# The generator invariant

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

Shen introduced a numerical invariant of II₁ factors that measures how economically a factor can be generated relative to finite partitions of the unit [Sh]. Dykema, Sinclair, Smith and White reformulated it as the *generator invariant* \(\mathcal G(M)\), with a self-adjoint version \(\mathcal G_{\rm sa}(M)\), and proved that a positive value for one separable II₁ factor would produce separable II₁ factors that are not singly generated [DSSW, Corollary 5.1]. Since every II₁ factor with separable predual is singly generated ([Theorem 5.1 of Single generation of II₁ factors](single-generation-of-ii1-factors.md#5-the-main-theorem)), the invariant vanishes, as [OpenAI-Gen, Section 5.1] observes.

**Theorem 3.1.** For every II₁ factor \(M\) with separable predual, \(\mathcal G(M)=\mathcal G_{\rm sa}(M)=0\).

The proof needs two estimates from [DSSW]: a factor generated by \(n\) self-adjoint elements has \(\mathcal G_{\rm sa}\le n-1\) (Lemma 2.1), and \(\mathcal G_{\rm sa}(M)\le t^2\mathcal G_{\rm sa}(pMp)\) for a projection \(p\) of trace \(t\) (Lemma 2.2). [DSSW] also prove the exact scaling formula \(\mathcal G(pMp)=t^{-2}\mathcal G(M)\) and the identity \(\mathcal G_{\rm sa}=2\mathcal G\); we do not need them.

We use: Theorem 5.1 of [Single generation of II₁ factors](single-generation-of-ii1-factors.md); from [Haar unitaries, freeness and tracial ultraproducts](haar-unitaries-freeness-and-ultraproducts.md), Lemma 2.3 (projections of equal trace are equivalent) and Corollary 2.4 (a II₁ factor has projections of every trace); that a corner \(pMp\) of a II₁ factor is a II₁ factor with separable predual (Lemma 1.1 of [Relative generation by one unitary](relative-generation-by-one-unitary.md) and the proof of Theorem 5.1 of [Single generation of II₁ factors](single-generation-of-ii1-factors.md)); the definition of type II from [Projections and types of von Neumann algebras, Definition 7.1](course:foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras#OA-FND-TY-10); and the Weierstrass approximation theorem, [Corollary 14.3 of the Stone–Weierstrass lesson](course:function-algebras-and-approximation/the-stone-weierstrass-theorem-for-functions-vanishing-at-infinity#OA-FND-SW-14).

## 1. Definitions

Let \(M\) be a II₁ factor with its faithful normal tracial state \(\tau\). Following [DSSW], for a subset \(X\subseteq M\) we write \(W^*_0(X)\) for the smallest \(\sigma\)-weakly closed \(*\)-subalgebra of \(M\) containing \(X\); it need not contain \(1\). A finite set \(X\) *generates* \(M\) in this section if \(W^*_0(X)=M\). Let \(\mathcal P\) be the set of finite families \(P\) of mutually orthogonal nonzero projections of \(M\) with sum \(1\). For \(x\in M\), a finite \(X\subseteq M\) and \(P\in\mathcal P\), put
\[
\begin{gathered}
\mathcal I(x;P)=\sum_{\substack{p,q\in P\\pxq\neq0}}\tau(p)\tau(q),\\
\mathcal I(X;P)=\sum_{x\in X}\mathcal I(x;P),\\
\mathcal I(X)=\inf_{P\in\mathcal P}\mathcal I(X;P).
\end{gathered}
\]
The *generator invariant* \(\mathcal G(M)\) is the infimum of \(\mathcal I(X)\) over the finite sets \(X\) with \(W^*_0(X)=M\), and the *self-adjoint generator invariant* \(\mathcal G_{\rm sa}(M)\) is the same infimum over the finite sets \(X\) of self-adjoint elements with \(W^*_0(X)=M\),
both \(+\infty\) if \(M\) has no finite generating set.

Three facts are immediate.

