11b. Type I algebras
In order to decompose our von Neumann algebras into factors, we must first make some upgrades to our terminology and notations regarding the factors, as follows:
The von Neumann algebras having trivial center, also called factors, can be divided into several types, as follows:
- The matrix algebra [math]M_N(\mathbb C)[/math] is of type [math]{\rm I}_N[/math].
- The operator algebra [math]B(H)[/math], with [math]H[/math] separable, is of type [math]{\rm I}_\infty[/math].
- The factors which are infinite dimensional and have a trace are of type [math]{\rm II}_1[/math].
- The tensor products [math]A\otimes B(H)[/math], with [math]A[/math] being a [math]{\rm II}_1[/math] factor, are of type [math]{\rm II}_\infty[/math].
- As for the factors left, these are called of type [math]{\rm III}[/math].
It is possible to be more abstract here, but in practice, this is how these factors are best remembered. Now back to reduction theory, we will present it gradually, by following the general type I, II, III hierarchy for the von Neumann algebras, coming from the above classification of factors. Let us first discuss the type I case. Here as starting point we have the following result, which is something that we know well, from chapter 5:
The finite dimensional von Neumann algebras [math]A\subset B(H)[/math] are exactly the direct sums of matrix algebras,
This is something that we know well from chapter 5. The center of [math]A[/math] is a finite dimensional commutative von Neumann algebra, of the following form:
Now let [math]P_i[/math] be the Dirac mass at [math]i\in\{1,\ldots,k\}[/math]. Then [math]P_i\in B(H)[/math] is an orthogonal projection, and these projections form a partition of unity. With [math]A_i=P_iAP_i[/math], it is elementary to check that we have a non-unital [math]*[/math]-algebra decomposition, as follows:
On the other hand, it follows from the minimality of each of the projections [math]P_i\in Z(A)[/math] that we have [math]A_i\simeq M_{n_i}(\mathbb C)[/math]. Thus, we are led to the conclusion in the statement.
It is possible to further build on the above result, in several directions, either by allowing the factors in the decomposition to be type [math]{\rm I}_\infty[/math] factors as well, that is, [math]A_x\simeq B(H)[/math], or by allowing the center to be an infinite measured space, [math]|X|=\infty[/math], or by allowing both. The first possible generalization is not very interesting. The second possible generalization, however, is something quite interesting, and we have here:
\begin{fact}[Reduction theory, type I finite case] Given a von Neumann algebra [math]A\subset B(H)[/math] which is of discrete type, and has a trace [math]tr:A\to\mathbb C[/math], we can write
with [math]X[/math] coming via [math]Z(A)=L^\infty(X)[/math], and the trace decomposes as well, as
with the fibers [math]A_x[/math] being usual matrix algebras, [math]A_x=M_{n_x}(\mathbb C)[/math], with [math]n_x\in\mathbb N[/math]. \end{fact} As a first observation, this statement generalizes both what we know about the commutative algebras, and the finite dimensional ones. However, having these two things jointly generalized is something quite technical, that we will not explain here in detail. The idea is of course first that of axiomatizing what “discrete” should mean in the above, say by looking at the finiteness properties of the projections [math]p\in A[/math], and then, once the statement properly formulated, to prove it by jointly generalizing what we know about the commutative algebras, and the finite dimensional ones.
Moving ahead, let us lift now the assumption that the factors in the decomposition are of type [math]{\rm I}_N[/math], with [math]N \lt \infty[/math]. We are led in this way to a general result, as follows:
\begin{fact}[Reduction theory, type I case]
Given a von Neumann algebra [math]A\subset B(H)[/math] which is of type I, in the sense that it is of a suitable discrete type, we can write
with [math]X[/math] coming via [math]Z(A)=L^\infty(X)[/math], and with the fibers [math]A_x[/math] being type [math]{\rm I}[/math] factors, meaning [math]A_x\simeq B(H_x)[/math], with each [math]H_x[/math] being either finite dimensional, or separable. \end{fact} As before with Fact 11.6, we will not attempt to explain this here. As a comment, however, this can only follow from Fact 11.6 applied to the “finite” part of the algebra, obtained by removing the infinite part, and after proving that this infinite part is something of type [math]L^\infty(Y)\otimes B(H)[/math], with [math]Y\subset X[/math], and with [math]H[/math] being separable.
