Measuring an inclusion through modules and corners

An index is useful only if it behaves predictably when we change the representation, pass to a corner, or insert an intermediate algebra. This lesson develops those rules. They also give a quick explanation for the special role of the number four.

We use A projection that remembers an inclusion, the actual normal-amplification theorem in Spatial tensor products, and the complete finite-factor arguments proved below. These give the representation, comparison, compression and coupling formulas without a separability assumption. At infinite dimension the number alone does not classify representations of arbitrary cardinality. The support-projection route and its finite compression proof are retained in (GM14), (GM16) and Proposition 2.9.

Basic references are [Anantharaman–Popa], [Jones] and [Murray–von Neumann].

The representation and trace ingredients

The following complete programme arguments are used at the indicated places. Normal modules and their orthogonal families may have arbitrary cardinality.

The faithful normal normalized trace used by is constructed for in Finite algebras and normal traces, TE10–TE14. Its consumed positive compactness and group fixed-point inputs have complete proofs in TE1–TE9. TE13 retains the general finite-algebra trace-separation statement; TE14 specializes it to every nonzero finite factor. TE15 proves uniqueness among all tracial states, including ones not assumed normal, by the same comparison argument as BC Lemma1.4a. These arguments impose no separability or representation-cardinality restriction. The module construction below retains its finite and infinite dimension scope.

A direct proof of the dimension and coupling rules

Proposition 2.0. The projection-trace definition is independent of its realization. It gives faithful intrinsic dimension traces, finite-module classification, the finite commutant criterion, both compression rules, reciprocal dimension, the scalar coupling constant, arbitrary orthogonal sums and tensor products. The full proof follows in A–E; Proposition 2.9 records the rules together.

Let be a finite factor equipped with a faithful normal tracial state . A normal module means a unital normal left representation; allow the zero Hilbert space. No separability assumption is made. On , let be the right algebra with its opposite trace . For any nonempty set , put

FT Theorem M10.1, SP Proposition 7.1 and AT Proposition 6.5 prove that is a factor and is a faithful normal semifinite trace. The sum means the supremum of finite subsums. SP Theorem 8.2 gives a representation for every normal -module. For the zero module, take in a singleton amplification; thus factor assertions always use a nonempty index set. Define

A. Independence, finite comparison and the finite commutant criterion

Two realizations can be placed in . Extend an intertwining unitary between their ranges by zero. It belongs to the commutant, has initial and final projections equal to the representing projections, and the trace identity gives equal -values. Thus (GM2) is independent of realization. The trace on the commutant is intrinsic by the same transport argument. For a commutant projection , its range is represented by in this same amplification, so

For projections of a semifinite factor carrying a faithful normal trace , with and , factor comparison PR Theorem 5.5 gives or . In the second case, , so . The assumed inequality forces equality, and faithfulness applied to gives . Hence . Equal finite traces give equivalence. This argument compares a finite trace projection with an infinite trace one too; it never classifies two infinite trace projections.

If , its corner has a faithful finite trace, so is finite: an isometry in the corner has complementary range of trace zero. Conversely suppose is finite. Semifiniteness and faithfulness give with . Explicitly choose a nonzero positive finite-trace element below , then a nonzero spectral cut at some positive threshold. Compare successive remainders of with . Until a remainder is subequivalent to , remove a subprojection equivalent to . This must terminate after finitely many steps: an infinite sequence of removed projections has a sum equivalent to its proper tail by the strong sum of their shift partial isometries, contradicting finiteness of . At termination is a finite orthogonal sum of copies of and one projection subequivalent to , so . Thus

Faithfulness also gives for . Equal finite dimensions imply unitary module equivalence by the comparison just proved in a common amplification. Nonzero infinite-dimensional modules are not asserted equivalent.

When is II₁, every nonzero finite commutant corner obtained here is II₁ too. Indeed let , whose corner in is , hence has no nonzero abelian projection. If had an abelian projection , PR Lemma 5.3 would give equivalent nonzero subprojections and . Their corners would be isomorphic, making abelian, a contradiction. Thus , and each of its nonzero corners, has no nonzero abelian projection. A finite such factor is II₁. This verifies the II₁ hypotheses when Theorem 2.1 is later applied to an inclusion of finite commutants.

B. Algebra compression at every dimension

Let , , and , with normalized trace . On , SP Proposition 6.1 and PR Proposition 3.5 identify the commutant of with . The map

is a normal faithful isomorphism: its central kernel is zero because has full central support in the factor . Transport through , obtaining a faithful normal semifinite trace on .

