Smooth representations and compression onto the tracial tower

A representation can enlarge both algebras in a subfactor inclusion. A finite common basis still controls its basic construction, even when the larger algebras are neither finite nor factors. We construct that expected tower explicitly. We then show how the canonical tracial tower can produce a compatible hypertrace on a smooth representation. The latter argument has two precise hypotheses: an injective projection onto the tracial tower and its bicommutant identity. Generation supplies the first hypothesis; the second remains a separate mathematical condition. Every nonzero physical corner and positive real amplification also preserves the full all-smooth representation amenability condition. Explicit finite matrix induction and return maps retain all tower levels and allow singular compatible hypertraces. An explicit downward expected tunnel gives a compatible hypertrace directly from generation. Together with SA.7, this proves the full original strong-amenability/generating-tunnel equivalence for proper finite-index inclusions with separable predual. The full original global-finite-approximation/strong-amenability equivalence also holds without ambient separability. Exact finite-prefix matching and a separate factorial-core construction preserve its original finite-set scope.

The finite partial basis is Theorem 3.2, and the ordinary trace and cup identities are Theorems 4.1–4.2 and Proposition 4.3. The generating inclusion used below is Theorem 60.5. Normal tracial expectations, standard left/right commutation, matrix positivity and the bimodularity of a unital positive projection onto a C*-subalgebra are the operator-algebra prerequisites already used in lessons 3, 49 and 60. For normal state extension we use the exact existing-course Theorem 10.1 of The double commutant theorem: every positive normal functional on a concrete von Neumann algebra is a sum of square-summable vector functionals. Its complete proof applies to arbitrary Hilbert spaces.

The human source is Sorin Popa, Classification of amenable subfactors of type II, Sections 2.1–2.3 and 4.5, printed pp. 191–197 and 223–225. We give the finite matrix construction and keep the GNS compression explicit. The injective projection constructed from finite-dimensional generation below also preserves the right trace, which permits an exact corner normalization.

Let be II₁ factors of finite index , with . A nondegenerate expected representation is a square

where is normal and faithful, and the ultraweak linear span of is . The algebras can be arbitrary von Neumann algebras. We denote the original tower by ; our cup implements . The represented tower is , containing the original tower and the same cups. Its construction is proved in Theorem 61.2.

The representation is smooth when

Commutants in this formula are taken in the indicated represented finite stages. Write

for the canonical completion in the compatible normalized tower trace. This is distinct from any completion of the represented tower.

A finite row gives full compatibility

Lemma 61.1. Let be a partial right -basis with supports , so that

Then the same row formula holds for every . If is an -bimodule conditional expectation with , it satisfies

Neither nor the state is required to be normal.

Proof. The bounded normal map fixes , , by the original basis identity and right -bimodularity of . It therefore fixes their ultraweak span, which is all . Put and . The finite row gives

Here , and only finite sums are used. Thus is -central, restricts to , and satisfies .

Constructing the expected tower without a trace on the representation

Theorem 61.2. The square (61.1) extends to normal faithful expected tower squares with the original cups, a common partial basis and scalar index normalization at every step.

Proof. Put and

The entries lie in , and . The row for , followed by , gives . The row for gives . Adjoint preservation gives , and (61.4) gives . Thus is a unital normal *-homomorphism. It is faithful because is the coefficient column , which recovers .

For , . Also . Consequently is a projection commuting with , and

The vector has entry in position and zero elsewhere. Hence is the matrix with entry in position . Such matrices span all of . Therefore

We identify with its image under . Define the dual map by

This finite row formula is normal and completely positive, and . The row identity gives , whence . The same identity and its adjoint prove -bimodularity.

To prove faithfulness, write . Set . Then . If this is zero, every . Taking adjoints and extracting its -th coefficient by gives ; its left support is . Thus . The row for also gives

The next common basis is . Directly from (61.8) and (61.11),

For the last equality, the right side at is , and these words span . The middle equality is the sum of the coordinate projections , multiplied by . This basis permits iteration of exactly the same construction. The original finite Jones tower is the identical model on the common basis with in place of ; its embeddings and restricted expectations are therefore the original ones. At each step the new row spans the represented stage over the preceding stage, proving nondegeneracy of each expected square.