All the above was quite abstract, and as something more concrete now, let us discuss the reduction theory for the group von Neumann algebras [math]L(\Gamma)[/math], in the finite case, [math]|\Gamma| \lt \infty[/math]. For this purpose, it is convenient to change a bit our terminology and notations, making them more in tune with the quantum group formalism from chapter 7. First, we will denote our finite group [math]\Gamma[/math], which is at the same time discrete and compact, by [math]F[/math], and we will think of it as being the dual of a finite quantum group [math]G=\widehat{F}[/math]. Also, since in the finite group case everything is automatically norm or weakly closed, we will use the more familiar notation [math]C^*(F)[/math] for the associated von Neumann algebra [math]L(F)[/math]. With these conventions, we have the following result, which is standard:
Given a finite group [math]F[/math], the center of the associated von Neumann algebra is isomorphic to the algebra of central functions on [math]F[/math],
In what concerns the first assertion, regarding the center, this is something that we already 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, in the case where [math]\Gamma=F[/math] is a finite group, the computation there gives the following formula for the center:
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:
Regarding now the second assertion, let us first recall that the Peter-Weyl theory applied to the finite group [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:
Thus by dualizing, which is a standard functional analysis procedure, to be explained more in detail below, in a more general setting, we obtain a direct sum decomposition of the group algebra, as follows, which is this time a [math]*[/math]-algebra isomorphism:
Our claim now, which will finish the proof, is that this is exactly what comes out from von Neumann's reduction theory, applied to the von Neumann algebra [math]L(F)=C^*(F)[/math]. Indeed, 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:
We conclude that von Neumann's reduction theory, applied to the von Neumann algebra [math]L(F)=C^*(F)[/math], gives a [math]*[/math]-algebra isomorphism of the following type:
But a careful examination of the fibers appearing in this decomposition, based on their very definition, shows that these are precisely the above matrix blocks coming from Peter-Weyl. That is, we have [math]n_r=\dim(r)[/math] for any [math]r\in Irr(F)[/math], and we are done.
Our next goal will be that of extending the above result to the finite quantum group setting. For this purpose, we will not really need the general compact quantum group formalism from chapter 7, and it is more convenient to start with:
A finite dimensional Hopf algebra is a finite dimensional [math]C^*[/math]-algebra, with comultiplication, counit and antipode maps, satisfying the conditions
In this definition everything is standard, except for the last axiom, [math]S^2=id[/math], which corresponds to the fact that, in the corresponding quantum group, we should have:
It is possible to prove that this condition is automatic, in the present [math]C^*[/math]-algebra setting. However, this is something non-trivial, and since all this is just an informative discussion, not needed later, we have opted for including [math]S^2=id[/math] in our axioms.
We say that an algebra [math]A[/math] as above is cocommutative if [math]\Sigma\Delta=\Delta[/math], where [math]\Sigma(a\otimes b)=b\otimes a[/math] is the flip. With this convention made, we have the following result, which summarizes the basic theory of finite quantum groups, and justifies the terminology and axioms:
The following happen:
- If [math]G[/math] is a finite group then [math]C(G)[/math] is a commutative Hopf algebra, with
[[math]] \Delta(\varphi)=(g,h)\to \varphi(gh)\quad,\quad \varepsilon(\varphi)=\varphi(1)\quad,\quad S(\varphi)=g\to\varphi(g^{-1}) [[/math]]as structural maps. Any commutative Hopf algebra is of this form.
- If [math]F[/math] is a finite group then [math]C^*(F)[/math] is a cocommutative Hopf algebra, with
[[math]] \Delta(g)=g\otimes g\quad,\quad \varepsilon(g)=1\quad,\quad S(g)=g^{-1} [[/math]]as structural maps. Any cocommutative Hopf algebra is of this form.
- If [math]G,F[/math] are finite abelian groups, dual to each other via Pontrjagin duality,
[[math]] C(G)=C^*(F) [[/math]]as Hopf algebras, coming via a Fourier transform type operation.
