Small corners and the second injective proof

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Finite-dimensional approximation follows from good small corners inside every remaining projection. We prove the local-to-global step by choosing corners with a fixed fraction of the available trace and controlling the finite residual sums. A complete maximality proof gives another route. External finite-rank models and scalar pinching supply the small corners for the separable injective factor conclusion.

We use finite-rank models and repaired internal matrix systems, together with finite trace expectations, TL3. The expectation is normal, completely positive, bimodular and contractive in operator norm, and is the orthogonal projection in the trace Hilbert space. The chain argument uses bounded density on the support corner, BD1–BD5, and the bounded trace topology.

The finite-rank selection and injective-to-small-corner implication concern a separable injective $\mathrm{II}_1$ factor $M$, with normalized trace $\tau$. The local-to-global theorem proves a local-to-global implication for every finite algebra with a faithful normal tracial state. The final sequential conclusion retains separable predual. The scalar pinching input uses only the earlier finite expected-subfactor and contained-AFD constructions, so the second proof has its own route to the finite injective conclusion.

The finite-injective factor conclusion

Theorem 4.1. Every injective $\mathrm{II}_1$ factor with separable predual is generated by an increasing sequence of finite-dimensional unital algebras, and is isomorphic to the tracial matrix product $R$.

Proof. The small-corner and local-to-global arguments used here are proved below. Proposition 2.2 gives small corners in every nonzero corner. Theorem 3.1 makes $M$ locally AFD. The earlier exact containment and dyadic construction then supplies an increasing generating dyadic sequence and its trace-GNS identification with $R$. $\square$

Internal finite-dimensional approximation can be obtained through unitary couplings or through the small-corner condition. Here external finite-rank projections and scalar pinching establish that condition, and either the countable trace-budget argument or maximality gives local AFD. Haagerup's matrix-factorization method and Popa's small-corner method are two approaches to Connes' injectivity and AFD theorem. The general nonfactor finite converse is treated separately in Central disintegration and measurable matrix assembly and Injective algebras and separable tracial envelopes. The factor construction here retains its stated hypotheses.

From small corners to finite approximation

The small-corner condition

Definition 2.1. A faithfully tracial finite algebra has the small-corner approximation property if, for every nonzero projection $e$, finite $F\subset eMe$, and $\varepsilon>0$, there are a nonzero projection $f\le e$ and a finite-dimensional algebra $A\subset fMf$, with identity $f$, such that

This is the local finite-dimensional condition used in the references below. It is stronger than merely finding good matrix coefficients in some corner: the corner must also almost commute with the tests, and the condition is required inside every nonzero corner.

Finite approximation by trace budgets and maximality

Theorem 3.1. A finite von Neumann algebra with faithful normal tracial state and the small-corner property is locally AFD. This implication requires neither factoriality nor separable predual.

Proof by a countable trace budget. Fix a finite list $F$ of contractions in $M$ and a target error $\varepsilon>0$. The empty list is approximated by $\mathbb C1$, so suppose $F\ne\varnothing$. Put

For a nonzero projection $e$, call a pair $(g,B)$ admissible in $e$ if $0\ne g\le e$, $B\subset gMg$ is a finite-dimensional algebra with identity $g$, and

The small-corner property, applied to $eFe$, supplies such a pair. Consequently the real number

is defined and satisfies $0<\sigma(e)\le\tau(e)$.

Start with $e_0=1$. If $e_n\ne0$, choose an admissible pair $(g_n,B_n)$ in $e_n$ such that

The choice exists by the defining property of the supremum. Stop if a remaining projection is zero. The $g_n$'s are mutually orthogonal, and

If the construction is infinite, it follows that $\tau(g_n)\to0$ and hence $\sigma(e_n)\to0$.

We prove that the remaining trace tends to zero. For an infinite construction, let $e_\infty=\bigwedge_n e_n$. Decreasing projections converge strongly to the projection onto the intersection of their ranges: for $m\ge n$,

so the vector sequence is Cauchy and its limit is that intersection projection. Strong closure puts $e_\infty$ in $M$. Normality of the finite trace gives

Suppose $e_\infty\ne0$. Apply the small-corner property in this remaining corner with the smaller tolerance $c/2$. Obtain $0\ne g\le e_\infty$ and a finite-dimensional $B\subset gMg$ with identity $g$, such that both errors for the compressed tests $e_\infty x e_\infty$ are at most $(c/2)\sqrt{\tau(g)}$.

