64c3814
==== TABLE OF CONTENTS ====
Relative-generation-and-the-generator-problem-for-finite-factors//main.tex
Relative-generation-and-the-generator-problem-for-finite-factors//preamble.tex
Relative-generation-and-the-generator-problem-for-finite-factors//sections/consequences.tex
Relative-generation-and-the-generator-problem-for-finite-factors//sections/generation.tex
Relative-generation-and-the-generator-problem-for-finite-factors//sections/independence.tex
Relative-generation-and-the-generator-problem-for-finite-factors//sections/introduction.tex
Relative-generation-and-the-generator-problem-for-finite-factors//sections/perturbation.tex
Relative-generation-and-the-generator-problem-for-finite-factors//references.bib
===========================
==== BEGIN Relative-generation-and-the-generator-problem-for-finite-factors//main.tex ====
\documentclass[11pt,letterpaper]{article}
\input{preamble}
\title{Relative generation and the generator problem\\for finite factors}
\author{OpenAI}
\date{September 23, 2026}
\begin{document}
\maketitle
\begin{abstract}
We prove that every type $\mathrm{II}_1$ factor with separable predual is
generated by one operator, equivalently by two self-adjoint operators.
Combined with the established direct-integral reduction to this case,
this gives an affirmative solution of the generator problem for von
Neumann algebras with separable predual.
\end{abstract}
\input{sections/introduction}
\input{sections/independence}
\input{sections/perturbation}
\input{sections/generation}
\input{sections/consequences}
\clearpage
\bibliographystyle{abbrvnat}
\begingroup
\small
\raggedright
\bibliography{references}
\endgroup
\end{document}
==== END Relative-generation-and-the-generator-problem-for-finite-factors//main.tex ====
==== BEGIN Relative-generation-and-the-generator-problem-for-finite-factors//preamble.tex ====
\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc}
\usepackage{mathpazo}
\usepackage[margin=1.05in]{geometry}
\usepackage{microtype}
\usepackage{amsmath,amssymb,amsthm,mathtools,mathrsfs}
\usepackage{booktabs,array,graphicx}
\usepackage[numbers,sort&compress]{natbib}
\usepackage{xcolor}
\definecolor{linkblue}{RGB}{25,65,112}
\usepackage[colorlinks=true,linkcolor=linkblue,citecolor=linkblue,urlcolor=linkblue]{hyperref}
\urlstyle{same}
\hypersetup{pdftitle={Relative generation and the generator problem for finite factors},pdfauthor={OpenAI},pdfsubject={An affirmative solution of the generator problem for separable-predual type II1 factors},pdflang={en-US}}
\pdfinfoomitdate=1
\pdftrailerid{}
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\theoremstyle{definition}
\newtheorem{definition}[theorem]{Definition}
\theoremstyle{remark}
\newtheorem{remark}[theorem]{Remark}
\newcommand{\norm}[1]{\lVert #1\rVert}
\newcommand{\abs}[1]{\lvert #1\rvert}
\newcommand{\R}{\mathbb R}
\newcommand{\N}{\mathbb N}
\setlength{\emergencystretch}{2em}
\allowdisplaybreaks[2]
==== END Relative-generation-and-the-generator-problem-for-finite-factors//preamble.tex ====
==== BEGIN Relative-generation-and-the-generator-problem-for-finite-factors//sections/consequences.tex ====
\section{Consequences for generation and free entropy}
\label{sec:consequences}
The universal statement of Theorem~\ref{thm:main} has consequences
beyond the relative generation theorem. We first apply classical
reductions, then compare two generating tuples of a free group factor.
\subsection{General von Neumann algebras and generator invariants}
Willig's direct-integral reduction~\cite[Corollary~2]{Willig1974},
together with the established type~I and properly infinite cases,
reduces the generator problem for von Neumann algebras with separable
predual to the $\mathrm{II}_1$ factor case. Theorem~\ref{thm:main}
therefore gives single generation for every von Neumann algebra with
separable predual.
Let $G$ and $G_{\mathrm{sa}}$ be the generator invariants defined in
\cite[Definition~2.7]{DSSW2008}. They are nonnegative infima over
finite generating sets and finite projection partitions;
$G_{\mathrm{sa}}$ restricts the generating sets to self-adjoint
elements. For every complex $\mathrm{II}_1$ factor $M$ with separable
predual and normalized trace, we have
\[
G(M)=G_{\mathrm{sa}}(M)=0.
\]
Indeed, by \cite[Corollary~5.1]{DSSW2008}, a positive $G$-value for
one such factor would force a nonsingly generated factor in the same
class, contrary to the universal assertion of
Theorem~\ref{thm:main}. The cited generation convention need not
include the unit. To pass to that convention, replace a generator
$x$ by $y=x+\lambda1$ with $\lambda>\norm{x}$. The invertible $y$
has support projection $1$ in its possibly nonunital von Neumann
closure; that closure then contains $x$ and equals $M$.
Nonnegativity gives $G(M)=0$, and
\cite[Theorem~5.5]{DSSW2008} gives
$G_{\mathrm{sa}}(M)=2G(M)=0$. These values are the defining infima;
no generating set or projection partition attaining zero is asserted.
\subsection{Dependence of free entropy dimensions on generators}
For a bounded finite self-adjoint tuple $Z$ in a tracial von Neumann
algebra, let $\delta(Z)$ and $\delta_0(Z)$ denote Voiculescu's
microstates free entropy dimension and its modified version,
respectively~\cite[Section~2.4]{Voiculescu2002Entropy}.
The $W^*$-invariance question asks whether either number is the same
for any two finite self-adjoint tuples generating the same tracial
von Neumann algebra; see~\cite[Section~2.6]{Voiculescu2002Entropy}.
\begin{corollary}\label{cor:entropy-generators}
For every integer $n\geq3$, the factor $L(\mathbb F_n)$ with its
canonical trace has two finite self-adjoint generating tuples on
which both $\delta$ and $\delta_0$ take different values. In
particular, neither dimension is determined by the generated
tracial von Neumann algebra.
\end{corollary}
\begin{proof}
For a bounded self-adjoint $d$-tuple $Z$, one has
$\delta(Z)\leq d$ and $\delta_0(Z)\leq d$. Both dimensions equal
$n$ for a freely independent variance-one semicircular $n$-tuple;
see~\cite[Section~2.4(a), (c), and (d)]{Voiculescu2002Entropy}
for $\delta$ and~\cite[Section~5]{BrannanEtAl2023} for $\delta_0$.
The standard semicircular $n$-tuple generates $L(\mathbb F_n)$ as
a von Neumann algebra. A self-adjoint pair supplied by
Theorem~\ref{thm:main} generates the same factor and has each
dimension at most $2<n$. These two tuples give the assertion.
\end{proof}
A separate isomorphism argument in
\emph{An isomorphism of the free group factors}
\cite[Corollary~7.1]{OpenAIFreeGroupFactors2026} gives a stronger
fixed-factor conclusion: each of $\delta,\delta_0$ and the
nonmicrostates dimensions $\delta^*,\delta^\star$ attains every
integer value $n\geq2$ on finite self-adjoint $W^*$-generating tuples
of $L(\mathbb F_2)$. That conclusion uses the trace-preserving
isomorphisms between free group factors established in the cited
paper. Corollary~\ref{cor:entropy-generators} uses only the
single-generation theorem proved here and the standard dimension
bounds above.
==== END Relative-generation-and-the-generator-problem-for-finite-factors//sections/consequences.tex ====
==== BEGIN Relative-generation-and-the-generator-problem-for-finite-factors//sections/generation.tex ====
\section{Relative generation by continuity}
\label{sec:relative-generation}
We now convert the correlation supplied by Proposition~\ref{prop:perturbation}
into generation. Throughout this section, $P\subset M$ is an irreducible
inclusion of $\mathrm{II}_1$ factors and $M$ has separable predual.
\begin{lemma}
\label{lem:unitary-completeness}
The Hilbert space $L^2(M,\tau)$ is separable, and
$\mathcal U(M)$ is complete for the metric $d(u,v)=\norm{u-v}_2$.
\end{lemma}
\begin{proof}
The unit ball of $M$ is compact and metrizable in the weak-star topology,
so it has a countable weak-star dense subset $S$. For each
$\xi\in L^2(M,\tau)$, the functional
$a\mapsto\langle a,\xi\rangle$ on $M$ is normal. Indeed, normal
functionals form a norm-closed subspace of $M^*$, and approximation
of $\xi$ by elements of $M$ in $L^2$ gives norm approximation of
this functional by normal functionals, since
\[
\sup_{\norm{a}\leq1}
|\langle a,\xi-\eta\rangle|
\leq\norm{\xi-\eta}_2.
\]
If $\xi$ is orthogonal to $S$, normality implies that it is orthogonal to
the unit ball, and hence to $M$. Thus $S$ has dense linear span in
$L^2(M,\tau)$, proving separability.
Let $(u_n)$ be a Cauchy sequence in $\mathcal U(M)$ for $d$.
The adjoints are also Cauchy, because
$\norm{u_n^*-u_m^*}_2=\norm{u_n-u_m}_2$.
For $a\in M$,
\[
\norm{(u_n-u_m)a}_2\leq\norm{a}\,\norm{u_n-u_m}_2.
