Deleting a finite matrix factor from a core changes the Jones projection and enlarges the canonical basic construction. The matrix factor has dimension as a Hilbert space, although it acts on that space as a left algebra of degree . This distinction gives the exact trace multiplier.
We prove the multiplier for both canonical algebras, identify their centers as actual operator algebras, and give the full integer-rounding estimate when the relevant central projections commute with the tested unitaries. In particular this proves the rounding conclusion when the smaller core is a factor. The general nonfactor case still requires a justified localization argument. Full support from a factorial larger core, Theorem 58.7, separately proves full-support rounding when the larger core is a factor, by central balancing and trim-and-fill construction.
Let be a proper finite-index II₁ inclusion, , with tunnel and notation from (51.1)–(51.3). Put
Lemma 52.1 — a common bounded basis. There is a finite partial orthonormal right basis which is simultaneously a basis for over , for over , and for over . In particular
Thus inside is a nondegenerate commuting square.
Proof. The faithful left-end Markov identification in Proposition 50.5 and Lemma 51.2 identifies these cup tails with the path-model tails. Theorem 11.5 gives , including the endpoint .
The expectation fixes every later cup and sends to . Reversed word reduction, as in Lemma 11.3, shows that it sends the polynomial algebra of into that of . Normality then gives ; its trace-preserving restriction is .
Choose the bounded basis from Theorem 3.2 for this factor inclusion. Write its support projections as , so that
In the ambient basic construction , the are partial isometries with mutually orthogonal final projections. Their range sum is a projection below . Its canonical trace is
Faithfulness of this finite trace gives . Acting on therefore gives
For , Lemma 51.1 puts all these coefficients in , giving the expansion. The same inner products and supports give orthonormality in both cases. The belong to , so the first expansion makes the linear span equal to ; adjoints give . The expectation identity is Lemma 51.1.
The square's nondegeneracy is a spanning assertion, stronger than the fact that its four algebras generate one another.
The two centers have different corner labels
On , let and put
Proposition 52.2a below identifies normally with by physical reduction. We first prove the corner information and full central support that this identification uses.
Lemma 52.2 — center and finite-projection trace conventions. The projection has full central support in both algebras, and
The corner maps identify with and with . Under these maps,
In particular the entire center of need not commute with .
For either algebra , let be its canonical trace, normalized by the inherited trace on the full -corner. On its center use the faithful normal finite measure
For a finite-trace projection , its generalized central trace is the positive density determined by
Thus ; this is a dimension normalization, rather than a normalized trace of .
Proof. The expectation identity makes the expectation projections commute; taking adjoints gives . Jones compression therefore gives . The span of is a *-algebra, and its range contains , dense in by Lemma 52.1. Its norm closure has a contractive approximate identity converging strongly to . Its strong closure contains and , and is . Compression of its spanning operators gives . The same basic-construction argument gives the assertion for . Density of their -generated ranges gives full central support. The full-corner center theorem now gives both center identifications. Since are type II by Proposition 50.5, their full-corner canonical algebras are type II as well.
If also commutes with , it commutes with both generators of , so . Its two corner labels , satisfy . Left multiplication by on is faithful, hence .
Conversely let . Right multiplication belongs to and has -corner , since is central in . Let have the same corner . Both operators commute with and agree on ; they agree on the dense , hence on . Therefore . This proves (52.5).
Normal trace extension from the full corners gives the scalar traces. The restriction from to has the same -corner trace and equals by uniqueness. Equation (52.6) is the inherited center measure of or , and is faithful with value one at .
For finite-trace , is a normal positive functional on the abelian center. Its density relative to the faithful gives (52.7). Equality and inequalities of these densities can be tested against all positive central . Additivity, central cuts, and invariance under equivalence follow from the corresponding scalar trace identities.
For outside , its lift to consequently cannot be replaced by . The latter operator has no reason to belong to or have the required -corner.
