[/math] is an equality. As before with the FC groups, there are many known things about the FC subgroups [math]F\subset\Gamma[/math], and we refer here to the group theory literature.

In connection now with our reduction theory questions, we have:

Theorem

Given a discrete group [math]\Gamma[/math], the center of the associated von Neumann algebra is isomorphic to the algebra of central functions on its FC subgroup [math]F\subset\Gamma[/math],

[[math]] Z(L(\Gamma))\simeq C(F)_{central} [[/math]]

and the reduction theory applied to this algebra, which is a formula of type

[[math]] L(\Gamma)\simeq\int_{r\in X}A_r [[/math]]

appears in relation with the representation theory of [math]\Gamma[/math], and of its FC subgroup [math]F\subset\Gamma[/math].

Show Proof

In what concerns the first assertion, regarding the center, this is something that we know from chapter 10, coming from our study there of the general group algebras [math]L(\Gamma)[/math], with [math]\Gamma[/math] being a discrete group. To be more precise, we know from there that:

[[math]] Z(L(\Gamma))=\left\{\sum_g\lambda_gg\Big|\lambda_{gh}=\lambda_{hg},\forall h\in F\right\}'' [[/math]]

Now since on the right we have central functions on the FC subgroup, [math]\lambda\in C(F)_{central}[/math], we obtain the isomorphism in the statement, namely:

[[math]] Z(L(\Gamma))\simeq C(F)_{central} [[/math]]

Regarding the second assertion, this is something more tricky, and we refer here to the relevant group theory and operator algebra literature, including Dixmier .

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As a last topic for this section, let us briefly discuss the reduction theory in the general case, type III. In order to get started, we must discuss the type III factors, which are new to us. According to our various conventions above, these factors are defined as follows:

Definition

A type [math]{\rm III}[/math] factor is a von Neumann algebra [math]A\subset B(H)[/math] which is a factor, [math]Z(A)=\mathbb C[/math], and satisfies one of the following equivalent conditions:

• [math]A[/math] is not of type [math]{\rm I}[/math], or of type [math]{\rm II}[/math].
• [math]A[/math] has no semifinite trace [math]tr:A\to\mathbb C[/math].
• [math]A[/math] has no trace [math]tr:A\to\mathbb C[/math], and is not of type [math]{\rm I}_\infty[/math] or [math]{\rm II}_\infty[/math].

In order to investigate such factors, the general idea will be that of looking at the crossed products of type II factors, which can be lacking traces [math]tr:A\to\mathbb C[/math], and so which allow us to exit the type II world. In order to get started, however, we have:

Theorem

Any locally compact group [math]G[/math] has a left invariant Haar measure [math]\lambda[/math], and a right invariant Haar measure [math]\rho[/math],

[[math]] d\lambda(x)=d\lambda(yx)\quad,\quad d\rho(x)=d\lambda(xy) [[/math]]

which are unique up to multiplication by scalars. These two measures are absolutely continuous with respect to each other, and the Radon-Nikodym derivative

[[math]] m:G\to\mathbb R\quad,\quad m(x)=\frac{d\lambda(x)}{d\rho(x)} [[/math]]

well-defined up to multiplication by scalars, is called modulus of the group. The unimodular groups, for which [math]m=1[/math], include all compact groups, and all abelian groups.

Show Proof

There are many things here, with everything being very classical, and the proof, along with comments, examples and more theory, especially in what regards the unimodular groups, can be found in any good measure theory book.

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As it has become customary in this book, whenever talking about groups we must make some comments about quantum groups too. Things are quite interesting in connection with Theorem 11.21, because it is possible “twist” things in the compact case, as to have a notion of modulus there as well. We refer here to Woronowicz and related papers. In relation now with our factor questions, we have the following result:

Theorem

The type [math]{\rm III}[/math] factors basically appear from the type [math]{\rm II}[/math] factors, via various crossed product constructions, and their generalizations.

Show Proof

This statement is obviously something quite informal, and we will certainly not attempt to explain the proof either. Here are however the main ideas, with the result itself being basically due to Connes , along with some historical details:

(1) First of all, Murray and von Neumann knew of course about such questions, but were quite evasive in their papers about type III, with the brief comment “we don't know. Whether they really worked or not on these questions, we'll never know.

(2) Inspired by Theorem 11.21, it is possible to develop a whole machinery for the study of the non-tracial states [math]\varphi:A\to\mathbb C[/math], the main results here being the Kubo-Martin-Schwinger (KMS) condition, and the Tomita-Takesaki theory. See Takesaki .

(3) On the other hand, looking at type II factors and their crossed products by automorphisms, which are not necessarily of type II, leads to a lot of interesting theory as well, leading to large classes of type III factors, appearing from type II factors.

(4) The above results are basically from the 50s and 60s, and Connes was able to put all this together, in the early 70s, via a series of quick, beautiful and surprising Comptes Rendus notes, eventually leading to his paper , which is a must-read.

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In equivalent terms, and also by remaining a bit informal, we have:

Theorem

The von Neumann algebra factors can be classified as follows,

[[math]] {\rm I_N},{\rm I}_\infty [[/math]]

[[math]] {\rm II}_1,{\rm II}_\infty [[/math]]

[[math]] {\rm III}_0,{\rm III}_\lambda,{\rm III}_1 [[/math]]

with the type [math]{\rm II}_1[/math] ones being the most important, basically producing the others too.

Show Proof

This follows by putting altogether what we have, results of Murray and von Neumann in type I and II, and then of Connes in type III. The last assertion is of course something quite informal, because the situation is not exactly as simple as that.

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Getting back now to our series of reduction theory results, we have:

Theorem

Given an arbitrary von Neumann algebra [math]A\subset B(H)[/math], write its center as follows, with [math]X[/math] being a measured space:

[[math]] Z(A)=L^\infty(X) [[/math]]

The whole algebra [math]A[/math] decomposes then over this measured space [math]X[/math], as a direct sum of fibers, taken in an appropriate sense,

[[math]] A=\int_XA_x\,dx [[/math]]

with the fibers [math]A_x[/math] being von Neumann factors, which can be of type [math]{\rm I},{\rm II},{\rm III}[/math].

Show Proof

As before with other such results, this is something heavy, generalizing our previous knowledge in type I, and type II. The proof however is quite similar, basically using the same ideas. We refer here to the literature, for instance to Dixmier .

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General references

Banica, Teo (2024). "Principles of operator algebras". arXiv:2208.03600 [math.OA].