Internal Sequential Commutation and Single Generation

If |$M$| is generated by a countable set of unitaries |$\{u_{i}\}_{i\in \mathbb{N}}$| such that |$u_{i}\sim ^{\mathcal{U}} u_{i+1}$| for all |$i$|⁠, then |$M$| is singly generated.

Let |$M$| be a II|$_{1}$| factor and let |$u,v\in M$| be Haar unitaries. Then |$u\sim ^{\mathcal{U}} v$| if and only if there exists |$k\in \mathbb{N}$| and a sequence |$(v_{n})_{n}$| of Haar unitaries in |$M$| converging in the SOT to |$v$| such that |$u\sim ^{+}_{k} v_{n}$| for all |$n.$|

If |$u\in \prod _{\mathcal{U}} (M_{n},\tau _{n})$| is a Haar unitary and each |$M_{n}$| is diffuse, then there are Haar unitaries |$u_{n}\in M_{n}$| such that |$u=(u_{n})_{n}$|⁠.

Let |$x\in \prod _{\mathcal{U}} M_{n}$| be a normal element. Suppose |$x = (x_{n})_{n}$| is a lift of |$x$| such that each |$x_{n}$| is normal and has the same moments (and thus the same spectral measure) as |$x.$| Then for any bounded Borel function on |$\sigma (x)$|⁠, |$(f(x_{n}))_{n}$| is a lift of |$f(x).$|

The result is immediate if |$f$| is a ^*-polynomial by the ultrapower construction. Otherwise, take a sequence of ^*-polynomials |$p_{m}$| such that |$\|p_{m}-f\|_{2} < 1/m$|⁠, where |$\|g\|_{2}^{2} = \int _{\sigma (x)} |g|^{2} d\mu ^{x} = \int _{\sigma (x_{n})}|g|^{2} d\mu ^{x_{n}}$| (the 2-norm with respect to |$\mu ^{x} = \mu ^{x_{n}}$|⁠).

Let |$(y_{n})_{n}$| be a lift of |$f(x)$|⁠. We note that for all |$m,$|  |$p_{m}(x_{n})$| is a lift of |$p_{m}(x).$| Then |$\lim _{n\to{\mathcal{U}}} \|y_{n} - p_{m}(x_{n})\|_{2} = \|f(x) - p_{m}(x)\|_{2} = \|f-p_{m}\|_{2} <1/m.$| But we also have that |$\|p_{m}(x_{n}) - f(x_{n})\|_{2} = \|p_{m}-f\|_{2} < 1/m$| so that |$\lim _{n\to{\mathcal{U}}} \|y_{n} - f(x_{n})\|_{2} < 2/m.$| Since we can take |$m$| arbitrarily large, we have that |$(f(x_{n}))_{n}$| is a lift of |$f(x).$|

Let |$(M_{n})_{n}$| be a sequence of diffuse tracial von Neumann algebras and let |$x\in \prod _{\mathcal{U}} M_{n}$| be normal. Then |$x$| lifts to a sequence |$x_{n}\in M_{n}$| such that |$\tau (x^{r}(x^{*})^{s}) = \tau (x_{n}^{r}(x_{n}^{*})^{s})$| for all nonnegative integers |$r,s,n.$|

|${\mathcal{M}} = \prod _{\mathcal{U}} M_{n}$| is diffuse, so there is a Haar unitary |$u\in{\mathcal{M}}$| so that |$x\in W^{*}(u).$| Furthermore, by the Borel functional calculus there is a bounded Borel function |$f$| such that |$f(u) = x.$| Use Lemma 2.1 to lift |$u$| to a sequence |$(u_{n})_{n}$| of Haars in |$M_{n}.$| By Lemma 2.2  |$(f(u_{n}))_{n}$| is a lift of |$x$| with all the same moments.

Let |$N$| be a countably decomposable von Neumann algebra on a Hilbert space |${\mathcal{H}}$| and let |$\xi \in{\mathcal{H}}$|⁠. If |$p,q \in N$| are equivalent projections then there is a partial isometry |$v\in N$| such that |$vv^{*} = p,$|  |$v^{*}v = q,$| and |$\|(v-p)\xi \| \leq 6\|(p-q)\xi \|$|⁠. If |$N$| is a II|$_{1}$| factor and |$p,q\in N$| are projections such that |$\tau (p)=\tau (q)$| then we can pick |$v\in N$| a partial isometry such that |$vv^{*}=p,$|  |$v^{*}v=q,$| and |$\|v-p\|_{2} \leq 3\|p-q\|_{2}$|⁠.