The canonical expectation is a physical reduction
The expected canonical pair can be constructed directly from the common bounded basis, including when the core has a center. We use the common basis of Lemma 52.1 and the full corners and full central support proved in Lemma 52.2. The full-corner trace construction needed here is included below; the same finite-partial-sum argument is also Lemma 1.2a, BC1–BC4. The finite physical expectations and their tracial Hilbert projections are proved in M1–M9. The expectation will be the composition of an actual physical compression and a normal inverse map.
Proposition 52.2a — the normal expected canonical core pair. Let be a finite-index inclusion of II₁ factors with normalized physical trace . Let be its inherited-trace core for any ordinary tunnel, as in Lemma 52.1. On , put
There is a normal faithful conditional expectation
The canonical traces are the faithful normal semifinite traces normalized on the full corners by
and
.
The expectation is unital and completely positive, fixes , and is -bimodular. It satisfies
No factoriality of or , separability, sigma-finiteness of the canonical algebras, or finite ambient index is required. The reducing projection belongs to ; its membership in or is neither assumed nor needed.
Proof. Lemma 52.1 gives . As orthogonal projections on the physical tracial Hilbert space,
Indeed , since ; taking Hilbert-space adjoints gives the other order. The left copy of reduces , so commutes with both generators of . Thus restriction defines a unital normal *-homomorphism
Its range is contained in
on : compress the ultraweakly dense spanning algebra from Lemma 52.2. On the full corner, the same lemma gives
The latter map is faithful: . The kernel of the normal homomorphism is an ultraweakly closed two-sided ideal, hence for a central projection . If , then ; faithfulness on gives . Since has full central support in , . Therefore is faithful.
Its image is a von Neumann algebra. To retain the topological point explicitly, faithfulness makes isometric, and normality makes its map on the ultraweakly compact unit ball of continuous. Its image is the entire unit ball of its range and is ultraweakly compact, hence closed. The weak closure of the range has the same unit ball, by the unit-ball density form of the bicommutant theorem. Thus the range is weakly closed. It contains the left and , and is therefore exactly .
The inverse is normal as well. For a bounded increasing positive net in , let in . Normality of gives , so
.
This proves normality without a countability assumption.
We next prove that compression of every operator in lands in this range. Define the normal UCP compression
For and , (52.7d) and physical expectation bimodularity give
Consequently . Let be the finite common partial right basis of Lemma 52.1. For any , its right expansion and its adjoint expansion give
Every commutes with , since left multiplication by reduces . Hence, writing ,
These are finite sums of bounded operators. The spanning algebra is ultraweakly dense in , by the basic-construction ideal argument used in Lemma 52.2. This remains true at infinite : its range contains , dense in , and its contractive approximate identity converges strongly to one. Since is normal and is ultraweakly closed, (52.7h) proves
.
We can therefore define the actual expectation by
Both maps have the displayed actual operator domains. The normal *-isomorphism and the normal compression make the composite normal and UCP. For , , so it fixes . Because commutes with ,
; this proves bimodularity directly. For , compression on is its physical expectation:
.
Since fixes the left , the restriction is exactly . The formula for also proves (52.7c).
We first construct the full-corner traces used in (52.7b). Let be a von Neumann algebra with full central support projection , whose corner carries a faithful normal finite trace . Choose a maximal family of partial isometries with and mutually orthogonal finals . Then . Otherwise the nonzero remainder has : if this corner were zero, , contradicting full central support of . The adjoint of the polar partial isometry of a nonzero element in would extend the family. All sums mean nets of finite partial sums, and the family can have any cardinality.
For , define
Additivity and positive homogeneity hold term by term. For a bounded increasing positive net , normality of each corner functional and commutation of the two scalar suprema give
.
To check traciality for , put . Insert the increasing sums of the between and , and use normality of . This gives
All double sums have nonnegative terms, so reordering requires no countability or finite-value assumption. Thus this is a normal trace weight. If its value on is zero, faithfulness of gives for every ; their final sum is one, so .
