# The generator problem

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

The *generator problem* asks whether every von Neumann algebra on a separable Hilbert space is generated by a single operator. The [previous lesson](single-generation-of-ii1-factors.md) proved this for factors of type II₁ with separable predual, following [OpenAI-Gen]. This lesson proves the general statement, as [OpenAI-Gen] deduces it from the II₁ case.

**Theorem 5.1.** Every von Neumann algebra \(M\) on a separable Hilbert space is generated by one operator: \(M=W^*(x)\) for some \(x\in M\). Equivalently, \(M\) is generated by two self-adjoint operators.

Corollary 5.2 gives the same conclusion for every von Neumann algebra with separable predual, in whatever representation.

The deduction has three steps.

- Properly infinite algebras are singly generated, by a theorem of Wogen [Wo]. Section 2 proves it with a sequence of matrix units.
- A finite factor on a separable Hilbert space is a matrix algebra or a II₁ factor with separable predual (Section 3), and so it is singly generated.
- If almost every fibre of a direct integral of von Neumann algebras is singly generated, so is the direct integral, by a theorem of Willig [Wi]. Section 4 chooses generators of the fibres in a Borel way and absorbs the diagonal algebra into one generator.

The central decomposition applies the third step to the finite part of \(M\), and the first step treats the properly infinite part (Section 5). The case of type I algebras, due to Pearcy [Pe], is contained in this argument.

We use the following results. From [Projections and types of von Neumann algebras](course:foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras#OA-FND-TY-10): Theorem 7.2 (a central projection separating the finite part from the properly infinite part), Proposition 8.4 (matrix units split off a matrix algebra), and [Proposition 15.3(3)](course:foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras#OA-FND-TY-19), which decomposes the unit of a properly infinite locally \(\sigma\)-finite algebra into countably many projections equivalent to \(1\). From [Abelian operator algebras](course:foundations-of-von-neumann-algebras/abelian-operator-algebras#OA-FND-AO-15): Theorem 8.2, that a countably generated abelian von Neumann algebra is generated by one self-adjoint element. From [The double commutant theorem](course:foundations-of-von-neumann-algebras/the-double-commutant-theorem#OA-FND-BI-02): Proposition 2.1, Theorem 4.4 and [Theorem 9.5](course:foundations-of-von-neumann-algebras/the-double-commutant-theorem#OA-FND-BI-12) (\(\sigma\)-finite algebras). Kaplansky's density theorem, [Theorem 7.1 of its lesson](course:foundations-of-von-neumann-algebras/kaplansky-s-density-theorem-and-its-consequences#OA-FND-KD-07). 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). From [The central decomposition and the types of the fibres](course:measurable-fields-and-direct-integrals/the-central-decomposition-and-the-types-of-the-fibres#1-conventions-and-results-used): the facts on measurable fields and direct integrals of von Neumann algebras collected in its Section 1 (Facts 1.1, 1.2, 1.7(a)), Theorem 4.1 (central decomposition), Lemma 5.1 (Borel choice) and Theorem 5.3 (types of the fibres). Finally, [Lemma 4.2 of Base changes and disintegration](course:OA-FOUND-REMAINDER/reader/base-changes-and-disintegration#BD-SEPARABLE-GNS) represents an algebra with separable predual on a separable Hilbert space, and from the lesson on compact and trace-class operators we use [Theorem 6.2(d)](course:foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies#OA-FND-LT-07) (finite sums of vector functionals are dense in the predual of \(B(H)\)) and [Theorem 9.4(a)](course:foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies#OA-FND-LT-10) (the predual of a von Neumann algebra is a quotient of that of \(B(H)\)).

## 1. Generators

For a subset \(S\) of a von Neumann algebra \(M\) on \(H\), \(W^*(S)=(S\cup S^*)''\) is the von Neumann algebra generated by \(S\); it contains \(1\). For a projection \(z\) in the centre of \(M\), the *central summand* \(Mz\) is the von Neumann algebra \(\{xz:x\in M\}\) on \(zH\), with unit \(z\); generation inside \(Mz\) is taken on \(zH\).

