# Single generation of II₁ factors

*Written by Claude Opus 5.5 (Anthropic), October 2026. Self-checked by the writing AI. Public domain (CC0).*

This lesson proves that every II₁ factor with separable predual is generated, as a von Neumann algebra, by one operator, equivalently by two self-adjoint operators. Three steps lead there. An irreducible copy of the hyperfinite II₁ factor \(R\) inside the factor is generated by two self-adjoint elements, because \(R\) is generated by commuting copies of \(2\times2\) matrices, and countably many commuting projections are recovered from one weighted sum. Adding a relative generator from [Relative generation by one unitary](relative-generation-by-one-unitary.md) gives four self-adjoint generators for every such factor. Finally, the factor is a \(2\times2\) matrix algebra over a corner of trace \(1/2\), and four self-adjoint generators of the corner can be packed into two self-adjoint matrices.

We use: Theorem 4.4 of [Relative generation by one unitary](relative-generation-by-one-unitary.md); Lemma 1.1 of [Free independence in ultraproducts](free-independence-in-ultraproducts.md) (corners of irreducible inclusions); Section 1, Lemma 2.3 and Corollary 2.4 of [Haar unitaries, freeness and tracial ultraproducts](haar-unitaries-freeness-and-ultraproducts.md); Theorem 2.1 of [MASAs in expected semifinite subfactors](course:OA-APPROX/masas-in-expected-semifinite-subfactors#theorem-2-1), which places a copy of the hyperfinite factor containing a maximal abelian subalgebra of the ambient algebra inside every irreducible subfactor; the infinite tensor product of matrix algebras and its factoriality, Section 5 and Corollary 5.3 of [Infinite tensor products and their reference states](course:OA-APPROX/infinite-tensor-products#5-commutants-factors-and-traces); spectral projections, [Section 5 of the spectral theorem lesson](course:foundations-of-von-neumann-algebras/the-spectral-theorem-for-bounded-self-adjoint-operators#OA-FND-ST-05); and normality of \(*\)-isomorphisms between von Neumann algebras, [Corollary 11.4 of its lesson](course:foundations-of-von-neumann-algebras/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras#OA-FND-WA-24).

## 1. Irreducible hyperfinite subfactors

Write \(R=\overline{\bigotimes}_{m\ge1}(M_2(\mathbb C),\operatorname{tr})\) for the von Neumann algebra generated, in the representation of the product trace, by mutually commuting unital copies \(M^{(m)}\cong M_2(\mathbb C)\), \(m\ge1\), whose finite products are full matrix algebras with their normalized traces. It is a II₁ factor (Corollary 5.3 and Section 12 of the tensor product lesson).

**Theorem 1.1.** Let \(N\subseteq M\) be an irreducible inclusion of II₁ factors with \(M\) of separable predual. There is a von Neumann subalgebra \(R_0\subseteq N\) isomorphic to \(R\) with \(R_0'\cap M=\mathbb C1\).

**Proof.** The trace-preserving conditional expectation \(E_N\colon M\to N\) is a normal expectation. Theorem 2.1 of the MASA lesson gives \(A\subseteq R_0\subseteq N\) with \(R_0\cong R\) and \(A\) maximal abelian in \(M\). Then \(R_0'\cap M\subseteq A'\cap M=A\subseteq R_0\), so \(R_0'\cap M\subseteq R_0'\cap R_0=\mathbb C1\), because \(R_0\) is a factor. \(\square\)

## 2. Two generators for the hyperfinite factor

**Lemma 2.1.** Let \((e_m)_{m\ge1}\) be commuting projections in a von Neumann algebra and \(b=\sum_{m\ge1}2\cdot3^{-m}e_m\), a norm convergent sum. Then every \(e_m\) lies in \(W^*(b)\).