- (F1) \(0\le\mathcal I(X;P)\le|X|\), since \(\sum_{p,q\in P}\tau(p)\tau(q)=1\).
- (F2) \(\mathcal G(M)\le\mathcal G_{\rm sa}(M)\), because a self-adjoint generating set is a generating set.
- (F3) If \(x\) commutes with every member of \(P\), then \(pxq=pqx=0\) for \(p\neq q\), so \(\mathcal I(x;P)\le\sum_{p\in P}\tau(p)^2\), which is at most \(\max_{p\in P}\tau(p)\).

The convention \(W^*_0\) is the one of [DSSW]. The von Neumann algebra \(W^*(X)=W^*_0(X\cup\{1\})\) used in the rest of the course is at least as large, so a generating set in the present sense also generates in the earlier sense.

**Lemma 1.1.** Let \(c\in M\) be self-adjoint with \(\sigma(c)\subseteq(0,\infty)\). Then \(1\in W^*_0(\{c\})\). Consequently, if \(W^*(X)=M\) for a finite set \(X\) and some \(c\in X\) is invertible and positive, then \(W^*_0(X)=M\).

**Proof.** The function equal to \(0\) at \(0\) and to \(1\) on \(\sigma(c)\) is continuous on the compact set \(\{0\}\cup\sigma(c)\), so by Weierstrass it is a uniform limit there of polynomials \(f_k\) with \(f_k(0)=0\). Then \(f_k(c)\to1\) in norm, and \(f_k(c)\) lies in the \(*\)-algebra generated by \(c\) without unit. The second statement follows, since \(W^*_0(X)\) then contains \(1\) and \(X\). \(\square\)

## 2. Two estimates

**Lemma 2.1** (few self-adjoint generators). If a II₁ factor \(M\) is generated, as a von Neumann algebra, by \(n\) self-adjoint elements \(x_1,\ldots,x_n\), then \(\mathcal G_{\rm sa}(M)\le n-1\).

**Proof.** Put \(x_n'=x_n+\lambda1\) with \(\lambda>\|x_n\|\); then \(x_n'\) is self-adjoint with spectrum in \((0,\infty)\), and \(W^*(x_1,\ldots,x_{n-1},x_n')=W^*(x_1,\ldots,x_n)\), which is \(M\). By Lemma 1.1 the set \(X=\{x_1,\ldots,x_{n-1},x_n'\}\) generates \(M\) in the sense of Section 1.

*A diffuse abelian algebra containing \(x_n'\).* By Zorn's lemma there is a maximal abelian von Neumann subalgebra \(A\) of \(M\) containing \(x_n'\): the abelian von Neumann subalgebras containing \(x_n'\), ordered by inclusion, are nonempty, and the von Neumann algebra generated by a chain is abelian. Maximality gives \(A'\cap M=A\). If \(e\) were a nonzero minimal projection of \(A\), every \(y\in eMe\) would commute with \(A\), since \(eAe=\mathbb Ce\) and \(y\) vanishes on the other corners; so \(eMe\subseteq A\) and \(eMe=\mathbb Ce\), and \(e\) would be a nonzero abelian projection of the type II algebra \(M\), which is impossible. So every nonzero projection of \(A\) is the sum of two nonzero orthogonal projections of \(A\). Repeatedly keeping the half of smaller trace, every nonzero projection of \(A\) has nonzero subprojections in \(A\) of arbitrarily small trace. As in the proof of Lemma 55.1 of [Finite Fourier bases produce one small quantized corner](course:OA-SUBFACTORS/html/finite-phase-local-quantization#a-diffuse-diagonal-supplies-exact-scalar-dimensions), a maximal subprojection of trace at most \(c\) then has trace exactly \(c\): every projection \(f\in A\) contains, for each \(0\le c\le\tau(f)\), a projection of \(A\) of trace \(c\).