The index bound is also explicit. For , , Parseval gives . The operator row has norm squared , so

For the dual bound take , as above and any supported column . Set . Schwarz for gives

The last equality uses . Matrix positivity proves , with viewed through . Since , compression by the nonzero shows that is optimal. For the original downward cup , with , gives the same optimality. Thus the actual normalization stays at every step.

Figure 61.1. The exact maps in Theorem 61.2 retain every partial support. Positions are schematic; the formulas give the maps, normalization and finite spanning relation.

Corollary 61.3. Smoothness propagates: for , ,

Proof. Such an element commutes with by initial smoothness, and with . Nondegeneracy gives commutation with . It also commutes with . These cups and generate . Bounded multiplication is ultraweakly continuous, so it commutes with all of .

A sufficient bicommutant test

Lemma 61.4. Let be a von Neumann algebra. Set and . If

then .

Proof. Put . Since , , so . Also . Each is a finite sum , , by the finite basic construction. Therefore

Hence . Since , induction gives . Normality and density give , so its range equals .

Taking gives the bicommutant identity when (61.16) has been proved. The source additionally states commuting-square conditions; the sufficient proof above needs only (61.16). We do not assert that those scalar expectations or commuting squares hold for an arbitrary .

Normal state extension and compression before projection

Lemma 61.5. Every normal state on extends to a normal state on , with no separability or sigma-finiteness assumption on .

Proof. Represent faithfully and normally on . The exact existing-course Theorem 10.1 stated in the introduction writes the given state as on , with . Use the same formula on . The positive vector functionals have summable norms and converge in the predual norm, so their sum is normal and positive, has value at , and restricts to the given state. This does not assert that the extension is faithful on .

Theorem 61.6. Suppose there is a UCP projection

onto the standard left action, and suppose

Every smooth nondegenerate expected representation has a conditional expectation with , , and the compatible -hypertrace .

Proof. Extend to a normal state on by Lemma 61.5. Compose it with the normal tower expectations from to . Their states agree on overlaps and restrict to the original tower traces, because the expected squares restrict to the original ones. They define a state on the norm closure of . Let be its GNS representation.

Each finite restriction is normal. For , , its coefficient at is , a normal functional on . Polarization and approximation by these dense vectors show that all coefficients are normal. No normality of an entire represented infinite tower is being assumed.

The subspace is the original tracial GNS space . Its projection commutes with every original , since this subspace reduces their *-algebra. Compression

is UCP, fixes , and is -bimodular. Initially . For , Corollary 61.3 gives . Since commutes with , so does ; the -bimodularity of gives .

The tracial expectations preserve commutation with . For , in , with uniform norm bound. Thus the union of the finite is ultraweakly dense in . The bounded element commutes with this whole algebra. Hypothesis (61.19) puts it in .

For , smoothness gives , since . The same compression argument gives . If commutes with , then . For ,

This follows from the canonical Markov trace. Thus implies , proving and . The left algebra element on has not been identified with the expectation operator on .

The UCP map fixes and has range in , so it is a conditional expectation. Lemma 61.1 gives its full expectation compatibility and the hypertrace.

Figure 61.2. The two hypotheses of Theorem 61.6 are drawn explicitly. The projection commutes with the original tower. It need not commute with the represented tower. The maps can be nonnormal.

Generation supplies the injective projection

Lemma 61.7. If each has a UCP projection , then has the projection (61.18).

Proof. Let project onto , and define , with output in . For every and ,

Consequently ultraweakly for every . The UCP maps have a point-ultraweak cluster by compactness of the output unit balls; positivity at every matrix size and linearity pass to this cluster. Its range lies in the closed algebra , and the equality just proved shows that it fixes all of . It is the desired projection. Fixing only the algebraic union would be insufficient for a possibly singular cluster; (61.22) covers every before taking the limit.

Theorem 61.8. In the separable generating case of Theorem 60.5, (61.18) holds. The construction of each finite-stage projection can preserve the right trace as well.