**Proof.** Suppose \(e_1,\dots,e_{m-1}\in W^*(b)\), and put \(r_m=b-\sum_{j<m}2\cdot3^{-j}e_j\in W^*(b)\). Then \(r_m=2\cdot3^{-m}e_m+\rho\) with \(\rho=\sum_{j>m}2\cdot3^{-j}e_j\), and \(0\le\rho\le3^{-m}\). All these operators commute, so \(e_m\) reduces \(r_m\): on the range of \(e_m\), \(2\cdot3^{-m}\le r_m\le3\cdot3^{-m}\), and on the range of \(1-e_m\), \(0\le r_m\le3^{-m}\). Hence the spectral projection of \(r_m\) for the interval \((\frac32\cdot3^{-m},\infty)\) equals \(e_m\), and it belongs to \(W^*(r_m)\subseteq W^*(b)\). Induction on \(m\) proves the lemma. \(\square\)

**Lemma 2.2.** \(R\) is generated by two self-adjoint elements.

**Proof.** In \(M^{(m)}\cong M_2(\mathbb C)\) let
\[
d_m=\begin{pmatrix}1&0\\0&0\end{pmatrix},\qquad q_m=\frac12\begin{pmatrix}1&1\\1&1\end{pmatrix}.
\]
Then \(2d_mq_m(1-d_m)\) is the matrix unit \(e_{12}\), so \(d_m\) and \(q_m\) generate \(M^{(m)}\). The projections \(d_m\), \(m\ge1\), commute with each other, and so do the \(q_m\). Put \(b=\sum_m2\cdot3^{-m}d_m\) and \(b'=\sum_m2\cdot3^{-m}q_m\). By Lemma 2.1, \(W^*(b,b')\) contains every \(d_m\) and \(q_m\), hence every \(M^{(m)}\), hence \(R\). \(\square\)

A \(*\)-isomorphism of von Neumann algebras is normal, so it maps \(W^*(S)\) onto \(W^*\) of the image of \(S\). Hence every von Neumann algebra isomorphic to \(R\) is generated by two self-adjoint elements.

## 3. Four generators

**Proposition 3.1.** Every II₁ factor \(M\) with separable predual is generated by four self-adjoint elements.

**Proof.** Theorem 1.1 with \(N=M\) (irreducible, as \(M\) is a factor) gives \(R_0\subseteq M\) isomorphic to \(R\) with \(R_0'\cap M=\mathbb C1\). It is a II₁ factor, so \(R_0\subseteq M\) is an irreducible inclusion of II₁ factors. By Lemma 2.2, \(R_0=W^*(a,b)\) with \(a,b\) self-adjoint. Theorem 4.4 of the relative generation lesson gives a unitary \(u\in M\) with \(W^*(R_0\cup\{u\})=M\). Since \(u=\operatorname{Re}u+i\operatorname{Im}u\),
\[
M=W^*(a,b,\operatorname{Re}u,\operatorname{Im}u).\qquad\square
\]

## 4. Packing four generators into two by two matrices

**Lemma 4.1.** If a von Neumann algebra \(N\) is generated by four self-adjoint elements, then \(M_2(N)\) is generated by two self-adjoint elements.