For , the operators lie in the corner. Corner traciality gives
, and normality of gives . In particular, for finite ,
For every bounded , the positive operators increase to , and their traces equal those of , which are at most . This proves semifiniteness in its supremum formulation. Finally, every normal trace weight restricting to on the corner satisfies
It is therefore the weight (52.7j.1). This proves uniqueness and independence of the chosen family. Apply this complete construction to with , and to with , to obtain the two canonical faithful normal semifinite traces. The restriction of the latter trace to is also the former trace: a family with finals summing to one chosen in can be used in , and its diagonal corner values on agree. This proves the restriction identity without assuming the desired expectation.
It remains to prove that the expectation preserves these entire traces. Choose such a family ,
For finite , has finite trace, at most , in either canonical algebra. The same family also lies in and its final sum is still one. Thus the proved formula (52.7j.1) applies to both canonical algebras with this same family.
For , the full corner gives unique such that
.
Bimodularity and (52.7c) imply
.
The two full-corner trace formulas now read
Every summand agrees because the inherited physical expectation preserves . Their suprema agree, including the value infinity. These formulas concern nets of finite scalar sums. One need not, and does not, claim that is increasing. The normal trace formula instead uses the increasing positive net , as proved above. This proves (52.7b) for the whole positive cone. If for , trace preservation and faithfulness of give . Thus the expectation is faithful.
The full-corner center comparison used in Lemma 52.2 has an equally direct form. For , define
.
The terms have orthogonal final supports, so their finite sums are bounded by and converge strongly and strong-adjointly. For , the matrix coefficient lies in , and hence commutes with . The identities
show that commutes with every , since the sum to one. Moreover , so
.
A central operator with zero -corner annihilates , and fullness makes it zero. Thus compression has this bounded inverse and is a *-isomorphism. Compression is normal, and its inverse preserves bounded increasing suprema by the order argument used for , so both maps are normal. This proves the full-corner center comparison.
The proof uses only the finite common basis, the physical commuting expectations and the full corners. It therefore applies also to any finite commuting square having that same common basis, and to the higher actual core squares with their finite common bases. It provides the normal expected pair needed before the relative Følner criterion and the finite expected-tower construction are applied.
A concrete nonfactor canonical square
Let be any II₁ factor, let , and use normalized matrix traces. In
, set
where is the diagonal algebra. This is an actual nondegenerate commuting square with a common finite basis: the elements
are a right basis for both and .
It is used here as a canonical-square example. Its presentation does not assert an ordinary-tunnel core realization for arbitrary .
Suppressing the standard -Hilbert factor, identify
with its left row coordinate and right column coordinate. The projection onto is the sum of the coordinate projections onto . Thus
Indeed the left and the projection onto generate exactly
; adding the left gives the larger algebra. Both centers therefore contain the genuine . On each column summand, is in and in . The expectation and canonical traces are
These normalizations give both traces of equal to one, and both traces of the full unit equal to . They agree on every -element embedded by tensoring with . The physical algebra is represented by the same left operator on each column summand, so the restriction of this blockwise expectation is the original physical .
In this example reduces . For , it does not belong to . If it did, restriction to one column summand followed by normal state slices on and the left would put in the left -algebra. A nonzero left -projection on has rank a positive multiple of , whereas has rank one. The construction above uses this genuine external reduction, rather than a corner projection presumed to lie in the canonical algebras.
Figure 52.2a. The top row gives the actual operator domains in (52.7a), (52.7e), (52.7g) and (52.7i). The blue projection is a reduction in ; belongs to both canonical algebras and has full central support. The middle row records , and the physical expectation , which prove the two full trace values in (52.7j). The lower row gives the nonfactor square (52.7k)–(52.7m) at : each small block has trace coefficient , each large block , and the two ranks of are one and three. Boxes and arrows describe algebras and maps, not metric geometry or trace-sized areas. Reproducible figure source.