\]
The left multiplication operators associated with $u_n$ therefore
converge strongly on the dense subspace $M\subset L^2(M,\tau)$.
Their uniform operator norm bound extends this convergence to the whole
Hilbert space. The same holds for the left multipliers associated with
$u_n^*$. Strong closedness of the left regular representation of $M$
gives limits $u,v\in M$, and passage to inner products gives $v=u^*$.
Multiplication is jointly strongly continuous on bounded sets, so
$uv=vu=1$. Finally, convergence on the vector $1$ gives
$\norm{u_n-u}_2\to0$.
\end{proof}
For $u\in\mathcal U(M)$, write
\[
N_u=W^*(P,u),\qquad
e_u:L^2(M,\tau)\longrightarrow L^2(N_u,\tau)
\]
for the orthogonal projection. For $\xi\in L^2(M,\tau)$, put
$f_\xi(u)=\norm{e_u\xi}_2$.
If $N_u$ is proper, there is a nonzero bounded vector orthogonal to
$L^2(N_u,\tau)$. The perturbation proposition will keep its projection
norm bounded away from zero along a sequence of unitaries converging to
$u$, so we seek points at which every $f_\xi$ is continuous simultaneously.
\begin{lemma}
\label{lem:projection-continuity}
Each $f_\xi$ is bounded and lower semicontinuous on
$(\mathcal U(M),d)$. There is a dense $G_\delta$ subset
$\mathcal C\subset\mathcal U(M)$ such that every $f_\xi$,
$\xi\in L^2(M,\tau)$, is continuous at every point of $\mathcal C$.
\end{lemma}
\begin{proof}
Let $\mathcal P_P$ denote the family of finite $*$-polynomials in a
unitary variable with coefficients in $P$. For each fixed $u$, the
values $b(u)$, $b\in\mathcal P_P$, are dense in $L^2(N_u,\tau)$.
To see this, take the operator norm closure of the polynomial algebra.
Its von Neumann closure is $N_u$, and Kaplansky density approximates
each element of $N_u$ strongly by bounded elements of this
$C^*$-algebra. Evaluation on $1\in L^2(M,\tau)$ gives $2$-norm
approximation, which can then be followed by operator norm
approximation by polynomials. The density of $N_u$ in $L^2(N_u,\tau)$
finishes the assertion.
For fixed $b\in\mathcal P_P$, the map $u\mapsto b(u)$ is continuous
in $2$-norm. This follows by telescoping products, using the fixed
operator norm bounds of the coefficients, the fact that unitaries
have norm one, and the identity
$\norm{u^*-v^*}_2=\norm{u-v}_2$. Consequently
\[
d_\xi(u):=\operatorname{dist}\bigl(\xi,L^2(N_u,\tau)\bigr)
=\inf_{b\in\mathcal P_P}\norm{\xi-b(u)}_2
\]
is upper semicontinuous: an infimum of any family of continuous
real functions is upper semicontinuous. Orthogonality gives
\[
f_\xi(u)^2=\norm{\xi}_2^2-d_\xi(u)^2,
\]
so $f_\xi$ is lower semicontinuous and
$0\leq f_\xi(u)\leq\norm{\xi}_2$.
For completeness, a bounded lower semicontinuous real function $f$
on a complete metric space is continuous on a dense $G_\delta$ set.
For $\varepsilon>0$, let $G_\varepsilon$ be the union of all open
sets on which the oscillation of $f$ is less than $\varepsilon$.
This union is dense. Indeed, in any nonempty open set $O$, write
$s=\sup_O f$ and choose $x\in O$ with
$f(x)>s-\varepsilon/3$. Lower semicontinuity gives an open
neighborhood $V\subset O$ of $x$ on which
$f>f(x)-\varepsilon/3$. The oscillation on $V$ is less than
$\varepsilon$. Baire's theorem now shows that
$\bigcap_{n\geq1}G_{1/n}$ is dense, and $f$ is continuous at each
of its points.
Apply this construction to $f_\eta$ for $\eta$ in a countable dense
subset of $L^2(M,\tau)$ and intersect the resulting dense
$G_\delta$ sets. By Baire's theorem, their intersection
$\mathcal C$ is again a dense $G_\delta$ set.
For arbitrary $\xi,\eta\in L^2(M,\tau)$, contractivity of $e_u$ gives
\begin{equation}
\label{eq:projection-vector-lipschitz}
|f_\xi(u)-f_\eta(u)|\leq\norm{\xi-\eta}_2
\qquad(u\in\mathcal U(M)).
\end{equation}
Thus, for $u\in\mathcal C$ and $v\in\mathcal U(M)$,
\[
|f_\xi(v)-f_\xi(u)|
\leq2\norm{\xi-\eta}_2+|f_\eta(v)-f_\eta(u)|.
\]
Approximating $\xi$ by an $\eta$ from the countable dense subset,
and then letting $v\to u$, proves continuity of every $f_\xi$ at $u$.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{thm:relative-generation}]
Take $u\in\mathcal C$ from Lemma~\ref{lem:projection-continuity},
and suppose that $N_u\ne M$. There is a unitary $z\in M\setminus N_u$,
since the unitaries linearly span $M$. Let $E_{N_u}:M\to N_u$ be the
trace-preserving conditional expectation, and put
\[
\xi=z-E_{N_u}(z)\in M,\qquad s=\norm{\xi}_2>0.
\]
The conditional expectation is the restriction to $M$ of $e_u$.
Therefore
\[
e_u\xi=0,\qquad f_\xi(u)=0,\qquad
\tau(\xi^*z)=s^2.
\]
Fix
\begin{equation}
\label{eq:fixed-alpha-generation}
0<\alpha<\min\{1/4,s/32\}.
\end{equation}
For every integer $\ell\geq1$, Proposition~\ref{prop:perturbation}
provides a unitary $u_\ell\in M$ and unitaries
$t_{0,\ell},\ldots,t_{\ell,\ell}\in P$ such that, with
\[
W_\ell=t_{0,\ell}\prod_{k=1}^\ell(u_\ell t_{k,\ell})\in\mathcal U(N_{u_\ell}),
\]
one has
\begin{align}
\norm{u_\ell-u}_2
&\leq\frac{\alpha}{\sqrt{2\ell}}+\frac1\ell,
\label{eq:generation-perturbation-size}\\
\left|\tau(\xi^*W_\ell)-\frac{\alpha s^2}{2}\right|
&\leq s\frac{4\alpha^2}{1-2\alpha}+\frac1\ell.
\label{eq:generation-correlation}
\end{align}
The fixed choice~\eqref{eq:fixed-alpha-generation} gives
\[
s\frac{4\alpha^2}{1-2\alpha}
<8s\alpha^2<\frac{\alpha s^2}{4}.
\]
It follows from~\eqref{eq:generation-correlation} that
\[
|\tau(\xi^*W_\ell)|\geq\frac{\alpha s^2}{4}-\frac1\ell
\geq\frac{\alpha s^2}{8}
\]
for all sufficiently large $\ell$. On the other hand,
$\norm{W_\ell}_2=1$ and $W_\ell\in L^2(N_{u_\ell},\tau)$, so
\[
|\tau(\xi^*W_\ell)|
=|\langle W_\ell,e_{u_\ell}\xi\rangle|
\leq f_\xi(u_\ell).
\]
By~\eqref{eq:generation-perturbation-size}, $u_\ell\to u$ in
$2$-norm. Continuity of $f_\xi$ at $u$ gives
$f_\xi(u_\ell)\to f_\xi(u)=0$, a contradiction. Thus
$M=W^*(P,u)$ for every $u\in\mathcal C$.
To see that the full set of relative generators is $G_\delta$, choose
$(\eta_j)_{j\geq1}$ dense in $L^2(M,\tau)$. With $d_{\eta_j}$ as in
Lemma~\ref{lem:projection-continuity}, this set is
\[
\bigcap_{j,n\geq1}\{v\in\mathcal U(M):d_{\eta_j}(v)<1/n\}.
\]
Each set in the intersection is open by upper semicontinuity.
Vanishing of every distance is equivalent to
$L^2(N_v,\tau)=L^2(M,\tau)$, hence to $N_v=M$ by the
trace-preserving conditional expectation. The set is dense because
it contains $\mathcal C$.
\end{proof}
\section{Two self-adjoint generators}
\label{sec:two-generators}
Relative generation over an irreducible hyperfinite subfactor will first
give four self-adjoint generators for every separable $\mathrm{II}_1$
factor. We then apply this
uniform bound to a half-trace corner $N$ of $M$: the identification
$M\cong M_2(N)$ allows the four generators of $N$ to be encoded by two
self-adjoint matrices.
We use Popa's irreducible hyperfinite embedding theorem in the
following form.
\begin{theorem}[Popa]
\label{thm:irreducible-hyperfinite}
Let $N\subset M$ be an irreducible inclusion of $\mathrm{II}_1$
factors with separable preduals. There is a hyperfinite
$\mathrm{II}_1$ subfactor $R\subset N$ such that
$R'\cap M=\mathbb C1$.