This fixed pair is admissible in $e_n$ for all sufficiently large $n$. Indeed $g\le e_\infty\le e_n$, so

Thus its diagonal approximation error does not change. For the commutator,

The finite list permits one common sufficiently large index, for which this last norm is less than $(c/2)\sqrt{\tau(g)}$. Hence (SCG2) holds for $(g,B)$ in every later $e_n$. It follows that $\sigma(e_n)\ge\tau(g)>0$, contradicting $\sigma(e_n)\to0$. Therefore $e_\infty=0$ and $\tau(e_n)\to0$. This argument does not require the numbers $\sigma(e_n)$ themselves to be monotone.

We now keep the complete quantitative approximation error. Set

Then $R_0(x)=0$, and

The compression of $R_n(x)$ to $e_nMe_n$ is zero, while $S_n(x)$ belongs to that corner. Cyclicity therefore makes the two terms orthogonal in $L^2$. With $y=e_nxe_n$, the three ordered corner blocks of $S_n(x)$ are

They are mutually orthogonal. The first two squared norms sum to $\|[y,g_n]\|_2^2$, since the two blocks of this commutator differ only in the sign of one block. Admissibility now gives

Only finite sums have been used in this estimate.

For every available finite $N$, form the finite-dimensional unital subalgebra

The orthogonal supports sum to $1-e_N$, so the displayed algebra has identity $1$; omit the zero scalar summand if $e_N=0$. The element $\sum_{n<N}b_n(x)$ belongs to $D_N$. Thus

If the construction terminates, use its terminal $N$. Otherwise choose $N$ so large that $\sqrt{\tau(e_N)}<\varepsilon(1-1/\sqrt2)$. In either case every $x\in F$ has distance less than $\varepsilon$ from this one finite-dimensional unital algebra. Its trace expectation is the $L^2$ nearest-point map, so $E_{D_N}(x)$ gives the same error bound and satisfies $\|E_{D_N}(x)\|\le\|x\|$.

Arbitrary finite bounded lists reduce to contractions by a common scalar rescaling. We have proved local AFD for every finite von Neumann algebra with the stated faithful normal tracial state. The countable construction is chosen for each finite test list; no countable dense subset, factoriality or separable-predual assumption is used. $\square$

Second proof, by maximality. Fix a finite contraction list $F\subset M$ and $\varepsilon>0$. Consider all locally AFD von Neumann subalgebras $A\subset M$, permitting identity $f\ne1$ and the zero algebra, which satisfy

Here $E_A(fxf)$ is the normal trace expectation in $fMf$, extended to $M$ by compression. Order these algebras by inclusion.

We verify the chain condition. For an increasing chain $A_i$, let $f_i$ be its supports and $f=\sup_i f_i$. Take $A$ to be the nonunital strong operator closure of $\bigcup_i A_i$ in $M$. If $f=0$, this is the zero algebra and already an upper bound. If $f\ne0$, the union acts nondegenerately on $fH$, since $f_i\to f$ strongly. Its bicommutant on $fH$, extended by zero on $(1-f)H$, is precisely $A$, with identity $f$. In particular we do not take the ambient bicommutant on all of $H$, which would adjoin the complementary scalar identity.

This $A$ is locally AFD: Kaplansky approximation for the nondegenerate algebra on $fH$ gives bounded $L^2$ approximants from the union to any finite list in $A$; approximate that finite list further inside one chain member. Add the scalar algebra $\mathbb C(f-f_i)$ to make the resulting finite algebra unital in $A$.

The closed spaces $L^2(A_i)$ increase to $L^2(A)$, so their orthogonal projections converge strongly. These projections are precisely $x\mapsto E_{A_i}(f_i x f_i)$. Also $(1-f_i)x(1-f_i)\to(1-f)x(1-f)$ in $L^2$, by bounded strong convergence and the finite trace. Therefore (13) passes to $A$. Every chain has an upper bound in the set, and Zorn's lemma supplies a maximal $A$.

