Semiregular MASAs and scalar pinching

Written by GPT-6.1 Sol (OpenAI), Ultra, September 2026. Self-checked by the writing AI. New original text: public domain (CC0).

A semiregular MASA has enough normalizers to match any two of its projections of equal trace. Compatible matches build finite matrix algebras whose diagonals approximate the entire MASA. Their closure is an irreducible AFD subfactor. A second construction, using decreasing irreducible AFD factors, turns finite pinching into simultaneous approximation by the ambient trace.

We use the relative pinching preparation, the expected-subfactor construction, and the already proved decreasing AFD subfactors. General trace expectations are the selected modular-course prerequisite. Factor and separability hypotheses are stated separately for each result.

1. Matching equal-trace projections by normalizers

Let $N$ be a $\mathrm{II}_1$ factor, $\tau$ its normalized trace, and $A\subset N$ a semiregular MASA. Put

By definition $P$ is a factor, and $A\subset P$ is a MASA. In this section separability is unnecessary.

A partial normalizer is a partial isometry $v\in P$ with initial and final projections in $A$ and

Lemma 1.1. For $p,q\in\operatorname{Proj}(A)$,

More generally equal-trace projections have a partial normalizer with precisely those initial and final projections.

Proof. Trace preservation gives necessity. For existence, first note that if $p,q\ne0$, some $u\in G$ has $qup\ne0$. Indeed

is a nonzero projection of $A$, invariant under $G$, hence central in $P$. Factorality gives $r=1$. If all $qup$ were zero, $q$ would be orthogonal to $r$.

For such a $u$, $qup$ is already a partial isometry: its initial projection is $p\wedge u^*qu$, and its final projection is $q\wedge upu^*$. Both are nonzero projections of $A$; this uses its abelianness. It is a partial normalizer.

Take a maximal family of these partial normalizers with mutually orthogonal initial projections under $p$ and mutually orthogonal final projections under $q$. A finite trace makes this family countable after removing zero members. Their strong sum $v$ is a partial isometry satisfying (2): the off-diagonal terms in $vav^*$ vanish because the initial supports are orthogonal projections in $A$, and similarly for $v^*av$.

The unmatched remainders have equal trace. If either were nonzero, both would be, and the preceding argument would supply another match. Maximality rules this out. Thus $v^*v=p$, $vv^*=q$.

Match $1-p$ with $1-q$ as well, obtaining $w$. Then $u=v+w$ is unitary and normalizes $A$, since its two domain and range partitions belong to $A$. It gives (3). $\square$

Lemma 1.2. If $0\ne e\in A$, then $Ae$ is a regular MASA in the factor $ePe$.

Proof. It is maximal abelian by the corner commutant lemma applied to $A\subset P$. The compressed operators $eue$, $u\in G$, span an ultraweakly dense subspace of $ePe$, because $G$ is a group generating $P$. Each $v=eue$ is a partial normalizer of $Ae$. Its initial and final supports have equal trace. Apply the matching proof of Lemma 1.1 to their complements inside $e$; the resulting partial normalizer lies in $ePe$. Adding it to $v$ gives a unitary normalizer $u_e$ of $Ae$, and $v=u_e(v^*v)$.

Thus every $eue$ lies in the algebra generated by the unitary normalizers of $Ae$ and its projections. The latter projections already belong to that algebra, since the unitaries of $Ae$ normalize it. Their ultraweak span gives all of $ePe$. $\square$

2. Choosing diagonals compatibly

A MASA in a $\mathrm{II}_1$ factor is diffuse. If $e$ were one of its atoms, maximal abelianness in the corner would give $eNe=\mathbb Ce$, a minimal projection in $N$, contradicting type II.

For later use, a diffuse abelian finite algebra has projections of any prescribed trace between zero and the trace of its identity. To see this, partially order the projections of trace at most $t$ by inclusion. Normality gives an upper bound to every chain. A maximal member whose trace is less than $t$ could be enlarged by a sufficiently small nonzero projection in its complement. Such small projections exist by repeatedly splitting a nonzero projection and choosing a piece of at most half its trace. This contradiction proves exact trace $t$.

Lemma 2.1. In a diffuse abelian finite algebra $B$, a finite set can be approximated arbitrarily well in normalized $2$-norm by a diagonal algebra generated by $2^k$ equal-trace atoms. Its size can be made arbitrarily large.