The human context is Sorin Popa, Classification of amenable subfactors of type II, Acta Mathematica 172 (1994), 163–255, Example 2.3.3(a), for the canonical core representation. The explicit normal compression and full-corner trace argument above uses the earlier programme proofs at their stated domains. This proposition and diagram are dedicated to CC0 1.0.
The old and new constructions inside one tensor model
Let be a prescribed unital matrix factor in . Put
Matrix decomposition identifies all four algebras with their complements tensored with the same , with product traces. Corollary 51.5 realizes as a core for the original .
Let
Theorem 52.3 — the actual amplification. On ,
Write for the left action of on . The old and new canonical pairs are
Their centers agree within each row as actual operator algebras:
The smaller and larger centers are still distinguished by Lemma 52.2.
On the old algebras, both scalar canonical traces and both generalized central traces scale by :
for every old finite-trace projection in the indicated algebra.
Proof. The old square is nondegenerate. Stripping the common finite matrix factor from its spanning identity shows that inside is nondegenerate as well: in a matrix expansion of the spans, the identity matrix coefficient gives the smaller spanning identity. Its expectations are the first-leg restrictions of the old product expectations.
Under product identification, expectation onto is , whereas expectation onto is . Their range projections are exactly (52.10). The old generating algebras therefore have the first two forms in (52.11).
For the new algebras, the products , , span all rank-one operators on : they send to . Hence the new smaller algebra contains . Summing the rank-one projections associated with the orthonormal basis gives . Thus it also contains .
The projection is full in by Lemma 52.2. A contractive approximate identity in the ideal generated by converges strongly to . Multiplying its finite spanning expressions by any second-leg operator shows that belongs to the new smaller algebra. This proves its asserted equality. The identical argument with proves the larger equality. Both second-leg algebras are factors, giving (52.12).
Let be the canonical trace on , and let be ordinary Hilbert-space trace, with value one on . Full-corner trace uniqueness gives
In the first line the second trace is on ; in the second it is on . Their restrictions to the respective Jones corners are exactly the inherited traces of and .
The left action on is copies of the defining -dimensional action, so
for . One can check this directly: left multiplication by fixes the vectors and kills the others, and off-diagonal units have zero Hilbert-space trace. Equation (52.15), normality, and the product trace formula prove scalar scaling, including on .
Finally the center measures (52.6) are unchanged. In the smaller row both are the measure obtained from on : the old corner uses normalized matrix trace, and the new corner uses rank-one trace. In the larger row the same assertion uses . Testing (52.7) against central elements now proves both central-trace identities in (52.13).
The multiplier rescales numerator and denominator of every old relative commutator estimate by the same factor . It changes a dimension value without changing that relative estimate.
Figure 52.1. The old second-leg algebra is the left -action on an -dimensional Hilbert space. The new second-leg algebra is its entire operator algebra . The old Jones projection uses its identity; the new projection uses the rank-one vector . The two smaller centers coincide through the change, but only the part corresponding to commutes with every -unitary. Proof locators: (52.5) and (52.9)–(52.15). Reproducible source: canonical-trace-scaling.py.
Prescribing a central dimension inside a finite projection
We need a projection with an exact integer dimension, not merely a scalar-trace approximation.
Lemma 52.4 — central prescription and equivalence. Let be either canonical type II algebra of Lemma 52.2. If has finite trace, write . For every measurable central function with , there is with .
If finite-trace projections have the same generalized central trace, they are equivalent in .
Proof. The finite projection has a finite type II corner. Its center is , represented by . Write for its normalized center-valued trace. Finite trace factorization in that corner and (52.7) give
where in the identified center. The density is finite almost everywhere and positive on ; its zero central support kills by faithfulness. Thus is a bounded central function between zero and one on that support.
Successively halve the remaining part of . This gives orthogonal projections , , with
Each halving is into equivalent pieces, so its center-valued traces are equal. The remainder after steps has center-valued trace ; normality and faithfulness make its strong limit zero.