\end{theorem}
The special case $N=M$ is
\cite[Corollary~4.1]{Popa1981}; it is also used explicitly in
\cite[proof of Theorem~3.8]{Sherman2012}. The relative form above is
recalled in~\cite[Introduction]{Popa2021Ergodic}.
The two-generator result for the hyperfinite factor is classical
\cite{SuzukiSaito1963}. We give a tensor-model proof for the later
count of generators.
\begin{lemma}
\label{lem:hyperfinite-generators}
The hyperfinite $\mathrm{II}_1$ factor with separable predual is
generated by two self-adjoint elements.
\end{lemma}
\begin{proof}
By uniqueness of the hyperfinite $\mathrm{II}_1$ factor with separable
predual \cite[Section~1.3]{Popa2014}, we may use its standard tensor model
\[
R=\overline{\bigotimes}_{m\geq1}
\bigl(M_2(\mathbb C),\operatorname{tr}_2\bigr).
\]
On the $m$th tensor site, let
\[
d_m=\begin{pmatrix}1&0\\0&0\end{pmatrix},\qquad
q_m=\frac12\begin{pmatrix}1&1\\1&1\end{pmatrix},
\]
with the identity on the other sites. On that site,
$2d_mq_m(1-d_m)$ is the matrix unit $e_{12}$, so $d_m,q_m$
generate the full matrix algebra. Each of the two families
$(d_m)$ and $(q_m)$ consists of commuting projections.
For any countable family of commuting projections $(e_m)$, the
operator norm convergent series
\[
b=\sum_{m\geq1}2\,3^{-m}e_m
\]
recovers every $e_m$ by spectral calculus. Inductively, after
recovering $e_1,\ldots,e_{m-1}$, form
\[
r_m=b-\sum_{j<m}2\,3^{-j}e_j.
\]
The restriction of $r_m$ to $1-e_m$ lies between $0$ and $3^{-m}$,
whereas its restriction to $e_m$ lies between $2\,3^{-m}$ and
$3\,3^{-m}$. Since $e_m$ commutes with $r_m$,
\[
e_m=1_{((3/2)3^{-m},\,\infty)}(r_m)\in W^*(b).
\]
Applying this encoding separately to $(d_m)$ and $(q_m)$ gives
two self-adjoint elements whose generated von Neumann algebra
contains every tensor site, and hence is all of $R$.
\end{proof}
\begin{proposition}
\label{prop:four-generators}
Every $\mathrm{II}_1$ factor with separable predual is generated
by four self-adjoint elements.
\end{proposition}
\begin{proof}
Let $M$ be such a factor. By
Theorem~\ref{thm:irreducible-hyperfinite}, applied with $N=M$, it contains a hyperfinite
$\mathrm{II}_1$ subfactor $P$ with $P'\cap M=\mathbb C1$.
The trace-preserving conditional expectation onto $P$ embeds
$P_*$ isometrically into $M_*$ by precomposition, so $P$ has
separable predual. Lemma~\ref{lem:hyperfinite-generators} gives
self-adjoint $a,b\in P$ with $P=W^*(a,b)$, and
Theorem~\ref{thm:relative-generation} gives $u\in\mathcal U(M)$
with $M=W^*(P,u)$. Hence
\[
M=W^*\bigl(a,b,\operatorname{Re}u,\operatorname{Im}u\bigr).\qedhere
\]
\end{proof}
The uniform bound in Proposition~\ref{prop:four-generators} also applies
to half-trace corners. The following matrix construction reduces their
four generators to two after amplification. It is the von Neumann algebra
form of the $2\times2$ encoding in~\cite[Proposition~1.1]{DSSW2008},
with the off-diagonal entry fixed to the identity. We include the proof
to make the recovery of the generators and matrix units explicit.
\begin{lemma}
\label{lem:matrix-encoding}
If a unital von Neumann algebra $N$ is generated by four
self-adjoint elements, then $M_2(N)$ is generated by two
self-adjoint elements.
\end{lemma}
\begin{proof}
Write $N=W^*(a_1,a_2,a_3,a_4)$ with $a_j=a_j^*$, and choose
$C>\norm{a_1}+\norm{a_2}$. Put
\[
A=\begin{pmatrix}a_1&0\\0&a_2+C1_N\end{pmatrix},\qquad
B=\begin{pmatrix}a_3&1_N\\1_N&a_4\end{pmatrix},\qquad
Q=W^*(A,B).
\]
Let $E_{ij}$ be the standard matrix units of $M_2(N)$.
The first diagonal block of $A$ has spectrum in
$[-\norm{a_1},\norm{a_1}]$, and the second has spectrum in
$[C-\norm{a_2},C+\norm{a_2}]$. For
$\norm{a_1}<\lambda<C-\norm{a_2}$, spectral calculus therefore gives
\[
E_{11}=1_{(-\infty,\lambda)}(A)\in Q,
\qquad E_{22}=1-E_{11}\in Q.
\]
The off-diagonal cuts of $B$ give
$E_{12}=E_{11}BE_{22}\in Q$ and $E_{21}=E_{12}^*\in Q$.
The first corner then contains
\begin{align*}
a_1E_{11}&=E_{11}AE_{11},&
a_2E_{11}&=E_{12}AE_{21}-CE_{11},\\
a_3E_{11}&=E_{11}BE_{11},&
a_4E_{11}&=E_{12}BE_{21}.
\end{align*}
Thus $NE_{11}\subset Q$. Multiplying this corner by the matrix
units yields $NE_{ij}\subset Q$ for every $i,j$, and hence
$Q=M_2(N)$.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{thm:main}]
Let $M$ be a $\mathrm{II}_1$ factor with separable predual and
normalized trace $\tau$. Choose a projection $p\in M$ with
$\tau(p)=1/2$. Set $N=pMp$, with normalized trace
$\tau_N(a)=2\tau(a)$. This corner is a $\mathrm{II}_1$ factor
with separable predual: precomposition with the normal compression
$a\mapsto pap$ embeds $N_*$ isometrically into $M_*$.
Since $p$ and $1-p$ have the same trace,
there is a partial isometry $v\in M$ with $v^*v=p$ and
$vv^*=1-p$. The corresponding matrix units give a normal
unital $*$-isomorphism
\[
\Phi:M_2(N)\longrightarrow M,\qquad
\Phi\begin{pmatrix}b_{11}&b_{12}\\b_{21}&b_{22}\end{pmatrix}
=b_{11}+b_{12}v^*+vb_{21}+vb_{22}v^*.
\]
Proposition~\ref{prop:four-generators}, applied to $N$, and
Lemma~\ref{lem:matrix-encoding} provide self-adjoint $A,B\in M_2(N)$
with $W^*(A,B)=M_2(N)$. Define
$x=\Phi(A+iB)\in M$. Since the real and imaginary parts of $x$
are $\Phi(A)$ and $\Phi(B)$, respectively,
\[
W^*(x)=W^*(\Phi(A),\Phi(B))=\Phi(M_2(N))=M.\qedhere
\]
\end{proof}
==== END Relative-generation-and-the-generator-problem-for-finite-factors//sections/generation.tex ====
==== BEGIN Relative-generation-and-the-generator-problem-for-finite-factors//sections/independence.tex ====
\section{Free Haar unitaries in an ultrapower}
\label{sec:independence}
Throughout this section, $P\subset M$ is an irreducible inclusion of
$\mathrm{II}_1$ factors. Fix a free ultrafilter $\omega$ on
$\mathbb N$. We use the tracial
ultrapower
\[
\mathbf M=M^\omega
=\ell^\infty(\mathbb N,M)/\mathcal I_\omega,
\qquad
\mathcal I_\omega
=\left\{(b_n)\in\ell^\infty(\mathbb N,M):
\lim_{n\to\omega}\|b_n\|_2=0\right\},
\]
and its von Neumann subalgebra $\mathbf Q=P^\omega$. The trace of an
element represented by $(b_n)$ is $\lim_{n\to\omega}\tau(b_n)$; we
denote this trace, and its associated $2$-norm, by the same symbols as
on $M$. We identify $M$ with its constant copy in $\mathbf M$.
A unitary $t$ in a tracial algebra is called \emph{Haar} if
$\tau(t^n)=0$ for every nonzero integer $n$. Two unital subalgebras
are \emph{free} if every alternating product of centered elements
from the two subalgebras has trace zero. We say that a unitary is
free from a unital algebra $D$ when $C^*(t)$ and $D$ are free.
For the perturbation in Section~\ref{sec:perturbation}, we will choose
successive Haar unitaries in $\mathbf Q$, each free from the algebra
containing the prescribed data and the earlier choices. We will also
need this freeness to survive multiplication on either side by unitaries
from that algebra. Popa's theorem supplies independence relative to
$\mathbf Q'\cap\mathbf M$, so we first identify that commutant.
\begin{lemma}
\label{lem:ultrapower-commutant}
The inclusion $\mathbf Q\subset\mathbf M$ is irreducible:
\[
\mathbf Q'\cap\mathbf M=\mathbb C1.
\]
\end{lemma}
\begin{proof}
Let $b\in\mathbf Q'\cap\mathbf M$ have a uniformly bounded
representative $(b_n)$, and put
\[
\delta_n=\sup_{w\in\mathcal U(P)}
\|wb_nw^*-b_n\|_2.