Proof. Joint finite spectral partitions approximate the prescribed elements by simple functions on finitely many atoms. Approximate their atom traces by positive dyadic weights with common denominator $2^k$. Move small subprojections from excess atoms to deficit atoms, using the trace cuts just proved. The total moved trace tends to zero, so the adjusted simple functions retain their $2$-norm approximations. Split an adjusted atom of trace $m_i2^{-k}$ into $m_i$ pieces of trace $2^{-k}$. Their diagonal contains the adjusted simple functions. Increasing $k$ supplies any required size. $\square$

Theorem 2.2. If $N$ has separable predual and $A\subset N$ is a semiregular MASA, there is an AFD $\mathrm{II}_1$ subfactor

The constructed $A$ is regular in $R$.

Proof. Choose a $2$-norm dense sequence $(a_j)$ in the unit ball of $A$. We construct increasing dyadic matrix factors $F_n\subset P$, with diagonal $A_n\subset A$, such that their matrix units are partial normalizers of $A$ and

Start with $F_0=\mathbb C1$. Suppose $F_n\cong M_d$, with units $e_{ij}$, is chosen. Put $e=e_{11}$. Every $a\in A$ is diagonal with respect to the old partition, and the partial normalizer property puts

Choose equal dyadic atoms $f_1,\ldots,f_s$ in $Ae$ whose diagonal $B$ approximates this finite list in normalized corner $2$-norm below $2^{-(n+1)}$, by Lemma 2.1.

Lemma 1.1 matches $f_1$ with each $f_t$ by a partial normalizer $v_t\in ePe$; take $v_1=f_1$. Then $f_{st}=v_s v_t^*$ are full matrix units with the chosen diagonal and retain the partial normalizer property. Propagate them across the old rows:

Products of partial normalizers retain (2). These units generate a dyadic factor $F_{n+1}$ containing $F_n$, with diagonal $A_{n+1}\subset A$. Reconstruct each $a_j$ by its diagonal entries (6), using their simple-function approximants in $B$. Their errors are orthogonal, and the ambient squared $2$-norm is $1/d$ times the sum of the normalized corner squared norms. This proves (5) at the next stage. Choose $s\ge2$.

Let $R=(\bigcup_n F_n)''$. Its increasing full matrix factors and restricted trace make it a factor, by the finite increasing-factor lemma. The dyadic trace-GNS tensor identification gives $R\cong R_{\mathrm{hyperfinite}}$. In the reconstruction for (5), use the trace expectation $E_B(b_{ij})$ for each entry; its nearest-point property preserves the error bound and its contractivity bounds the reconstructed operator by $\|a_j\|$. Equation (5) therefore puts $A\subset R$, since bounded $2$-norm convergence is strong convergence on the finite trace representation and $R$ is closed.

Every finite permutation of the matrix diagonal is a sum of the partial normalizers in (7); it is therefore a unitary normalizer of the whole $A$. Together with diagonal phases these permutations generate each $F_n$, hence $R$. Thus $A$ is regular in $R$.

Finally, if $x\in R'\cap N$, it commutes with $A$, so $x\in A\subset R$. Factorality of $R$ gives $x\in Z(R)=\mathbb C$, proving (4). $\square$

The diagonals in this construction are chosen in the current first corner and then transported to the other rows. This is the compatibility needed for induction. An arbitrary list of previously chosen Bernoulli coordinate projections need not remain in the commutant of earlier matrix units; no such membership is assumed here.

3. Scalar pinching in an irreducible finite subfactor

Theorem 3.1. Let $M$ be a $\mathrm{II}_1$ factor with separable predual, let $\tau$ be its normalized trace, and let $N\subset M$ be a unital subfactor with $N'\cap M=\mathbb C$. For a finite $F\subset M$ and $\varepsilon>0$, there is a finite partition $e_1,\ldots,e_r$ of $1$ in $N$ such that

Neither $N$ nor $M$ is assumed AFD.

Proof. Trace-preserving expectations onto unital subalgebras exist by the modular expectation theorem. If $N$ is type I, irreducibility and its matrix decomposition force $M=N$, which is impossible for type II. Thus $N$ is $\mathrm{II}_1$. The expected-subfactor construction gives an AFD $R_0\subset N$ containing a MASA of $M$; consequently $R_0'\cap M=\mathbb C$. Equivalently one can use that MASA and Theorem 2.2.

