Removing a projection produces a subfactor

The path models give factors generated by Jones projections. Removing the first generator gives a smaller factor, and removing a second gives its predecessor. The first removed projection then recognizes the larger algebra as a basic construction. This converts a positive model of relations into an inclusion with a computed index.

We assume Paths, local projections and a faithful trace, The trace and the tail of a path, and Going up and down the Jones tower. The models considered here are for , with , and for . Put . Basic references are [Jones] and [Anantharaman–Popa].

Construction and proof sources: The weighted path matrices and faithful Markov trace are Propositions 9.1–9.2 and Theorem 9.3 and Theorems 9.4–9.5 of Paths, local projections and a faithful trace, with factoriality proved in The trace and the tail of a path. Lemma 11.1 and Proposition 11.2 below prove word reduction and the trace-preserving normal shift. The remaining compression and full-corner arguments culminate in Theorem 11.5, which recognizes the actual basic construction and computes the index using Going up and down the Jones tower. Corollary 11.6 keeps the separate identity endpoint. Anantharaman–Popa retains its comparison credit for the factor and basic-construction setting.

Reducing the last occurrence of a generator

Let denote the algebraic unital *-algebra generated by . An empty interval means .

Lemma 11.1. In any algebra satisfying (9.9),

There is also the reversed formula

Proof. Induct on the interval length. Between two nearest occurrences of , a word uses only lower generators. By the induction hypothesis, that middle word is a sum of words in and words with in that algebra. These coefficients commute with . In the first case, ; in the second,

Each replacement removes one occurrence of . Repeating leaves at most one, proving (11.1). The same argument with the order of indices reversed proves (11.2).

This reduction also proves finite dimensionality of every algebra generated by finitely many of the projections: its dimension is bounded inductively by , where bounds the preceding interval's dimension. We use the reduction for traces rather than this coarse bound.

The Markov trace is unchanged by a shift

Proposition 11.2. Replacing every by , for a fixed , gives an isometric trace-preserving -isomorphism from the path C-algebra onto the C*-algebra of that tail. It extends to an isomorphism of their tracial von Neumann algebras.

Proof. A normalized trace on words in a finite interval, satisfying

is determined recursively. On the first interval, the trace of is . For the inductive step use (11.1); cyclicity evaluates , and all terms on the right belong to the preceding interval. The path trace satisfies (11.3) by Theorem 9.5. Each shifted interval satisfies precisely the same rule. Therefore every word and its shifted word have the same trace.

For a polynomial , this applies to . Faithfulness of the path trace shows that if and only if . Hence substitution is well defined on the algebraic generated algebras, with inverse onto its image, and preserves the trace of every power of .

For a positive element of a C*-algebra with faithful tracial state,

The upper bound is immediate. For the lower bound, any interval below and sufficiently close to the spectral maximum has positive trace spectral mass: a nonzero continuous nonnegative function supported there has positive trace by faithfulness. Its mass bounds below by a constant times that interval's lower endpoint to the -th power. This proves the formula. Apply it to to obtain norm preservation. The substitution extends to the C*-closures.

Finally, trace preservation identifies the two GNS Hilbert spaces by a unitary intertwining their left multiplication representations. Conjugation by that unitary gives the required normal isomorphism of the von Neumann closures.

Write

The preceding proposition and the factoriality theorems show that all three are II₁ factors, with the restrictions of the same normalized trace. They are each approximately finite dimensional.

Lemma 11.3. The Markov trace also obeys its left-end version:

Proof. First use a finite interval and induct on its length. On , the identity follows from and , by the right-end Markov rule. For a longer interval, (11.1) reduces its elements to the preceding interval and terms . For such a term, the right-end rule and the induction hypothesis give

The trace of is . This proves the left-end identity on every finite interval. Normality of the two functionals extends it to the von Neumann closure.

Compression removes the next generator

Proposition 11.4. For ,

Proof. The second identity follows immediately by testing the expectation with (11.5). For the first, put and . The reversed reduction (11.2) writes every element of as a sum of terms and with . Because commutes with ,

The map on is a faithful *-homomorphism into the corner . Indeed, (11.5) gives , so its kernel is zero. It is therefore isometric.

Compression thus determines a well-defined map by . It is unital, completely positive, -bimodular on the algebraic domain, and contractive. These assertions follow from compression and the inverse of the isometric *-isomorphism on . It fixes . Moreover,

For , bimodularity yields . Density in characterizes the trace-preserving expectation, so on . Bounded strong approximation from that algebra and normality of extend the compression identity to .

Recognizing the basic construction

Theorem 11.5. The inclusion is a Jones basic-construction triple, and

Proof. In , consider the closed left-invariant space . The reversed reduction at and (11.6) show algebraically that

where is the algebraic union generated by all the projections. Thus . Equation (11.5) makes

a unitary: its norm squared on is . It intertwines the left -actions.

The left representation of on is normal and nonzero. Since is a factor, its kernel, a weakly closed ideal, is zero. Under (11.8), left multiplication by sends to , by (11.6). It is therefore the Jones projection for . Since , this faithful representation identifies with .

The canonical semifinite trace on this basic construction has value one on . Its full corner there has the normalized trace of . The trace has the same corner trace, because (11.5) gives . Uniqueness of trace extension from a full corner therefore identifies the canonical trace with . Its value on the identity is , so . Equality of consecutive tower indices, proved in Theorem 4.1, gives .

Corollary 11.6. Every number in

occurs as the index of an inclusion of separable approximately finite-dimensional II₁ factors.

Proof. For , use the finite graph and Theorem 11.5. For , use the infinite graph with . For , the value is one; take the identity inclusion of the separable hyperfinite II₁ factor. Countably many finite-dimensional stages give separability in the nontrivial models.

Together with the positivity restriction, this proves both necessity and realizability of the index set. A generating tail realizes the path invariant identifies the finite-graph construction's full standard invariant, proving that both its principal graphs are the intended rooted path. Computing a general inclusion's standard invariant still requires its own relative commutants.

Exercises

Exercise 11.1 — introductory. Reduce , and then compute its trace.

Solution. Distant commutation and the adjacent relation give . The Markov rule gives , so the requested trace is .

Exercise 11.2 — intermediate. Show that the map in Proposition 11.4 cannot acquire a kernel even though has trace smaller than one.

Solution. If , then . Here commutes with , and the last equality is (11.5). Since and is faithful, . A small trace projection can carry a faithful copy of another factor in its corner.

Exercise 11.3 — intermediate. For the model, determine the index of each adjacent tail inclusion and the canonical trace of the identity in its basic construction.

Solution. We have , so both are . The normalized factor trace gives the Jones projection value ; multiplying that trace by gives the canonical trace with projection value one and identity value .

Exercise 11.4 — advanced. For , find a parameter for the infinite path model and state the index of its tail inclusion.

Solution. Here . Solving gives , so . Use the weights . The tail inclusion has index by Theorem 11.5.

References

• Vaughan F. R. Jones, Index for subfactors, Inventiones Mathematicae 72 (1983), 1–25.
• Vaughan F. R. Jones, The Jones polynomial for dummies, lecture notes, 2014, Sections 5–6.
• Claire Anantharaman and Sorin Popa, An introduction to II₁ factors, open lecture notes.

Written by GPT-6.1 Sol (OpenAI), Ultra reasoning effort, September 2026. Self-checked by the writing AI. Public domain (CC0).