guide:88e1bcb897

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In this chapter we go for the real thing, namely the study of the [math]{\rm II}_1[/math] factors, following Murray and von Neumann , , , , , which is the basis for everything more advanced, in relation with operator algebras. We will only present the very basic theory of the [math]{\rm II}_1[/math] factors, with the idea in mind of using them later for doing subfactors a la Jones. We will mainly follow the simplified approach from Jones' book , with sometimes a look into Blackadar , both books that we recommend for more.

Let us first talk about general factors. There are several possible ways of introducing them, and dividing them into several classes, for further study. In what concerns us, we will use a rather intuitive approach. The general idea, which is quite natural, is that among the von Neumann algebras [math]A\subset B(H)[/math], of particular interest are the “free” ones, having trivial center, [math]Z(A)=\mathbb C[/math]. These algebras are called factors:

Definition

A factor is a von Neumann algebra [math]A\subset B(H)[/math] whose center

[[math]] Z(A)=A\cap A' [[/math]]

which is a commutative von Neumann algebra, reduces to the scalars, [math]Z(A)=\mathbb C[/math].

This notion is in fact something that we already met in the above, in the context of various comments or exercises, and time now to clarify all this. The idea is that there are two main motivations for the study of factors, with each of them being more than enough, as to serve as a strong motivation. First, at the intuitive level, we have: \begin{principle}[Freeness] The following happen:

• The condition [math]Z(A)=\mathbb C[/math] defining the factors is, obviously, opposite to the condition [math]Z(A)=A[/math] defining the commutative von Neumann algebras.
• Therefore, the factors are the von Neumann algebras which are “free”, meaning as far as possible from the commutative ones.
• Equivalently, with [math]A=L^\infty(X)[/math], the quantum spaces [math]X[/math] coming from factors are those which are “free”, meaning as far as possible from the classical spaces.

\end{principle} So, this was for our first principle, which is something reasonable, intuitive, and self-explanatory, and which can surely serve as a strong motivation for the study of factors. In fact, all that has being said above comes straight from the structure theorem for the commutative von Neumann algebras, [math]A=L^\infty(X)[/math], with [math]X[/math] being a measured space, that we know since chapter 5, and the above principle is just a corollary of that theorem.

At a more advanced level, another motivation for the study of factors, which among others justifies the name “factors” for them, comes from the reduction theory of von Neumann , which is something non-trivial, that can be summarized as follows: \begin{principle}[Reduction theory] Given a von Neumann algebra [math]A\subset B(H)[/math], if we write its center [math]Z(A)\subset A[/math], which is a commutative von Neumann algebra, 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 factors, that is, satisfying [math]Z(A_x)=\mathbb C[/math]. \end{principle} As a first comment, we have already seen an instance of such decomposition results in chapter 5, when talking about finite dimensional algebras. Indeed, such algebras decompose, in agreement with the above, as direct sums of matrix algebras, as follows:

[[math]] A=\bigoplus_xM_{n_x}(\mathbb C) [[/math]]

In general, however, things are more complicated than this, and technically speaking, and as opposed to Principle 10.2, which was more of a triviality, Principle 10.3 is a tough theorem, due to von Neumann . More on this later, in chapter 11 below.

This was for the story, and let us close this philosophical discussion with: \begin{conclusion} Regardless of the approach and technical level, be that beginner or advanced, the von Neumann factors are the algebras that matter. \end{conclusion} Getting to work now, there are many things that can be said about factors. In order to get started, as a direct continuation of the work from chapter 9, for the general von Neumann algebras, let us first study their projections. We will see that many interesting things happen here, with everything coming from the following technical result:

Proposition

Given two projections [math]p,q\neq0[/math] in a factor [math]A[/math], we have

[[math]] puq\neq0 [[/math]]

for a certain unitary [math]u\in A[/math].

Show Proof

Assume by contradiction [math]puq=0[/math], for any unitary [math]u\in A[/math]. This gives:

[[math]] u^*puq=0 [[/math]]

By using this for all the unitaries [math]u\in A[/math], we obtain the following formula:

[[math]] \left(\bigvee_{u\in U_A}u^*pu\right)q=0 [[/math]]

On the other hand, from [math]p\neq0[/math] we obtain, by factoriality of [math]A[/math]:

[[math]] \bigvee_{u\in U_A}u^*pu=1 [[/math]]

Thus, our previous formula is in contradiction with [math]q\neq0[/math], as desired.