The decreasing AFD theorem supplies

Their trace GNS projections in $L^2(M,\tau)$ decrease to the scalar projection. To identify the intersection, use the bounded cluster argument of the pinching lesson: for $x\in M$, $E_{R_n}(x)$ is bounded, its GNS vectors converge, and any ultraweak cluster point belongs to every $R_n$; the normal pairings identify the vector limit. It follows that

Choose $n$ so that the square sum of these errors over $F$ has square root below $\varepsilon/4$.

Put $y_x=E_{R_0}(x)$ and $z_x=x-y_x$. Within $R_0$, apply the simultaneous pinching theorem to $R_n$ and

Its relative commutant in $R_0$ is scalar, so that theorem applies to this whole kernel. It gives a finite partition $f_1,\ldots,f_s\in R_n$ with

Here $T_f(b)=\sum_j f_jbf_j$. For example choose its common relative error below $\varepsilon/(4(1+Y))$, where $Y$ is the square sum norm of the finite list. Its $2$-norm contractivity, (10) and (11) yield

It remains to suppress $z_x$, the part outside $R_0$. In each nonzero corner $f_jMf_j$, the algebra $f_jR_0f_j$ is irreducible by the corner commutant lemma. Its trace expectation is the restriction of $E_{R_0}$, so $f_jz_xf_j$ belongs to its kernel. Apply simultaneous pinching there and refine $f_j$ into a finite partition $(e_{jk})\subset f_jR_0f_j$.

Use a common relative error less than $\varepsilon/(2(1+Z))$, where $Z=(\sum_{x\in F}\|z_x\|_2^2)^{1/2}$. Normalized corner norms are converted to ambient squared norms by multiplying by $\tau(f_j)$. Corner orthogonality and contractivity of $T_f$ consequently give, for the combined partition $e=(e_{jk})$,

If $Z=0$, no additional refinements are needed for this estimate.

Since $e$ refines $f$, $T_e=T_eT_f$, and $T_e$ fixes scalars. Thus (12) stays valid with $T_e$ in place of $T_f$. Add that bound and (13) in the Hilbert direct sum over $F$. This gives (8). Every selected projection lies in $R_0\subset N$. $\square$

The two refinement stages have different jobs. The first suppresses the nonscalar part already inside $R_0$, after a decreasing expectation has become nearly scalar. The second removes the part orthogonal to $R_0$. Irreducibility inside $R_0$ in the first stage is not replaced by an unsupported irreducibility assertion about $R_n$ in $M$.

4. Exercises with complete solutions

Exercise 1. Prove that the orbit join $r$ in Lemma 1.1 belongs to $Z(P)$.

Solution. Every orbit projection belongs to the abelian algebra $A$, so their join belongs to $A\subset P$. A normalizer permutes the orbit and fixes the join, hence $r$ commutes with every normalizer. Those unitaries generate $P$, so $r\in P'\cap P=Z(P)$.

Exercise 2. Compute the initial and final projections of $qup$.

Solution. Since $p$ and $u^*qu$ are projections of the same abelian algebra, their product is their meet. Thus $(qup)^*(qup)=p u^*qu p=p\wedge u^*qu$. The reverse product is $q u p u^*q=q\wedge upu^*$. These products are projections, making $qup$ a partial isometry.

Exercise 3. Why does a finite trace make the nonzero members of the matching family countable?

Solution. Their nonzero initial supports have positive traces and are mutually orthogonal under $p$. For each integer $k$, only finitely many can have trace at least $1/k$. Every positive trace is at least $1/k$ for some $k$, so the family is a countable union of finite sets.

Exercise 4. Show that the strong sum of orthogonal partial normalizers still satisfies (2).

Solution. If $v_i^*v_i=p_i$ are orthogonal in $A$, then for $a\in A$ the cross term $v_i a v_j^*=v_i p_i a p_jv_j^*=0$ when $i\ne j$. Thus $vav^*$ is the bounded strong sum of elements $v_i a v_i^*\in A$, and lies in $A$. Orthogonal final supports give the reverse inclusion in (2) by the same argument.

Exercise 5. Explain why a MASA of a type II finite factor cannot have an atom.

Solution. If $e$ were an atom, $Ae=\mathbb Ce$. The corner relative commutant identity gives $(Ae)'\cap eNe=Ae$. But a scalar corner algebra commutes with all of $eNe$, so $eNe=\mathbb Ce$. That makes $e$ a minimal projection of $N$, contrary to type II.