**Lemma 1.1** (operations on generators). Let \(M\) be a von Neumann algebra on a separable Hilbert space \(H\).

1. For self-adjoint \(a,b\in M\), \(W^*(a+ib)=W^*(a,b)\).
2. Let \(q_1,q_2,\ldots\in M\) be mutually orthogonal projections with \(\sum_kq_k=1\), and \(\lambda_1,\lambda_2,\ldots\) distinct real numbers converging to \(0\), none equal to \(0\). Then \(a=\sum_k\lambda_kq_k\) converges in norm, and every \(q_k\) lies in \(W^*(a)\).
3. (*Merging a central element.*) If \(M=W^*(x,c)\) for some \(x\in M\) and some self-adjoint \(c\) in the centre of \(M\), then \(M=W^*(y)\) for one \(y\in M\).
4. (*Two central summands.*) Let \(z\) be a central projection of \(M\). If \(Mz=W^*(x_1)\) and \(M(1-z)=W^*(x_2)\), with \(x_1\in Mz\) and \(x_2\in M(1-z)\), then \(M=W^*(y)\) for one \(y\in M\).

**Proof.** (1) \(a+ib\) and its adjoint generate the same \(*\)-algebra as \(a\) and \(b\), because \(a=\frac12((a+ib)+(a+ib)^*)\) and \(b=\frac1{2i}((a+ib)-(a+ib)^*)\).

(2) The partial sums converge in norm because \(\|\sum_{k>K}\lambda_kq_k\|=\sup_{k>K}|\lambda_k|\to0\). The spectrum of \(a\) lies in the closure \(\Lambda=\{0\}\cup\{\lambda_k\}\): for \(\mu\notin\Lambda\), \(\sum_k(\lambda_k-\mu)^{-1}q_k\) is a bounded inverse of \(a-\mu\). For a polynomial \(p\), \(p(a)=\sum_kp(\lambda_k)q_k\), by orthogonality. A continuous \(f\) on \(\Lambda\) is a uniform limit of polynomials \(p_n\) there (Weierstrass), and both \(p_n(a)\to f(a)\) and \(\sum_kp_n(\lambda_k)q_k\to\sum_kf(\lambda_k)q_k\) in norm; so \(f(a)=\sum_kf(\lambda_k)q_k\). Each \(\lambda_k\) is an isolated point of \(\Lambda\), so its indicator is continuous on \(\Lambda\), and \(q_k=1_{\{\lambda_k\}}(a)\) lies in the norm-closed algebra generated by \(a\), hence in \(W^*(a)\).

(3) Write \(x=a+ib\) with \(a,b\) self-adjoint. Since \(c\) is central, \(a\) and \(c\) commute, and \(W^*(a,c)\) is an abelian von Neumann algebra generated by two elements. By Theorem 8.2 of the lesson on abelian operator algebras, \(W^*(a,c)=W^*(d)\) for a self-adjoint \(d\). Then \(M=W^*(a,b,c)\), which is \(W^*(d,b)=W^*(d+ib)\) by (1).

(4) Put \(x=x_1+x_2\in M\) and \(N=W^*(x,z)\). Then \(z\in N\), so \(N\ni zx=x_1\), and the von Neumann algebra \(Nz\) on \(zH\) contains \(x_1\) and its unit \(z\); hence \(Nz\supseteq W^*(x_1)=Mz\). Likewise \(N(1-z)\supseteq M(1-z)\). Since \(z\in N\), both \(Nz\) and \(N(1-z)\) lie in \(N\), so \(M=Mz+M(1-z)\subseteq N\). By (3), applied with the central self-adjoint element \(z\), \(M\) is singly generated. \(\square\)

**Lemma 1.2** (separability). Let \(M\) be a von Neumann algebra on a separable Hilbert space \(H\).

1. \(M\) is generated by a sequence of self-adjoint elements of norm at most \(1\).
2. \(M\) is \(\sigma\)-finite, and every central summand of \(M\) is \(\sigma\)-finite.
3. \(M\) has separable predual.