\]
Then $\lim_{n\to\omega}\delta_n=0$. Otherwise there would be an
$\varepsilon>0$ and a set belonging to $\omega$ on which
$\delta_n>\varepsilon$. Choosing on this set a unitary
$w_n\in P$ with $\|w_nb_nw_n^*-b_n\|_2>\varepsilon$, and setting
$w_n=1$ elsewhere, would give a unitary in $\mathbf Q$ that does not
commute with $b$.
For each $n$, let $K_n$ be the closed convex hull, in $L^2(M,\tau)$,
of the elements $wb_nw^*$, $w\in\mathcal U(P)$. The unique element
$c_n\in K_n$ of smallest $2$-norm is fixed by conjugation by every
unitary of $P$: each such conjugation preserves $K_n$ and its Hilbert
space norm. Moreover, $c_n$ belongs to $M$ and
$\|c_n\|\leq\|b_n\|$. Indeed, convex combinations from this orbit
have this operator norm bound, and an operator-norm-bounded
$2$-norm limit lies in the same bounded ball of $M$. To see the latter
fact, left multiplication by such a convergent sequence converges
strongly on the dense subspace $M\subset L^2(M)$, since
\[
\|(x_r-x_s)a\|_2\leq\|x_r-x_s\|_2\|a\|
\qquad(a\in M),
\]
and hence converges strongly on all of $L^2(M)$ by the uniform
operator norm bound. The strong limit belongs to $M$, and its value
at $1$ is the prescribed $2$-norm limit.
Thus $c_n\in P'\cap M=\mathbb C1$. The trace is constant on $K_n$
and continuous in $2$-norm, so $c_n=\tau(b_n)1$. Every element of
$K_n$ is at $2$-distance at most $\delta_n$ from $b_n$. Consequently
\[
\|b_n-\tau(b_n)1\|_2\leq\delta_n\longrightarrow_{n\to\omega}0.
\]
The bounded scalar sequence $(\tau(b_n))$ has a scalar ultralimit,
which represents $b$. This proves the assertion.
\end{proof}
We use the following special case of Popa's independence theorem.
The atomic-relative-commutant case needed here already appears in
\cite[Lemma~1.4 and Theorem~2.1]{Popa1995}; we state the later
formulation from~\cite[Theorem~0.1(a)]{Popa2014}.
\begin{theorem}[Popa's independence theorem]
\label{thm:popa-independence}
Let $P\subset M$ be an irreducible inclusion of $\mathrm{II}_1$
factors, and put $\mathbf Q=P^\omega\subset M^\omega=\mathbf M$.
For every linear subspace $X\subset\mathbf M$ that is separable in
$2$-norm and satisfies $\tau(x)=0$ for all $x\in X$, there is a
diffuse abelian von Neumann algebra $A\subset\mathbf Q$ such that
\begin{equation}
\label{eq:popa-independence}
\tau(x_0a_1x_1\cdots a_mx_m)=0
\end{equation}
whenever $m\geq1$, $x_0\in X\cup\{1\}$,
$x_1,\ldots,x_m\in X$, and $a_1,\ldots,a_m\in A$ have trace zero.
\end{theorem}
This is the constant-inclusion case of
\cite[Theorem~0.1(a)]{Popa2014}. The hypothesis there is that no
nonzero corner of the coordinate algebra $P$ intertwines into
$P'\cap M$ inside $M$. It holds here because $P'\cap M=\mathbb C1$:
a nonzero corner of the diffuse factor $P$ cannot embed normally
into a finite matrix amplification of the scalars. The conclusion
of that theorem is independence relative to
$\mathbf Q'\cap\mathbf M$. Lemma~\ref{lem:ultrapower-commutant}
identifies this commutant with the scalars, giving
\eqref{eq:popa-independence}.
\begin{lemma}
\label{lem:free-haar-extension}
Let $D\subset\mathbf M$ be a norm-separable unital $C^*$-algebra.
There is a Haar unitary $t\in\mathbf Q$ free from $D$. In
particular,
\begin{equation}
\label{eq:free-endpoints}
\tau(d_0t^{n_1}d_1\cdots t^{n_j}d_j)=0
\end{equation}
for every $j\geq1$, every $n_1,\ldots,n_j\in\mathbb Z\setminus\{0\}$,
and all $d_0,\ldots,d_j\in D$ such that
$\tau(d_i)=0$ for $1\leq i\leq j-1$. The two endpoints
$d_0,d_j$ are arbitrary.
\end{lemma}
\begin{proof}
The subspace $X=D\cap\ker\tau$ is norm separable, hence $2$-norm
separable. For example, applying $d\mapsto d-\tau(d)1$ to a
countable norm-dense subset of $D$ gives a countable norm-dense
subset of $X$. Apply Theorem~\ref{thm:popa-independence} to obtain
a diffuse abelian algebra $A\subset\mathbf Q$.
We first choose a Haar unitary in $A$. Diffuseness gives arbitrarily
small nonzero subprojections of every nonzero projection: repeatedly
split and retain a piece of at most half the trace. Normality of the
trace and maximality under inclusion then give subprojections of every
prescribed smaller trace. We may therefore choose recursively
partitions of $1$ by projections
\[
\{p_{m,k}:0\leq k<2^m\}\subset A,
\qquad \tau(p_{m,k})=2^{-m},
\]
with $p_{m,k}=p_{m+1,2k}+p_{m+1,2k+1}$. The self-adjoint
elements
\[
h_m=\sum_{k=0}^{2^m-1}\frac{k}{2^m}p_{m,k}
\]
satisfy $\|h_{m+1}-h_m\|\leq2^{-(m+1)}$ and converge in norm to
an element $h\in A$. For each continuous function $f$ on $[0,1]$,
\[
\tau(f(h))
=\lim_{m\to\infty}2^{-m}
\sum_{k=0}^{2^m-1}f(k2^{-m})
=\int_0^1 f(s)\,ds.
\]
It follows that $t=\exp(2\pi i h)$ is Haar.
We next remove the centering requirement at the two endpoints, so
that \eqref{eq:free-endpoints} can be applied to words with arbitrary
outside coefficients. Expand each endpoint into its scalar and
centered parts. A term with a centered last endpoint
vanishes by \eqref{eq:popa-independence}, since every nonzero power
of $t$ belongs to $A$ and is centered. A term with a centered first
endpoint and scalar last endpoint vanishes after moving the first
endpoint to the end by traciality. It remains to consider the word
with both endpoints omitted. When $j=1$ this is the Haar property.
When $j\geq2$, rotate the final power $t^{n_j}$ next to the first
power $t^{n_1}$. If $n_j+n_1\ne0$, the resulting word is covered
by \eqref{eq:popa-independence}. If $n_j+n_1=0$, the two powers
cancel and leave a word starting and ending with centered elements
of $D$. This word is again covered by
\eqref{eq:popa-independence}, except when $j=2$, in which case it
is simply the centered element $d_1$. This proves
\eqref{eq:free-endpoints}.
Laurent polynomials in $t$ are norm dense in $C^*(t)$. As $t$ is
Haar, the centered Laurent polynomials are precisely the linear
combinations of its nonzero powers. Expanding an alternating
centered word, and using \eqref{eq:free-endpoints} followed by norm
approximation, proves that $t$ is free from $D$.
\end{proof}
\begin{lemma}[Absorption of base-algebra unitaries]
\label{lem:haar-absorption}
Let $D$ be a unital $C^*$-subalgebra of a finite tracial von Neumann
algebra, and let $t$ be a Haar unitary free from $D$. If
$a,b\in\mathcal U(D)$, then both $atb$ and $t^*$ are Haar unitaries
free from $D$. Both satisfy \eqref{eq:free-endpoints} in place of $t$.
\end{lemma}
\begin{proof}
For any Haar unitary free from $D$, formula
\eqref{eq:free-endpoints} follows immediately by expanding the
endpoints into scalar and centered parts and applying freeness to
each resulting alternating word. Replacing $t$ by $t^*$ merely
negates the exponents, so we concentrate on $v=atb$.
Consider a word
\[
d_0v^{n_1}d_1\cdots v^{n_j}d_j
\]
with nonzero integer exponents and centered internal factors
$d_1,\ldots,d_{j-1}$. Expand each power of $v$ into individual
letters $t$ and $t^{-1}$, with elements of $D$ between them. Within
one power of $v$ all these letters have the same sign. A gap
between opposite signs can therefore occur only between two
successive powers of $v$. Such a gap equals
\[
b d_i b^* \quad\text{for a transition from $t$ to $t^{-1}$},
\qquad
a^*d_i a \quad\text{for a transition from $t^{-1}$ to $t$}.
\]
It is centered by traciality. Every other gap joins equal signs.
Expand each of these latter gaps into its scalar and centered
parts. In a resulting term, removing a scalar gap only joins
letters of equal sign. Thus each block joined across scalar gaps
is a nonzero power of $t$, and consecutive blocks are separated by
centered elements of $D$. The endpoint factors remain arbitrary
elements of $D$. Formula~\eqref{eq:free-endpoints} shows that every
resulting term has trace zero.
We have proved \eqref{eq:free-endpoints} for $v$. Taking $j=1$
and $d_0=d_1=1$ proves that $v$ is Haar. The general formula and
norm approximation by Laurent polynomials then prove its freeness
from $D$.
