guide:07d91e0a4a

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We are now in position of constructing the coupling constant. The idea here, following as usual the key paper of Murray and von Neumann , will be that given a representation of a [math]{\rm II}_1[/math] factor [math]A\subset B(H)[/math], we can try to understand how far is this representation from the standard form, where [math]H=L^2(A)[/math], from “above” or from “below”.

In order to discuss this, which is something quite technical, let us start with:

Proposition

Given a [math]{\rm II}_1[/math] factor [math]A\subset B(H)[/math], with its embedding into [math]B(H)[/math] being represented as above, in terms of an isometry

[[math]] u:H\to L^2(A)\otimes l^2(\mathbb N)\quad,\quad ux=(x\otimes1)u [[/math]]

the following quantity does not depend on the choice of this isometry [math]u[/math]:

[[math]] C=tr(uu^*) [[/math]]

Moreover, for the standard form, where [math]H=L^2(A)[/math], this constant takes the value [math]1[/math].

Show Proof

Assume indeed that we have an isometry [math]u[/math] as in the statement, and that we have as well a second such isometry, of the same type, namely:

[[math]] v:H\to L^2(A)\otimes l^2(\mathbb N)\quad,\quad vx=(x\otimes1)v [[/math]]

We have then [math]uu^*=uv^*vu^*[/math], and by using this, we obtain:

[[math]] \begin{eqnarray*} C_u &=&tr(uu^*)\\ &=&tr(uv^*vu^*)\\ &=&tr(vu^*uv^*)\\ &=&tr(vv^*)\\ &=&C_v \end{eqnarray*} [[/math]]

Thus, we are led to the conclusion in the statement. As for the last assertion, regarding the standard form, this is clear from definitions, because here we can take [math]u=1[/math].

■

As a conclusion to all this, given a [math]{\rm II}_1[/math] factor [math]A\subset B(H)[/math], we know from Theorem 10.29 that [math]H[/math] must appear as an “inflated” version of [math]L^2(A)[/math]. The corresponding inflation constant is a certain number, that we can call coupling constant, as follows:

Definition

Given a representation of a [math]{\rm II}_1[/math] factor [math]A\subset B(H)[/math], we can talk about the corresponding coupling constant, as being the number

[[math]] \dim_AH\in(0,\infty] [[/math]]

constructed as follows, with [math]u:H\to L^2(A)\otimes l^2(\mathbb N)[/math] isometry satisfying [math]ux=(x\otimes1)u[/math]:

[[math]] \dim_AH=tr(uu^*) [[/math]]

For the standard form, where [math]H=L^2(A)[/math], this coupling constant takes the value [math]1[/math].

This definition might seem a bit complicated, but things here are quite non-trivial, and there is no way of doing something substantially simpler. Alternatively, we can define the coupling constant via the following formula, after proving first that the number on the right is indeed independent of the choice on a nonzero vector [math]x\in H[/math]:

[[math]] \dim_AH=\frac{tr_A(P_{A'x})}{tr_{A'}(P_{Ax})} [[/math]]

This latter formula was in fact the original definition of the coupling constant, by Murray and von Neumann . However, technically speaking, things are slightly easier when using the approach in Definition 10.32. We will be back to this key formula of Murray and von Neumann, with full explanations, in a moment.

Let us start our study of the coupling constant with some basic results, coming from definitions and from what we already have, as results, as follows:

Proposition

The coupling constant [math]\dim_AH\in(0,\infty][/math] associated to a [math]{\rm II}_1[/math] factor representation [math]A\subset B(H)[/math] has the following properties:

• For the standard form, [math]H=L^2(A)[/math], we have [math]\dim_AH=1[/math].
• For the usual representation on [math]H=L^2(A)\otimes l^2(\mathbb N)[/math], we have [math]\dim_AH=\infty[/math].
• We have [math]\dim_AH \lt \infty[/math] precisely when [math]A'[/math] is a [math]{\rm II}_1[/math] factor.
• We have additivity, [math]\dim_A(\oplus_iH_i)=\sum_i\dim_AH_i[/math].
• We have [math]\dim_A(L^2(A)p)=tr(p)[/math], for any projection [math]p\in A[/math].
• The coupling constant can take any value in [math](0,\infty][/math].

Show Proof

All these assertions are elementary, the idea being as follows:

(1) This is something that we already know, coming from definitions.

(2) This is something that comes from definitions too.

(3) This comes from the general properties of the [math]{\rm II}_\infty[/math] factors, and their traces.

(4) Again, this is clear from the definition of the coupling constant.

(5) This follows by using [math]u(x)=x\otimes\xi[/math], with [math]\xi\in l^2(\mathbb N)[/math] being of norm 1.

(6) This follows by starting with (5), and then making direct sums, as in (4).

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At a more advanced level now, in relation with projections and compressions, and getting towards the above-mentioned Murray-von Neumann approach, we have:

Proposition

We have the compression formula

[[math]] \dim_{pAp}(pH)=\frac{\dim_AH}{tr_A(p)} [[/math]]

valid for any projection [math]p\in A[/math].