Proof. Let a finite factor be generated by increasing unital finite-dimensional . On , Haar-average conjugation by the right unitary group . This UCP projection has range the commutant of right , namely left . This statement includes direct sums of matrix blocks. If is the orthogonal projection, put

For left , compression is left , so . For right multiplication , compression is right . Averaging its unitary conjugates gives

Here the central expectation is inside . The elements have norm at most , trace , and commute with every fixed once . Any ultraweak cluster is therefore central in the generating factor , and has trace . Thus the entire net tends ultraweakly to . A point-ultraweak cluster of gives a UCP projection satisfying, for every ,

This extra right-trace identity is proved for this projection; it is not assumed for an arbitrary injective projection.

For , order its standard space as , with the last index the right matrix leg. Apply normalized trace on that last leg and to each operator entry on . The resulting UCP projection fixes left , and (61.25) gives . All matrix sums are finite.

Now take a nonzero , let , and . The space is the standard corner space: the normalized-trace unitary sends to . Extend an operator on this space by zero, and set

Bimodularity and the right-trace identity give , so . The map is UCP and unital with corner identity . For , its left corner action is , and . Its range is the corner, so it is a conditional expectation. The right corner action of is , and

Thus the right-trace identity survives every nonzero corner. The operator normalizer is ; the scalar support mass is a different quantity.

Both in Theorem 60.5 have increasing finite-dimensional generating algebras. They therefore have these projections. Theorem 61.2 realizes as a finite matrix corner over , as one over , and each further stage as one over the stage two steps earlier. Finite matrix amplification and corner compression supply all . The identifications preserve the unique normalized factor traces and hence their standard spaces. Lemma 61.7 gives (61.18).

Figure 61.3. The left and right support define the standard corner space. The displayed maps and constants are exact; positions are schematic. Generation gives tower injectivity here, while the bicommutant condition needed for Theorem 61.6 is separate.

The nonextremal physical bicommutant argument in Popa, Section 4.5, passes to a two-step inclusion, modifies downward cups and invokes a trace-preserving comparison carrying them to shifted upward cups. That comparison is distinct from the specified reflection refuted in lesson 47, and its full construction remains a separate requirement for that route. The downward represented-tunnel construction below proves the generating-to-amenability implication directly. The general physical bicommutant is not established by that state construction.

Every smooth representation can be induced through a corner

We now address the quantifier over every smooth representation. The finite constructions below work at every finite index, including , and impose no separability, extremality or finite-depth hypothesis. Their trace-pairing input is Lemma 60.1; their physical corner tower is Theorem 17.11. The human source for the heredity statement is Popa, Classification of amenable subfactors of type II, Proposition 3.2.3(iii), printed pp. 206–208.

Fix , put , and abbreviate , , with normalized trace . The finite frame of (17.33) consists of partial isometries satisfying

The unit in every entry algebra is . The unit of a matrix corner by is .

Theorem 61.9. Compression takes every smooth nondegenerate expected representation of to one of . Conversely, every such representation of induces one of . Both operations preserve all the represented finite tower levels and their smoothness inclusions.

Proof. For any von Neumann algebra , the maps

are inverse unital normal *-isomorphisms. Indeed, proves multiplicativity and both inverse identities; gives the target corner and its identity. All entries are bounded multiplications, so both maps are normal. These are compatible maps for all nested choices of .

Start with a representation of the original pair; in this proof is the ambient algebra denoted by in (61.1), whereas in (61.29) is the finite row. Compress to , where . This expectation is normal and faithful and restricts to . For ,

The first factor is in , and the second is in . Compression is ultraweakly continuous; the nondegenerate span of therefore proves nondegeneracy of the compressed square.

Its represented Jones tower is exactly , with cups . Here and below the lower tower level is . To check the full range, use from Theorem 61.2, the commutation of with , and the identity

Thus the preceding corner and its cup generate the entire next corner. The restricted normal faithful dual expectation sends to . These maps and the common original basis identify the finite matrix construction of Theorem 61.2 at every level, rather than only a dense algebra of words. The original physical tower is , by Theorem 17.11. That theorem also gives

If , original smoothness makes it commute with . Hence commutes with . This is exactly smoothness of the compressed representation at every .

In the other direction, let be any smooth nondegenerate expected representation of , with represented tower . Define