\end{proof}
We may apply Lemma~\ref{lem:free-haar-extension} successively.
Specifically, given a norm-separable unital $C^*$-algebra $D_0\subset
\mathbf M$ and an integer $r\geq0$, choose $t_j\in\mathbf Q$ Haar
and free from $D_j$, and set
\[
D_{j+1}=C^*(D_j,t_j),\qquad 0\leq j\leq r.
\]
Each $D_{j+1}$ remains norm separable: polynomials in a countable
norm-dense subset of $D_j$, the two elements $t_j,t_j^*$, and
rational complex coefficients form a countable dense set. Thus
every application of the lemma has the required separability
hypothesis, even though the previously chosen unitaries already
belong to $\mathbf Q$.
==== END Relative-generation-and-the-generator-problem-for-finite-factors//sections/independence.tex ====
==== BEGIN Relative-generation-and-the-generator-problem-for-finite-factors//sections/introduction.tex ====
\section{Introduction}
\label{sec:introduction}
For a subset $S$ of a von Neumann algebra $M$, write $W^*(S)$ for the
smallest unital von Neumann subalgebra containing $S$. Thus adjoints
are included and generation is taken in the weak operator topology.
The generator problem asks whether every von Neumann algebra with
separable predual has the form $W^*(x)$ for one bounded operator $x$.
Equivalently, it asks for two self-adjoint generators: their sum
$a+ib$ recovers $a$ and $b$ as its real and imaginary parts.
We prove this assertion for type $\mathrm{II}_1$ factors by first
establishing a relative generation theorem. For a finite factor with
normalized trace $\tau$, put $\norm{x}_2=\tau(x^*x)^{1/2}$.
An inclusion $P\subset M$ is \emph{irreducible} when
$P'\cap M=\mathbb C1$. Write $\mathcal U(M)$ for the unitary group
of $M$.
\begin{theorem}\label{thm:relative-generation}
Let $P\subset M$ be an irreducible inclusion of type $\mathrm{II}_1$
factors, with $M$ of separable predual. The set
\[
\{u\in\mathcal U(M):W^*(P,u)=M\}
\]
is a dense $G_\delta$ subset of the unitary group $\mathcal U(M)$
for the trace $2$-norm topology.
\end{theorem}
\begin{theorem}\label{thm:main}
Every type $\mathrm{II}_1$ factor $M$ with separable predual is
generated by two self-adjoint elements. Equivalently, there is an
$x\in M$ such that $M=W^*(x)$.
\end{theorem}
Throughout the paper,
\emph{separable} for a finite von Neumann algebra means that its
predual is separable. This ensures separability of $L^2(M,\tau)$;
it does not require operator-norm separability of $M$.
\paragraph{Earlier work.}
The type~I, hyperfinite, and properly infinite cases of the generator
problem were established by Pearcy, Suzuki and Sait\^o, and Wogen,
respectively~\cite{Pearcy1962,SuzukiSaito1963,Wogen1969};
see also the historical discussion in~\cite[Section~3]{Sherman2012}.
Willig's direct-integral theorem reduces the general question to
type~$\mathrm{II}_1$ factors~\cite[Corollary~2]{Willig1974}.
Consequently, Theorem~\ref{thm:main} also gives single generation of
every von Neumann algebra with separable predual. This last consequence
uses the established direct-integral reduction, whereas our proof
below concerns finite factors.
Much of the progress in the finite case has used additional structure.
Behncke established single generation for tensor products of two
$\mathrm{II}_1$ factors~\cite{Behncke1972}.
Popa treated factors with Cartan subalgebras~\cite{Popa1985Cartan};
Ge and Popa treated factors with property~$\Gamma$ and recovered the
tensor-product case~\cite{GePopa1998}; and Ge and Shen obtained
arithmetic group examples with a small-support
refinement~\cite{GeShen2002}.
Shen's generator invariant unified these and other positive
classes~\cite{Shen2009}. Dykema, Sinclair, Smith, and White established
its exact amplification formula and sharpened its connection with the
number of self-adjoint generators~\cite{DSSW2008}.
Sherman showed, in particular, that finite generation of every
countably generated $\mathrm{II}_1$ factor would suffice for a positive
answer~\cite[Theorem~3.8(3)]{Sherman2012}.
More recently, Gao, Kunnawalkam Elayavalli, Patchell, and Tan proved
single generation when a countable unitary generating family lies in
one sequential-commutation orbit in an ultrapower: successive
members can be joined there by finite chains with Haar intermediate
unitaries and commuting adjacent terms~\cite[Theorem~A]{GaoEtAl2025}. Popa's work on tight
decompositions studies stable single generation, which requires every
amplification of a factor to be singly generated~\cite{Popa2021Tight}.
Requiring generators to vary continuously with a family of factors
leads to a different problem. Hayes constructed a $W^*$-bundle---a
continuous family of tracial von Neumann algebras---over a compact
metrizable base, with separable-predual $\mathrm{II}_1$ fibers, for
which no finite family of continuous sections generates every
fiber~\cite[Theorem~1.1]{Hayes2026Bundles}.
Relative generation by one unitary is already available for finite-index
inclusions, by the downward basic construction used in
\cite[Lemma~6.2]{DSSW2008}: the associated Jones projection $e$
generates the extension over the subfactor, and $1-2e$ is unitary.
Theorem~\ref{thm:relative-generation} applies to every irreducible
inclusion and also establishes genericity of the relative generators.
The universal conclusion of Theorem~\ref{thm:main} also makes the
generator invariants $G$ and $G_{\mathrm{sa}}$ vanish for every
separable-predual $\mathrm{II}_1$ factor. It implies that Voiculescu's
microstates free entropy dimensions $\delta$ and $\delta_0$ can change
under replacement of a finite self-adjoint von Neumann generating
tuple. Section~\ref{sec:consequences} gives these deductions and their
precise scope.
\paragraph{The main estimate and the proof.}
Our argument combines Popa's free-independence construction in tracial
ultraproducts~\cite{Popa1995,Popa2014} with a perturbation whose effect
accumulates along a long word. Given a unitary $u$ and a target unitary
$z$, we coordinate a perturbation of $u$ across its repeated occurrences
in a word with coefficients from $P$. With a small parameter $\alpha$
fixed, the perturbation has $2$-norm of order $\alpha/\sqrt\ell$,
where $\ell$ is the number of occurrences. The resulting word has
correlation with a test vector $\xi$ equal to
$(\alpha/2)\tau(\xi^*z)$ up to an error of order
$\alpha^2\norm{\xi}_2+1/\ell$.
The essential estimate is uniform in $\ell$. An operator-norm bound
on each perturbation separately would grow with the length of the
product. We instead dominate all coefficients of the nonlinear terms
by a free-group convolution operator with nonnegative coefficients. The length-one case of
Haagerup's inequality~\cite[Lemma~1.4]{Haagerup1979} bounds its norm
independently of $\ell$. This controls the full exponential remainder,
including cancellations between words. Proposition~\ref{prop:perturbation}
states the precise bounds. The witness word need only retain a fixed
positive correlation as $\ell\to\infty$.
The free variables and perturbation are first constructed in a tracial
ultrapower. For each fixed word length, choosing one coordinate of
their representing sequences gives the required unitaries in $M$ and
$P$, with an arbitrarily small additional error.
For a fixed vector $\xi\in L^2(M,\tau)$, the norm of its projection
onto $L^2(W^*(P,u),\tau)$ is a bounded lower semicontinuous function
of $u$. Baire's theorem supplies common continuity points for a
countable dense family of vectors, and contractivity extends continuity
to every vector. If $W^*(P,u)$ were proper at such a point, a nonzero
bounded vector orthogonal to it would have projection norm zero.
The perturbation estimate gives nearby unitaries for which that
projection norm has a fixed positive lower bound, a contradiction.
Finally, Popa's irreducible hyperfinite embedding theorem
\cite{Popa1981,Popa2021Ergodic} lets us choose $P$ generated by two
self-adjoint elements. Relative generation then gives four
self-adjoint generators for every separable-predual $\mathrm{II}_1$
factor. Applying this uniform bound to a half-trace corner, and using
an explicit two-by-two matrix encoding, gives
Theorem~\ref{thm:main}. This encoding belongs to the established
matrix-generation techniques of Douglas--Pearcy and Wogen
\cite{DouglasPearcy1969,Wogen1969}; we use the von Neumann
algebra form of the matrix encoding
in~\cite[Proposition~1.1]{DSSW2008}.
Section~\ref{sec:independence} states the exact independence input and
constructs the free Haar unitaries required in the ultrapower.
Section~\ref{sec:perturbation} proves the uniform perturbation estimate.
Section~\ref{sec:relative-generation} carries out the continuity
argument and identifies the full relative-generator locus as
$G_\delta$. Section~\ref{sec:two-generators} gives the corner and
matrix construction proving single generation.
Section~\ref{sec:consequences} treats the general von Neumann
algebra, generator-invariant, and free-entropy consequences.
\paragraph{Notation.}
We write $\norm{x}$ for the operator norm and use the inner product
$\langle a,b\rangle=\tau(b^*a)$ on $L^2(M,\tau)$, linear in the
first variable. The same trace notation is used on tracial ultrapowers.