Choose one amplification coordinate , and put . The range of is in that coordinate. The unitary

identifies it with the standard -module. Moreover . The projection is nonzero in the factor , hence full. The dimension trace of the -module , defined by (GM2) for the finite factor , also gives this standard projection mass one. On its corner both traces are the unique normalized trace of , by FT Theorem M10.1 and BC Lemma 1.4a. BC Lemma 1.2a extends a trace uniquely from this full finite corner. Therefore that dimension trace equals .

If , its compressed module is represented inside by . Evaluating the preceding trace proves, including ,

For both sides are zero. The argument uses neither coupling reciprocity nor a classification theorem, so it can precede them.

Figure 2.0. Algebra compression, (GM5)–(GM7). The lower right module is identified by the unitary . The arrows are restrictions and corner selections, not inclusions of into the commutant. Nonzero and make both commutants factors. FT Theorem M10.1, SP Proposition 6.1, and BC Lemmas 1.2a and 1.4a supply the standard commutant and full-corner uniqueness. The trace may have infinite total mass; its standard corner has mass one.

C. Reciprocal dimension without a coupling premise

Let have . Choose an integer . By A, is unitarily equivalent to , where

The normalized trace of the finite factor is , and its action on is normal. The standard -module is the orthogonal sum of copies of : consists of its matrix columns, and acts standardly on via the opposite multiplication. Rescaling each column by the scalar required by gives a unitary. The dimension of the standard module is one by (GM2), and block-diagonal trace evaluation is additive, so

Apply the already proved algebra compression (GM7) to . Its corner is , denoted , and its normalized trace gives weight . Thus

The commutant is a finite factor by A. No unproved coupling constant has been used to obtain (GM10).

D. Cyclic modules and equality with the scalar coupling definition

Every cyclic normal module of a finite factor embeds into . To prove this, let be its cyclic vector and . This is a positive normal functional. The faithful standard representation and its inverse are normal: an order bijection preserves bounded increasing suprema, and FT's normality criterion applies. The standard trace vector is separating, so PR Proposition 18.2 gives with . The map preserves inner products and extends to a unitary onto the reducing subspace . Consequently a cyclic module has dimension at most one. PR Proposition 18.2's complete proof uses transfer PR Theorem 17.5 and a positive normal vector series; that series is supplied here by the independently proved SP Proposition 3.1(4), rather than a citation-only background label.

Now let , finite, and . For , let

where brackets denote the projections onto those closed subspaces. Both fix . On , the two finite factors and , acting by restriction, are mutual commutants by SP Proposition 6.1 and full central support. They both have as a cyclic vector:

Let be the normalized trace of . Since , BC Lemma 1.4a gives . First commutant compression (GM3), then algebra compression (GM7), gives

Cyclicity gives . Cyclicity for the other algebra gives . Reciprocity (GM10), applied to these two mutual finite commutants, makes the latter dimension . Thus , proving

For , both sides vanish. Equation (GM14) is exactly the scalar support-projection coupling relation, so its constant is . Uniqueness follows from any nonzero vector, because . At an infinite commutant (GM4) gives infinite projection dimension, agreeing with the extended definition. This proves finite-factor coupling agreement, reciprocity, both compression rules and finite-module classification in their full hypotheses, without importing the general centre-valued coupling theorem.

E. Sums, tensors, uniqueness and the representation extension consumer

Arbitrary sums have block-diagonal representing projections in an amplification over the disjoint union of the index sets. Normality of (GM1) evaluates their dimension as the supremum of finite partial sums. Thus dimension is additive for every orthogonal family, the standard module has dimension one, and (GM3) is valid at every dimension. For finite , it becomes ; no normalized commutant trace is used at infinite dimension.

For another finite factor and representing projections , SP Theorem 10.1(1),(4) supplies the faithful normal product trace, tracial by separate normal continuity from algebraic tensors; SP Theorem 11.4 and Corollary 11.5(3) supplies the tensor commutant and scalar centre. The standard Hilbert identification comes from the product trace pairing. After reordering the amplification factors, the representing projection is , and the diagonal formula is

Every sum is nonnegative, so suprema over finite subsets prove this identity at infinite trace too. A zero projection gives zero directly, with the convention . This supplies all tensor clauses, including arbitrary scalar Hilbert amplifications.

Finite comparison also proves uniqueness of the dimension: the complete rational squeeze is Theorem 2.10 below.