To justify the projection limit directly, split a Hilbert vector into its projection onto the closed union and its orthogonal remainder. Every projection in the chain annihilates the remainder. Approximate the first part by a vector in one chain subspace; every later orthogonal projection fixes that approximant, and contractivity bounds the remaining error by its distance from the first part. This proves strong convergence for the whole directed chain. Bounded density on the support corner, applied to the trace vector, proves the required density of the union in the trace Hilbert space. Thus the argument uses nets and does not assume a countable chain.

Suppose $f\ne1$, and put $e=1-f$. Apply (12) in $eMe$ to $eFe$, with tolerance $\varepsilon/2$. Obtain $g\ne0$, $g\le e$, and a finite-dimensional $B\subset gMg$. For $y=exe$, set

Every block of $R$ has at least one $f$ coordinate, whereas $S\in eMe$; they are orthogonal in $L^2$. The three blocks of $S$ are

They are mutually orthogonal. The first two squared norms sum to $\|yg-gy\|_2^2$. Condition (12) therefore gives

For $L=A\oplus B$, identity $h=f+g$, its residual in (13) is $R+S$, so

The direct sum is locally AFD and strictly contains $A$, contradicting maximality. Thus $f=1$, and (13) gives $\|x-E_A(x)\|_2\le\varepsilon$ for all $x\in F$.

Approximate the finitely many $E_A(x)$'s by a finite-dimensional unital algebra of $A$. Choosing both this new error and the original $\varepsilon$ arbitrarily small proves local AFD of $M$. The expectation onto the final finite algebra supplies approximants of norm at most one. $\square$

For a separable $\mathrm{II}_1$ factor, local AFD also implies the small-corner property. The finite amplification and corner permanence proof is valid without separability. Inside a nonzero corner approximate the finite tests by a unital finite-dimensional algebra and take $f=e$; its commutator error is zero. Thus the source's LAFD and AFD conditions are equivalent at their stated hypotheses.

Obtaining small corners in injective factors

Proposition 2.2. Every injective $\mathrm{II}_1$ factor with separable predual has this property.

Proof. The almost invariant matrix-corner selection used here is proved in the next section. For finite unitary tests in $M$, Lemma 1.1, (11) and the two-parameter choice give (12), since $\|v s-sv\|_2=\|v s v^*-s\|_2$.

Every bounded element is a scalar linear combination of at most four unitaries. For a self-adjoint contraction $h$, write $h=(u+u^*)/2$ with $u=h+i(1-h^2)^{1/2}$; apply this to the real and imaginary parts and rescale. For a finite operator list, take the union of these unitary lists. Divide the unitary tolerance by the largest sum of absolute coefficients, using $1$ if that sum is zero. Linearity, the triangle inequality and linearity of $E_A$ give (12) for the operator list.

Finally every nonzero $eMe$ is an injective $\mathrm{II}_1$ factor with separable predual, by corner compression of a retraction and finite factor comparison. Repeat the argument with its normalized trace $\tau_e=\tau/\tau(e)$. Both sides of (12) acquire the same factor $\sqrt{\tau(e)}$ when converted to the ambient trace. $\square$

Here the corner assertions have separate elementary reasons. The center-of-a-corner proof makes a nonzero corner of a factor a factor. It has no minimal projection: a minimal projection in the corner would also be minimal in the original algebra. Its restricted trace is faithful, normal and finite. Every normal functional on the corner extends normally to the ambient algebra by compression, so restriction makes its predual a quotient of the separable ambient predual. Injectivity itself is the completely positive extension and compression argument of Lemma 6.1; it does not require a normal retraction.

Selecting an almost invariant matrix model

Fix a finite $F\subset\mathcal U(M)$ containing $1$, and first choose a finite-rank model

with $x_i\in M$ orthonormal in $L^2$. The number $a>0$ is the first approximation parameter. This choice fixes $m$ and $B\ge\max(1,\|x_i\|)$. Only after that choice do we make the Gram and pinching parameter $b$ small.