These results are all elementary, the idea being as follows:
(1) The fact that [math]\Delta,\varepsilon,S[/math] satisfy the axioms is clear from definitions, and the converse follows from the Gelfand theorem, by working out the details, regarding [math]\Delta,\varepsilon,S[/math].
(2) Once again, the fact that [math]\Delta,\varepsilon,S[/math] satisfy the axioms is clear from definitions. For the converse, we use a trick. Let [math]A[/math] be an arbitrary finite dimensional Hopf algebra, as in Definition 11.9, and consider its comultiplication, counit, multiplication, unit and antipode maps. The transposes of these maps are then linear maps as follows:
It is routine to check that these maps make [math]A^*[/math] into a Hopf algebra. Now assuming that [math]A[/math] is cocommutative, it follows that [math]A^*[/math] is commutative, so by (1) we obtain [math]A^*=C(G)[/math] for a certain finite group [math]G[/math], which in turn gives [math]A=C^*(G)[/math], as desired.
(3) This follows indeed from the discussion in the proof of (2), and from the general theory of Pontrjagin duality for finite abelian groups, explained in chapter 7.
There are many other things that can be said about the finite dimensional Hopf algebras, and in what follows we will be particularly interested in the notion of corepresentation. These corepresentations can be introduced as follows:
A unitary corepresentation of a finite dimensional Hopf algebra [math]A[/math] is a unitary matrix [math]u\in M_n(A)[/math] satisfying the following conditions:
Observe the similarity with the notions introduced in chapter 7, for the Woronowicz algebras. In fact, by using left regular representations we can see that any finite dimensional Hopf algebra in the sense of Definition 11.9 is a Woronowicz algebra in the sense of chapter 7. Thus, we can freely use here the results established in chapter 7, and in particular, we can use the Peter-Weyl type theory developed there.
In relation now with our von Neumann algebra questions, we have the following result, coming from that Peter-Weyl type theory, which generalizes Theorem 11.8:
Given a finite quantum group [math]F[/math], the center of the associated von Neumann algebra is isomorphic to the algebra of central functions on [math]F[/math],
The proof here is nearly identical to the proof of Theorem 11.8. To be more precise, with the more familiar notation [math]A=C^*(F)[/math], the proof goes as follows:
(1) In what concerns the first assertion, regarding the center, we recall from Woronowicz [1] that [math]A_{central}[/math] is by definition the subalgebra of [math]A[/math] appearing as follows:
But this shows, first by dualizing, and then by doing some computations similar to those that we did in chapter 10, when computing the centers of the usual group von Neumann algebras, that we have an isomorphism as in the statement, namely:
(2) Regarding now the second assertion, we recall that the Peter-Weyl theory applied to Hopf algebra [math]A^*[/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:
Thus by dualizing, we obtain a direct sum decomposition of the group algebra, as follows, which is this time a [math]*[/math]-algebra isomorphism:
(3) Our claim now, which will finish the proof, is that this is exactly what comes out from von Neumann's reduction theory, applied to the algebra [math]A[/math]. Indeed, by using the standard identification between corepresentations [math]u[/math] of [math]A^*[/math] and their characters [math]\chi_u[/math], which belong to the algebra [math](A^*)_{central}[/math], the center computation that we did above reads:
We conclude that von Neumann's reduction theory, applied to the von Neumann algebra [math]A[/math], gives a [math]*[/math]-algebra isomorphism of the following type:
But a careful examination of the fibers shows that these are precisely the matrix blocks coming from Peter-Weyl. That is, [math]n_u=\dim(u)[/math] for any [math]u\in Irr(A^*)[/math], and we are done.
All this is quite interesting, and it is possible to say more about it. However, when it comes to type I algebras, in general, the following comment is unavoidable: \begin{comment} The most interesting type I algebras are probably those having an isotypic decomposition, and so which can be written as follows:
But these are precisely the random matrix algebras, that we investigated in great detail in chapter 6, right after introducing the von Neumann algebras. So, job done. \end{comment} Needless to say, this is something subjective. But, in any case, whether you agree or not with this, now you know more on the organization of the present book.
General references
Banica, Teo (2024). "Principles of operator algebras". arXiv:2208.03600 [math.OA].