Exercise 6. Why are the entries in (6) in $Ae$?

Solution. The matrix unit $e_{1i}$ is a partial normalizer of $A$, so $e_{1i}a_j e_{i1}\in A$. Its left and right support are below $e_{11}=e$. Hence it belongs to $Ae$. This is stronger than membership in the ambient corner algebra.

Exercise 7. Check that a finite matrix permutation built from partial normalizers normalizes all of $A$.

Solution. For $u=\sum_i e_{\sigma(i),i}$, its initial supports partition $1$ in $A$. In $uau^*$, the mixed terms vanish because $a$ commutes with those supports. Each diagonal term belongs to $A$ by the partial normalizer property, so $uAu^*\subset A$. Apply the same argument to $u^*$ to obtain equality.

Exercise 8. Why can the next diagonal not be chosen as an arbitrary previously fixed binary coordinate?

Solution. It must lie in the current first corner, and its propagated copies must commute with the old matrix algebra in the specified tensor embedding. Old normalizing unitaries can act nontrivially on any previously fixed unused coordinate. Choosing the partition after pulling the finite approximation targets into the current corner, as in (6), supplies the needed compatibility.

Exercise 9. Identify the limit of the decreasing projections in (10).

Solution. For a fixed bounded $x$, their expected operators remain bounded. A cluster point lies in every $R_n$, hence in $\mathbb C$. Trace preservation makes it $\tau(x)1$. Normal GNS pairings identify its vector with the strong projection limit. Density of bounded vectors gives the scalar projection on all of $L^2(M,\tau)$.

Exercise 10. Why is the first application of simultaneous pinching made inside $R_0$?

Solution. The known relative commutant is $R_n'\cap R_0=\mathbb C$, which makes its relevant join equal to $R_n$ there. The operators $y_x-E_{R_n}(x)$ belong to $R_0$ and have zero expectation onto $R_n$. The proof does not establish $R_n'\cap M=\mathbb C$, so an ambient application at that stage would not be justified.

Exercise 11. Verify that refinement preserves (12).

Solution. If $e$ refines $f$, every fine projection is under one $f_j$, so $T_eT_f=T_e$. Since $T_e$ is an orthogonal projection on trace GNS space and fixes $1$, $\|T_e(y_x)-\tau(x)1\|_2\le\|T_f(y_x)-\tau(x)1\|_2$. Squaring and summing gives the assertion.

Exercise 12. Convert a normalized corner error to an ambient one in (13).

Solution. In the corner $f_jMf_j$, the normalized trace is $\tau(f_j)^{-1}\tau$. An error of squared normalized norm $c^2$ therefore has squared ambient norm $\tau(f_j)c^2$. Multiply each corner estimate by this weight before adding. The total is controlled by the ambient squared norms of the original orthogonal diagonal corners, not by the number of corner partitions.

Reading and prerequisites

Claire Anantharaman and Sorin Popa, An introduction to II₁ factors, author draft. Lemma 12.1.2, printed p.192, proves extension of partial normalizers by a maximal orthogonal family. Lemmas 1.1–1.2 above supply the complete orbit-join matching argument in the normalizer factor and its corner, with no separability assumption. Lemma 2.1 also supplies exact diffuse abelian trace cuts and dyadic movements; no equal-atom approximation is left as an exercise.

Sorin Popa, On a problem of R. V. Kadison on maximal abelian *-subalgebras in factors, INCREST preprint 41/1981, May 1981, second version. Theorem 1 and its compatible matrix construction, printed pp.10–13, build a particular regular MASA inside an irreducible AFD subfactor. Here Theorem 2.2 starts with an arbitrary given semiregular MASA, matches its projections by its own partial normalizers, and proves that its entire algebra is contained in the constructed factor. These are distinct hypotheses, justified by the matching and first-corner induction above.

The scalar pinching proof combines the freely accessible finite tracial pinching mechanism with the explicit decreasing AFD construction. That construction uses finite outer actions and Bernoulli mixing. Both refinement stages, their relative commutants, normalized corner errors and the scalar expectation limit are given above, retaining arbitrary irreducible subfactors of a separable finite factor. No AFD assumption on either given factor is added. Trace expectations, projection comparison and bounded trace-GNS topology remain declared foundations; their transitive source verification is pending.