Scalar commutants and corrected tracial reflection

A scalar expectation of each Jones projection gives a way to collapse the tower and recover the original factor as a bicommutant. We prove this for a generating tunnel with trace-preserving finite reflection, including every finite-depth generating tunnel. We then correct the finite reflection explicitly in the nonextremal weighted-spin example of Lesson 47. The correction extends normally, carries a modified lower cup to the upper cup, and supplies the missing scalar identities in that example.

We begin with the Pimsner–Popa construction of a projection with scalar expectation onto the relative commutant. Its generally non-scalar expectation onto the smaller factor explains why the two expectation targets must be kept separate.

We use the local-index formula, Theorems 2.4–2.5; downward construction and recognition, Theorems 4.4–4.5; coherent finite reflection and finite-depth trace agreement, Proposition 14.2 and Theorem 14.4; tracial extension, Theorem 40.1; the actual weighted-spin tunnel, 47.1–47.5; and smooth representations and tower compression, 61.4 and 61.6–61.8. Smooth representations retain all the expected, nondegenerate hypotheses of Lesson 61.

A projection with scalar relative expectation

Theorem 62.1. Every finite-index inclusion of II₁ factors has a projection with the relative expectation and exact smaller-factor expectation in (62.5).

Proof.

Let be II₁ factors of finite index , let , and put . Choose orthogonal minimal projections in with sum one, including every diagonal minimal projection of every matrix block. Set

Here has unit , and its normalized trace is . The isomorphism , , preserves normalized traces. The local formula gives

There exist orthogonal projections , of traces , summing to one. Put . Then .

For each , downward construction for gives a projection and a II₁ subfactor with

The diffuse factor has a projection of normalized trace . Projection comparison in provides a unitary carrying to that projection. Thus commutes with and still has the displayed expectation. The product

is a projection. The are orthogonal, since . They have equal trace in the factor . Choose partial isometries with and , taking . Set . These satisfy and . In particular .

Define

The coefficient matrix is the rank-one orthogonal projection onto the unit vector . Consequently , , and

For the relative-commutant identity, observe that , so . Every element of commutes with all . Tracial cyclicity therefore gives if . Minimality of gives ; if , then

This characterizes . This argument handles full matrix blocks of , including their off-diagonal entries.

For the -expectation, when : pair it with , use , and move to the front of the trace. For ,

Pairing both sides with proves this formula, since the normalized corner expectation preserves . Apply it to to obtain the final formula in (62.5).

The operator is positive and invertible in , because the finite nonzero projections partition one and every coefficient is strictly positive. Its trace is . The equality holds precisely when

These equations say that the normalized trace on the left commutant of agrees with on , by Theorem 2.4 and the matrix-block trace property. Thus the scalar -expectation occurs precisely in the extremal case. When it occurs, Theorem 4.5 recognizes as a downward Jones projection. In general (62.5) alone does not place in the commutant of a specified deeper tunnel algebra.

This construction follows Pimsner–Popa's Entropy and index for subfactors, proof of Theorem 4.4.

Figure 62.1. The numbers are exact for the locally trivial corner inclusion of Lesson 40. The two smaller-factor spectral projections have traces ; the two relative-commutant projections have traces . Their expectation coefficients are (62.1)–(62.7). Editable source.

The trace in the cup functional

Lemma 62.2. The cup functional is given by (62.9). Its relative expectation is scalar precisely under the trace equality (62.10).

Proof.

Suppose is a tracial basic construction of II₁ factors, with consecutive index . Let . Represent on , so . For , write , with . The map is a linear anti-isomorphism, and

Indeed acts by right multiplication by . Its compression to is right multiplication by , since commutes with and is a factor. The normalized trace on satisfies , so

The restriction is unchanged if we replace this representation by the standard representation of , or any normal -representation with finite commutant. Here is the needed representation argument. Finite-module classification makes another such representation a common corner of a finite amplification of this one by a projection . For , the amplified commutes with this factor. The trace-preserving expectation from the amplified to the amplified therefore sends to : its image commutes with the whole factor, and its trace fixes the scalar. Hence the normalized corner trace evaluates as . This proves the assertion, including arbitrary finite amplification and a nonzero common corner.

It follows that

The right side is exactly extremality of . The relative commutant is finite dimensional, so equality on its projections or matrix units already suffices. Formula (62.9) identifies the correct commutant trace before any equality with the inherited trace is invoked.

Every fixed endpoint of a generating reflection

Theorem 62.3. Actual generation (62.11) and finite trace agreement (62.12) give the normal anti-isomorphism (62.13), every endpoint (62.14), and extremality of every lower consecutive inclusion.

Proof.