Fix |$t>0$| and |$0<\kappa < \pi /2.$| Then for all |$\varepsilon>0$| there is |$\delta>0$| such that whenever |$u_{1},u_{2} \in M$| are unitaries in a II|$_{1}$| factor with the same purely atomic spectral measure |$\mu $| where each atom of |$\mu $| has measure at least |$t$| and all distinct eigenvalues of |$u_{i}$| have radian distance between them at least |$\kappa $|⁠, and |$\|u_{1}-u_{2}\|_{2} < \delta $| then there is a unitary |$v\in M$| such that |$\|v-1\|_{2} < \varepsilon $| and |$v^{*}u_{1}v = u_{2}.$|

Let |$\varepsilon>0$| and set choose |$\delta> 0$| such that |$\delta < \frac{\varepsilon ^{2}}{6}\sqrt{t(1-\cos (\kappa )}$|⁠.

By Lemma 3.1 there are partial isometries |$v_{j}$| such that |$v_{j}v_{j}^{*} = p_{1j}$|⁠, |$v_{j}^{*}v_{j} = p_{2j}$|⁠, and |$\|v_{j}-p_{1j}\|_{2} \leq 6\|p_{1j}-p_{2j}\|_{2}$|⁠. Set |$v = \sum _{j=1}^{n} v_{j}$|⁠. Then |$v$| is a unitary such that |$v^{*}u_{1}v = u_{2}$|⁠.

Let |$M$| be a II|$_{1}$| factor. If |$u,v \in M^{\mathcal{U}}$| are commuting Haar unitaries, then for any lift |$u_{n}$| of |$u$| to Haar unitaries and any |$\varepsilon>0$| there is a lift of |$v$| to Haar unitaries |$v_{n}$| and there are finite-dimensional unitaries |$w_{n} \in M$| s.t. all atoms of the spectral measure of |$w_{n}$| have measure less than |$\varepsilon $| and |$[u_{n},w_{n}]=[w_{n},v_{n}] = 0.$| Moreover, for any |$N$| with |$\frac{1}{N} < \varepsilon $|⁠, we may choose each |$w_{n}$| so that |$w_{n} \in W^{*}(u_{n})$|⁠, |$\|u_{n}-w_{n}\|<\varepsilon \pi $| and all atoms of |$w_{n}$| have measure exactly |$\frac{1}{N}$|⁠.

Since |$w$| and |$v$| commute in |$M^{\mathcal{U}},$| they generate a diffuse abelian subalgebra. Lemma 2.3 guarantees commuting unitary lifts |$x_{n}$| and |$v_{n}$| of |$w,v$| respectively such that |$x_{n}$| has the same moments as |$w$| and the |$v_{n}$| are all Haar.

Set |$t = 1/N$| and |$\kappa = 2\pi /N.$| For |$\varepsilon = 1/k$|⁠, |$k$| an integer, choose |$\delta _{k}>0$| as guaranteed by Lemma 3.2. Without loss of generality we can choose the |$\delta _{k}$| to be decreasing to 0. Since |$(w_{n})_{n}$| and |$(x_{n})_{n}$| are both lifts of |$w$|⁠, the decreasing sequence of sets |$A_{k} = \{n\in{\mathbb{N}}: \|w_{n}-x_{n}\|_{2} < \delta _{k}\}$| are all in |${\mathcal{U}}.$| Since |${\mathcal{U}}$| is countably incomplete, we may choose a sequence of sets |$B_{k} \in{\mathcal{U}}$| decreasing to |$\varnothing $|⁠, so |$A^{\prime}_{k} = A_{k} \cap B_{k}$| is a decreasing sequence of sets in |${\mathcal{U}}$| with |$\cap _{k} A^{\prime}_{k} = \varnothing $|⁠. For |$n \in A^{\prime}_{k} \setminus A^{\prime}_{k+1}$|⁠, apply Lemma 3.2 to get unitaries |$y_{n}$| such that |$\|y_{n}-1\|_{2} < 1/k$| and |$y_{n}^{*}x_{n}y_{n} = w_{n}.$| For |$n \notin A_{1}$|⁠, let |$y_{n} = 1$|⁠. Then as |$\cap _{k} A^{\prime}_{k} = \varnothing $|⁠, we have defined |$y_{n}$| for all |$n$|⁠. We note that |$\|y_{n}-1\|_{2} \to 0$| as |$n\to{\mathcal{U}}$| and so |$(y_{n}^{*}v_{n}y_{n})_{n}$| is a Haar lifting of |$v.$| Then the sequences of unitaries |$(w_{n})_{n}$| and |$(y_{n}^{*}v_{n}y_{n})_{n}$| satisfy the conclusion of the lemma.