**Proof.** Let \(N=W^*(a_1,a_2,a_3,a_4)\) with \(a_i=a_i^*\), choose \(C>\|a_1\|+\|a_2\|\), and put
\[
\begin{gathered}
A=\begin{pmatrix}a_1&0\\0&a_2+C\end{pmatrix},\\
B=\begin{pmatrix}a_3&1\\1&a_4\end{pmatrix},\qquad Q=W^*(A,B).
\end{gathered}
\]
Let \(E_{ij}\) be the matrix units of \(M_2(N)\). The two diagonal blocks of \(A\) have spectra in \([-\|a_1\|,\|a_1\|]\) and \([C-\|a_2\|,C+\|a_2\|]\), which are separated by any \(\lambda\) with \(\|a_1\|<\lambda<C-\|a_2\|\). Since \(A\) acts on the two summands separately, its spectral projection for \((-\infty,\lambda)\) is \(E_{11}\). So \(E_{11},E_{22}=1-E_{11}\in Q\), \(E_{12}=E_{11}BE_{22}\in Q\) and \(E_{21}=E_{12}^*\in Q\). The corner \(E_{11}QE_{11}\) contains
\[
\begin{aligned}
E_{11}AE_{11}&=\begin{pmatrix}a_1&0\\0&0\end{pmatrix},\\
E_{12}AE_{21}-CE_{11}&=\begin{pmatrix}a_2&0\\0&0\end{pmatrix},\\
E_{11}BE_{11}&=\begin{pmatrix}a_3&0\\0&0\end{pmatrix},\\
E_{12}BE_{21}&=\begin{pmatrix}a_4&0\\0&0\end{pmatrix}.
\end{aligned}
\]
The map \(x\mapsto\begin{pmatrix}x&0\\0&0\end{pmatrix}\) is a normal \(*\)-isomorphism of \(N\) onto \(E_{11}M_2(N)E_{11}\), and \(E_{11}QE_{11}\) is a von Neumann algebra on that corner containing the images of \(a_1,\dots,a_4\). So it contains the image of \(N\). Every matrix is \(\sum_{i,j}E_{i1}\begin{pmatrix}x_{ij}&0\\0&0\end{pmatrix}E_{1j}\), so \(Q=M_2(N)\). \(\square\)

## 5. The main theorem

**Theorem 5.1.** Every II₁ factor \(M\) with separable predual is generated by two self-adjoint elements. Equivalently, \(M=W^*(x)\) for one \(x\in M\).

**Proof.** By Corollary 2.4 of the first lesson there is a projection \(p\in M\) with \(\tau(p)=1/2\). The corner \(N=pMp\) is a II₁ factor (Lemma 1.1 of the second lesson, for the inclusion \(M\subseteq M\)). Its predual is separable: \(\psi\mapsto\psi(p\,\cdot\,p)\) maps the normal functionals of \(N\) isometrically into those of \(M\), because \(x\mapsto pxp\) is normal and maps the unit ball of \(M\) onto that of \(N\).

By Lemma 2.3 of the first lesson there is \(v\in M\) with \(v^*v=p\) and \(vv^*=1-p\). Then \(f_{11}=p\), \(f_{12}=v^*\), \(f_{21}=v\), \(f_{22}=1-p\) are matrix units in \(M\): \(f_{ij}f_{kl}=\delta_{jk}f_{il}\), \(f_{ij}^*=f_{ji}\), \(f_{11}+f_{22}=1\). The map
\[
\begin{aligned}
&\Phi\colon M_2(N)\to M,\\
&\Phi\bigl((b_{ij})\bigr)=\sum_{i,j}f_{i1}b_{ij}f_{1j}=b_{11}\\
&\quad +b_{12}v^*+vb_{21}+vb_{22}v^*
\end{aligned}
\]
is a unital \(*\)-isomorphism. It is multiplicative and \(*\)-preserving by the matrix unit relations and \(b_{ij}=pb_{ij}p\); it is injective because \(f_{1i}\Phi((b_{kl}))f_{j1}=b_{ij}\); and it is onto, because every \(m\in M\) equals \(\sum_{i,j}f_{i1}(f_{1i}mf_{j1})f_{1j}\) with \(f_{1i}mf_{j1}\in pMp\). As a \(*\)-isomorphism of von Neumann algebras it is normal.

Proposition 3.1 for \(N\) and Lemma 4.1 give self-adjoint \(A,B\in M_2(N)\) with \(W^*(A,B)=M_2(N)\). Put \(x=\Phi(A)+i\Phi(B)\). Its real and imaginary parts are \(\Phi(A)\) and \(\Phi(B)\), so
\[
\begin{aligned}
W^*(x)&=W^*\bigl(\Phi(A),\Phi(B)\bigr)\\
&=\Phi\bigl(M_2(N)\bigr)=M .
\end{aligned}
\]
Conversely, if \(M=W^*(x)\), then \(M=W^*(\operatorname{Re}x,\operatorname{Im}x)\). \(\square\)

The preprint [OpenAI-Gen] also draws consequences that combine Theorem 5.1 with further results. The [next lesson](the-generator-problem.md) proves the first of them: with the properly infinite case of Wogen and Willig's direct-integral reduction, every von Neumann algebra with separable predual is singly generated. The [lesson after it](the-generator-invariant.md) proves the second: the generator invariant of Dykema, Sinclair, Smith and White [DSSW] vanishes for every II₁ factor with separable predual. The third uses Voiculescu's free entropy dimension: two generating tuples of a free group factor on which this dimension differs.

