Canonical rescaling of Jones cups

A Jones cup can be rescaled inside its actual basic construction so that its expectation onto the upper relative commutant becomes scalar. The resulting projection retains its deeper commutation and implements a faithful normal expectation with the original index. We prove the density formula, its inverse under finite duality, and all Jones relations for the modified cups throughout a prescribed tunnel. The general trace-preserving shifted comparison remains a separate requirement.

We use the local-index formula, Theorem2.4; finite common bases, Lesson3; the tracial basic construction and Markov expectation, Lesson4; normal finite expected constructions, Theorem61.2; and the cup functional and representation independence, Lemma62.2. Every representation used below is a faithful normal representation of a finite factor.

The two traces and their central density

Proposition 63.1. The normalized commutant trace has the central density (63.3)–(63.4), and the original cup has relative expectation (63.5).

Proof.

Let be II₁ factors of index , let , and write

We represent on . Put

This is a linear *-anti-isomorphism , represented by right multiplication. Define

It is the restriction to of the normalized trace of the commutant of left on . Indeed ; the anti-isomorphism between these finite factors preserves their unique normalized traces.

Both and are faithful normalized traces on the finite-dimensional algebra . Therefore there is a unique positive invertible central element with

To construct it explicitly, on each matrix block choose a minimal projection , put , and put . The local formula 2.4 gives . Hence the coefficient of on that block is

Equivalent minimal projections in a matrix block have the same two trace values, so this coefficient is independent of the choice. Formula (63.3) follows on every matrix unit. We may instead list all diagonal minimal projections in all blocks; this convention will be used for sums below.

The cup functional 62.2 gives . Pairing with every and using (63.3) now yields

For example, . The normalized trace of the commutant of on is under , whereas the inherited -trace is under . These traces are identified only when , namely when is extremal.

Rescaling inside the actual basic construction

More generally let be positive invertible with . Define

Theorem 63.2. is a faithful normal UCP conditional expectation onto . The operator is an actual projection in , of trace , satisfying

Its two inherited tracial expectations are

In particular has scalar -expectation . It retains the deeper commutation and implements the generally nontracial expectation .

Proof. Since commutes with , is central in , and its trace makes it . This proves that is unital, fixes and is -bimodular. Normality and complete positivity follow from its formula. If and , faithfulness of gives ; invertibility gives .

The basic relation gives . Self-adjointness is immediate. Both factors in its formula commute with , so . The Markov formula gives and the first identity in (63.8).

For , compress by . Because commutes with and , this gives (63.7). The bounded inverse gives , so the generated upper algebra is exactly .

For , right multiplication by and left multiplication by agree on , since commutes with . Taking adjoints of this relation shows

Bimodularity of , followed by (63.5), proves its formula in (63.8). Although reverses products, commutes with the central , so no order issue arises in that formula.

The expectation also has the useful pairing expression

For each , cyclicity and commutation with make the three pairings with equal. Faithfulness of the -trace proves the equality. This does not assert that is positive.

Finally . Thus preserves precisely when . The scalar relative expectation in (63.8) occurs precisely when . Consequently the two properties hold together precisely in the extremal case.

A finite basis and the unchanged scalar index

Theorem 63.3. The rescaled expectation has the common basis (63.14) and exact positive index (63.15). The canonical choice retains the original index. The minimum within the stated rescaling family is (63.16).

Proof.

Choose a finite common basis for the original tracial :

No orthogonality of this basis is required. For , set .

The value is central in . To see this for , put , and expand with (63.11):

The second equality uses ; the last equality is the adjoint reconstruction identity for . Hence is scalar.

Its scalar value is

Indeed commutes with every . Apply to (63.11) times . Cyclicity and compression by give

For the compression identity used here, if , then ; it follows first on and then by continuity on .