**Proof.** (1) Let \((\zeta_j)\) be dense in \(H\). On the unit ball of \(B(H)\), strong convergence is convergence of \(b\zeta_j\) for every \(j\), so the strong topology on the unit ball is that of the map \(b\mapsto(b\zeta_j)_j\) into the separable metrizable space \(H^{\mathbb N}\). Every subset of a separable metrizable space is separable, so the unit ball of \(M\) contains a countable strongly dense subset \(S\). The strongly closed algebra \(W^*(S)\) contains the strong closure of \(S\), which is the unit ball of \(M\); so \(W^*(S)=M\). Replacing each element of \(S\) by its real and imaginary parts, which have norm at most \(1\), gives self-adjoint generators.

(2) A dense sequence in \(H\) is separating for \(M\), and Theorem 9.5 of the double commutant lesson gives \(\sigma\)-finiteness. A central summand \(Mz\) acts on the separable space \(zH\).

(3) By Theorem 6.2(d) of the lesson on compact and trace-class operators, the finite sums of vector functionals \(\omega_{\xi,\eta}(x)=\langle x\xi,\eta\rangle\) are norm dense in \(B(H)_*\), and \(\|\omega_{\xi,\eta}-\omega_{\xi',\eta'}\|\) is at most \(\|\xi-\xi'\|\,\|\eta\|+\|\xi'\|\,\|\eta-\eta'\|\). So the finite sums with \(\xi,\eta\) in a countable dense subset of \(H\) form a countable dense subset of \(B(H)_*\). By Theorem 9.4(a) there, restriction maps \(B(H)_*\) onto the predual \(M_*\) and induces an isometric isomorphism of \(B(H)_*/M_\perp\) onto \(M_*\); the image of a dense set is dense. \(\square\)

## 2. Properly infinite algebras

**Theorem 2.1** (Wogen). A properly infinite von Neumann algebra \(M\) on a separable Hilbert space \(H\) is generated by two self-adjoint elements.

**Proof.** *Matrix units.* \(M\) is \(\sigma\)-finite (Lemma 1.2(2)), hence locally \(\sigma\)-finite. Proposition 15.3(3) of the projections lesson, with \(p=1\), gives mutually orthogonal projections \(p_1,p_2,\ldots\) with \(\sum_np_n=1\) and \(p_n\sim1\), so \(p_n\sim p_1\). Choose partial isometries \(u_n\in M\) with \(u_n^*u_n=p_1\) and \(u_nu_n^*=p_n\), \(u_1=p_1\), and put \(w_{ij}=u_iu_j^*\). These are matrix units: \(w_{ij}w_{kl}=\delta_{jk}w_{il}\), \(w_{ij}^*=w_{ji}\), \(w_{ii}=p_i\), and \(w_{11}=p_1\). Let \(e=p_1\).

*\(M\) is generated by the matrix units and the corner \(eMe\).* Let \(x\in M\). For a finite set \(F\subseteq\mathbb N\), put \(P_F=\sum_{i\in F}w_{ii}\). Then
\[
P_FxP_F=\sum_{i,j\in F}w_{i1}\,(w_{1i}xw_{j1})\,w_{1j},
\]
because \(w_{i1}w_{1i}=w_{ii}\), and \(w_{1i}xw_{j1}\in eMe\). As \(F\) increases, \(P_F\to1\) strongly, so \(P_FxP_F\to x\) strongly. Hence \(M=W^*(\{w_{ij}\}\cup eMe)\).