For use in Proposition 2.6, every positive finite trace value is realized by a projection in an amplification of a II₁ factor. Here is the construction. By PR Proposition 13.3, split the identity into two equivalent projections, then split the unused half into two equivalent projections, and continue. With the given faithful trace this gives mutually orthogonal projections of traces . For , choose a binary expansion , including the expansion with all digits one when . The strong sum of those projections for which has trace , by normality. Adding finitely many standard coordinates realizes any positive finite number. Equal finite trace projections are equivalent by A. An arbitrary normal representation is an orthogonal sum of cyclic reducing representations: a maximal orthogonal family of cyclic subspaces, obtained by Zorn's lemma, exhausts the Hilbert space because a nonzero orthogonal complement would contain another cyclic subspace. By D each summand embeds in the standard module. This proves the two concrete representation ingredients used in Proposition 2.6 for every Hilbert cardinality.

The support relation (GM14) gives another useful proof of finite commutant compression in coupling notation. Suppose is finite, is nonzero and is its normalized trace. For , put and . Then . On , the new support projections are and , because . The faithful restriction identifies with ; the normalized trace on is . Applying (GM14) gives

Thus the classical support-projection compression route is retained alongside the projection-trace proof. General centre-valued coupling theory is not needed for this scalar result.

Matching left dimension to the basic construction

Lemma 2.0b — the normalization at the standard submodule. Let be II₁ factors, , and . Write . Then

Here is normalized by , as in BC Theorem 1.4.

Proof. FT Theorem M10.1 and BC Theorem 1.2 give , , , and full central support of in . Therefore is full in too. For positive , set . Antiunitary conjugation preserves order, bounded increasing suprema and the trace identity, so is a faithful normal semifinite trace. On , the standard module has commutant ; conjugation by identifies this corner with the left standard corner . Thus is its opposite normalized trace.

The intrinsic dimension trace on , supplied by Proposition 2.0, gives mass one, because is the standard -module. Its restriction to this finite factor corner is the same normalized trace by BC Lemma 1.4a. Full-corner uniqueness, BC Lemma 1.2a, now gives . Evaluating at proves the asserted equality at finite or infinite index. In particular the index defined by the basic construction is the same left dimension used below.

The cyclic standard submodule also proves , and equality is equivalent to , by BC Corollary 1.5. Consequently a ratio of restricted module dimensions will always have a positive index multiplier; a finite nonzero denominator is required.

Dimension and restriction of scalars

For any finite factor , including , a normal left module has the form

Its dimension is

The representation theorem and projection comparison make this independent of the choice of realization. Equivalently, the commutant carries a dimension trace: a projection has trace equal to the dimension of its range. The standard module has dimension one. A zero module has dimension zero; every nonzero normal module has strictly positive dimension.

Trace normality gives additivity over arbitrary orthogonal direct sums. Tensor-product traces give

For modules of finite positive -dimension, the normalized trace on the commutant gives

The reciprocal formula is when both and are finite. Proposition 2.9 below reconciles these formulas with the coupling definition for every finite factor. For an infinite module we use its unnormalized dimension trace on the commutant; a normalized trace there may not exist. The inclusions in Theorems 2.1–2.6 are specifically inclusions of II₁ factors.

Theorem 2.1 — restriction formula. Let be II₁ factors, and put . For every normal left -module ,

For the zero module the right side means zero, including when .

Proof. First assume . On , the trace on has total mass . Its restriction to is therefore , by uniqueness of the trace on a finite factor. In an amplification, each diagonal standard summand has the same restriction. On finite matrix corners the restricted trace is consequently . Normality extends the equality to the whole amplification. Evaluating on a representing projection for proves the formula.

Suppose . A nonzero representing projection contains a nonzero finite projection in the semifinite factor . Projection comparison makes a standard summand subequivalent to a finite direct sum of copies of . If had finite -dimension, that standard summand would too, contradicting . Thus every nonzero -module has infinite -dimension.

Alternative proof of representation independence. Let be normal -modules with finite commutants, and put , . Choose an integer with . The finite commutant is II₁, by Proposition 2.0A. Its projection-halving construction supplies of normalized trace . Commutant compression and finite-module classification give

This proves the finite-amplification realization appearing in the classical ratio proof; it uses classification only at finite positive dimension.

If , the normalized trace of restricts to on , by BC Lemma 1.4a. Hence the same commutant-compression and direct-sum rules give

When , Theorem 2.1 makes both sides infinite; the multiplier is positive and finite. Dividing the two formulas by the finite positive -dimensions proves representation independence of the ratio at every index, including an infinite numerator. The direct trace proof of Theorem 2.1 proves more: restriction on every normal module.

