Split Inclusions, Type-I Intermediates, and Statistical Independence

A strict inclusion $\mathcal N\subset\mathcal M$ is split when a type-I factor lies between the two algebras. Under the standard factorial hypotheses this is equivalent to a normal tensor-product realization of the commuting pair $(\mathcal N,\mathcal M')$, and it licenses normal product states with independently chosen marginals. The type-I factor depends on the collar and is generally noncanonical; it does not turn either sharp local algebra into a type-I algebra. The separated free-scalar construction is applied on the split property and approximate tensor products.

Required background. Modular theory, nuclearity, and the split property gives the implication chain; representation types, factors, and local-algebra structure distinguishes type I from type III; and phase-space nuclearity and compactness maps supplies the quantitative hypothesis.

Helpful background. The split property and approximate tensor products gives the subsystem interpretation, while local preparation, state dependence, and operational independence develops its operational consequences.

Split inclusions and the spatial tensor product

For von Neumann factors $\mathcal N\subset\mathcal M\subset\mathcal B(\mathcal H)$, the inclusion is split if

$$\mathcal N\subset\mathcal F\subset\mathcal M$$

for a type-I factor $\mathcal F$. In a local net one takes

$$\mathcal A(\mathcal O_1) \subset\mathcal F \subset\mathcal A(\mathcal O_2), \qquad \mathcal O_1\Subset\mathcal O_2.$$

The strict containment of regions provides a spacetime collar. Locality makes $\mathcal N$ commute with $\mathcal M'$. For a standard split inclusion, the algebraic multiplication map

$$\eta_0:\mathcal N\odot\mathcal M' \longrightarrow\mathcal N\vee\mathcal M', \qquad \eta_0(A\otimes B')=AB',$$

extends to a normal spatial isomorphism

$$\eta:\mathcal N\,\overline\otimes\,\mathcal M' \overset{\simeq}{\longrightarrow} \mathcal N\vee\mathcal M'.$$

Equivalently, there is a unitary tensor-product implementation in which $\mathcal N$ acts on the first factor and $\mathcal M'$ on the second. The corresponding $\mathcal B(\mathcal H_1)\otimes1$ pulls back to an intermediate type-I factor. Doplicher and Longo establish these equivalences and the standard-inclusion structure in Doplicher and Longo 1984, pp. 493–536.

The word “spatial” matters. Algebraic commutativity alone always defines $A\otimes B'\mapsto AB'$ on finite sums, but the map need not extend normally to the von Neumann tensor product in the vacuum representation.

Product states and local preparation

Let $\omega_1$ and $\omega_2$ be normal states on $\mathcal N$ and $\mathcal M'$. Their tensor product is normal on $\mathcal N\overline\otimes\mathcal M'$. Pulling it through $\eta$ gives

$$\omega(AB')=\omega_1(A)\omega_2(B'), \qquad A\in\mathcal N,\quad B'\in\mathcal M'.$$

Thus arbitrary normal marginals have a normal product extension on the joined algebra. Convex combinations give normally correlated states, and the type-I realization supports local preparation statements. These are exact algebraic consequences for the separated pair.

They are not claims that the vacuum itself factorizes. Vacuum correlations generally remain nonzero. Nor is the intermediate factor unique. Different collars, tensor-product unitaries, or choices of product vector can produce different $\mathcal F$.

Normality is the decisive strengthening over purely algebraic independence. On finite sums, commuting algebras always permit the formal multiplication $A\otimes B'\mapsto AB'$. A split inclusion says this map is continuous in the relevant von Neumann topology, so separately chosen normal marginals remain normal after combination. Without that extension, a positive algebraic product functional may lie outside the vacuum folium and cannot represent a normally preparable state there.