Now put . Equation (63.10) and commutation of with give

For example the reconstruction sum equals . Taking adjoints gives the left reconstruction identity. Thus the finite normal expected basic-construction argument of 61.2 applies, with scalar index

This is the scalar common-basis index. The same finite coefficient/Schwarz argument of 61.2 gives for . Here is an actual witness proving optimality without assuming a tracial . Downward construction 4.4 gives and a cup with . In its finite representation on , the cup compression of is the normalized -trace times . Representation independence in 62.2 identifies that trace with , so . Thus

is a nonzero projection in , and . Any inequality for all , applied to and compressed by it, forces . This proves the exact positive index as well as the common-basis index.

For the canonical choice , (63.3) yields . Hence the canonical modified expectation has precisely the original scalar index. Its trace density may be nonconstant even though its index is unchanged.

Within this positive-invertible rescaling family, the smallest index is

Apply the finite tracial Cauchy–Schwarz inequality to and ; their trace pairing is , and . Equality holds precisely for the displayed proportionality. The final formula follows from (63.4), listing every diagonal minimal projection in every block. This is a minimum over the stated rescaling family; no classification of all normal expectations is imported here.

Adjacent densities are inverse under finite reflection

Proposition 63.4. The two adjacent densities satisfy (63.17).

Proof.

For the dual inclusion , let be the density of its normalized commutant trace relative to . Then

The finite representation of on has normalized -trace under . By the representation-independence proof of 62.2, its restriction to is the same normalized commutant trace used to define in the standard representation of . Its inherited trace is under . Equations (63.3)–(63.5) therefore prove (63.17) on every element of .

In particular this is a statement about each finite dual inclusion. It does not extend the infinite tracial tower normally to a fixed -space.

Canonical modified cups throughout a prescribed tunnel

Theorem 63.5. Every prescribed Jones tunnel has the actual modified cups (63.18), with all properties (63.19) and every Jones relation (63.20).

Proof.

Let be any actual Jones tunnel of consecutive index . Its cup , for , implements the tracial expectation of onto . Let be the density of 63.1 for the inclusion . Define

Theorem 63.2 and (63.15) prove, without changing the tunnel factors,

They also satisfy the same adjacent and distant Jones relations:

Proof of (63.20). Represent the finite triple on . Its cup is . Formula (63.17) says . As in (63.9), right and left multiplication by agree on the cup range. Thus

The first equality is (63.10); the last uses the tracial Markov expectation for this finite triple.

The projection commutes with , so it commutes with . Compressing by (63.19) gives

For the reverse relation, has and . Both projections have inherited trace in the same finite upper factor. Faithfulness of its trace gives , proving the second identity. If , then , which commutes with ; this gives distant commutation.

No actual generating hypothesis or extremality was used in 63.1–63.5. The placements and scalar relative expectations needed in the modified-cup criterion 62.4 are therefore available for every prescribed tunnel. Applying that criterion in the general nonextremal case still requires the normal trace-preserving comparison with every fixed endpoint and the stated cup images. Neither (63.19) nor the Jones relations alone supplies that map.

Exact worked examples

1. Weighted spin. In the lower relative commutant at the second site, give , , and . On the basis , the second-site has coefficients . Conjugating the old cup with diagonal gives the modified cup with diagonal and the same off-diagonal . The first-site relative expectation is , while the second-site inherited expectation is . The implemented expectation has reversed weights , and its scalar index remains .

2. Original versus canonical versus minimizing density. In the same inclusion, gives the original trace expectation of index . The canonical also has index , now with scalar upper relative cup expectation. The minimizing density is , since ; (63.16) gives index . Its scalar relative cup condition fails: in the reflected order , (63.8) gives cup-expectation coefficients . Thus minimal index and scalar relative cup are distinct requirements.

3. An extremal matrix example. For , with a II₁ factor, , , and all diagonal minimal have . Formula (63.4) gives . Thus the canonical cup is the original cup, and both expectations are . Formula (63.16) also gives ; there is no smaller index within this rescaling family.

4. Noncommutative rescaling inside an extremal block. In the preceding example take , and let have eigenvalues in . Its normalized trace is one. Formula (63.15) gives . In the maximally entangled cup model the rescaled rank-one vector has coordinates proportional to ; the actual normalized cup trace stays . The upper relative expectation is , with eigenvalues . Positivity, finite index and the deeper commutation all survive, although neither expectation-preserved trace nor the scalar relative condition survives. Conjugating by any unitary of gives the same calculations, showing that need not be central.

