Canonical densities multiply along finite chains

The two normalized traces on a relative commutant determine a positive central density. For any finite-index chain of II₁ factors, that density for the whole inclusion is the product of the adjacent densities. This proves the general identification left open after the two-step calculation in Lesson 64, including its centrality. The argument uses concrete finite bases and the previously proved duality maps; it does not assume a statistical-dimension theorem.

The density identity also identifies the lower blocked cup in the finite-comparison criterion,65.14. That criterion still requires the general finite pair anti-isomorphisms with their designated cup images.

We use finite-module traces and index multiplicativity, Lesson 2; common bases, Lesson 3; fusion units and concrete duality,6.1–6.5; and cup densities and their transfer formula,63.1–63.4. The elementary interchange argument is related to Roberto Longo and John E. Roberts, A Theory of Dimension, Theorem 3.8, printed p.15 in the arXiv version (PDF zero index14). We prove the required statement here with all three factor endpoints specified. The statistical-dimension standardness criterion in their Lemma 3.9 is not a prerequisite of this argument.

The two closures of the inclusion module

Let have finite index . Write

The conjugate is identified by . A linear bimodule endomorphism of is left multiplication , with ; the corresponding endomorphism of is right multiplication . The map reverses products.

Choose any finite common tracial basis for . Let be inclusion, and let

By 6.4, . The maps of 6.5 are

Their two conjugate equations hold with the specified fusion units and associators. The endomorphisms of the standard - and - units are scalars.

For , define scalar closure functionals by

Let be the normalized trace of the commutant of left on , restricted to . Let be its density from 63.1:

Strict positivity here includes bounded invertibility.

Proposition 66.1. The closures in (66.3) are

For , the modified maps

still satisfy the conjugate equations, and their two closures agree:

Proof. The fusion unit identifies with . Thus the first closure is . Since commutes with , .

For the second closure, evaluate on a bounded vector . The formula for and gives

The transfer formula 63.13 makes . This proves (66.5) first on bounded vectors and then everywhere. No basis orthogonality was used.

The modification rule in (66.6) preserves duality. In the first conjugate equation, the factor acts on the output and the factor acts on the input . The composite is therefore

In the second equation the two inverse factors meet on the middle leg and cancel. Its remaining composite is the original identity on . This calculation uses ; bounded inverse and functoriality justify every displayed map.

The new closures are

Since is a central function of , both equal the value in (66.7). This proves equality on the entire endomorphism algebra, including all its matrix units. In particular both modified maps have squared norm .

The fourth root balances duality closures. The square root in the cup has a different role. We retain both normalizations.

Why balanced closures survive fusion

Here the module endpoints need not be the same. Let be an - correspondence and a - correspondence. Suppose they have bounded duality maps

Here , and similarly for . Define as in (66.3). Assume on and on . These equations compare scalar values even though their unit operators belong to different factors.

Lemma 66.2 — typed partial-closure interchange. For , with conjugate , put

These maps satisfy the conjugate equations. Their scalar closures agree on every .

Proof. For the first conjugate equation, interchange of the disjoint bounded maps rewrites the expanded composite as the product of

Both bracketed maps are the respective first conjugate identities. Thus . For the second equation the expanded composite similarly factors into

The second conjugate equations make both factors identity operators, giving . All bracket changes use the given associator and unit maps; no untyped algebraic tensor product is introduced.

For , define partial closures

For example has source and target . The -action at its output is the final right -action of . Every map in that composite intertwines the outer actions. The analogous check gives the claimed type of .

Expanding (66.10) gives

The two scalars and coincide. To see this without invoking a trace theorem, write

Interchange of bounded fusion maps also gives . Both scalar closures expand to precisely

The operator in (66.13) is a - endomorphism of , hence a scalar. This is the partial-closure interchange.

Now use the two assumed balanced identities, with the types in (66.11):

This proves the equality on the whole endomorphism algebra of , whether or not is irreducible.

Transitivity of the canonical density

Let be any finite-index chain of II₁ factors. Set

The densities are defined separately by their own commutant trace as in (66.4). The inclusion need not be a Jones tower, extremal or of finite depth.

Theorem 66.3 — canonical density transitivity. Inside the actual factor ,

Consequently the canonical expectations compose:

Proof. The two densities commute because commutes with and . Put

Then is positive invertible in ; its centrality there has not been assumed.

Take and . Their fusion is . On bounded coordinates, its unit map is . The conjugate fusion identifies with by . Both are the previously proved standard fusion unit maps.