*The estimate.* Let \(k\ge1\). Splitting off projections of trace \(1/k\) one at a time gives \(P=\{p_1,\ldots,p_k\}\in\mathcal P\) in \(A\) with \(\tau(p_i)=1/k\). Each \(p_i\) commutes with \(x_n'\in A\), so \(\mathcal I(x_n';P)\le1/k\) by (F3), and \(\mathcal I(x_i;P)\le1\) for \(i<n\) by (F1). Hence \(\mathcal G_{\rm sa}(M)\le\mathcal I(X;P)\le n-1+1/k\) for every \(k\). \(\square\)

**Lemma 2.2** (compression). Let \(M\) be a II₁ factor, \(p\in M\) a projection with \(\tau(p)=t\in(0,1)\), and \(pMp\) the corner with its trace \(\tau_p=t^{-1}\tau|_{pMp}\). Then \(\mathcal G_{\rm sa}(M)\le t^2\mathcal G_{\rm sa}(pMp)\).

**Proof.** We may assume that \(pMp\) has a finite generating set of self-adjoint elements \(X\) (in the sense of Section 1, inside \(pMp\)), and we fix one and \(\varepsilon>0\). Choose \(E=\{e_1,\ldots,e_r\}\in\mathcal P(pMp)\) with \(\mathcal I_{pMp}(X;E)<\mathcal I_{pMp}(X)+\varepsilon\). Since \(\tau_p=t^{-1}\tau\) on \(pMp\), the identity (2.1) holds:
\[
\begin{aligned}
&\mathcal I_{pMp}(X;E)\\
&\quad=t^{-2}\sum_{x\in X}\sum_{e_ixe_j\neq0}\tau(e_i)\tau(e_j).
\end{aligned}
\]

*Small pieces.* Corners of a II₁ factor are II₁ factors, and a II₁ factor has projections of every trace (Corollary 2.4 of the lesson on Haar unitaries). Splitting off projections of trace \(\delta<\varepsilon/m\) one at a time inside the corners \(e_jMe_j\), where \(m\ge1\) is the largest integer with \(mt\le1\), we write each \(e_j\) as a finite sum of mutually orthogonal nonzero projections of trace at most \(\delta\). Since \(1-mt<t=\tau(p)\), we can add these pieces one at a time until the sum first reaches trace at least \(1-mt\); if it exceeds \(1-mt\), split the last piece added into two projections, the first of which brings the sum to trace exactly \(1-mt\). The resulting family \(F\in\mathcal P(pMp)\) refines \(E\), its members have trace less than \(\varepsilon/m\), and some subfamily of \(F\) has a sum \(f_0\) with \(\tau(f_0)=1-mt\) (if \(mt=1\), \(f_0=0\)).

*Copies of \(p\).* Choose mutually orthogonal projections \(p_1=p,p_2,\ldots,p_{m+1}\) with sum \(1\), \(\tau(p_i)=t\) for \(i\le m\) and \(\tau(p_{m+1})=1-mt\) (again by Corollary 2.4, in the corners of the remaining projections). By Lemma 2.3 of the lesson on Haar unitaries there are partial isometries \(v_2,\ldots,v_{m+1}\) with \(v_iv_i^*=p_i\) for all \(i\), \(v_i^*v_i=p\) for \(2\le i\le m\), and \(v_{m+1}^*v_{m+1}=f_0\). Put \(v_1=p\).

*Generators of \(M\).* Let \(Y\) be the union of \(X\) and the elements \(v_i+v_i^*\) and \(i(v_i-v_i^*)\) for \(2\le i\le m+1\),
a finite set of self-adjoint elements. \(W^*_0(Y)\) contains \(pMp=W^*_0(X)\) and every \(v_i\), hence every \(p_i=v_ipv_i^*\) and \(1=\sum_ip_i\). For \(y\in M\), \(p_iyp_j=v_i(v_i^*yv_j)v_j^*\) with \(v_i^*yv_j\in pMp\) (here \(v_1=p\)), so \(p_iMp_j\subseteq W^*_0(Y)\) and \(W^*_0(Y)=M\).