Corollary 2.2. Index is invariant under isomorphisms of inclusions. For II₁ factors ,

For two inclusions ,

Proof. Isomorphisms preserve the unique normalized traces and hence their standard Hilbert spaces. Apply Theorem 2.1 to , first restricting to and then to . The tensor identity follows from and the tensor-product dimension formula. All indices are at least one, so no undefined product occurs.

Corollary 2.3. If acts normally and faithfully on with finite commutant and , then is finite and

Proof. Put . Then . The reciprocal formula and restriction formula for give

Finiteness of the two commutants follows from finite positive dimension.

Local index without a normalization trap

Let , now with the left action of . Denote the dimension trace on by . Thus . For a nonzero projection , the algebra is a unital II₁ subfactor of . Define

Theorem 2.4 — local formula. For arbitrary index,

If , write , the normalized trace on . Then

If is an orthogonal family of nonzero projections in summing to one, then

Proof. The module has -dimension . The faithful isomorphism identifies this with its -dimension. Its -dimension is , by algebra compression of the standard left -module. Taking the ratio in Theorem 2.1 gives the local formula. Finally, normality gives . A faithful finite trace permits only countably many nonzero orthogonal projections, and the formula also holds as a sum in .

The traces and can have different restrictions to . We must not replace by without an additional argument. The unnormalized formula remains meaningful at infinite index.

Why reducibility costs at least four

Theorem 2.5. If , then is finite dimensional. If , then .

Proof. Suppose is a partition into nonzero relative-commutant projections. Put . Each corner inclusion has index at least one. Theorem 2.4 and Cauchy–Schwarz give

An infinite-dimensional von Neumann algebra has partitions of its identity into arbitrarily many nonzero projections. Here is the finite-dimensionality argument explicitly. If a von Neumann algebra had a finite bound on those partition sizes, choose a partition of maximal size. Each is minimal, because a split would produce a larger partition. Spectral calculus gives : every spectral projection in that corner is either or , so every self-adjoint element is scalar, and every element is a complex linear combination of two self-adjoint elements. If , choose in that corner and take its polar decomposition . Minimality makes and . For any , , and . Thus each matrix corner has dimension at most one. Since , it has dimension at most , a contradiction if is infinite dimensional. Thus must be finite dimensional. A nontrivial relative commutant has a partition with , forcing .

The same calculation yields a sharper estimate when local indices are known:

Equality holds exactly when is proportional to . This explains how trace weights affect a reducible inclusion.

Extending representations

Proposition 2.6. If , every normal representation of extends to a normal representation of on the same Hilbert space. If , a nonzero module of finite -dimension cannot support such an extension.

Proof. Let a nonzero -module have finite dimension . Choose a projection in an amplification of with trace ; the halving and binary expansion construction after (GM15) realizes any positive finite trace value. The associated -module has -dimension , so finite-module classification identifies its restriction with the prescribed module. Transport the -action through this unitary.

For an arbitrary normal -representation, decompose the Hilbert space into orthogonal cyclic subrepresentations. Each cyclic subrepresentation embeds into a standard module and has finite dimension. Extend each one by the preceding argument, and take their direct sum. This argument preserves arbitrary cardinalities and does not classify all infinite modules by the single symbol . The infinite-index assertion follows directly from Theorem 2.1.

Examples with different mechanisms

Example 2.7 — subgroup index. Let be an ICC discrete group, and let also be ICC. The group factors act on . Decompose into right cosets . Each is invariant under the left -action and is unitarily equivalent to by . Additivity therefore gives

including infinite subgroup index. If , the ICC assumption on follows from that on : a finite -conjugacy class would produce a finite -conjugacy class by taking a finite union over coset representatives.

For a concrete example, let be the group of permutations of with finite support, and let be its even-permutation subgroup. A nonidentity permutation has infinitely many distinct conjugates: move its finite support to infinitely many disjoint sets using finitely supported permutations. Hence is ICC. The sign homomorphism is onto , so ; the finite-index argument above makes ICC too. Thus .

Example 2.8 — diagonal corners. Let be a II₁ factor, let have trace , and suppose there is an isomorphism

Set . This is a factor with identity one, and . Both corner inclusions have index one. The local formula gives

More generally, suppose the trace-scaling factors of automorphisms of fill . Let , and fix a rank-one projection in . Choose an automorphism with . Then and have the same finite trace , so projection comparison gives a partial isometry with these initial and final projections. The map

is a normal isomorphism of their corners. Identifying the rank-one corners with and gives the required . Since a II₁ factor has projections of every trace , the full trace-scaling hypothesis realizes every value by this diagonal construction.