Conditional expectations from a chosen split

A split realization also gives conditional expectations, but not a canonical one. Suppose $W:\mathcal H\to\mathcal H_1\otimes\mathcal H_2$ implements the split and

$$\mathcal F =W^*(\mathcal B(\mathcal H_1)\otimes1)W.$$

Choose a normal state $\chi$ on $\mathcal B(\mathcal H_2)$. The slice map defines

$$E_\chi(W^*XW) =W^*\left[ (\operatorname{id}\otimes\chi)(X)\otimes1 \right]W.$$

Then $E_\chi:\mathcal B(\mathcal H)\to\mathcal F$ is a normal conditional expectation and is faithful when $\chi$ is faithful. It fixes $\mathcal F$ and satisfies the $\mathcal F$-bimodule identity. Its dependence on $\chi$ displays the nonuniqueness.

The split property alone does not provide a conditional expectation from $\mathcal M$ onto $\mathcal N$, nor a preferred one from the global algebra onto $\mathcal F$. A state-preserving expectation onto a subalgebra requires the modular-invariance hypothesis of Takesaki’s theorem.

Two separated free-scalar double cones

Let $\mathcal O_1$ and $\mathcal O_3$ be spacelike separated double cones in the massive free scalar net. Choose a larger double cone $\mathcal O_2$ with $\mathcal O_1\Subset\mathcal O_2\subset\mathcal O_3'$. Energy nuclearity for $\mathcal A(\mathcal O_1)\subset\mathcal A(\mathcal O_2)$ gives

$$\mathcal A(\mathcal O_1) \subset\mathcal F_\delta \subset\mathcal A(\mathcal O_2) \subset\mathcal A(\mathcal O_3)',$$

where $\delta$ is a positive collar width. The Buchholz–Wichmann phase-space condition supplies this split conclusion for the free-field setting in Buchholz and Wichmann 1986, pp. 321–344.

Given normal states $\omega_1$ on $\mathcal A(\mathcal O_1)$ and $\omega_3$ on $\mathcal A(\mathcal O_3)$, use the spatial isomorphism for the commuting pair to define their normal product extension. An independent check is immediate:

$$\omega(A1)=\omega_1(A), \qquad \omega(1B)=\omega_3(B), \qquad \omega(AB)=\omega_1(A)\omega_3(B).$$

This verifies the prescribed marginals and vanishing connected correlations in the prepared state. It does not erase vacuum correlations or select $\mathcal F_\delta$ uniquely.

Adversarial test: collapse the collar

Send $\delta\downarrow0$ while pretending that one type-I factor and one nuclear-norm bound remain uniform. High-frequency modes localized in the shrinking collar remove the needed phase-space suppression; free-field nuclearity estimates deteriorate as the separation scale vanishes. The family $\mathcal F_\delta$ need not converge to a type-I factor associated with the sharp boundary.

At zero collar, the sharp local algebra remains type III in the usual continuum models. The strongest licensed result is a family of split inclusions for every fixed positive separation, not an exact inside–outside tensor product at a common boundary.

Exercises

Show that the slice map $E_\chi$ is an $\mathcal F$-bimodule projection.

Solution

In the tensor representation, let $F_1=Y_1\otimes1$ and $F_2=Y_2\otimes1$. Then

$$(\operatorname{id}\otimes\chi)(F_1XF_2) =Y_1(\operatorname{id}\otimes\chi)(X)Y_2.$$

After re-embedding with $\otimes1$ and conjugating by $W$, this gives $E_\chi(F_1XF_2)=F_1E_\chi(X)F_2$. If $X\in\mathcal F$, the slice returns $X$, so $E_\chi^2=E_\chi$.

References

• Buchholz, Detlev, and Eyvind H. Wichmann. 1986. “Causal Independence and the Energy-Level Density of States in Local Quantum Field Theory.” Communications in Mathematical Physics 106: 321–344. DOI.
• Doplicher, Sergio, and Roberto Longo. 1984. “Standard and Split Inclusions of von Neumann Algebras.” Inventiones Mathematicae 75: 493–536. DOI.