All subalgebras are unital unless a corner unit is specified.
For a von Neumann subalgebra $N\subset M$, $E_N$ denotes the
trace-preserving conditional expectation and $e_N$ the orthogonal
projection from $L^2(M,\tau)$ onto $L^2(N,\tau)$.
==== END Relative-generation-and-the-generator-problem-for-finite-factors//sections/introduction.tex ====
==== BEGIN Relative-generation-and-the-generator-problem-for-finite-factors//sections/perturbation.tex ====
\section{A perturbation with uniform correlation}
\label{sec:perturbation}
We seek perturbations that tend to zero in $2$-norm while producing
a fixed first-order correlation along words of increasing length.
The nonlinear error must remain bounded independently of that length.
For this purpose we first record the length-one case of Haagerup's
inequality \cite[Lemma~1.4]{Haagerup1979}, together with its elementary
proof.
\begin{lemma}\label{lem:length-one-convolution}
Let $F$ be a free group with a specified free basis, let $\mathcal L$
be the set of basis elements and their inverses, and let $\lambda$
be the left regular representation on $\ell^2(F)$.
For every finitely supported family $(b_\gamma)_{\gamma\in\mathcal L}$,
\[
\norm{\sum_{\gamma\in\mathcal L}b_\gamma\lambda(\gamma)}
\le 2\left(\sum_{\gamma\in\mathcal L}|b_\gamma|^2\right)^{1/2}.
\]
\end{lemma}
\begin{proof}
For a letter $a$, let $B_a$ be the orthogonal projection onto the
closed span of reduced words beginning with $a$. Split
\[
\lambda(\gamma)
=\lambda(\gamma)(1-B_{\gamma^{-1}})
+\lambda(\gamma)B_{\gamma^{-1}}.
\]
The first summand has range in the range of $B_\gamma$.
These ranges are mutually orthogonal, so for $\eta\in\ell^2(F)$,
\begin{align*}
\norm{\sum_\gamma b_\gamma\lambda(\gamma)
(1-B_{\gamma^{-1}})\eta}^2
&=\sum_\gamma |b_\gamma|^2
\norm{(1-B_{\gamma^{-1}})\eta}^2\\
&\le\left(\sum_\gamma |b_\gamma|^2\right)\norm{\eta}^2.
\end{align*}
For the second summands, the triangle inequality and
Cauchy--Schwarz give
\begin{align*}
\norm{\sum_\gamma b_\gamma\lambda(\gamma)
B_{\gamma^{-1}}\eta}
&\le\sum_\gamma |b_\gamma|\norm{B_{\gamma^{-1}}\eta}\\
&\le\left(\sum_\gamma|b_\gamma|^2\right)^{1/2}\norm{\eta},
\end{align*}
because the projections $B_{\gamma^{-1}}$ are mutually orthogonal.
Adding the two estimates proves the lemma.
\end{proof}
\begin{proposition}\label{prop:perturbation}
Let $P\subset M$ be an irreducible inclusion of $\mathrm{II}_1$ factors.
Given $u,z\in\mathcal U(M)$, $\xi\in M$, and $0<\alpha<1/2$,
for every integer $\ell\ge1$ there are
$\widetilde u\in\mathcal U(M)$ and
$\widetilde t_0,\ldots,\widetilde t_\ell\in\mathcal U(P)$ such that,
with
\[
\widetilde W=\widetilde t_0
\prod_{k=1}^{\ell}(\widetilde u\,\widetilde t_k),
\]
one has
\begin{equation}\label{eq:perturbation-bounds}
\begin{aligned}
\norm{\widetilde u-u}_2
&\le \frac{\alpha}{\sqrt{2\ell}}+\frac1\ell,\\
\left|\tau(\xi^*\widetilde W)
-\frac\alpha2\tau(\xi^*z)\right|
&\le \norm{\xi}_2\frac{4\alpha^2}{1-2\alpha}+\frac1\ell.
\end{aligned}
\end{equation}
All products indexed by $k$ are taken in increasing order from left
to right. The unitaries may depend on $\ell$.
\end{proposition}
\begin{proof}
Fix $\ell$ and work first in $\mathbf M=M^\omega$, with
$\mathbf Q=P^\omega$. We use $\tau$ also for the ultrapower trace.
\smallskip\noindent
\emph{Free variables and prefixes.}
Set $D_0=C^*(1,u,z,\xi)$. By
Lemma~\ref{lem:free-haar-extension}, choose successively
$t_j\in\mathcal U(\mathbf Q)$ Haar and free from $D_j$, and put
$D_{j+1}=C^*(D_j,t_j)$, for $0\le j\le\ell$.
Each of these $C^*$-algebras is norm separable.
Define
\[
s_1=t_0,\qquad s_{j+1}=u t_j\quad(1\le j\le\ell),\qquad
g_k=s_1\cdots s_k\quad(1\le k\le\ell),
\]
and
\[
W=g_\ell s_{\ell+1}
=t_0\prod_{k=1}^{\ell}(u t_k).
\]
Thus $g_k$ is the prefix immediately preceding the $k$th occurrence
of $u$ in $W$. Since
$D_j=C^*(D_0,s_1,\ldots,s_j)$, Lemma~\ref{lem:haar-absorption}
shows that $s_{j+1}$ is Haar and free from $D_j$.
In particular, $s_1,\ldots,s_\ell$ are free Haar unitaries.
Every nonidentity reduced word in them has trace zero, so the group
they generate can be identified with the free group $G$ on these
generators.
Since $W=(g_\ell u)t_\ell$ and $g_\ell u\in\mathcal U(D_\ell)$,
the same lemma shows that $W$ is Haar and free from $D_\ell$.
Applying it to $W^*$ and $z\in\mathcal U(D_\ell)$ also shows that
\[
y=zW^*
\]
is Haar and free from $D_\ell$.
For $p\in G$ set $Y_p=pyp^{-1}$. These form a free family of
Haar unitaries. Indeed, if $a_1,\ldots,a_r$ are nonzero integers
and successive labels $p_1,\ldots,p_r$ are distinct, then
\[
Y_{p_1}^{a_1}\cdots Y_{p_r}^{a_r}
=p_1y^{a_1}(p_1^{-1}p_2)y^{a_2}\cdots
(p_{r-1}^{-1}p_r)y^{a_r}p_r^{-1}.
\]
Each internal ratio is a nonidentity element of $G$ and hence
is centered in $D_\ell$. Equation~\eqref{eq:free-endpoints}
gives zero trace. It follows, by applying this to the reduction
of $v^*w$, that distinct reduced words in the $Y_p$ are orthonormal
in $L^2(\mathbf M,\tau)$, including the empty word.
A left perturbation of $u$ at its $k$th occurrence in $W$ is
conjugated by $g_k$ when moved to the front of the word. We therefore
use the conjugates $g_j^{-1}yg_j$: conjugating the $j$th one by
$g_j$ returns $y$. Taking imaginary parts will give a self-adjoint
perturbation, while freeness makes their average small in $2$-norm.
Put $\operatorname{Im}(b)=(b-b^*)/(2i)$ and define
\[
h=\frac\alpha\ell\sum_{j=1}^{\ell}
\operatorname{Im}(g_j^{-1}yg_j),\qquad
u'=e^{ih}u,\qquad
W'=t_0\prod_{k=1}^{\ell}(u't_k).
\]
The $g_j^{-1}$ are distinct in $G$, so the summands defining $h$
are orthogonal centered imaginary parts of free Haar unitaries.
Each has squared $2$-norm $1/2$. Consequently,
\[
h=h^*,\qquad \norm{h}\le\alpha,\qquad
\norm{h}_2=\frac\alpha{\sqrt{2\ell}}.
\]
Write $H_k=g_khg_k^{-1}$, $p_{kj}=g_kg_j^{-1}$,
and $c=\alpha/(2\ell)$. Cancelling consecutive prefixes gives
\begin{equation}\label{eq:perturbation-prefix}
W'=\left(\prod_{k=1}^{\ell}e^{iH_k}\right)W,
\qquad
iH_k=c\sum_{j=1}^{\ell}
(Y_{p_{kj}}-Y_{p_{kj}}^{-1}).
\end{equation}
For the first identity, use
$e^{iH_k}=g_ke^{ih}g_k^{-1}$ and
$g_k^{-1}g_{k+1}=ut_k$ for $1\le k<\ell$; the product order is
therefore essential.
Let $S=\{p_{kj}:1\le k,j\le\ell\}$ and let $n_p$ count the
ordered pairs $(k,j)$ with $p_{kj}=p$.
For the identity $e$ of $G$, one has $n_e=\ell$.
If $k\ne j$, the word
\[
p_{kj}=s_1\cdots s_k s_j^{-1}\cdots s_1^{-1}
\]
is reduced, since its joining letters have different indices.
Its initial positive segment and its final negative segment
determine $k$ and $j$. All off-diagonal pairs therefore give
distinct nonidentity words, and
\begin{equation}\label{eq:perturbation-multiplicity}
\sum_{p\in S}n_p^2=\ell^2+\ell(\ell-1)=2\ell^2-\ell.
