11a. Reduction theory

[math] \newcommand{\mathds}{\mathbb}[/math]

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Welcome to advanced von Neumann algebra theory, and in the hope that we will survive, both you reader, and me author. Our main purpose here will be to discuss some key decomposition methods for the von Neumann algebras [math]A\subset B(H)[/math], in terms of von Neumann factors, [math]Z(A)=\mathbb C[/math], which altogether are called “reduction theory”.

The reduction theory, due to von Neumann himself, is something quite fundamental, to the point that it would have made sense to get into it right after the von Neumann algebra basics from chapter 5. However, the subject being quite technical, we have not done this after chapter 5, and nor in fact we will really do it here, with the presentation below being just a modest introduction to all this. So, this is the situation, and in the hope that the gods of mathematics, Bourbaki and others, will pardon us.

The story, first. Von Neumann started to work on operator algebras in the 1930s, and became increasingly convinced that these should be subject to a reduction theory theorem, with his interest in factors, which is obvious in his papers , , , , , basically coming from this. However, he was not able to come at that time, during his prime years of work, with a complete proof, and paper, on reduction theory. He only did that much later, after a break involving other things, like the Manhattan Project, game theory, computers and more, in his 1949 paper , written towards the end of his career. His main theorem in , which is quite easy to formulate, is as follows: \begin{fact}[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 von Neumann algebra factors, [math]Z(A_x)=\mathbb C[/math]. \end{fact} 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 Fact 11.1, as direct sums of matrix algebras, as follows:

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

More generally, it is possible to axiomatize a certain class of “type I algebras”, and then show that these algebras appear as direct integrals of matrix algebras:

[[math]] A=\int_XM_{n_x}(\mathbb C)\,dx [[/math]]

Observe in particular that in the case where the decomposition is isotypic, [math]n_x=N[/math] for some [math]N\in\mathbb N[/math], we obtain the random matrix algebras studied in chapter 6:

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

Beyond type I, however, things become quite complicated. Next in the hierarchy is the general “finite case”, where the algebra is assumed to have a trace:

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

Here the existence of the trace simplifies a bit things, although these still remain fairly complicated, and actually adds to the final result, in the form of the supplementary formula, regarding its decomposition, the precise statement being as follows: \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 factors which are “finite”, in the sense that they have traces, which in practice means that they can be usual matrix algebras, or [math]{\rm II}_1[/math] factors. \end{fact} As already mentioned, while some tricks are potentially available here, coming from the presence of the trace [math]tr:A\to\mathbb C[/math], this remains something complicated. As for the most general case, where the von Neumann algebra [math]A\subset B(H)[/math] is taken arbitrary, corresponding to Fact 11.1 in full generality, this is something even more complicated, with the only possible tools coming from advanced operator theory, and functional analysis.

So, this is the situation, and what to do now. We cannot explain the above, because it is too complicated, but we cannot skip it either, because these are fundamentals. This situation has been known to generations of mathematicians, starting with von Neumann himself, who finished and published his reduction theory paper long after developing the basics of operator algebra theory, as mentioned above. The various books written afterwards, including Blackadar , Connes , Dixmier , Jones , Kadison-Ringrose , Sakai , Str\u atil\u a-Zsido and Takesaki did not arrange things, being either evasive, or way too technical, not to say unreadable, on this subject.

The present book won't be an exception to the rule. Our plan in what follows will be that of discussing a bit all this, reduction theory, notably with a study of examples:

(1) First we have the type I algebras, which are direct integrals of matrix algebras [math]M_{n_x}(\mathbb C)[/math], with the case [math]n_x=1[/math] corresponding to commutativity, the case [math]n_x\in\mathbb N[/math] corresponding to the “type I finite case”, and with the general case being [math]n_x\in\mathbb N\cup\{\infty\}[/math]. At the level of main examples, these come from finite groups and quantum groups.

(2) Then we have the type II algebras, where we can have both type I and type II factors in the decomposition. Of particular interest is the “finite” case, where the algebra is simply assumed to come with a trace, [math]tr:A\to\mathbb C[/math], and where the reduction theory result is Fact 11.2, with the factors being matrix algebras [math]M_N(\mathbb C)[/math], or [math]{\rm II}_1[/math] factors.

(3) Finally, we have the general type III case, with no assumption on the algebra [math]A\subset B(H)[/math], corresponding to Fact 11.1. Here the factors in the decomposition can be of type I, or of type II, or neither of type I or II, which are called by definition of type III. The interesting questions here regard the structure of the type III factors.

In order to get started, let us look at the commutative von Neumann algebras. Here we have the following result, that we basically know from chapter 5:

Theorem

The commutative von Neumann algebras are the algebras of type

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

with [math]X[/math] being a measured space. Thus, we formally have for them the formula

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

with the fibers [math]A_x[/math] being trivial in this case, [math]A_x=\mathbb C[/math], for any [math]x\in X[/math].

Show Proof

We have several assertions to be proved, the idea being as follows:

(1) In one sense, we must prove that given a measured space [math]X[/math], we can realize the commutative algebra [math]A=L^\infty(X)[/math] as a von Neumann algebra, on a certain Hilbert space [math]H[/math]. But this is something that can be done via multiplicity operators, as follows:

[[math]] L^\infty(X)\subset B(L^2(X)) [[/math]]

(2) In the other sense, given a commutative von Neumann algebra [math]A\subset B(H)[/math], we must construct a certain measured space [math]X[/math], and an identification [math]A=L^\infty(X)[/math]. But this can be done by writing our von Neumann algebra as follows:

[[math]] A= \lt T_i \gt [[/math]]

Indeed, no matter what particular family of generators [math]\{T_i\}[/math] we choose for our algebra [math]A[/math], these generators [math]T_i[/math] will be commuting normal operators. Thus the spectral theorem for such families of operators, from chapter 3, applies and gives the result.

(3) In fact, by using the theory of projections from chapters 9-10, we can write our commutative von Neumann algebra [math]A\subset B(H)[/math] in singly generated form:

[[math]] A= \lt T \gt [[/math]]

But this simplifies the situation, because the basic spectral theorem, for single normal operators, from chapter 3, applies to our generator [math]T[/math], and gives the result.

(4) Finally, the last assertion, regarding the validity of the reduction theory result in this case, is something trivial, and of course without much practical interest.

■

Moving forward, the above result is not the end of the story with the commutative von Neumann algebras, because we still have to understand how a given such algebra [math]A=L^\infty(X)[/math], or rather the weak topology isomorphism class of such an algebra, can be represented as an operator algebra, over the various Hilbert spaces [math]H[/math]:

[[math]] L^\infty(X)\subset B(H) [[/math]]

But this can be again solved by writing our algebra as [math]A= \lt T \gt [/math], and then applying the spectral theorem for normal operators, with the conclusion that the commutative von Neumann algebras are, up to spatial isomorphism, the algebras of the following form, with [math]X[/math] being a measured space, and with all this being up to a multiplicity:

[[math]] L^\infty(X)\subset B(L^2(X)) [[/math]]

With these results in hand, we are now in position of better understanding the idea behind von Neumann's reduction theory. Indeed, given an arbitrary von Neumann algebra [math]A\subset B(H)[/math], the idea is to consider its center, and write it as follows:

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

The point is then that everything will decompose over the measured space [math]X[/math], and in particular, the whole algebra [math]A[/math] itself will decompose as a direct integral of fibers:

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

As already mentioned, we will only partly explain this in what follows, and by insisting on examples. Also, we will do this slowly, following the type I, II, III hierarchy.

General references

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