guide:Fc7125860d

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

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Let us go back now to the general theory of the [math]{\rm II}_1[/math] factors, with the aim of talking about representations of such [math]{\rm II}_1[/math] factors, inside the category of the [math]{\rm II}_1[/math] factors, [math]A\subset B[/math]. For this purpose we will need a key notion, called coupling constant.

In order to discuss the construction of the coupling constant, we will need some further results on the type [math]{\rm II}[/math] factors, complementing those that we already have. The point indeed is that the class of [math]{\rm II}[/math] factors, to be axiomatized later, and with this being not something urgent, comprises, besides the [math]{\rm II}_1[/math] factors discussed above, the [math]{\rm II}_\infty[/math] factors as well:

Definition

A [math]{\rm II}_\infty[/math] factor is a von Neumann algebra of the form

[[math]] B=A\otimes B(H) [[/math]]

with [math]A[/math] being a [math]{\rm II}_1[/math] factor, and with [math]H[/math] being an infinite dimensional Hilbert space.

We should mention that there are several possible ways of defining the [math]{\rm II}_\infty[/math] factors, and the above definition is something rather intuitive, the point being that, once you learn the theory of the [math]{\rm II}_\infty[/math] factors, as we will do here, what you remember at the end of the day is what has been said above, [math]B=A\otimes B(H)[/math], with [math]A[/math] being a [math]{\rm II}_1[/math] factor.

Getting started now, as a useful characterization of such factors, we have:

Proposition

For an infinite factor [math]B[/math], the following are equivalent:

• There exists a projection [math]p\in B[/math] such that [math]pBp[/math] is a [math]{\rm II}_1[/math] factor.
• [math]B[/math] is a [math]{\rm II}_\infty[/math] factor.

Show Proof

This is something elementary, as follows:

[math](1)\implies(2)[/math] Assume indeed that [math]p\in B[/math] is a projection such that [math]pBp[/math] is a [math]{\rm II}_1[/math] factor. We choose a maximal family of pairwise orthogonal projections [math]\{p_i\}\subset B[/math] satisfying [math]p_i\simeq p[/math], for any [math]i[/math], and we consider the following projection, which satisfies [math]q\preceq p[/math]:

[[math]] q=1-\sum_ip_i [[/math]]

Since the indexing set for our set of projections [math]\{p_i\}[/math] must be infinite, we can use a strict embedding of this index set into itself, as to write a formula as follows:

[[math]] \begin{eqnarray*} 1 &=&q+\sum_ip_i\\ &\preceq&p_0+\sum_{i\neq0}p_i\\ &\preceq&1 \end{eqnarray*} [[/math]]

Thus we have [math]\sum_ip_i\simeq1[/math], and we may further suppose that we have in fact:

[[math]] \sum_ip_i=1 [[/math]]

Thus the family [math]\{p_i\}[/math] can be used in order to construct a copy [math]B(H)\subset B[/math], with [math]H=l^2(\mathbb N)[/math], and we must have [math]B=A\otimes B(H)[/math], with [math]A[/math] being a [math]{\rm II}_1[/math] factor, as desired.

[math](2)\implies(1)[/math] This is clear, because when assuming [math]B=A\otimes B(H)[/math], as in Definition 10.24, we can take our projection [math]p\in B[/math] to be of the form [math]p=1\otimes q[/math], with [math]q\in B(H)[/math] being a rank 1 projection, and we have then [math]pBp=A[/math], which is a [math]{\rm II}_1[/math] factor, as desired.

■

Getting back now to the original interpretation of the [math]{\rm II}_\infty[/math] factors, from Definition 10.24, the tensor product writing there [math]B=A\otimes B(H)[/math] suggests tensoring the trace of the [math]{\rm II}_1[/math] factor [math]A[/math] with the usual operator trace of [math]B(H)[/math]. We are led in this way to:

Definition

Given a [math]{\rm II}_\infty[/math] factor [math]B[/math], written as [math]B=A\otimes B(H)[/math], with [math]A[/math] being a [math]{\rm II}_1[/math] factor and with [math]H[/math] being an infinite dimensional Hilbert space, we define a map

[[math]] tr:B_+\to[0,\infty]\quad,\quad tr((x_{ij}))=\sum_itr(x_{ii}) [[/math]]

where we have chosen a basis of [math]H[/math], as to have [math]H\simeq l^2(\mathbb N)[/math], and so [math]B(H)\subset M_\infty(\mathbb C)[/math].

As an important observation, to start with, unlike in the [math]{\rm II}_1[/math] factor case, that of the factor [math]A[/math], or in the [math]{\rm I}_\infty[/math] factor case, that of the factor [math]B(H)[/math], it is not possible to suitably normalize the trace constructed above. This follows indeed from the results below.

On the positive side now, the trace that we constructed has all sorts of good properties, that we can use for various purposes, which can be summarized as follows:

Proposition

The [math]{\rm II}_\infty[/math] factor trace that we constructed above

[[math]] tr:B_+\to[0,\infty] [[/math]]

has the following properties:

• [math]tr(x+y)=tr(x)+tr(y)[/math], and [math]tr(\lambda x)=\lambda tr(x)[/math] for [math]\lambda\geq0[/math].
• If [math]x_i\nearrow x[/math] then [math]tr(x_i)\to tr(x)[/math].
• [math]tr(xx^*)=tr(x^*x)[/math].
• [math]tr(uxu^*)=tr(x)[/math] for any [math]u\in U_B[/math].