*Generators.* The corner \(eMe\) is a von Neumann algebra on the separable space \(eH\), so by Lemma 1.2(1) it is generated, with unit \(e\), by self-adjoint elements \(y_1,y_2,\ldots\in eMe\) of norm at most \(1\). Put
\[
\begin{aligned}
&a=\sum_{k\ge1}2^{-k}w_{kk},\\
&b=\sum_{k\ge1}2^{-k}\big(w_{1,k+1}+w_{k+1,1}\big)\\
&\quad +\sum_{k\ge1}2^{-k}\,w_{k+1,1}\,y_k\,w_{1,k+1}.
\end{aligned}
\]
Both series converge in norm, and \(a,b\) are self-adjoint. By Lemma 1.1(2), every \(w_{kk}\) lies in \(W^*(a,b)\). The products of matrix units give
\[
\begin{gathered}
w_{11}\,b\,w_{k+1,k+1}=2^{-k}w_{1,k+1},\\
w_{1,k+1}\,b\,w_{k+1,1}=2^{-k}y_k\qquad (k\ge1).
\end{gathered}
\]
Indeed, in the first product only the term \(2^{-k}w_{1,k+1}\) survives, because \(w_{11}w_{j+1,1}=0\) for \(j\ge1\); in the second, \(w_{1,k+1}w_{1,j+1}=0\) and \(w_{1,k+1}w_{j+1,1}w_{k+1,1}=\delta_{jk}w_{11}w_{k+1,1}=0\), while \(w_{1,k+1}w_{j+1,1}y_jw_{1,j+1}w_{k+1,1}=\delta_{jk}\,ey_je\), which is \(\delta_{jk}y_j\). So \(W^*(a,b)\) contains \(w_{1,k+1}\) for every \(k\), hence every \(w_{ij}=w_{1i}^*w_{1j}\), and every \(y_k\). The von Neumann algebra \(W^*(a,b)\) contains \(e\) and the \(y_k\); the \(*\)-algebra they generate inside \(eMe\) is weakly dense in \(eMe\), because \(eMe\) is generated by the \(y_k\) with unit \(e\). So \(eMe\subseteq W^*(a,b)\), and by the previous paragraph \(M=W^*(a,b)\). \(\square\)

## 3. Finite factors

**Lemma 3.1.** Let \(F\) be a finite factor on a separable Hilbert space, \(F\neq\{0\}\). Then \(F\) is generated by one operator.

**Proof.** \(F\) is \(\sigma\)-finite (Lemma 1.2(2)), so it has a faithful normal tracial state \(\tau\) (Fact 1.7(a) of the central decomposition lesson). A nonzero finite factor is of type I or of type II₁.

*Type I.* There is a nonzero abelian projection \(e\in F\), and \(eFe=\mathbb Ce\) because \(F\) is a factor. Any two nonzero projections \(f,g\) of a factor are comparable, and a nonzero projection below \(e\) equals \(e\); so \(e\precsim g\) for every nonzero projection \(g\). Choose a maximal family of mutually orthogonal projections equivalent to \(e\). It has at most \(1/\tau(e)\) members, because they have equal trace \(\tau(e)>0\). Its sum \(q\) satisfies \(1-q=0\): otherwise \(1-q\) contains a copy of \(e\), contradicting maximality. So \(1=e_1+\cdots+e_n\) with \(e_i\sim e\), and Proposition 8.4 of the projections lesson gives \(F\cong eFe\otimes M_n(\mathbb C)\cong M_n(\mathbb C)\). In \(M_n(\mathbb C)\), let \(a=\operatorname{diag}(1,2,\ldots,n)\) and \(b=\sum_{i<n}(E_{i,i+1}+E_{i+1,i})\). The diagonal matrix units are polynomials in \(a\), and \(E_{ii}bE_{i+1,i+1}=E_{i,i+1}\), so all matrix units lie in \(W^*(a,b)\), and \(M_n(\mathbb C)=W^*(a+ib)\). An isomorphism of von Neumann algebras carries generators to generators.