Use the preceding lesson's Corollary 3.1, and abbreviate its moduli by $D=D_m(b)$, $d=d_m(b)$. It gives orthogonal initial projections $e_k$, repaired vectors $w_i(k)$, matrix units $E_{ij}(k)$ and their supports $s_k$. Write $p=\sum_k e_k$. By making $b$ small with $m,B$ fixed, arrange

Then each $\|x_i p\|_2^2\ge1-B^2\tau(1-p)>1/2$.

Lemma 1.1. For some $k$, putting $s=s_k$ and

we have, for every $v\in F$,

Proof. Let $Q_k=\sum_i K_{x_i e_k,x_i}=R_{e_k}q$, where $R_{e_k}$ means right multiplication on the output vector. The operator $R_{e_k}$ is an orthogonal projection on $H$, commuting with every $L_v$. For each $v$, the operators $R_{e_k}(v q v^*-q)$ are orthogonal in Hilbert–Schmidt space. Thus

Orthonormality of the second vectors in $Q_k$ gives

Consequently there is $k$ with summed error less than $2a^2\|Q_k\|_{\mathrm{HS}}^2$. This $Q_k$ is nonzero.

Put $V=\sum_i K_{w_i(k),x_i}$. The same orthonormality and the repaired vector errors give

The triangle inequality, applied to each test and to its conjugate, therefore gives

The first inequality in (3) is the matrix coefficient estimate already proved. To obtain the second, expand the rank-one Hilbert–Schmidt pairing:

Indeed its cross term is $\sum_{i,j}\tau(w_i^*v w_j)\overline{\tau(x_i^*v x_j)}$, equal to $\tau(C_v^*v)$. Both $C_v$ and $s v s$ are supported by $s$. Hence

Also

Combining (7)–(10) proves the second bound in (3). $\square$

Let $A_k=\operatorname{span}\{E_{ij}(k)\}$, a matrix algebra with identity $s$. The $L^2$ nearest-point property of its trace expectation gives

Given any desired small tolerance $c>0$, first choose $a$ so $\sqrt2\,a<c/2$. Then fix the model (1), and decrease $b$ until (2) holds and both $\sqrt m\,d<c$ and $\sqrt{\eta^2+2\sqrt m\,d}<c$. This is possible because $m,B$ are now fixed and $D,d\to0$. Allowing the dimension of the first model to vary while taking the second limit would not justify these inequalities.

5. Exercises with complete solutions

Global approximation and residual budgets

Exercise 7. Why is chain closure of the algebras alone insufficient for the Zorn argument?

Solution. An upper bound must also satisfy the quantitative residual condition (13). The increasing $L^2$ subspaces give strong convergence of their expectation projections, while the increasing supports give convergence of the complementary compressions. These two limits and normality of the trace pass the inequality to the generated algebra.

Exercise 8. Verify the squared error budget (15).

Solution. The two off-diagonal blocks have squared sum $\|yg-gy\|_2^2\le(\varepsilon^2/4)\tau(g)$. The diagonal approximation error has the same squared bound. Orthogonality gives their sum at most $(\varepsilon^2/2)\tau(g)$. Its normalization uses the new support $g$.

Exercise 9. Why does the maximal algebra supply bounded finite-dimensional approximants to the original $x$'s?

Solution. At full support its expectation approximates the $x$'s within the first tolerance. Its local AFD property supplies a finite algebra $D$ approximating those finitely many expectations. The triangle inequality bounds the distance of each $x$ to $D$. The tracial nearest-point property bounds $\|x-E_D(x)\|_2$ by that distance, and contractivity gives $\|E_D(x)\|\le\|x\|$.

Exercise 10. Which nonfactor statement is proved here, and what remains open?

Solution. The small-corner-to-local-AFD implication of Theorem 3.1 holds in every finite algebra with faithful normal trace. The proof that an injective algebra has small corners uses scalar pinching in a separable $\mathrm{II}_1$ factor. Thus this route proves the full second factor argument, while the general nonfactor injective converse and its central decomposition consequences still need their separate proof.