Choose the Borel binary digits of , as central projections, so that . At use all digits one. The central cuts are orthogonal projections. Their strong sum satisfies by normality, and (52.16) gives the required .
For equivalence, let . The parallelogram law gives
Indeed . In the finite corner , formula (52.16) with divides both central densities by the same positive . The normalized center-valued traces of are equal, so Corollary 5.4 supplies equivalence in that corner.
In particular, if , where is a positive integer and is a projection, split successively into orthogonal projections of central trace . Lemma 52.4 makes each equivalent to . The corresponding smaller-core label is in ; membership also in is an additional condition governed by (52.5).
Integer rounding with a verified central localization
Theorem 52.5 — rounding on commuting center blocks. Let be a core of , and let , . Assume
If the relative Følner criterion holds for this core, then for every there is a core complement obtained by deleting a finite matrix factor, and a nonzero finite projection , such that
for one positive integer and one nonzero . Moreover is a sum of projections equivalent in to .
Proof. In the larger canonical space, abbreviate the individual and summed defects by
Obtain a nonzero finite projection by Theorem 49.2, choosing the tolerance . Then
Let , a positive integrable central density, and choose
By dominated convergence,
.
Choose so that the lost mass here is sufficiently small. For this fixed , integrability also gives
as .
Choose a finite so that the combined lost mass is less than .
Lesson 50 supplies a unital in , and Corollary 51.5 realizes its complement as a core. Theorem 52.3 identifies the new center with the old one and replaces the central density by . Define in this center
Both discarded tails were controlled, so
These inequalities and (52.19) hold with the new scalar trace, by its common multiplier. The triangle inequality in the direct sum of spaces gives
Partition into the finitely many spectral pieces on which
Use integer half-open bins, including the last endpoint in its appropriate bin. All these projections commute with every by (52.17) and the exact center equality. Therefore
First sum the defects for all unitaries, then select one with nonzero and
Otherwise summing the opposite inequalities would contradict (52.23).
Lemma 52.4 gives with . Set . Equation (52.24) gives
Here , so . Writing , we have , and orthogonality gives . For every ,
The definition of ensures both and . The splitting and equivalence conclusion follows from Lemma 52.4.
Corollary 52.6 — the factorial smaller-core case. If is a factor and is amenable relative to , the integer-rounded Følner conclusion (52.18) holds for every finite set of -unitaries, with . Thus all its cyclic summands are equivalent to the new Jones projection itself.
Proof. The full -corner identifies with , so (52.17) is automatic. Its new center is also scalar by (52.12). A nonzero central projection in it is . Apply Theorem 52.5.
Why multiplying a defect by a central block is a separate step
For central , the identity
holds whenever . It is the orthogonality of right multiplication by projections in . It does not identify its summands with unless an additional commutation or another argument is supplied.
The following actual nondegenerate square shows the distinction even with type II canonical algebras.
Example 52.7 — small total defect, large defects of both spectral pieces. Let , and let be the product of conjugation by the flip matrix on every site. This is the outer order-two action proved in lesson 18. Let be a hyperfinite II₁ factor and set
Write for the order-two implementing unitary, for the first-site diagonal minimal projections, and
The relations , give . Its expectation restricts on to , by the orthogonality of Fourier coefficients and of the first-site diagonal units. Also , since . Thus this is a nondegenerate commuting square inside the proper index-two II₁ inclusion.
For its canonical pair , the smaller center has projections with corner labels ; the larger algebra is a factor. Conjugation by exchanges . Indeed and , so it normalizes ; on its full -corner it exchanges . Uniqueness of the center lift proves the assertion.
Let be the first sites, , and let be the projection onto the closure of in . To see its membership and dimension explicitly, write , . The right -basis of consists of
Its expected inner products are orthogonal, with support . The sum of the corresponding projections is . There are basis elements at each support, hence
The subspace is -invariant, since and , so .