*Type II₁.* \(F\) has separable predual by Lemma 1.2(3), and Theorem 5.1 of the previous lesson applies. \(\square\)

## 4. Direct integrals

In this section we use the setting of the central decomposition lesson: \(\Gamma\) is a standard Borel space with a \(\sigma\)-finite measure \(\mu\), \((H(\gamma))\) is a measurable field of Hilbert spaces with direct integral \(\mathcal H\), \(\mathcal A\) is the diagonal algebra, \(\Gamma_d=\{\gamma:\dim H(\gamma)=d\}\) for \(d\in\{0,1,2,\ldots,\infty\}\), \(\ell^2_d\) has standard basis \((\varepsilon_i)\), and for \(\gamma\in\Gamma_d\) the unitaries \(U(\gamma):H(\gamma)\to\ell^2_d\) of Fact 1.1(c) there identify the fibre with \(\ell^2_d\) in a measurable way.

**Theorem 4.1** (Willig). Let \(M\) be a measurable field of von Neumann algebras on \((H(\gamma))\), and \(Q=\int^\oplus M\). Suppose that \(M(\gamma)\) is generated by one operator for almost every \(\gamma\). Then there is a decomposable \(T\in Q\) with \(Q=W^*(\mathcal A\cup\{T\})\), and \(Q\) is generated by one operator.

**Proof.** *Dense sequences.* By Fact 1.2(a) of the central decomposition lesson, applied to \(M\) and \(M'\), there are measurable operator fields \(x_k\), \(y_k\) (\(k\ge1\)) of norm at most \(1\) and a null set \(N_0\) such that for \(\gamma\notin N_0\) the \(x_k(\gamma)\) lie in \(M(\gamma)\) and are \(\sigma\)-weakly dense in its unit ball, and the \(y_k(\gamma)\) lie in \(M(\gamma)'\) and are \(\sigma\)-weakly dense in its unit ball. Enlarge \(N_0\) by a null set outside of which \(M(\gamma)\) is singly generated. Fix \(d\ge1\). For \(\gamma\in\Gamma_d\) put \(\tilde x_k(\gamma)=U(\gamma)x_k(\gamma)U(\gamma)^*\), \(\tilde y_k(\gamma)=U(\gamma)y_k(\gamma)U(\gamma)^*\) and \(\tilde M(\gamma)=U(\gamma)M(\gamma)U(\gamma)^*\); the matrix entries of \(\tilde x_k\) and \(\tilde y_k\) are Borel functions on \(\Gamma_d\).

*A Borel set of generators.* Let \(\mathfrak B\) be the closed unit ball of \(B(\ell^2_d)\) with the \(\sigma\)-weak topology, a compact metrizable space (Fact 1.2(h) there), and let \(\mathcal F\) be the countable set of noncommutative polynomials \(f(Z,Z^*)\) with coefficients in \(\mathbb Q(i)\). As in part (e) of the proof of Proposition 5.2 of the central decomposition lesson, the matrix entries of a finite product of factors, each equal to the variable \(T\in\mathfrak B\), to \(T^*\), or to one of the operators \(\tilde x_k(\gamma)\), \(\tilde y_k(\gamma)\), are Borel functions of \((\gamma,T)\in\Gamma_d\times\mathfrak B\). Consequently so are the numbers \(\|g\,\varepsilon_i\|^2=\sum_j|\langle g\,\varepsilon_i,\varepsilon_j\rangle|^2\) for every linear combination \(g\) of such products, and the norm \(\|g\|\), which is the supremum of \(\|gv\|\) over the countable set of unit vectors \(v\) with finitely many nonzero coordinates in \(\mathbb Q(i)\), suitably normalized. Let \(G_d\) be the set of \((\gamma,T)\in(\Gamma_d\setminus N_0)\times\mathfrak B\) such that

- (H1) \(T\tilde y_k(\gamma)=\tilde y_k(\gamma)T\) for every \(k\);
- (H2) for all \(k,i,m\ge1\) there is \(f\in\mathcal F\) with \(\|f(T)\|\le1\) and with the sum over \(i'\le\min(i,d)\) of \(\|(f(T)-\tilde x_k(\gamma))\varepsilon_{i'}\|\) less than \(1/m\).