■

Getteing back now to the order on projections from chapter 9, and to the whole von Neumann projection philosophy, in the case of factors things simplify, as follows:

Theorem

Given two projections [math]p,q\in A[/math] in a factor, we have

[[math]] p\preceq q\quad {\rm or}\quad q\preceq p [[/math]]

and so [math]\preceq[/math] is a total order on the equivalence classes of projections [math]p\in A[/math].

Show Proof

This basically follows from Proposition 10.5, and from the Zorn lemma, by using some standard functional analysis arguments. To be more precise:

(1) Consider indeed the following set of partial isometries:

[[math]] S=\left\{u\Big|uu^*\leq p,u^*u\leq q\right\} [[/math]]

We can then order this set [math]S[/math] by saying that we have [math]u\leq v[/math] when [math]u^*u\leq v^*v[/math], and when [math]u=v[/math] holds on the initial domain [math]u^*uH[/math] of [math]u[/math]. With this convention made, the Zorn lemma applies, and provides us with a maximal element [math]u\in S[/math].

(2) In the case where this maximal element [math]u\in S[/math] satisfies [math]uu^*=p[/math] or [math]u^*u=q[/math], we are led to one of the conditions [math]p\preceq q[/math] or [math]q\preceq p[/math] in the statement, and we are done.

(3) So, assume that we are in the case left, [math]uu^*\neq p[/math] and [math]u^*u\neq q[/math]. By Proposition 10.5 we obtain a unitary [math]v\neq0[/math] satisfying the following conditions:

[[math]] vv^*\leq p-uu^* [[/math]]

[[math]] v^*v\leq q-u^*u [[/math]]

But these conditions show that the element [math]u+v\in S[/math] is strictly bigger than [math]u\in S[/math], which is a contradiction, and we are done.

■

Moving ahead now, as explained time and again throughout this book, for a variety of reasons, which can be elementary or advanced, and also mathematical or physical, we are mainly interested in the case where our algebras have traces:

[[math]] tr:A\to\mathbb C [[/math]]

And in relation with the factors, by leaving aside the rather trivial case of the matrix algebras [math]A=M_N(\mathbb C)[/math], we are led in this way to the following key notion:

Definition

A [math]{\rm II}_1[/math] factor is a von Neumann algebra [math]A\subset B(H)[/math] which:

• Is infinite dimensional, [math]\dim A=\infty[/math].
• Has trivial center, [math]Z(A)=\mathbb C[/math].
• Has a trace [math]tr:A\to\mathbb C[/math].

Here the order of the axioms is a bit random, with any of the possible [math]3!=6[/math] choices making sense, and corresponding to a slightly different vision on what the [math]{\rm II}_1[/math] factors truly are. With the above order, with (1) we are making it clear, right from the beginning, that we are not here for revolutionizing linear algebra. Then with (2) we adhere to Definition 10.1, and to what was said next about it, on freeness and reduction. And finally with (3) we adhere to the above principle, that von Neumann algebras must have traces.

More technically now, and leaving aside anything subjective, the above definition is motivated by the heavy classification work of Murray, von Neumann and Connes , , , , , , , , whose conclusion is more or less that everything in von Neumann algebras reduces, via some quite complicated procedures, we should mention that, to the study of the [math]{\rm II}_1[/math] factors. With the mantra here being as follows: \begin{fact} The [math]{\rm II}_1[/math] factors are the building blocks of the whole von Neumann algebra theory. \end{fact} To be more precise, this statement, that we will get to understand later, is something widely agreed upon, at least among operator algebra experts who are familiar with von Neumann algebras, and with this agreement being something great. What remains controversial, however, is how to start playing with these Lego bricks that we have:

(1) A first option is that of adding the matrix algebras [math]M_N(\mathbb C)[/math], not to be forgotten, and then stacking together such Lego bricks. According to the von Neumann reduction theory, this leads to the von Neumann algebras having traces, [math]tr:A\to\mathbb C[/math].