Write , , and let be an actual Jones tunnel. Let

This is the generating hypothesis. On , use the coherent finite tower of Proposition 14.2. Let be its separate canonical tracial completion. Assume

Set , . Theorem 40.1 extends the specified finite reflection to a normal trace-preserving anti-isomorphism

For every fixed , this same extension satisfies

To prove this, take . The expectation sends onto , because . Density in (62.11) and normality of the expectation show that the union of these images is weakly dense in . Finite reflection (14.3) maps its -th algebra onto . Expectations onto preserve commutation with for and converge in . Their union is therefore weakly dense in . Take the normal closures through (62.13).

Every inclusion , , is extremal. Indeed the normalized trace on , in the coherent representation on , is carried by to the normalized trace on . Equation (62.12) equates it with on , and hence on . The representation argument in 62.2 identifies that restriction with the standard commutant trace for . This is the claimed extremality.

Recovering the initial bicommutant

Theorem 62.4.

Under (62.11)–(62.12), put . For ,

where implements .

The same sufficient test allows modified cups: suppose the actual generating hypothesis (62.11) holds, and a normal trace-preserving anti-isomorphism satisfies . For every , suppose there is a projection with and . Then (62.15) and (62.18) still follow. No scalar expectation onto is needed in this test.

Proof. First . Since commutes with , bimodularity implies that also commutes with , and hence lies in . More generally the restriction of to has range , fixes that algebra, and preserves the trace. It is therefore its trace-preserving expectation.

By (62.14), the anti-isomorphism carries

It carries the lower cup to , by Proposition 14.2. The lower inclusion is extremal by 62.3, so (62.10) gives . Trace-preserving anti-isomorphisms intertwine trace-preserving expectations, as follows by pairing with every element of their range and using tracial cyclicity. Apply this to (62.16). The restriction observation above proves (62.15).

For the modified test, replace in (62.16) by the stated . Its fixed endpoints give exactly the same domain and range in that formula. The stated relative expectation of , transported through , gives (62.15) by the same restriction argument. The rest of the proof uses only (62.15) and the actual generating union.

Now apply Lemma 61.4 to the tower starting at , with and the cup conditions (62.15). The intersection equals for . The stage-collapse proof therefore gives

Since , we have . Let . In each finite , , it commutes with the reflected algebra . Represent this finite relation on . By (62.11), the union of the has weak closure , so the reflected union has weak closure . Commutation is weakly closed for the fixed bounded . Thus commutes with , and . We conclude

This last argument uses only the faithful normal representation of the finite algebra . It does not extend the whole tracial tower normally to . Once (62.18) is known, the remaining initial scalar identity is .

The compatible hypertrace

Corollary 62.5. An actual generating tunnel and the bicommutant identity (62.18) give the compatible projection (62.19) in every smooth expected representation. In particular this holds for every finite-depth generating tunnel.

Proof.

Hypothesis (62.11) supplies increasing finite-dimensional algebras generating . Their expectations onto , or directly the tunnel relative commutants in , generate : for . Both algebras are factors. The explicit projection construction of Theorem 61.8 therefore applies to and , and its finite matrix corners give projections for every . Lemma 61.7 gives the canonical projection .

Combine this projection with (62.18) and Theorem 61.6. Every nondegenerate smooth expected representation , in the precise sense of Lesson 61, has a UCP projection fixing , satisfying

The state has in its centralizer and is -invariant. This proves the smooth-hypertrace direction under the actual generating and trace-consistent tunnel hypotheses.

At finite depth, Theorem 14.4 supplies (62.12). Thus every actual generating Jones tunnel of a finite-depth inclusion gives (62.18) and (62.19), with no additional injectivity hypothesis. The finite-depth result does not supply an actual generating tunnel when one has not yet been constructed.

For a general nonextremal generating tunnel, Theorem 47.5 shows that (62.12) can fail. Theorem 62.1 constructs a scalar relative expectation, but its prescribed deeper commutation and a general trace-preserving shifted comparison still require proof. The modified-cup criterion in Theorem 62.4 states the exact input needed.

Figure 62.2. The arrows are normal trace-preserving anti-isomorphisms with the stated endpoints. Equations (62.15)–(62.18) first collapse the tower above , then recover using the actual generating union in its finite representation. Corollary 62.5 supplies the compatible hypertrace. Diagram positions are schematic. Editable source.

Correcting reflection in the weighted-spin tunnel

Theorem 62.6. For every with , the finite corrected maps (62.23) extend to the normal trace-preserving anti-isomorphism (62.24). The modified cups (62.25) implement (62.28) and satisfy (62.26)–(62.27). This inclusion has the bicommutant identity (62.18) and compatible smooth hypertraces.