These are countably many conditions of the kind just described, so \(G_d\) is a Borel subset of \(\Gamma\times\mathfrak B\).

*\(G_d\) consists of generators.* Let \(\gamma\in\Gamma_d\setminus N_0\) and \(T\in\mathfrak B\).

If \((\gamma,T)\in G_d\), then \(T\) commutes with a \(\sigma\)-weakly dense subset of the unit ball of \(\tilde M(\gamma)'\) by (H1), hence with \(\tilde M(\gamma)'\), and \(T\in\tilde M(\gamma)''=\tilde M(\gamma)\). By (H2), each \(\tilde x_k(\gamma)\) is the strong limit of a sequence of elements \(f(T)\) of norm at most \(1\): on a bounded sequence, convergence on every basis vector is strong convergence. So \(\tilde x_k(\gamma)\in W^*(T)\) for every \(k\), and since these are \(\sigma\)-weakly dense in the unit ball of \(\tilde M(\gamma)\), \(\tilde M(\gamma)\subseteq W^*(T)\). Hence \(W^*(T)=\tilde M(\gamma)\).

Conversely, suppose \(W^*(T)=\tilde M(\gamma)\). Then (H1) holds. Let \(\mathcal P\) be the unital \(*\)-algebra of polynomials in \(T\) and \(T^*\); its weak closure is \(W^*(T)\) (Theorem 4.4 of the double commutant lesson). By Kaplansky's density theorem, every \(\tilde x_k(\gamma)\) is a strong limit of elements of \(\mathcal P\) of norm at most \(1\). Given \(k,i,m\), choose \(p\in\mathcal P\) with \(\|p\|\le1\) and \(\sum_{i'\le\min(i,d)}\|(p-\tilde x_k(\gamma))\varepsilon_{i'}\|<1/(2m)\); write \(p=g(T)\) for a complex noncommutative polynomial \(g\). Approximating the finitely many coefficients of \(g\) by elements of \(\mathbb Q(i)\) gives \(h\in\mathcal F\) with \(\|h(T)-p\|<\delta\), and then \(f=h/(1+\delta)\) (with \(1/(1+\delta)\) rational) lies in \(\mathcal F\), has \(\|f(T)\|\le1\), and \(\|f(T)-p\|\le2\delta\). For \(\delta\) small, (H2) holds for \((k,i,m)\). So \((\gamma,T)\in G_d\).

*Borel choice.* Every \(\gamma\in\Gamma_d\setminus N_0\) has a generator of \(M(\gamma)\), which we may divide by a positive number to have norm at most \(1\); transported by \(U(\gamma)\), it lies in \(\mathfrak B\) and generates \(\tilde M(\gamma)\). So the projection \(A_d\) of \(G_d\) to \(\Gamma\) contains \(\Gamma_d\setminus N_0\). Lemma 5.1 of the central decomposition lesson gives a Borel set \(X_d\subseteq A_d\), with \(A_d\setminus X_d\) contained in a null set, and a Borel map \(s_d:X_d\to\mathfrak B\) with \((\gamma,s_d(\gamma))\in G_d\) for \(\gamma\in X_d\). Put \(T(\gamma)=U(\gamma)^*s_d(\gamma)U(\gamma)\) for \(\gamma\in X_d\), for each \(d\ge1\), and \(T(\gamma)=0\) elsewhere. The matrix entries of \(s_d\) are Borel, so \(T\) is a measurable operator field (Fact 1.1(c), (d) there), with \(\|T(\gamma)\|\le1\). For almost every \(\gamma\), \(T(\gamma)\in M(\gamma)\) and \(W^*(T(\gamma))=M(\gamma)\): this holds on each \(X_d\setminus N_0\), and \(\Gamma_0\) has \(H(\gamma)=0\).