The following are three notions of sequentially commutation in a tracial von Neumann algebra |$(M,\tau )$|⁠. The first two conditions appear in [14]. Throughout, let |$u,v\in M$| be Haar unitaries.

• (a)

  We write |$u\sim _{k} v$| if there exist Haar unitaries |$u=w_{0},w_{1},\ldots ,w_{k}=v$| such that |$w_{i}\in M$| for each |$i$| and |$[w_{i},w_{i+1}]=0$| for |$i=0,\ldots ,k-1.$| We write |$u\sim v$| if there exists |$k$| such that |$u\sim _{k} v$|⁠.

• (b)

  We write |$u\sim _{k}^{\mathcal{U}} v$| if there exist Haar unitaries |$u=w_{0},w_{1},\ldots ,w_{k}=v$| such that |$w_{i}\in M^{\mathcal{U}}$| for each |$i$| and |$[w_{i},w_{i+1}]=0$| for |$i=0,\ldots ,k-1.$| We write |$u\sim ^{\mathcal{U}} v$| and say that |$u,v\in M$|  sequentially commute if there exists |$k$| such that |$u\sim _{k}^{\mathcal{U}} v$|⁠.

• (c)

  We write |$u\sim _{k}^{+}v$| if there exists |$k$| so that for all |$\varepsilon>0$| there exist Haar unitaries |$u=w_{0},w_{1},\ldots , w_{k}=v$| and unitaries |$x_{1},\ldots ,x_{k-1}$|⁠, all in |$M,$| such that, for each |$i$|⁠, |$\|x_{i}-w_{i}\|<\varepsilon $|⁠, |$x_{i}$| is finite-dimensional, each atom of each |$x_{i}$| have the same measure, |$[u,x_{1}]=[x_{k-1},v]=0,$| and |$[x_{i},x_{i+1}] = 0.$| We write |$u\sim ^{+} v$| and say that |$u,v\in M$|  internally sequentially commute if there exists |$k$| such that |$u\sim _{k}^{+} v$|⁠.

Let |$M$| be a II|$_{1}$| factor and let |$u,v\in M$| be Haar unitaries. Then |$u\sim ^{\mathcal{U}} v$| if and only if there exists |$k\in \mathbb{N}$| and a sequence |$(v_{n})_{n}$| of Haar unitaries in |$M$| converging in the SOT to |$v$| such that |$u\sim ^{+}_{k} v_{n}$| for all |$n.$|

Suppose there are Haar unitaries |$v_{n}$| converging to |$v$| in the SOT and |$k\in{\mathbb{N}}$| so that |$u\sim _{k}^{+} v_{n}$| for all |$n.$| Then for each |$n\geq 1$| and |$i=1,\ldots ,k-1$| there are finite-dimensional unitaries |$x_{i,n}$| such that |$[u,x_{1,n}] = [x_{i,n},x_{i+1,n}]=[x_{k-1,n},v_{n}] = 0$| and |$\|x_{i,n}-w_{i,n}\| < 1/n$| for some Haar unitary |$w_{i,n}\in M$|⁠. Set |$x_{i} = (x_{i,n})_{n} \in M^{\mathcal{U}}.$| Then each |$x_{i}$| is a Haar unitary, and since |$v = (v_{n})_{n}$| as elements of |$M^{\mathcal{U}},$| we have that |$[u,x_{1}] = [x_{i},x_{i+1}] = [x_{k-1},v]$| as required. Hence, |$u\sim ^{\mathcal{U}} v.$|

Conversely, suppose |$u\sim ^{\mathcal{U}} v$|⁠. Then there is |$k$| and Haar unitaries |$u=w_{0},w_{1},\ldots ,w_{k}=v$| in |$M^{\mathcal{U}}$| as in Definition 3.4(b). Fix |$\varepsilon>0$| and apply Lemma 3.3 repeatedly to get Haar lifts |$w_{i} = (w_{i,n})_{n}$| and unitaries |$x_{i,n} \in W^{*}(w_{i,n})$| so that |$\|w_{i,n}-x_{i,n}\|<\varepsilon $| and |$[x_{i,n},w_{i+1,n}] = 0$| for all |$n$| and |$i=0,\ldots ,k-1.$| In particular, we note that |$[x_{i,n},x_{i+1,n}]=0$| too. The lift of |$u=w_{0}$| can be chosen to be the constant sequence |$(u)_{n}$| and the lift of |$v=w_{k}$| gives Haar unitaries |$w_{k,n}$| which converge to |$v$| in the SOT.