Show Proof

We can prove this result in two steps, as follows:

(1) Assume that [math]H[/math] is as follows, with [math]q\in A[/math] being a projection satisfying [math]q\leq p[/math]:

[[math]] H=L^2(A)q [[/math]]

We can use the following unitary, intertwining the left and right actions of [math]pAp[/math]:

[[math]] L^2(pAp)\to pL^2(A)p\quad,\quad pxp\Omega\to p(x\Omega)p [[/math]]

Indeed, we obtain that the following algebras are unitarily equivalent:

[[math]] pAp\subset B(pL^2(A)q)\quad,\quad pAp\subset B(L^2(pAp)q) [[/math]]

Thus, by using the formula (5) in Proposition 10.33 we obtain, as desired:

[[math]] \begin{eqnarray*} \dim_{pAp}(pH) &=&tr_{pAp}(q)\\ &=&\frac{tr_A(q)}{tr_A(p)}\\ &=&\frac{\dim_AH}{tr_A(p)} \end{eqnarray*} [[/math]]

(2) In the general case now, where [math]H[/math] is arbitrary, the result follows from what we proved above, and from the additivity property from Proposition 10.33 (4).

■

With all these properties established, we can now recover, as a theorem, the original definition of the coupling constant, due to Murray and von Neumann, as follows:

Theorem

Given a [math]{\rm II}_1[/math] factor [math]A\subset B(H)[/math], with the commutant [math]A'\subset B(H)[/math] assumed to be finite, the corresponding coupling constant is finite, given by

[[math]] \dim_AH=\frac{tr_A(P_{A'x})}{tr_{A'}(P_{Ax})} [[/math]]

with the number on the right being independent of the choice on a nonzero vector [math]x\in H[/math]. In the case where [math]A'[/math] is infinite, the corresponding coupling constant is infinite.

Show Proof

There are several things to be proved here, the idea being as follows:

(1) We know from Proposition 10.33 (3) that we have [math]\dim_AH \lt \infty[/math] precisely when the commutant [math]A'\subset B(H)[/math] is finite. Thus, we may assume that we are in this case.

(2) Assuming so, we have the following formula, valid for any projection [math]p\in A'[/math], which follows from the basic properties of the coupling constant, established above:

[[math]] \dim_{Ap}(pH)=tr_{A'}(p)\dim_AH [[/math]]

(3) Now with this formula in hand, the formula in the statement follows as well, once again by doing a number of standard amplification and compression manipulations.

■

As an illustration for all this, given an inclusion of ICC groups [math]\Lambda\subset\Gamma[/math], whose group algebras are both [math]{\rm II}_1[/math] factors, we have the following formula:

[[math]] \dim_{L(\Lambda)}L^2(\Gamma)=[\Gamma:\Lambda] [[/math]]

There are many other examples of explicit computations of the coupling constant, all leading into interesting mathematics. We will be back to this.

As a last topic for this chapter, given a [math]{\rm II}_1[/math] factor [math]A[/math], let us discuss now the representations of type [math]A\subset B[/math], with [math]B[/math] being another [math]{\rm II}_1[/math] factor. This is a quite natural notion, perhaps even more natural than the representations [math]A\subset B(H)[/math], because we have previously decided that the [math]{\rm II}_1[/math] factors [math]B[/math], and not the full operator algebras [math]B(H)[/math], are the correct infinite dimensional generalization of the usual matrix algebras [math]M_N(\mathbb C)[/math].

This was for the philosophy, and one can of course agree or not with this. Or at least agree or not at the present point of the presentation, because once we will get into the structure of the subfactors [math]A\subset B[/math], which is something amazing, there is no way back.

In practice now, given an inclusion of [math]{\rm II}_1[/math] factors [math]A\subset B[/math], a first question is that of defining its index, measuring how big is [math]B[/math] compared to [math]A[/math]. The first thought here goes into defining the index of [math]A\subset B[/math] as being a purely algebraic quantity, as follows:

[[math]] N=\dim_AB [[/math]]

However, this is non-trivial, due to the fact that we are in the “continuous dimension” setting, and so our algebraic intuition, where indices are always integers, will not help us much. We will be back to this question later, with a technical solution to it.

In order to solve our index problem, a much better approach is by using the ambient operator algebra [math]B(H)[/math], or rather the ambient Hilbert space [math]H[/math], as follows:

Theorem

Given an inclusion of [math]{\rm II}_1[/math] factors [math]A\subset B[/math], the number

[[math]] N=\frac{\dim_AH}{\dim_BH} [[/math]]

is independent of the ambient Hilbert space [math]H[/math], and is called index.

Show Proof

The fact that the index of the subfactor [math]A\subset B[/math], as defined by the above formula, is indeed independent of the ambient Hilbert space [math]H[/math], comes from the various basic properties of the coupling constant, established above.

■

There are many examples of subfactors coming from groups, and every time we obtain the intuitive index. More suprisingly now, Jones proved in that the index, when small, is in fact “quantized”, subject to the following unexpected restriction:

[[math]] N\in\left\{4\cos^2\left(\frac{\pi}{n}\right)\Big|n\geq3\right\}\cup[4,\infty] [[/math]]

This is in fact part of a series of non-trivial results about the subfactors, due to Jones, and also Ocneanu, Popa, Wassermann and others, and involving as well the Temperley-Lieb algebra , and many more. We will be back to this later, with the whole last part of the present book, chapters 13-16 below, being dedicated to subfactor theory.

General references

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