Let be a common tracial basis for and one for . Their ordered products are a common tracial basis for . Indeed successive reconstruction gives

The left reconstruction follows by taking adjoints, and the index sum is .

First use the unmodified maps (66.2) on . Under the two unit identifications, the product maps (66.10) are exactly the unmodified maps for . For , evaluate on : insertion of gives . This is the inclusion map with the normalization in (66.2). For , evaluating on a bounded gives

By (66.19) and 6.4 this is precisely . Density and boundedness extend both identifications to the full Hilbert spaces.

Next use the balanced maps (66.6), with on and on . Their product , on , becomes . Hence

For the other product map the bounded-coordinate expression is

Since commutes with , its second leg equals . Therefore

This step does not move across ; such commutation has not been asserted.

Proposition 66.1 balances the closures of each input module. Lemma 66.2 therefore balances the closures of on every . Applying the original closure formulas (66.5) for to (66.21)–(66.22) yields

The positive element is central in , so it commutes with . Trace cyclicity changes (66.23) into

The coefficient is self-adjoint and belongs to . Choose . Faithfulness gives , hence . Multiply by to obtain . In particular the product is central. Thus centrality was a consequence of the full closure calculation, rather than an inference from the adjacent commutation alone.

Finally, , and . Bimodularity of gives

These are the faithful normal expectations already constructed in 63.2.

The same proof also gives the identity

For a direct check, by (66.16), and . This is an identity for the explicitly balanced closure masses. No classification of all expectations is inferred.

The general two-step density and its actual cup

Now take the actual consecutive basic-construction chain of 64.1:

Let be its adjacent canonical densities. The finite dual identification 63.4 gives , with on .

Corollary 66.4. The product in 64.2 is the canonical blocked density:

It belongs to , and the projection in 64.2 is exactly the canonical scalar-relative-expectation cup for :

Proof. Apply Theorem 66.3 to . The finite right-multiplication identification gives the second expression in (66.26). The operator equality and projection assertion in (66.27) are already proved in 64.2. With , the general relative-expectation formula 64.21, or 63.8 for the blocked inclusion, becomes . All these statements concern the same prescribed factors and actual projections.

For the signed course convention, write and . The cup for the blocked inclusion is therefore

In 65.14 take . Its fixed lower scalar cup can now be written explicitly as

The upper target remains the tracial . We have identified the lower cup, not constructed a finite map carrying it to that target.

Corollary 66.5 — arbitrary finite chains. For of finite-index II₁ factors,

Proof. A newer density commutes with every earlier containing factor, so the factors in the product commute pairwise and are positive invertible. Inductively apply Theorem 66.3 to ; its base case is the definition for one step. The same induction proves the expectation identity. Every equality is at a finite endpoint.

Figure 66.1. The two partial closures of an - endomorphism meet in the - unit and yield the same scalar (66.11–66.14). For an actual finite inclusion chain, the fourth roots give ; equality of the full closures forces (66.23–66.24). The last panel records the exact signed lower and upper cups in 65.14. The density proof supplies no arrow between their finite algebras. Human source for interchange: Longo–Roberts, Theorem 3.8; concrete duality and densities:6.5,63.1–63.4; actual blocked word:64.1–64.2. Reproducible figure source.

Examples

Example 66.6 — three diagonal corners. Suppose three successive inclusions have local corner indices one and diagonal trace weights , each summing to one. For the first inclusion, and the canonical coefficient on corner is , by 63.4. Write and . On a nonzero simultaneous diagonal corner , (66.30) gives coefficient

The inherited joint weight is under the stated factor hypotheses. To see this, write the chain as , with successive diagonal projections , and . Expectation bimodularity puts in , and trace preservation makes it . Likewise . Therefore

Thus the left dimension weight of this joint projection is , using its displayed canonical-density coefficient. This calculation does not require an independent tensor-product construction. The joint projection need not be minimal in the full outer relative commutant; its trace and density coefficient are still the values just computed.

Example 66.7 — unequal weights do not disappear in a block. In the weighted two-spin family of 64.4, , , and on the two adjacent sectors has values . Its dual has reciprocal right-multiplication values. On the four joint sectors the blocked density is

in the ordering used there. At , its values are . The mixed sectors have value one, while the two unmixed sectors retain the distortion. The general proof 66.4 now identifies this product with the canonical blocked density outside that particular family as well.