Canonical capacity and optimal dimension repair

An integer target must fit the actual canonical algebra before rounding can succeed. We identify a structural restriction on that capacity: its finite part lives on at most countably many central atoms, and its value on each atom is at least the reciprocal of the atom's inherited trace. Every diffuse central region has infinite capacity. We then prove that trim and fill repairs any feasible bounded dimension with the smallest possible squared Hilbert–Schmidt distance. The resulting rounding estimate uses an profile error and allows a varying capacity and a diffuse center.

The exact local inputs are 52.2–52.4 (actual centers, trace measures, matrix deletion and prescription in a finite projection), 58.16 and the proof of 58.5 (extended central dimensions and bounded prescription under an arbitrary projection), and 68.12 (the larger dimension of a projection in the expected smaller algebra). The bounded-prescription proof in Lesson 58 does not use its surrounding assumption that the larger core is a factor; we check its inputs below. Factor trace uniqueness, corner centers and projection comparison retain the declared programme prerequisites of Lesson 52. We prove the new capacity restriction, optimal distance and quantitative consequences here.

Human-source context is Sorin Popa, Classification of amenable subfactors of type II, DOI 10.1007/BF02392646, Theorem4.2.2, printed pp.213–214. The estimates below identify a sufficient dimension input for its rounding step. General relative amenability has not yet supplied that input. No unrestricted rounding, near-one common bounded frame, full partition or generation conclusion is inferred.

Unit capacity in either canonical row

Let be an actual core for a proper finite-index inclusion of II₁ factors. Use either of the two rows

The canonical trace on is . Its full Jones corner is , and the inherited probability trace on is . For , write for the canonical lift. Thus , and . Physical and represented are distinguished throughout.

For arbitrary projections, the generalized dimension is extended, as in (58.16). The unit capacity is the possibly infinite central function

The notation means a positive extended measurable function, not that is a bounded operator. Its measure is the finite measure on . Semifiniteness and normality give this extension: take the directed family of finite projections under a given projection and the increasing supremum of their central densities. Finite joins keep the family directed. The finite center measure permits a countable subfamily with the same essential supremum, by maximizing the integrals of bounded truncations; this is precisely the construction in (58.16). It works in either row without factoriality of . Since and , the lower bound in (85.2) follows.

Lemma 85.1 — every finite central unit has trace at least one. If is a projection and , then

Proof. Put . The functional is a finite normal trace on the ambient factor : positivity and normality follow from the trace, and traciality uses that commutes with . Factor trace uniqueness therefore gives . For the physical projection , this yields .

The Jones projection commutes with every physical element of . The two commuting projections have product , which majorizes . Indeed this Jones-corner projection is fixed by both of them. Monotonicity gives . Faithfulness of the inherited trace gives , so divide by it. This proves (85.3). Neither nor commutation of the entire smaller represented center with is assumed.

Theorem 85.2 — the finite capacity part is atomic. If a central projection has finite unit trace , then has at most atoms. In particular:

Proof. Every nonzero subprojection of has a finite represented central unit trace and hence trace at least 1 by Lemma85.1. An orthogonal family under can consequently have at most members. Repeatedly split any nonatom in a finite partition of . The bound forces this process to end in an atomic partition. Each atom has scalar abelian corner by spectral calculus, so the asserted dimension bound follows.

For each positive integer , the spectral set has . A nonzero such set is therefore a finite sum of atoms of . Their increasing union is the finite-capacity region, proving countability. On an atom , the extended function is a scalar, and . When finite, Lemma85.1 gives .

Finally the diffuse part is the complement of the sum of all atoms of . There are at most countably many atoms: for each positive integer , at most have trace at least , and every positive atom trace exceeds such a threshold for some . None of the finite-capacity spectral sets meets the diffuse part. Thus there. Infinite-capacity atoms are also allowed.

This theorem describes actual canonical capacity. It does not conclude that the inherited center measure is atomic, and it does not bound the normalized dimensions of arbitrary finite projections on an infinite center.

General prescription with a varying capacity

Proposition 85.3 — bounded prescription below an arbitrary projection. If is a projection and a bounded positive central function satisfies , there is a finite-trace projection with dimension .

Proof by the checked factor-free argument of 58.5. Choose an integer . In , normalize the central dimension by . The finite projection has dimension . Its type II corner and 52.4 give a finite projection of dimension .