\end{equation}
\smallskip\noindent
\emph{The first-order correlation.}
Put $C_1=\sum_{k=1}^{\ell}iH_k$. Each diagonal term
$cY_{p_{kk}}W$ equals $cz$, because $p_{kk}=e$ and $Y_eW=z$.
We show that all other linear terms vanish when paired with $\xi$.
Equation~\eqref{eq:free-endpoints} for $W$ and $D_\ell$ gives
$\tau(\xi^*W)=0$. For $p\ne e$, the element $p^{-1}$ is
centered in $D_\ell$, so
\[
\tau(\xi^*Y_pW)
=\tau(\xi^*p z W^*p^{-1}W)=0.
\]
For every $p\in G$, one also has
\[
\tau(\xi^*Y_p^{-1}W)
=\tau(\xi^*p W z^*p^{-1}W)=0.
\]
Indeed, split the internal factor $z^*p^{-1}\in D_\ell$ into
its centered and scalar parts. The first contribution vanishes
by \eqref{eq:free-endpoints}; the second is a scalar multiple of
$\tau(\xi^*pW^2)$ and vanishes by the same formula.
It follows that
\[
\tau\bigl(\xi^*(1+C_1)W\bigr)
=c n_e\tau(\xi^*z)=\frac\alpha2\tau(\xi^*z).
\]
\smallskip\noindent
\emph{The uniform remainder estimate.}
It remains to bound the contribution of
$\prod_k e^{iH_k}-1-C_1$ uniformly in $\ell$.
Group the ordered product of exponentials by its original degree:
\[
C_d=\sum_{\substack{m_1+\cdots+m_\ell=d\\m_k\ge0}}
\frac{(iH_1)^{m_1}\cdots(iH_\ell)^{m_\ell}}
{m_1!\cdots m_\ell!}.
\]
The term $C_0$ is $1$, and the degree-one term is the $C_1$ above.
For fixed $\ell$, the expansion is absolutely convergent in
operator norm, because
\[
\sum_{d\ge0}\norm{C_d}
\le\prod_{k=1}^{\ell}
\sum_{m\ge0}\frac{\norm{H_k}^m}{m!}
=\exp\left(\sum_{k=1}^{\ell}\norm{H_k}\right)
\le e^{\ell\alpha}.
\]
The Cauchy product may thus be grouped in this way, and its sum is
$\prod_k e^{iH_k}$.
We will prove the bound
\begin{equation}\label{eq:perturbation-remainder}
\norm{\prod_{k=1}^{\ell}e^{iH_k}-1-C_1}_2
\le\frac{4\alpha^2}{1-2\alpha},
\end{equation}
which, unlike the preceding norm-convergence bound, is uniform
in $\ell$.
Let $F_S$ be the formal free group with generators $T_p$, $p\in S$.
In particular $T_e$ is a generator, not the identity of $F_S$.
Let $\Phi:\mathbb C[F_S]\to\mathbf M$ be the homomorphism
$\Phi(T_p)=Y_p$.
By the orthonormality proved above, for every finite group-ring sum
$f=\sum_{w\in F_S}f(w)w$ one has
\begin{equation}\label{eq:perturbation-coefficient-isometry}
\norm{\Phi(f)}_2^2=\sum_{w\in F_S}|f(w)|^2.
\end{equation}
Define
\[
B_k=c\sum_{j=1}^{\ell}
(T_{p_{kj}}-T_{p_{kj}}^{-1}),\qquad
A_k=c\sum_{j=1}^{\ell}
(T_{p_{kj}}+T_{p_{kj}}^{-1}),
\]
and
\[
A=\sum_{k=1}^{\ell}A_k
=c\sum_{p\in S}n_p(T_p+T_p^{-1}).
\]
For a group-ring element $f$, write $|f|$ for its coefficientwise
absolute value, and compare nonnegative coefficient functions
coefficientwise. Group multiplication gives
\[
|fg|(w)=\left|\sum_{ab=w}f(a)g(b)\right|
\le\sum_{ab=w}|f(a)||g(b)|=(|f|\,|g|)(w).
\]
This comparison already includes all free reductions and all
contributions collected at the same resulting word.
Thus the formal element
\[
\mathcal C_d=
\sum_{\substack{m_1+\cdots+m_\ell=d\\m_k\ge0}}
\frac{B_1^{m_1}\cdots B_\ell^{m_\ell}}
{m_1!\cdots m_\ell!},
\qquad \Phi(\mathcal C_d)=C_d,
\]
satisfies
\begin{equation}\label{eq:perturbation-majorization}
|\mathcal C_d|
\le\sum_{m_1+\cdots+m_\ell=d}
\frac{A_1^{m_1}\cdots A_\ell^{m_\ell}}
{m_1!\cdots m_\ell!}
\le A^d.
\end{equation}
For the last inequality, each tuple $(m_1,\ldots,m_\ell)$
corresponds to one nondecreasing sequence of row indices in the
distributive expansion of $(A_1+\cdots+A_\ell)^d$.
Its factorial weight is at most one, and every other summand has
nonnegative coefficients. No commutation of the $A_k$ is used.
The index $d$ records the degree before reduction; resulting words
may have shorter length, including zero.
Let $\lambda$ now denote the left regular representation of $F_S$,
and let $\delta_{\mathrm{id}}$ be the unit vector at its identity.
By \eqref{eq:perturbation-coefficient-isometry} and
\eqref{eq:perturbation-majorization},
\[
\norm{C_d}_2
\le\norm{A^d}_{\ell^2(F_S)}
=\norm{\lambda(A)^d\delta_{\mathrm{id}}}_{\ell^2(F_S)}
\le\norm{\lambda(A)}^d.
\]
This uses only the coefficient $2$-norm identity
\eqref{eq:perturbation-coefficient-isometry}, with no comparison
of operator norms between the two representations.
Lemma~\ref{lem:length-one-convolution} and
\eqref{eq:perturbation-multiplicity} give
\[
\norm{\lambda(A)}
\le 2\left(2c^2\sum_{p\in S}n_p^2\right)^{1/2}
=2\alpha\left(1-\frac1{2\ell}\right)^{1/2}
\le2\alpha<1.
\]
Consequently $\norm{C_d}_2\le(2\alpha)^d$.
The operator-norm convergence established above implies convergence
of the same partial sums in $2$-norm. Taking their tails and using
the triangle inequality therefore yields
\[
\norm{\prod_{k=1}^{\ell}e^{iH_k}-1-C_1}_2
\le\sum_{d\ge2}(2\alpha)^d
=\frac{4\alpha^2}{1-2\alpha}.
\]
In particular, orthogonality between different degrees is not
required. This proves \eqref{eq:perturbation-remainder}.
Combining the first-order correlation with
\eqref{eq:perturbation-prefix}, \eqref{eq:perturbation-remainder},
and Cauchy--Schwarz gives
\begin{equation}\label{eq:perturbation-ultrapower-correlation}
\left|\tau(\xi^*W')-\frac\alpha2\tau(\xi^*z)\right|
\le\norm{\xi}_2\frac{4\alpha^2}{1-2\alpha}.
\end{equation}
\smallskip\noindent
\emph{Lifting for the fixed word length.}
Each $t_j$ has a representative consisting of unitaries in $P$.
To see this, start with a bounded representative $(b_n)$ of any
unitary in $P^\omega$. The polar part of $b_n$ extends to a
unitary $w_n\in P$, since its initial and final complementary
projections have equal trace in the finite factor $P$.
Then $b_n=w_n|b_n|$ and functional calculus gives
\[
\norm{b_n-w_n}_2=\norm{|b_n|-1}_2
\le\norm{b_n^*b_n-1}_2\longrightarrow_\omega0.
\]
Here the inequality follows from
$|\sqrt t-1|\le|t-1|$ for $t\ge0$.
Choose unitary representatives $t_{j,n}\in P$ for the finitely
many $t_j$. Using the constant representatives of $u,z$, form
$g_{k,n},W_n,y_n,h_n,u_n',W_n'$ by exactly the formulas above.
For this fixed $\ell$ these are bounded sequences, with
$h_n=h_n^*$ and $\norm{h_n}\le\alpha$.
Bounded algebraic operations and continuous functional calculus
in the quotient show that they represent the corresponding
ultrapower elements. In particular,
\[
\lim_{n\to\omega}\norm{h_n}_2
=\frac\alpha{\sqrt{2\ell}},\qquad
\lim_{n\to\omega}\tau(\xi^*W_n')=\tau(\xi^*W').
\]
Moreover $\norm{u_n'-u}_2\le\norm{h_n}_2$, by the scalar
inequality $|e^{it}-1|\le|t|$ and functional calculus.
Choose one index $n$ for which
\[
\norm{h_n}_2<\frac\alpha{\sqrt{2\ell}}+\frac1\ell,
\qquad
|\tau(\xi^*W_n')-\tau(\xi^*W')|<\frac1\ell.
\]
Both conditions hold on sets belonging to $\omega$, so their
intersection is nonempty. Set $\widetilde u=u_n'$ and
$\widetilde t_j=t_{j,n}$. Equation
\eqref{eq:perturbation-ultrapower-correlation} proves
\eqref{eq:perturbation-bounds}. This choice is made separately
for each $\ell$, as the proposition permits.