*Generation.* Let \(T=\int^\oplus T(\gamma)\,d\mu(\gamma)\), an element of \(Q\) by Fact 1.2(b) there. By Fact 1.2(f) there, \(W^*(\mathcal A\cup\{T\})=\int^\oplus M_T\) with \(M_T(\gamma)=W^*(T(\gamma))\), and \(M_T(\gamma)=M(\gamma)\) for almost every \(\gamma\); so \(W^*(\mathcal A\cup\{T\})=\int^\oplus M=Q\) (Fact 1.2(b), (d)). The diagonal algebra \(\mathcal A\) is an abelian von Neumann algebra on the separable space \(\mathcal H\) (Fact 1.1(e)); by Theorem 8.2 of the lesson on abelian operator algebras, \(\mathcal A=W^*(c)\) for a self-adjoint \(c\). It lies in the centre of \(Q\) (Fact 1.2(c)). So \(Q=W^*(T,c)\), and Lemma 1.1(3) gives one generator. \(\square\)

Willig also proves the converse: if \(Q\) is singly generated, so is almost every fibre [Wi, Theorem 1]. We do not need it.

## 5. The theorem

**Theorem 5.1.** Every von Neumann algebra \(M\) on a separable Hilbert space \(H\) is generated by one operator, equivalently by two self-adjoint operators.

**Proof.** If \(H=\{0\}\) there is nothing to prove. By Theorem 7.2 of the projections lesson, \(1=z+(1-z)\) for a central projection \(z\) such that \(Mz\) is finite and \(M(1-z)\) is properly infinite (the finite part \(e\) and the properly infinite part \(f\) of \(1\) given there have orthogonal central supports and \(e+f=1\), so \(e\le c(e)\le1-c(f)\le1-f=e\), and \(e=c(e)\) is central).

*The properly infinite part.* \(M(1-z)\) is a properly infinite von Neumann algebra on the separable space \((1-z)H\), and Theorem 2.1 gives two self-adjoint generators; if \(z=1\) this part is absent.

*The finite part.* Let \(Q=Mz\) on \(K=zH\), and assume \(K\neq\{0\}\). By Theorem 4.1 of the central decomposition lesson, there are a measurable field \((H(t))\) over \(\mathbb R\), a unitary \(W:K\to\mathcal H\) and a measurable field \(L\) of factors with \(WQW^*=\int^\oplus L\) and \(W\mathcal Z(Q)W^*=\mathcal A\). By Theorem 5.3 there, \(L(t)\) is finite for almost every \(t\), since \(Q\) is finite. Each \(L(t)\) acts on \(H(t)\), which has a countable orthonormal basis (Fact 1.1(c) there). By Lemma 3.1, almost every \(L(t)\) is generated by one operator, or is \(\{0\}\) on \(H(t)=\{0\}\). Theorem 4.1 shows that \(\int^\oplus L\) is singly generated, and so is \(Q=W^*(\int^\oplus L)W\).

By Lemma 1.1(4), \(M\) is singly generated. Lemma 1.1(1) gives the equivalence with two self-adjoint generators. \(\square\)

**Corollary 5.2.** Every von Neumann algebra with separable predual is generated by one operator.

**Proof.** Let \(M\neq0\) have separable predual. By Lemma 4.2 of the lesson on base changes and disintegration, there is a faithful normal state whose GNS representation \(\pi\) acts on a separable Hilbert space, with image a von Neumann algebra and \(\pi\) a normal \(*\)-isomorphism onto it with normal inverse. By Theorem 5.1, \(\pi(M)=W^*(\pi(x))\) for some \(x\in M\). The inverse isomorphism is normal, so it maps the von Neumann algebra generated by \(\pi(x)\) onto the von Neumann algebra generated by \(x\); hence \(M=W^*(x)\). \(\square\)

With the theorem of the previous lesson, this answers the generator problem positively.