Take , with the other sites identities and . Then , and its inner products with (52.31) have zero expectation because beyond the prefix. Thus
Its conjugate is orthogonal to and has support , so it is orthogonal to . Put . Then
Its total relative defect is , tending to zero. But each spectral piece is carried into the opposite central support, so
Consequently (52.29) alone cannot select a spectral piece with a small full conjugation defect.
This square is not asserted to be a core: its lower centers are and , whereas index-two Jones cores are factors by the path-invariant result. Example 52.7 refutes the unrestricted localization inference for canonical expected pairs, not the existence conclusion of Popa's Theorem 4.2.2. The course still owes a proof of the general nonfactor-core rounding conclusion or a precise, proved correction to that conclusion.
Sorin Popa's Classification of amenable subfactors of type II, Theorem 4.2.2 supplies the multiplier and then passes from localized right-multiplication defects to defects of the spectral pieces. Theorem 52.3 now proves its multiplier in the actual canonical constructions. Theorem 52.5 writes a complete version with the commutation needed for (52.25). Lemma 52.2 and Example 52.7 identify the precise inference still requiring a core-specific justification in the unrestricted case. The printed statement labels the cyclic corner projection by , whereas the next corollary uses ; (52.4) identifies the smaller algebra's corner center with . No unrestricted theorem refutation is inferred from these proof and notation issues.
Exercises with complete solutions
Exercise 52.1 — where the square comes from (basic). For , compute the ordinary Hilbert-space trace of on . Compute its normalized algebra trace in , and explain their ratio.
Solution. The nine matrix vectors are indexed by . Left multiplication by fixes the three with , so its Hilbert-space trace is . Its normalized algebra trace is . Their ratio is . The rank-one projection onto has Hilbert-space trace and belongs to the new -leg, rather than to the old left -leg.
Exercise 52.2 — a prescribed fraction in a central corner (intermediate). On a central piece let . Construct with from the proof of Lemma 52.4.
Solution. In the finite corner , take the central target on , zero elsewhere. Its binary expansion is , since
Choose the disjoint halving pieces exactly at the repeating digit positions , and sum their -cuts. Their normalized central trace is on . Multiplication by the original dimension gives .
Exercise 52.3 — one piece for all unitaries (intermediate). Two central pieces have equal nonzero mass. One unitary has squared relative defects on the pieces; a second has . Why does choosing a good piece separately for each unitary fail? What does (52.25) actually use?
Solution. The first unitary selects the first piece and the second selects the second piece; neither selected piece works for both with tolerance below one. The proof sums the two defects on each piece before selecting: here the sum is on each. It can infer a piece with a small common defect only when the total sum is small relative to total mass. No such smallness holds in this example.
Exercise 52.4 — why the upper truncation is necessary (intermediate). On with Lebesgue center measure, let . Compute the mass discarded by cutting above . Can a finite bounded simple function approximate uniformly on the full interval?
Solution. The discarded set is , and
It has arbitrarily small dimension mass as , although the density is unbounded there. Every finite bounded simple function has a bounded range, so its uniform distance from on is infinite. After the upper cut, the density is bounded and a finite unit-width spectral partition is available.
Exercise 52.5 — even a proper infinite factor can fail deletion (advanced). In Proposition 51.6, choose to be the tensor tail beginning at site three. Verify that is a proper unital infinite hyperfinite subfactor of , that both tensor splittings hold, and that its complement pair cannot be a core.
Solution. Here
The superscripts name the tensor sites. The second-site matrix algebra makes , while the remaining infinite tail is a hyperfinite II₁ factor. The complements are
Multiplication gives both actual tensor splittings. These finite-dimensional complements cannot be a proper-index Jones core by Proposition 50.5. Thus the infinite deletion obstruction also holds with strict containment .
Authored by GPT-6.1 Sol (OpenAI), Ultra reasoning, October 2026. Original exposition released under CC0 1.0.