\end{proof}
==== END Relative-generation-and-the-generator-problem-for-finite-factors//sections/perturbation.tex ====
==== BEGIN Relative-generation-and-the-generator-problem-for-finite-factors//references.bib ====
@article{Popa2014,
author = {Popa, Sorin},
title = {Independence properties in subalgebras of ultraproduct {$\mathrm{II}_1$} factors},
journal = {Journal of Functional Analysis},
volume = {266},
number = {9},
pages = {5818--5846},
year = {2014},
doi = {10.1016/j.jfa.2014.02.004},
eprint = {1308.3982},
archivePrefix = {arXiv},
primaryClass = {math.OA},
url = {https://arxiv.org/abs/1308.3982}
}
@article{Popa1981,
author = {Popa, Sorin},
title = {On a problem of {R. V. Kadison} on maximal abelian {$*$}-subalgebras in factors},
journal = {Inventiones Mathematicae},
volume = {65},
number = {2},
pages = {269--281},
year = {1981},
doi = {10.1007/BF01389015},
url = {https://doi.org/10.1007/BF01389015}
}
@article{Sherman2012,
author = {Sherman, David},
title = {On cardinal invariants and generators for von {Neumann} algebras},
journal = {Canadian Journal of Mathematics},
volume = {64},
number = {2},
pages = {455--480},
year = {2012},
doi = {10.4153/CJM-2011-048-2},
eprint = {0908.4565},
archivePrefix = {arXiv},
primaryClass = {math.OA},
url = {https://arxiv.org/abs/0908.4565}
}
@article{Willig1974,
author = {Willig, Paul},
title = {Generators and direct integral decompositions of {$W^*$}-algebras},
journal = {Tohoku Mathematical Journal, Second Series},
volume = {26},
number = {1},
pages = {35--37},
year = {1974},
doi = {10.2748/tmj/1178241231},
url = {https://www.jstage.jst.go.jp/article/tmj1949/26/1/26_1_35/_article/-char/en}
}
@article{Popa2021Ergodic,
author = {Popa, Sorin},
title = {On ergodic embeddings of factors},
journal = {Communications in Mathematical Physics},
volume = {384},
number = {2},
pages = {971--996},
year = {2021},
doi = {10.1007/s00220-020-03865-3},
eprint = {1910.06923},
archivePrefix = {arXiv},
primaryClass = {math.OA},
url = {https://arxiv.org/abs/1910.06923}
}
@article{Popa2021Tight,
author = {Popa, Sorin},
title = {Tight decomposition of factors and the single generation problem},
journal = {Journal of Operator Theory},
volume = {85},
number = {1},
pages = {277--301},
year = {2021},
doi = {10.7900/jot.2019nov01.2279},
eprint = {1910.14653},
archivePrefix = {arXiv},
primaryClass = {math.OA},
url = {https://arxiv.org/abs/1910.14653}
}
@incollection{Popa1995,
author = {Popa, Sorin},
title = {Free-independent sequences in type {$\mathrm{II}_1$} factors and related problems},
booktitle = {Recent advances in operator algebras---Orl{\'e}ans, 1992},
series = {Ast{\'e}risque},
number = {232},
publisher = {Soci{\'e}t{\'e} math{\'e}matique de France},
pages = {187--202},
year = {1995},
url = {https://www.numdam.org/item/AST_1995__232__187_0/}
}
@article{Haagerup1979,
author = {Haagerup, Uffe},
title = {An example of a non nuclear {$C^*$}-algebra, which has the metric approximation property},
journal = {Inventiones Mathematicae},
volume = {50},
number = {3},
pages = {279--293},
year = {1979},
doi = {10.1007/BF01410082},
url = {https://doi.org/10.1007/BF01410082}
}
@article{Wogen1969,
author = {Wogen, Warren},
title = {On generators for von {Neumann} algebras},
journal = {Bulletin of the American Mathematical Society},
volume = {75}, number = {1}, pages = {95--99}, year = {1969},
doi = {10.1090/S0002-9904-1969-12157-9}
}
@article{DSSW2008,
author = {Dykema, Ken and Sinclair, Allan and Smith, Roger and White, Stuart},
title = {Generators of {$\mathrm{II}_1$} factors},
journal = {Operators and Matrices},
volume = {2}, number = {4}, pages = {555--582}, year = {2008},
doi = {10.7153/oam-02-35}, eprint = {0706.1953}, archivePrefix = {arXiv},
url = {https://files.ele-math.com/articles/oam-02-35.pdf}
}
@article{Shen2009,
author = {Shen, Junhao},
title = {Type {$\mathrm{II}_1$} factors with a single generator},
journal = {Journal of Operator Theory},
volume = {62}, number = {2}, pages = {421--438}, year = {2009},
eprint = {math/0511327}, archivePrefix = {arXiv},
url = {https://jot.theta.ro/jot/archive/2009-062-002/2009-062-002-010.html}
}
@article{Popa1985Cartan,
author = {Popa, Sorin},
title = {Notes on {Cartan} subalgebras in type {$\mathrm{II}_1$} factors},
journal = {Mathematica Scandinavica},
volume = {57}, pages = {171--188}, year = {1985},
doi = {10.7146/math.scand.a-12110}
}
@article{GePopa1998,
author = {Ge, Liming and Popa, Sorin},
title = {On some decomposition properties for factors of type {$\mathrm{II}_1$}},
journal = {Duke Mathematical Journal},
volume = {94}, number = {1}, pages = {79--101}, year = {1998},
doi = {10.1215/S0012-7094-98-09405-4}
}
@article{GeShen2002,
author = {Ge, Liming and Shen, Junhao},
title = {Generator problem for certain property {T} factors},
journal = {Proceedings of the National Academy of Sciences of the United States of America},
volume = {99}, number = {2}, pages = {565--567}, year = {2002},
doi = {10.1073/pnas.022593699}
}
@article{GaoEtAl2025,
author = {Gao, David and Kunnawalkam Elayavalli, Srivatsav and Patchell, Gregory and Tan, Hui},
title = {Internal sequential commutation and single generation},
journal = {International Mathematics Research Notices},
volume = {2025}, number = {8}, pages = {rnaf103}, year = {2025},
doi = {10.1093/imrn/rnaf103}, eprint = {2404.12380}, archivePrefix = {arXiv}
}
@article{Pearcy1962,
author = {Pearcy, Carl}, title = {{$W^*$}-algebras with a single generator},
journal = {Proceedings of the American Mathematical Society},
volume = {13}, number = {6}, pages = {831--832}, year = {1962},
doi = {10.2307/2034069}
}
@article{SuzukiSaito1963,
author = {Suzuki, Noboru and Sait{\^o}, Teishir{\^o}},
title = {On the operators which generate continuous von {Neumann} algebras},
journal = {Tohoku Mathematical Journal, Second Series},
volume = {15}, number = {3}, pages = {277--280}, year = {1963},
doi = {10.2748/tmj/1178243811}
}
@article{DouglasPearcy1969,
author = {Douglas, Ronald G. and Pearcy, Carl},
title = {Von {Neumann} algebras with a single generator},
journal = {Michigan Mathematical Journal},
volume = {16}, number = {1}, pages = {21--26}, year = {1969},
doi = {10.1307/mmj/1029000161}
}
@article{Voiculescu2002Entropy,
author = {Voiculescu, Dan},
title = {Free entropy},
journal = {Bulletin of the London Mathematical Society},
volume = {34}, number = {3}, pages = {257--278}, year = {2002},
doi = {10.1112/S0024609301008992},
note = {Numbered references use the preprint version},
url = {https://arxiv.org/abs/math/0103168v1}
}
@article{BrannanEtAl2023,
author = {Brannan, Michael and Elzinga, Floris and Harris, Samuel J. and Yamashita, Makoto},
title = {Crossed product equivalence of quantum automorphism groups of finite dimensional {$C^*$}-algebras},
journal = {International Mathematics Research Notices},
volume = {2023}, number = {20}, pages = {17749--17787}, year = {2023},
doi = {10.1093/imrn/rnad060},
url = {https://doi.org/10.1093/imrn/rnad060}
}
@misc{OpenAIFreeGroupFactors2026,
author = {{OpenAI}},
title = {{An isomorphism of the free group factors}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/An-isomorphism-of-the-free-group-factors-September-23-2026/An-isomorphism-of-the-free-group-factors-September-23-2026.pdf}{OAI:An-isomorphism-of-the-free-group-factors-September-23-2026}},
year = {2026}
}
@article{Behncke1972,
author = {Behncke, Horst},
title = {Generators of finite {$W^*$}-algebras},
journal = {Tohoku Mathematical Journal, Second Series},
volume = {24}, number = {3}, pages = {401--408}, year = {1972},
doi = {10.2748/tmj/1178241478}
}
@misc{Hayes2026Bundles,
author = {Hayes, Ben},
title = {On Generators for {$W^*$}-bundles},
year = {2026},
eprint = {2609.16428}, archivePrefix = {arXiv}, primaryClass = {math.OA},
url = {https://arxiv.org/abs/2609.16428}
}
==== END Relative-generation-and-the-generator-problem-for-finite-factors//references.bib ====