LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-6-luna)

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Ordinary finite Jones index supplies a bounded finite Parseval frame

Statement

Assume AC. Under the ordinary finite-index hypotheses of Finite Jones index gives bounded vector pull-down there are a finite integer $k$ , elements $u_{1}, \dots, u_{k} \in M$ and a projection $p \in M_{k} (N)$ such that

$\sum u_{j} e u_{j}^{*} = 1$ on $L^{2} (M)$ ;
$x = \sum u_{j} E_{N} (u_{j}^{*} x)$ for all $x \in M$ ;
$\sum τ (u_{j}^{*} u_{j}) = λ$ and $\sum u_{j} u_{j}^{*} = λ 1$ ;
$U : M \to p N^{k}$ , $U (x) = (E_{N} (u_{j}^{*} x))_{j}$ , is an onto right-module unitary for the $N$ -valued inner product $E_{N} (x^{*} y)$ ;
$p_{i j} = E_{N} (u_{i}^{*} u_{j})$ and $({Tr}_{k} \otimes τ_{N}) (p) = λ$ .

In particular $M$ is complete in the norm $∥ E_{N} (x^{*} x) ∥^{1 / 2}$ and the bounded coordinate representatives are exactly $u_{j} = U^{- 1} (p ε_{j})$ . Furthermore $∥ x ∥ \leq \sqrt{λ} ∥ E_{N} (x^{*} x) ∥^{1 / 2}$ for $x \in M$ .

The map $U$ also extends to a right-module Hilbert-space unitary $L^{2} (M) ≅ p L^{2} (N)^{k}$ . Consequently ${End}_{N} (L^{2} (M)) ≅ p M_{k} (N) p$ , every such endomorphism preserves $M$ , and for its matrix $a = (a_{i j}) \in p M_{k} (N) p$ one has $T (x) = \sum_{i, j} u_{i} a_{i j} E_{N} (u_{j}^{*} x)$ for every $x \in M$ .

Facts & Assumptions

Given: AC and the algebras, traces, Hilbert modules, and finite dimension hypotheses in the Statement.

[F1]

AC permits Zorn maximality and implies Countable Choice and the ultrafilter lemma required by the Hilbert and compactness suppliers. (The Axiom of Choice)

[F2]

A concrete von Neumann algebra is a WOT-closed unital star-algebra; commutants are WOT closed and multiplication by a fixed bounded operator is WOT continuous. (Von Neumann algebras and commutants)

[F3]

A faithful normal tracial state is positive, faithful, tracial, normalized, and WOT continuous on the operator unit ball. (States, tracial states and faithful normal traces on a von Neumann algebra)

[F4]

In a factor projections are comparable; for finite-trace projections, trace inequalities characterize subequivalence. Diffuse factor corners are diffuse factors; a diffuse factor equipped with a faithful normal normalized trace has a unique normalized tracial state. (Projection comparison and unique normalized traces in finite factors)

[F5]

For ordinary finite Jones index, the commutant of right N is the basic construction, has finite canonical trace of total mass lambda, and satisfies bounded pull-down: Te=xe for unique x in M. All right-N bounded vectors of L2(M) belong to M. (Finite Jones index gives bounded vector pull-down)

[F6]

The ordinary-index contract supplies a unital II1 subfactor inclusion $N \subseteq M$ , with $λ = \dim_{N} L^{2} (M) < \infty$ . Extremality is unused in this lemma; the module model is derived in the proof. (Finite-index extremal II1 subfactor data)

[F7]

Every Hilbert right module has its matrix-entry commutant algebra; in a finite projection model this is the corner $p M_{k} (N) p$ . (Ordinary Hilbert-module dimension and finite projection models)

Proof

Given: The hypotheses of the Statement and the stated prerequisite contracts.