Compare with by the maximal partial-isometry argument of 58.5. Its only inputs are central support comparison, polar decomposition, normality of dimension and faithfulness of a finite projection's dimension. If a residual initial part survives, its central support is disjoint from the residual final support. On , all of is the final range of the partial isometry already used, with dimension . That contradicts , since is positive on its nonzero central support. Thus the initial residual is zero. The final range lies under , identifies with in its first corner and has dimension . It has finite trace .

Every hypothesis just listed holds for both type II canonical rows of 52.2. No constant unit capacity and no factor center of the larger core enter this proof. Thus the proof in 58.5 supplies precisely the broader statement above; we have not imported its separate constant-capacity Lemma58.4.

For a finite projection , write . Additivity of extended dimension gives

Here is finite almost everywhere because it is integrable. On finite-capacity regions the subtraction is ordinary; on infinite-capacity regions it has the indicated extended meaning.

The exact least-cost repair

Theorem 85.4 — optimal dimension repair. Let be finite trace, with dimension , and let be a bounded positive central function. A projection of dimension exists if and only if . Under this feasibility condition there is a finite projection commuting with such that

This is the least squared distance among all projections of dimension , including those that do not commute with .

Proof of existence and the attained cost. Necessity of follows from . For sufficiency, 52.4 gives a projection of dimension . Equation (85.5) and show that . This missing dimension is bounded by , so Proposition85.3 gives of dimension . Put . Its parts are orthogonal, and both commute with . They give exactly dimension .

The orthogonal removed and added pieces have dimensions and , respectively. Thus

Both pieces are finite trace because and are integrable.

Proof of optimality. For any finite projection with dimension , the positive trace-class operator defines a positive normal functional on the center. Let its density be . The operator inequalities and , together with cyclicity in every central pairing, give

Indeed pairing with each positive bounded central test gives the two upper bounds, and finite-measure duality gives the density inequalities. Cyclicity also makes real and nonnegative. Expanding the Hilbert–Schmidt square therefore yields

Equation (85.7) attains this bound. The optimal projection need not be unique.

Corollary 85.5 — direct commutator cost. For from (85.6), put , , and . For every ambient unitary , in the smaller row,

Proof. The triangle inequality gives . Unitary invariance bounds the last term by . Divide by . All norms use the larger canonical trace, whose restriction to the smaller algebra is the same trace by 52.2. This proof requires no commutation of with the smaller center: it compares the entire repaired projection with .

Matrix amplification preserves feasibility

Return to the smaller row , and denote its capacity by . Deleting a common gives the actual new core and canonical pair of 52.3. Its unit and old projection dimensions satisfy

For the unit identity, apply the finite-projection scaling of 52.3 to the directed finite projections under , then use normality of the extended dimension. Both algebras have the same unit and center. Relative ambient commutator defects of old projections are unchanged.

Fix a nonzero projection and a scalar satisfying . The scalar profile has mass . Define

There is no assumption here that is in . The profile error includes the entire mass of outside , so an additional noncommuting central cut is unnecessary. The feasibility inequality and (85.11) imply .

Theorem 85.6 — rounding from a feasible scalar profile. Under (85.12), optimal repair in produces a finite projection with . For ,

The projection is a sum of orthogonal pieces equivalent inside to . If additionally , its larger dimension is also .

Proof. The positive masses differ by at most their density difference, so . The floor gives . By the triangle inequality,

Theorem85.4 gives the exact minimum squared distance . Apply (85.10) with the new traces, then substitute (85.14), using the unchanged relative defect of . This proves (85.13). Since is a feasible bounded dimension, every application of prescription is justified.

Divide the dimension into orthogonal pieces of dimension by 52.4. The final zero-dimension remainder is zero by faithfulness. The new Jones corner has that same dimension, so each piece is equivalent to it. The larger dimension is by (68.12). It equals when the additional common-center membership holds.

For a concrete tolerance, let . The following conditions suffice:

Here is any prescribed finite set and . The last two conditions are choices of after a feasible profile is available. They give . The first square-root factor in (85.13) is less than , since . Its contribution is less than . The second contribution is at most , by squaring the latter inequality. Thus the resulting defect is strictly smaller than .