## 6. Exercises

**Exercise 6.1.** Let \(a=\operatorname{diag}(1,2,\dots,n)\) and let \(b=s+s^*\), where \(s\) is the matrix with ones directly above the diagonal and zeros elsewhere. Show that \(M_n(\mathbb C)=W^*(a,b)\), so that \(M_n(\mathbb C)\) is generated by the single matrix \(a+ib\).

**Exercise 6.2.** Show that \(\ell^\infty(\mathbb N)\), acting on \(\ell^2(\mathbb N)\), is generated by one self-adjoint element.

**Exercise 6.3.** Show that Lemma 4.1 holds for every constant \(C\) for which the spectra of \(a_1\) and \(a_2+C\) are disjoint.

**Exercise 6.4.** Show that if a von Neumann algebra \(N\) is generated by \(k\ge1\) self-adjoint elements, then so is every finite direct sum \(N\oplus\dots\oplus N\).

## 7. Solutions

**6.1.** The spectral projections of \(a\) are the diagonal matrix units \(e_{ii}\). Then \(e_{ii}be_{i+1,i+1}=e_{i,i+1}\), and the matrix units \(e_{i,i+1}\) and their adjoints generate all \(e_{ij}\) as products. So \(W^*(a,b)\) contains every matrix unit.

**6.2.** Let \(\delta_m\) be the projection onto the \(m\)th coordinate. These commute, so Lemma 2.1 puts every \(\delta_m\) in \(W^*(b)\) for \(b=\sum_m2\cdot3^{-m}\delta_m\). A bounded sequence \(c\) is the strong limit of the finite sums \(\sum_{m\le k}c_m\delta_m\), and \(W^*(b)\) is strongly closed.

**6.3.** If the spectra \(\sigma(a_1)\) and \(\sigma(a_2+C)\) are disjoint compact sets, choose a continuous \(g\) equal to \(1\) on the first and \(0\) on the second. Then \(g(A)=E_{11}\), and the rest of the proof is unchanged.

**6.4.** Let \(N=W^*(c_1,\dots,c_k)\) and let \(N^{\oplus r}=N\oplus\dots\oplus N\). Choose constants \(\gamma_1,\dots,\gamma_r\) such that the intervals \([\gamma_i-\|c_1\|,\gamma_i+\|c_1\|]\) are disjoint, and put \(c_1'=(c_1+\gamma_1)\oplus\dots\oplus(c_1+\gamma_r)\) and \(c_j'=c_j\oplus\dots\oplus c_j\) for \(j\ge2\). Spectral projections of \(c_1'\) give the central projections onto the summands, and in the \(i\)th summand the elements \(c_1'-\gamma_i\) and \(c_j'\) cut down to \(c_1,\dots,c_k\). So \(W^*(c_1',\dots,c_k')=N^{\oplus r}\).

## References

- [OpenAI-Gen] OpenAI, Relative generation and the generator problem for finite factors, preprint, 23 September 2026. https://github.com/openai/math/tree/main/preprints/Relative-generation-and-the-generator-problem-for-finite-factors-September-23-2026
- [DSSW] K. Dykema, A. Sinclair, R. Smith, S. White, Generators of II₁ factors, Operators and Matrices 2 (2008), 555–582. https://arxiv.org/abs/0706.1953
- [Popa-Ind] S. Popa, Independence properties in subalgebras of ultraproduct II₁ factors, Journal of Functional Analysis 266 (2014), 5818–5846. https://arxiv.org/abs/1308.3982