Proof.

Use the actual weighted spin inclusion of Lesson 47. Its parameters are , , , . Its finite balanced-word algebras use spin weights . Number the sites from zero. Write for their shifted tracial closure on sites . The tunnel is , and its finite relative commutant

Its matrix units are indexed by binary words of length with equal numbers of ones. A minimal word projection , with ones, has inherited trace .

For every such ,

To check this on a dense finite algebra, multiply a balanced word matrix unit on both sides by . Its first letters on both sides must be . The remaining two words have equal charge and give a matrix unit of . Take normal closures. The local corner index is thus one. Theorem 2.4, with , gives for the normalized trace of the left commutant of

Let , where a bar complements every letter. This is a unital *-automorphism of . It is compatible with the embeddings , and preserves the position of every tensor site. Finite reflection pulls the upward tower trace back to . Consequently

is a compatible trace-preserving anti-isomorphism. Indeed (62.22) evaluates as , and normalized traces on every finite matrix block are determined by their values on minimal projections. On the dense finite unions, is therefore a well-defined isometry onto a dense subspace. Its unitary extension carries right multiplication by to left multiplication by . Conjugating the weak closures gives a normal anti-isomorphism from onto , exactly as in the tracial-extension proof of 40.1. Thus

For the second equality, preserves the subalgebra supported on sites ; repeat the fixed-endpoint density proof of 62.3. The extension in (62.24) is the composite of the finite maps before completion. It does not require a normal extension of or of the specified separately. When , already differs from .

The original lower Jones projection lies at sites , with its nonzero block

Define the modified projection on the same two sites by

All other word blocks are zero. It belongs to . The local relative commutant is the diagonal algebra at site . The inherited trace-preserving expectation onto it averages the site , whose weights are . Its two diagonal values at are and . Thus

Complementing both sites gives , and finite tower reflection gives . Hence

For completeness, implements a faithful normal expectation , whose state on the first site has the reversed weights . With the two diagonal projections at that site, put

The positive invertible commutes with , and its inherited tracial expectation onto that factor is one. This formula therefore defines a faithful normal UCP conditional expectation. On finite words it is exactly partial state with weights . Rank-one compression on the block gives

For a finite tensor, this is the reduced state of the vector on its second site; its second-site probabilities are and its off-diagonal expectations vanish. Linear extension and normality give (62.28). The positive invertible element

satisfies . Its inverse gives . Thus together with generates the same upper factor . The expectation preserves the inherited trace only when .

The modified-cup test of 62.4 now applies with : (62.24) gives its fixed endpoints, (62.26) gives its relative scalar condition, and (62.27) gives the cup images. It follows that for , , and . The injection argument of 62.5 uses the actual generating tunnel, so every smooth expected representation of this explicit weighted-spin inclusion has the compatible hypertrace (62.19).

In particular (62.24) gives a normal trace-preserving anti-isomorphism of its downward and upward pairs at this infinite depth, with the modified finite maps (62.23). It is consistent with Theorem 47.5, which refutes extension of the different specified maps . No construction for all nonextremal standard invariants is inferred from this example.

Figure 62.3. All displayed weights and coefficients use . The table gives minimal-projection weights, whose block multiplicities are . Complementation exchanges the charge blocks and corrects their reflected traces. The two cup matrices and their expectations are (62.25)–(62.28). The fixed-endpoint extension is (62.24). Editable source.

Exercises with complete solutions

Exercise 62.1 — local budgets (basic)

Take the locally trivial corner inclusion of Lesson 40, with and . Compute , the common trace of the , and both expectations of the projection in Theorem 62.1.

Solution. The local formula gives , so . Then . Both have trace . The resulting projection has and , with . Its trace is . The smaller-factor expectation is invertible but is not scalar.

Exercise 62.2 — a full matrix relative commutant (intermediate)

Let be a II₁ factor, , and , with normalized matrix traces. Put . Compute , its trace, and its expectations onto and .

Solution. The first two legs of are the rank-one projection onto . Multiplying its matrix units gives , since the middle sum has equal contributions with coefficient . The normalized trace is . Partial normalized trace on either matrix leg gives times the identity on the other. Since , both requested expectations are . The index is , by the tensor and module formula. Theorem 4.5 therefore recognizes as a downward Jones projection. At , each expectation is .

Exercise 62.3 — the trace in the original cup (intermediate)

For , let be the first-site zero projection and the original lower cup of Lesson 47. Compute using Lemma 62.2 and the normalized commutant trace. Compare it with .

Solution. Here , , and the normalized commutant trace is . Thus , while . The relative expectation of has coefficient on and on . Averaging it with the inherited first-site weights gives .