The scalar-profile and capacity conditions in (85.12)/(85.15) have not been obtained from general amenability. Theorem84.4 supplies a different proved route under finite smaller center. A diffuse center alone guarantees capacity, not a nearly scalar dimension profile.

Capacity obstruction survives amplification. If , a positive central subset and some scalar satisfy on . Whenever , the proposed integer satisfies

Indeed . No projection of dimension then exists. Such follow by taking a positive-measure level set of . Thus increasing the matrix size does not repair an infeasible limiting scalar target; the floor can only hide a discrepancy at small sizes.

Finite central partitions and one simultaneous extraction

A preselected finite projection need not have almost all its dimension near one scalar. We can approximate that dimension by finitely many scalar levels, repair every positive cell, and select one cell for the entire finite target set. The additional quantity that must be controlled is the actual pinching energy between the represented central cells. We first prove its exact identity.

Exact pinching identity. Let be a finite-trace projection in , and let be orthogonal projections summing to . For every unitary ,

Proof of the identity. For any finite projection , the two operators and have squared norm . Their inner product is , a real nonnegative number. The middle equality is trace cyclicity; the factors are bounded and the products are trace class. Expanding the square gives .

The finite expansion is orthogonal in . Indeed the inner product of and is . It is zero if . If and , cyclicity places the orthogonal factors next to each other and again gives zero. Thus ; the same orthogonality applies to the off-diagonal sum. Subtracting the formula for from the sum of the formulas for , and using , gives both equalities in (85.17). Every product used is supported on finite projections, so no finite trace on the unit of is needed.

Theorem 85.7 — finite central partition extraction certificate. Use an actual core and its smaller canonical row , the common canonical trace and the inherited probability measure on , as in (85.1) and 52.2. Fix ambient unitaries , where , and an integer . Let be a finite-trace projection, and let be an orthogonal finite partition of . Omit zero-measure positive cells; the zero cell is allowed to be zero. Suppose

Put , including , where is the represented lift in . Define the whole-projection and pinching errors by

These numbers are finite, since every has finite trace. Make the common matrix deletion of 52.3, obtaining an actual core for the same inclusion and canonical pair . Choose its size so that

Then there is an and a nonzero projection with , , and

It splits into orthogonal projections, each equivalent inside to . The selected support belongs to . Its larger dimension is also when the additional membership holds, by (68.12). Throughout, the target unitaries are arbitrary elements of ; no invariance of the smaller center under them is assumed.

Proof of feasible trimming and its exact total cost. By 52.3, the old has new smaller dimension , and the centers and their probability measure are unchanged. Since , prescription in the finite projection , proved in 52.4, supplies of dimension . Put and . Since every , each difference is a projection. Summing its trace gives the exact cost, including the discarded zero cell:

For the inequalities, . Multiply by and sum to obtain . Then gives the displayed cost bound and both ratios. The lower bound makes every trim feasible below the original projection. Unit capacity, including its infinite part, therefore introduces no additional prescription hypothesis.

Proof of the simultaneous commutator estimate and selection. Apply (85.17) to each , then sum over all targets. Equation (85.19) gives . All these commutators belong to the old larger algebra, where the new trace scales by . The triangle inequality in the finite Hilbert direct sum indexed by now yields

Here unitary invariance gives ; summing its square over gives the factor . Divide the first line by and use (85.22) to obtain the second line. The term for vanishes.

The second line is a trace-weighted average of the nonzero 's summed squared relative defects. At least one therefore satisfies

Otherwise the sum would exceed . Each individual target defect is bounded by this summed defect, which proves (85.21) on one and the same cell. Finally divide the central dimension into dimensions by 52.4. Their sum exhausts , since a residual projection of zero dimension is zero by faithfulness. Each piece and the new Jones-corner projection have the same finite central dimension, so they are equivalent by the comparison argument of 52.4.

A concrete tolerance for all targets. For , put . It suffices that

Indeed . The first term in (85.21) is strictly below . The second is at most : squaring the last comparison reduces to . Thus every selected relative commutator defect is below . After are fixed, the finitely many positive 's have positive minimum. Arbitrarily large common matrix factors in the type II core , as used in 52.3, permit both and arbitrarily small . For this purpose even the matrix sizes obtained by repeated type II halving suffice.