*The partition.* Let \(G\) be the set of the projections \(v_ifv_i^*\) with \(1\le i\le m+1\), \(f\in F\) and \(f\le v_i^*v_i\). For \(i\le m\) the members with index \(i\) sum to \(v_ipv_i^*=p_i\), and for \(i=m+1\) to \(v_{m+1}f_0v_{m+1}^*=p_{m+1}\); so \(G\) is a family of mutually orthogonal nonzero projections with sum \(1\), \(G\in\mathcal P(M)\), and \(G\supseteq F\). For \(x\in X\subseteq pMp\), the only members \(g,g'\in G\) with \(gxg'\neq0\) lie in \(F\), because \(p_ix=0\) for \(i\neq1\). A refinement does not increase \(\mathcal I\): if \(f\le e\) and \(f'\le e'\) with \(f,f'\in F\) and \(e,e'\in E\), then \(fxf'=f(exe')f'\), so \(fxf'\neq0\) forces \(exe'\neq0\), and \(\sum_{f\le e,\,f'\le e'}\tau(f)\tau(f')=\tau(e)\tau(e')\). So, by (2.1),
\[
\begin{aligned}
&\mathcal I_M(X;G)=\sum_{x\in X}\sum_{\substack{f,f'\in F\\fxf'\neq0}}\tau(f)\tau(f')\\
&\quad \le t^2\,\mathcal I_{pMp}(X;E)<t^2\mathcal I_{pMp}(X)+t^2\varepsilon .
\end{aligned}
\]
For \(2\le i\le m+1\), if \(v_jfv_j^*\,v_i\,v_kf'v_k^*\neq0\), then \(j=i\) (otherwise \(v_j^*v_i=0\)) and \(k=1\) (otherwise \(v_iv_k=0\)), and then \(gv_ig'=v_i\,f(v_i^*v_i)f'\), where \(f(v_i^*v_i)f'=ff'\) because \(f\le v_i^*v_i\); so \(gv_ig'\neq0\) forces \(f=f'\). So
\[
\begin{aligned}
&\mathcal I_M(v_i;G)\le\sum_{f\in F}\tau(f)^2\\
&\quad \le\max_{f\in F}\tau(f)<\varepsilon/m,
\end{aligned}
\]
and the same bound holds for \(v_i^*\). By the subadditivity \(\mathcal I(a+b;G)\le\mathcal I(a;G)+\mathcal I(b;G)\), which holds because \(g(a+b)g'\neq0\) implies \(gag'\neq0\) or \(gbg'\neq0\), the two self-adjoint elements built from \(v_i\) contribute at most \(4\varepsilon/m\). Hence
\[
\begin{aligned}
&\mathcal G_{\rm sa}(M)\le\mathcal I_M(Y;G)\\
&\quad <t^2\mathcal I_{pMp}(X)+t^2\varepsilon+4\varepsilon .
\end{aligned}
\]
Since \(\varepsilon\) and the self-adjoint generating set \(X\) of \(pMp\) were arbitrary, \(\mathcal G_{\rm sa}(M)\le t^2\mathcal G_{\rm sa}(pMp)\). \(\square\)

## 3. The invariant vanishes

**Theorem 3.1** (OpenAI, after Dykema–Sinclair–Smith–White). For every II₁ factor \(M\) with separable predual, \(\mathcal G(M)=\mathcal G_{\rm sa}(M)=0\).

**Proof.** Let \(0<t<1\) and let \(p\in M\) be a projection of trace \(t\) (Corollary 2.4 of the lesson on Haar unitaries). The corner \(pMp\) is a II₁ factor with separable predual, so by Theorem 5.1 of the lesson on single generation it is generated by two self-adjoint elements. Lemma 2.1 gives \(\mathcal G_{\rm sa}(pMp)\le1\), and Lemma 2.2 gives \(\mathcal G_{\rm sa}(M)\le t^2\). As \(t\) is arbitrary, \(\mathcal G_{\rm sa}(M)=0\), and \(\mathcal G(M)=0\) by (F2) and nonnegativity. \(\square\)

The values are infima and are not attained: for a nonempty generating set \(X\) and \(P\in\mathcal P\), some \(x\in X\) is nonzero, some \(pxq\) with \(p,q\in P\) is nonzero, and \(\mathcal I(X;P)\ge\tau(p)\tau(q)>0\).