**C\*-algebras.** The corresponding question for C\*-algebras, whether every simple separable unital C\*-algebra is generated by one element, has a negative answer: simple unital AH algebras that are not singly generated have been constructed (see [ELN, Section 1] and [BCC, Section 1.1]). The theorem for II₁ factors of the previous lesson has since been used on the C\*-side. Elliott, Li and Niu consider a separable unital C\*-algebra \(A\) whose set of extreme tracial states is nonempty, compact and zero-dimensional. If \(A\) has strict comparison of positive elements by tracial states, it is generated by two elements; if \(A\) has real rank zero, comparison of projections by tracial states makes it singly generated, and its single generators form a norm-dense \(G_\delta\) subset [ELN, Theorems 1.2 and 1.3]. A third criterion asks for a unitary \(u\in A\) whose spectral distribution at every extreme trace is atomless with full support on the circle, together with a small-compression condition relative to the von Neumann algebra generated by \(u\) [ELN, Theorem 1.4]; it shows that the reduced group C\*-algebra \(C^*_{\mathrm r}(\mathbb F_n)\) is singly generated for every \(n\in\{2,3,\ldots,\infty\}\) [ELN, Corollary 1.6]. For \(n=2\) the same authors had proved this before with a direct construction [ELN2]. Bouwen, Castillejos and Curda proved that every simple separable tracially AF algebra has generator rank at most one, and so is singly generated [BCC, Theorem B].

## 6. Exercises

**Exercise 6.1** (easy). Show directly that \(B(\ell^2(\mathbb N))\) is generated by two self-adjoint operators, by applying the proof of Theorem 2.1 with \(eMe=\mathbb C e\).

**Exercise 6.2** (easy). Show that \(\ell^\infty(\mathbb N)\otimes M_2(\mathbb C)\), acting on \(\ell^2(\mathbb N)\otimes\mathbb C^2\), is generated by one operator, by following the proof of Theorem 5.1 for this finite algebra of type I.

**Exercise 6.3** (medium). In Lemma 1.1(4), show by an example that \(W^*(x_1+x_2)\) can be smaller than \(M\), so that the central element \(z\) is needed.

## 7. Solutions

**6.1.** Take \(M=B(\ell^2(\mathbb N))\), \(p_n\) the projection onto \(\mathbb C\delta_n\) and \(w_{ij}\) the matrix units \(\delta_j\mapsto\delta_i\). Then \(eMe=\mathbb Ce\) needs no generators \(y_k\) (take \(y_k=0\)), and \(a=\sum_k2^{-k}w_{kk}\), \(b=\sum_k2^{-k}(w_{1,k+1}+w_{k+1,1})\) generate all \(w_{ij}\), whose span is weakly dense in \(B(\ell^2(\mathbb N))\).

**6.2.** The centre is \(\ell^\infty(\mathbb N)\otimes1\), generated by \(c=\sum_n2^{-n}\delta_n\otimes1\) (diagonal multiplication). The fibre over \(n\) is \(M_2(\mathbb C)\), generated by \(T=\operatorname{diag}(1,2)+i(E_{12}+E_{21})\) for every \(n\), and \(T\) defines the constant decomposable operator \(1\otimes T\). Then \(W^*(1\otimes T,c)\) contains the projections \(\delta_n\otimes1\) and their products with \(1\otimes M_2(\mathbb C)\), so it is the whole algebra. Lemma 1.1(3) merges \(c\) into one generator: with \(a=1\otimes\operatorname{diag}(1,2)\), the abelian algebra \(W^*(a,c)\) is generated by one self-adjoint \(d\), and \(d+i(1\otimes(E_{12}+E_{21}))\) generates.

**6.3.** Let \(M=\mathbb C\oplus\mathbb C\) on \(\mathbb C^2\), \(z=1\oplus0\), and \(x_1=1\oplus0\), \(x_2=0\oplus1\) (the units of the two summands, which generate them). Then \(x_1+x_2=1\) generates only \(\mathbb C1\neq M\).

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