Finite-rank selection and corner scales

Exercise 1. Why is the right multiplication projection $R_{e_k}$ orthogonal on $L^2(M,\tau)$?

Solution. It is idempotent because $e_k^2=e_k$. Its adjoint is itself: $\tau((\xi e_k)^*\zeta)=\tau(e_k\xi^*\zeta)=\tau(\xi^*\zeta e_k)$. For orthogonal $e_k,e_l$, the corresponding projections have zero product. Left multiplication commutes with them, since $v(\xi e_k)=(v\xi)e_k$.

Exercise 2. Prove the lower bound in (5).

Solution. Orthonormality of the second rank-one vectors $x_i$ removes all cross terms, giving $\|Q_k\|_{\mathrm{HS}}^2=\sum_i\|x_i e_k\|_2^2$. Summing over the orthogonal $e_k$'s gives $\sum_i\|x_i p\|_2^2$. Each is $1-\tau(x_i^*x_i(1-p))\ge1-B^2\tau(1-p)>1/2$.

Exercise 3. Justify selection of $k$ between (4) and (5).

Solution. If every nonzero $Q_k$ had summed error at least $2a^2\|Q_k\|_{\mathrm{HS}}^2$, the total error would exceed $a^2m$ by (5), contradicting (4). A zero $Q_k$ has zero conjugation error and contributes nothing, so it does not affect this argument.

Exercise 4. Check the conjugates in the cross term (8).

Solution. The rank-one pairing gives $\tau(w_i^*v w_j)\tau((v x_j)^*x_i)=\tau(w_i^*v w_j)\overline{\tau(x_i^*v x_j)}$. Since $E_{ji}=w_jw_i^*$, cyclicity gives $\tau(E_{ji}v)=\tau(w_i^*v w_j)$. Summing yields $\tau(C_v^*v)$.

Exercise 5. Explain the two independent parameters in the proof.

Solution. The finite-rank model chosen at tolerance $a$ determines $m$ and $B$. The Gram repair moduli tend to zero with $b$ only for those fixed values. First choose $a$ to make $\sqrt2a$ small, then fix the model, then make $b$ small enough to control $D_m(b)$, $\sqrt m\,d_m(b)$ and the uncovered trace. This proves every bound in (3) simultaneously.

Exercise 6. Convert (12) from a normalized corner trace to the ambient trace.

Solution. In $eMe$, $\|z\|_{2,e}=\|z\|_2/\sqrt{\tau(e)}$, while $\tau_e(f)=\tau(f)/\tau(e)$. Multiply an inequality $\|z\|_{2,e}\le\varepsilon\sqrt{\tau_e(f)}$ by $\sqrt{\tau(e)}$. Both factors cancel, leaving (12).

References and proof scope

Sorin Popa, A short proof of “injectivity implies hyperfiniteness” for finite von Neumann algebras, Journal of Operator Theory 16 (1986), 261–272, Proposition 2.2 and Section 3. The small-corner and maximality methods are credited to this work. Popa constructs finite approximation in general finite algebras through an abelian-module argument. The external Hilbert-space finite-rank selection used here instead retains the separable factor hypothesis of its scalar-pinching input.

Claire Anantharaman and Sorin Popa, An introduction to II₁ factors, author draft, Definition 11.1.4 and Theorem 11.1.5, pp.174 and 182–184, develop the abelian-module small-corner method. Theorem 11.1.17, pp.184–185, uses maximal orthogonal families to pass from local corners to finite approximation. The complete maximality proof here supplies support-corner closure, bounded density, the expectation-projection limit and the precise orthogonal residual budget.

The countable proof chooses each new corner with more than half the supremum of the traces available at the fixed tolerance. A hypothetical nonzero remaining corner gives a fixed smaller-tolerance candidate which eventually works at every finite stage. This forces the remaining trace to vanish. Finite orthogonal residual sums then give an actual finite-dimensional unital approximating algebra. This proof requires neither factoriality nor separable predual.

The final increasing dyadic sequence and the identification with $R$ use the complete exact-containment and dyadic construction. They are separate from the arbitrary finite local-to-global implication. The general nonfactor injective converse has its own central-disintegration and tracial-envelope proofs.