Shen showed that \(\mathcal G(M)<\frac14\) implies generation by a projection and a self-adjoint element ([Sh, Theorem 4.1], as recalled in [DSSW, Section 3]), and [DSSW, Theorem 3.1] shows that \(\mathcal G(M)<\frac n2\) implies generation by \(n+1\) self-adjoint elements. Before [OpenAI-Gen], single generation of a II₁ factor was known in many cases through the vanishing of this invariant; the argument above runs in the opposite direction.

## 4. Exercises

**Exercise 4.1** (easy). Show that \(\mathcal I(X^*;P)=\mathcal I(X;P)\), where \(X^*=\{x^*:x\in X\}\).

**Exercise 4.2** (medium). Let \(R\) be a II₁ factor and \(u\in R\) a unitary. Show that \(\mathcal I(uXu^*;uPu^*)=\mathcal I(X;P)\), and deduce that \(\mathcal G\) and \(\mathcal G_{\rm sa}\) are invariants of the factor up to isomorphism.

**Exercise 4.3** (medium). Show directly from Lemma 2.1 that a II₁ factor generated by two self-adjoint elements has \(\mathcal G(M)\le1\), and explain why Lemma 2.2 alone, without Theorem 5.1 of the lesson on single generation, does not give \(\mathcal G(M)=0\).

## 5. Solutions

**4.1.** \(px^*q=(qxp)^*\), so \(px^*q\neq0\) exactly when \(qxp\neq0\); the sum defining \(\mathcal I(x^*;P)\) is the sum defining \(\mathcal I(x;P)\) with \(p\) and \(q\) exchanged.

**4.2.** \((upu^*)(uxu^*)(uqu^*)=u(pxq)u^*\) vanishes exactly when \(pxq\) does, and \(\tau(upu^*)=\tau(p)\). So the generating sets and partitions of \(R\) correspond under \(\operatorname{Ad}u\) with the same values of \(\mathcal I\). More generally, a \(*\)-isomorphism \(\pi:R\to S\) of II₁ factors carries the trace of \(R\) to that of \(S\) (the trace of a II₁ factor is unique), maps \(W^*_0\) of a set to \(W^*_0\) of its image, and preserves the relations \(pxq\neq0\); so \(\mathcal G(S)=\mathcal G(R)\) and \(\mathcal G_{\rm sa}(S)=\mathcal G_{\rm sa}(R)\).

**4.3.** By Lemma 2.1 with \(n=2\), \(\mathcal G_{\rm sa}(M)\le1\), and \(\mathcal G(M)\le\mathcal G_{\rm sa}(M)\) by (F2). Lemma 2.2 bounds \(\mathcal G_{\rm sa}(M)\) by \(t^2\mathcal G_{\rm sa}(pMp)\); without information on the corners, \(\mathcal G_{\rm sa}(pMp)\) could be large or infinite for small \(t\), and the bound gives nothing. Theorem 5.1 of the lesson on single generation supplies \(\mathcal G_{\rm sa}(pMp)\le1\) uniformly in \(t\).

## References

- [DSSW] K. Dykema, A. Sinclair, R. Smith and S. White, *Generators of II₁ factors*, Operators and Matrices 2 (2008), no. 4, 555–582; arXiv:0706.1953. https://arxiv.org/abs/0706.1953. Definitions 2.1, 2.3 and 2.7, Lemmas 4.3 and 5.3 (Lemmas 2.2 and 2.1 here), Theorem 3.1, Corollary 5.1 and Theorem 5.5.
- [OpenAI-Gen] OpenAI, *Relative generation and the generator problem for finite factors*, OpenAI Math Release preprint, 23 September 2026, Section 5.1. https://github.com/openai/math/blob/main/preprints/Relative-generation-and-the-generator-problem-for-finite-factors-September-23-2026
- [Sh] J. Shen, *Type II₁ factors with a single generator*, J. Operator Theory 62 (2009), 421–438; arXiv:math/0511327. https://arxiv.org/abs/math/0511327