Exercises with complete solutions

Exercise 63.1 — the two density columns (basic)

In the weighted-spin example, put , and label the lower second-site projections . Compute , the original cup's expectation onto in the order , and both expectations of the canonical cup.

Solution. The two traces give . Since , formula (63.5) gives original coefficients in the reflected order. The canonical cup has , while . Under the actual spin reflection is the upper first-site . Thus the original expectation in first-site order is , consistently with Exercise62.3. The labeling accounts for the apparent reversal.

Exercise 63.2 — normalization makes a projection (basic)

Let be positive invertible, without assuming its trace is one. Determine when is a projection, and compute its inherited trace.

Solution. , because commutes with the factor . Hence . Invertibility of and make , so it is a projection exactly when . The Markov trace gives . With normalization this is , even when is not tracial.

Exercise 63.3 — three different expectations (intermediate)

For the same spin example, write , where . Compute . Identify the tracial, scalar-relative-cup and minimizing choices of .

Solution. The normalization is . Using , formula (63.15) gives

The original tracial choice is , of index . The canonical relative-scalar choice is , also of index . The minimum occurs at , either by (63.16) or by ; its index is . Its cup has -coefficients in the reflected order and therefore fails the scalar-relative test.

Exercise 63.4 — a noncentral density and the sharp witness (advanced)

In , with normalized trace and , put

Use the actual inclusion , of index . Compute , and the value of on the optimal-bound witness in Theorem63.3.

Solution. has positive eigenvalues , and

Its eigenvalues are , and its normalized trace is one. The normalized trace of is . Thus . For any downward cup , the witness is , a nonzero projection, and . This forces the upper bound on every possible positive-index constant. The canonical density is still ; this noncentral is an allowed rescaling but is not the canonical choice.

Exercise 63.5 — inverse dual density and an adjacent cup (intermediate)

Take . Find in reflected order, verify its inherited trace is one, and compute when has those coefficients and .

Solution. Equation (63.17) gives . The inherited weights of these reflected projections are ; their weighted sum is . Equation (63.21) gives . Multiplication by on both sides cancels its inverse density and yields , which is precisely the cancellation in the adjacent Jones relation.

Exercise 63.6 — placement and comparison (advanced)

Explain which hypotheses of the modified-cup criterion62.4 are supplied by Theorem63.5, and which still require a separate proof for a general nonextremal generating tunnel.

Solution. For every , Theorem63.5 gives an actual projection , with the stronger commutation , and . It also proves the actual upper factor is unchanged. To use62.4, one still needs a normal trace-preserving anti-isomorphism with every fixed endpoint and , together with the actual generating hypothesis. Neither the Jones relations nor the scalar relative expectations alone establishes those maps. Theorem62.6 supplies them for its explicit weighted-spin family; Theorem47.5 still refutes the different specified unmodified reflection in that family.

Sources and exact scope

Figure 63.1. The triple and operator mechanisms are proved in 63.1–63.5; positions are schematic. The table uses the actual weighted-spin inclusion in the first worked example and identifies its lower relative commutant with the second site. Its inherited trace, normalized commutant trace and density are distinct columns. The upper first-site relative expectation of the modified cup is scalar, while its expectation onto has the displayed second-site coefficients. Reproducible source.

The human source for the modified-projection requirement is Sorin Popa, Classification of amenable subfactors of type II, Section4.5.1, printed p.224. The projection construction there is credited to Pimsner and Popa, Entropy and index for subfactors. Propositions63.1 and63.4 and Theorems63.2–63.3 and63.5 give the full finite rescaling proof using Lemma62.2. Popa cites a separate [Po12] comparison for the shifted trace-preserving maps; the cup construction does not supply those maps.

The modified cups now exist in every prescribed tunnel. To infer the general bicommutant conclusion from Theorem62.4, one still needs a normal trace-preserving comparison with every fixed endpoint and the required cup images. The Jones relations alone do not prove that comparison.