Show Proof

All this is elementary, the idea being as follows:

(1) This is clear from definitions.

(2) This is again clear from definitions.

(3) This is something which is elementary as well.

(4) This comes from (3), via the formula [math]uxu^*=u\sqrt{x}\cdot\sqrt{x}u^*[/math].

■

As a main result now regarding the [math]{\rm II}_\infty[/math] factor trace, we have:

Theorem

The [math]{\rm II}_\infty[/math] factor trace [math]tr:B_+\to[0,\infty][/math] constructed above, when restricted to the projections

[[math]] tr:P(B)\to[0,\infty] [[/math]]

induces an isomorphism between the totally ordered set of equivalence classes of projections in [math]B[/math] and the interval [math][0,\infty][/math].

Show Proof

We have several things to be checked here, as follows:

(1) Our first claim is that a projection [math]p\in B[/math] is finite precisely when [math]tr(p) \lt \infty[/math].

-- Indeed, in one sense, assume that we have [math]tr(p) \lt \infty[/math]. If our projection [math]p[/math] was to be infinite, we would have a subprojection [math]q\leq p[/math] having the same trace as [math]p[/math], and so [math]r=p-q[/math] would be a projection of trace [math]0[/math], which is impossible. Thus [math]p[/math] is indeed finite.

-- In the other sense now, assuming [math]tr(p)=\infty[/math], we have to prove that [math]p[/math] is infinite. For this purpose, let us pick a projection [math]q\leq p[/math] having finite trace. Then [math]r=p-q[/math] satisfies [math]tr(r)=\infty[/math], and so we can iterate the procedure, and we end up with an infinite sequence of pairwise orthogonal projections, which are all smaller than [math]p[/math]. But this shows that [math]p[/math] dominates an infinite projection, and so that [math]p[/math] itself is infinite, as desired.

(2) Our second claim is that if [math]p,q\in B[/math] are projections, with [math]p[/math] finite, then:

[[math]] p\preceq q\iff tr(p)=tr(q) [[/math]]

But this follows exactly as in the [math]{\rm II}_1[/math] factor case, discussed above.

(3) Our third and final claim, which will finish the proof, is that any infinite projection is equivalent to the identity. For this purpose, assume that [math]p\in B[/math] is infinite. By definition, this means that we can find a unitary [math]u\in B[/math] such that:

[[math]] uu^*=p\quad,\quad u^*u\leq p\quad,\quad uu^*\neq p [[/math]]

But these conditions show that [math](u^n)^iu^n[/math] is a strictly decreasing sequence of equivalent projections, and by using this sequence we conclude that we have [math]1\preceq p[/math], as desired.

■

Moving ahead now, in order to further investigate the [math]{\rm II}_\infty[/math] factors, we will need:

Theorem

Given a [math]{\rm II}_1[/math] factor [math]A\subset B(H)[/math], there exists an isometry

[[math]] u:H\to L^2(A)\otimes l^2(\mathbb N) [[/math]]

such that [math]ux=(x\otimes1)u[/math], for any [math]x\in A[/math].

Show Proof

We use a standard idea, that we used many times before, namely an amplification trick. Given a [math]{\rm II}_1[/math] factor [math]A\subset B(H)[/math], consider the following Hilbert space:

[[math]] K=H\oplus L^2(A)\otimes l^2(\mathbb N) [[/math]]

Consider, as operators over this space [math]K[/math], the following projections:

[[math]] p=id\oplus 0\quad,\quad q=0\oplus id [[/math]]

Both these projections [math]p,q[/math] belong then to [math]A'[/math], which is a type [math]{\rm II}_\infty[/math] factor. Now since [math]q\in A'[/math] is infinite, by Theorem 10.28 we can find a partial isometry [math]u\in A'[/math] such that:

[[math]] u^*u=p\quad,\quad uu^*\leq q [[/math]]

Now let us represent this partial isometry [math]u\in B(K)[/math] as a [math]2\times2[/math] matrix, as follows:

[[math]] u=\begin{pmatrix}a&b\\ c&d\end{pmatrix} [[/math]]

The above conditions [math]u^*u=p[/math] and [math]uu^*\leq q[/math] reformulate then as follows:

[[math]] b^*b+d^*d=0\quad,\quad aa^*+bb^*=0 [[/math]]

We conclude that our partial isometry [math]u\in B(K)[/math] has the following special form:

[[math]] u=\begin{pmatrix}0&0\\ c&0\end{pmatrix} [[/math]]

But the operator [math]c:H\to l^2(A)\otimes l^2(\mathbb N)[/math] that we found in this way must be an isometry, and from [math]u\in A'[/math] we obtain [math]ux=(x\otimes1)u[/math], for any [math]x\in A[/math], as desired.

■

As a basic consequence of the above result, which is something good to know, and that we will use many times in what follows, we have:

Theorem

The commutant of a [math]{\rm II}_1[/math] factor is a [math]{\rm II}_1[/math] factor, or a [math]{\rm II}_\infty[/math] factor.

Show Proof

This follows indeed from the explicit interpretation of the operator algebra embedding [math]A\subset B(H)[/math] of our [math]{\rm II}_1[/math] factor [math]A[/math], found in Theorem 10.29.

■

Summarizing, we have an extension of the general theory of the [math]{\rm II}_1[/math] factors, developed before, to the general case of the type [math]{\rm II}[/math] factors, which comprises by definition the [math]{\rm II}_1[/math] factors and the [math]{\rm II}_\infty[/math] factors. All this is of course technically very useful.

General references

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