1.1F1F2F3F4F5F6given

In the finite factor $B$ choose orthogonal projections $q_{1}, \dots, q_{k}$ summing to $1$ with $q_{j} ≾ e$ . This is a finite construction: while the remainder has trace greater than $1$ , comparison lets one remove a projection equivalent to $e$ ; after at most $⌊ λ ⌋$ such removals, the remainder has trace at most $1$ and hence is subequivalent to $e$ . If $λ$ is an integer, the final remainder can be zero and is discarded. Since ${Tr}_{B} (e) = 1 \leq {Tr}_{B} (1) = λ$ , this also proves $λ \geq 1$ . Choose partial isometries $v_{j}$ with $v_{j} v_{j}^{*} = q_{j}$ and $v_{j}^{*} v_{j} \leq e$ ; then $v_{j} = v_{j} e$ . Pull-down writes $v_{j} = u_{j} e$ with $u_{j} \in M$ . Thus $\sum u_{j} e u_{j}^{*} = 1$ . Applying this to $x \in M$ proves the frame identity. Taking ${Tr}_{B}$ gives $\sum τ (u_{j}^{*} u_{j}) = λ$ ; applying $F$ to the same identity gives $\sum u_{j} u_{j}^{*} = λ 1$ .

2.1F1F2F3F4F5F6step 1.1

Let $V : L^{2} (N)^{k} \to H$ be $V ((n_{j})) = \sum u_{j} n_{j}$ ; then $V V^{*} = 1$ and $V^{*} V = p = (E_{N} (u_{i}^{*} u_{j}))_{i j}$ . Hence $p = p^{*} = p^{2}$ and $U = V^{*}$ is an isometry on Hilbert spaces with image $p L^{2} (N)^{k}$ . On bounded elements, $U (M) \subseteq p N^{k}$ , and for each $a \in p N^{k}$ , $x = \sum u_{j} a_{j} \in M$ satisfies $U (x) = p a = a$ , proving bounded-module surjectivity. Directly, $⟨ U (x), U (y) ⟩_{N} = \sum E_{N} (x^{*} u_{j}) E_{N} (u_{j}^{*} y) = E_{N} (x^{*} y)$ by the frame identity. Thus this is an $N$ -valued isometry, not merely an $L^{2}$ isometry. The finite column module $p N^{k}$ is complete: its norm is the operator norm of its column in the C*-algebra of $(k + 1)$ -by- $(k + 1)$ matrices over $N$ , and its range is closed because $p$ is a bounded idempotent. This proves the required completeness. Matrix trace gives $({Tr}_{k} \otimes τ_{N}) (p) = \sum τ (u_{j}^{*} u_{j}) = λ$ , and $V (p ε_{j}) = V V^{*} V ε_{j} = u_{j}$ gives the specified coordinate compatibility.

3.1F1F2F3F4F5F6step 2.1

Finally, $x e x^{*} \leq ∥ E_{N} (x^{*} x) ∥ 1_{B}$ : its norm is $∥ V_{x} V_{x}^{*} ∥ = ∥ V_{x}^{*} V_{x} ∥ = ∥ E_{N} (x^{*} x) ∥$ , and it is positive. Applying positive $F$ yields $λ^{- 1} x x^{*} \leq ∥ E_{N} (x^{*} x) ∥ 1_{M}$ , proving the norm estimate. This also gives an independent completeness proof by norm equivalence, since $∥ E_{N} (x^{*} x) ∥^{1 / 2} \leq ∥ x ∥$ and $M$ is operator-norm complete.

4.1F1F2F3F4F5F6F7step 3.1∎

The scalar trace of the $N$ -valued inner-product identity proves that $U$ preserves the Hilbert-space norm. Its bounded range $p N^{k}$ is dense in $p L^{2} (N)^{k}$ , so it extends to an onto Hilbert-space unitary. Under this unitary every bounded right- $N$ endomorphism is a matrix in $p M_{k} (N) p$ by the matrix-entry commutant calculation of Ordinary Hilbert-module dimension and finite projection models. Such a matrix preserves $p N^{k}$ . Transporting it back by $U^{- 1} ((a_{i})) = \sum_{i} u_{i} a_{i}$ gives the asserted finite formula and preservation of $M$ .

Depends on

Used by

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Sources

Claire Anantharaman and Sorin Popa, An Introduction to II1 Factors (standard reference, not scraped)