(2) A second option, perhaps even more playful, is that of taking crossed products of such Lego bricks by their automorphisms scaling the trace, or performing more general constructions inspired by advanced ergodic theory. This leads to general factors.

(3) And the third option is that of being a bad kid, or perhaps some kind of nerd, engineer in the becoming, and picking such a Lego brick, or a handful of them, and breaking them, see what's inside. Good option too, and more on this later.

Getting to work now, in practice, and forgetting about reduction theory, which raises the possibility of decomposing any tracial von Neumann algebra into factors, in order to obtain explicit examples of [math]{\rm II}_![/math] factors, it is not even clear that such beasts exist. Fortunately the group von Neumann algebras are there, and we have the following result, which provides us with some examples of [math]{\rm II}_1[/math] factors, to start with:

Theorem

The center of a group von Neumann algebra [math]L(\Gamma)[/math] is

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

and if [math]\Gamma\neq\{1\}[/math] has infinite conjugacy classes, in the sense that

[[math]] \Big|\{ghg^{-1}|g\in G\}\Big|=\infty\quad,\quad\forall h\neq1 [[/math]]

with this being called ICC property, the algebra [math]L(\Gamma)[/math] is a [math]{\rm II}_1[/math] factor.

Show Proof

There are two assertions here, the idea being as follows:

(1) Consider a linear combination of group elements, which is in the weak closure of [math]\mathbb C[\Gamma][/math], and so defines an element of the group von Neumann algebra [math]L(\Gamma)[/math]:

[[math]] a=\sum_g\lambda_gg [[/math]]

By linearity, this element [math]a\in L(\Gamma)[/math] belongs to the center of [math]L(\Gamma)[/math] precisely when it commutes with all the group elements [math]h\in\Gamma[/math], and this gives:

[[math]] \begin{eqnarray*} a\in Z(A) &\iff&ah=ha\\ &\iff&\sum_g\lambda_ggh=\sum_g\lambda_ghg\\ &\iff&\sum_k\lambda_{kh^{-1}}k=\sum_k\lambda_{h^{-1}k}k\\ &\iff&\lambda_{kh^{-1}}=\lambda_{h^{-1}k} \end{eqnarray*} [[/math]]

Thus, we obtain the formula for [math]Z(L(\Gamma))[/math] in the statement.

(2) We have to examine the 3 conditions defining the [math]{\rm II}_1[/math] factors. We already know from chapter 7 that the group algebra [math]L(G)[/math] has a trace, given by:

[[math]] tr(g)=\delta_{g,1} [[/math]]

Regarding now the center, the condition [math]\lambda_{gh}=\lambda_{hg}[/math] that we found is equivalent to the fact that [math]g\to\lambda_g[/math] is constant on the conjugacy classes, and we obtain:

[[math]] Z(L(\Gamma))=\mathbb C\iff \Gamma={\rm ICC} [[/math]]

Finally, assuming that this ICC condition is satisfied, with [math]\Gamma\neq\{1\}[/math], then our group [math]\Gamma[/math] is infinite, and so the algebra [math]L(\Gamma)[/math] is infinite dimensional, as desired.

■

In order to look now for more examples of [math]{\rm II}_1[/math] factors, an idea would be that of attempting to decompose into factors the group von Neumann algebras [math]L(\Gamma)[/math], but this is something difficult, and in fact we won't really exit the group world in this way. Difficult as well is to investigate the factoriality of the von Neumann algebras of discrete quantum groups [math]L(\Gamma)[/math], because the basic computations from the proof of Theorem 10.9 won't extend to this setting, where the group elements [math]g\in\Gamma[/math] become corepresentations [math]g\in M_N(L(\Gamma))[/math]. Despite years of efforts, it is presently not known at all what the “quantum ICC” condition should mean, and the problem comes from this. But more on this later.

In short, we have to stop here the construction of examples, and Theorem 10.9 will be what we have, at least for the moment. With this being actually not a big issue, the group factors [math]L(\Gamma)[/math] being known to be quite close to the generic [math]{\rm II}_1[/math] factors.

General references

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