Let |$(M,\tau )$| be a II|$_{1}$| factor. Then |$u\sim ^{\mathcal{U}} v$| for all Haar unitaries |$u,v\in M$| if and only if there is an SOT-dense subset |$D\subset{\mathcal{H}}(M)$| of the Haar unitaries of |$M$| such that |$u\sim ^{+}v$| for all |$u,v\in D.$| Furthermore, |$D$| can be chosen to contain any particular Haar unitary |$u_{0} \in M.$|

If |$M$| is a separable II|$_{1}$| factor, then |$\sim $| has continuum many orbits. Moreover, there are continuum many Haar unitaries in |$M$| which are pairwise non-conjugate.

Moreover, we observe that if |$W^{*}(u) = A,$| then |$\lambda u$| and |$\mu u$| are non-conjugate for all |$\lambda \neq \mu \in{\mathbb{T}}.$| We also observe that if |$u\in A$| is any Haar unitary and |$w\in M$| is Haar such that |$u\sim w$|⁠, then necessarily |$w\in A.$|

Now let |$v$| be a Haar unitary in |$M\setminus A$|⁠. Write |$v = e^{ih}$| for some self-adjoint operator |$h\in M\setminus A.$| Define |$v_{t} = e^{ith}$| for |$t\in{\mathbb{R}}.$| Since |$v\not \in A,$| by continuity |$v_{t} \not \in A$| for all |$t$| in some non-trivial interval |$(1-1/N,1+1/N)$|⁠. Since |$v_{s+t}=v_{s}v_{t},$| we see that |$v_{t}\not \in A$| for all |$0\neq t\in (-1/N,1/N).$| Therefore, for all |$s,t \in (-1/(2N),1/(2N))$|⁠, |$v_{s}^{*}v_{t}\in A$| only if |$s=t.$| Therefore, |$v_{s}^{*}v_{t} u v_{t}^{*}v_{s} \not \in A$| for any such distinct |$s,t,$| by the coarseness of |$A$| in |$M$|⁠. This implies that for |$t\in (-1/(2N),1/(2N))$|⁠, the |$\sim $|-orbits of |$v_{t} u v_{t}^{*}$| are all distinct.

Let |$M$| be a II|$_{1}$| factor and |$u \in M$| a Haar unitary. Then for all |$1 \geq \varepsilon> 0$| there is |$R \subset M$| irreducible hyperfinite subfactor such that |$W^{*}(R,u)$| is generated by a unitary |$v$| contained in a matrix algebra |${\mathbb{M}}_{N}({\mathbb{C}}) \subset R$| and a self-adjoint |$T$| whose support is majorized by a projection |$p \in{\mathbb{M}}({\mathbb{C}})$| with trace less than |$\varepsilon .$|

Note that |$T_{1}$| is a self-adjoint with support majorized by |$p_{1} + p_{2} + p_{3}$|⁠, which has trace |$\frac{3}{N} < \frac{\varepsilon }{2}$|⁠.

|$T$| has support contained within |$p = p_{1} + p_{2} + p_{3} + P$|⁠. Since |$\tau (p_{1} + p_{2} + p_{3}) < \frac{\varepsilon }{2}$| and we easily see that |$\tau (P) \leq \frac{\varepsilon }{2}$|⁠, we have |$\tau (p) < \varepsilon $|⁠, as desired.

For all |$\varepsilon> 0$| and |$n> 0$| there is |$N> 0$| such that if |$u_{1},\ldots ,u_{n}$| are unitaries in a II|$_{1}$| factor |$M$| satisfying

• (1)

  |$u_{i+1}^{*}u_{i}u_{i+1} \in W^{*}(u_{1},\ldots ,u_{i})$| for all |$i = 1, \cdots , n-1$|⁠; and

• (2)

  For each |$i$|⁠, |$u_{i}$| is either diffuse or is finite-dimensional, has all its atoms having the same trace, and has the number of atoms being a multiple of |$N$|⁠;

then for any irreducible hyperfinite subfactor |$R \subset M$|⁠, |$W^{*}(R,u_{1},\ldots ,u_{n})$| is generated by |$R,$|  |$u_{1},$| and a self-adjoint whose support is majorized by a projection |$p \in R$| with trace less than |$\varepsilon .$|

By direct calculations, we see that the positive solutions to the equation |$\frac{m(m-1)}{2} \geq k$| are given by |$m \geq \frac{1+\sqrt{1+8k}}{2} = \frac{1}{2} + \sqrt{\frac{1}{4} + 2k}$|⁠. We also see that |$\sqrt{\frac{1}{4} + 2k} - \sqrt{2k} \leq \frac{1}{2}$| for all |$k \geq 0$|⁠. Note that |$\sqrt{\frac{1}{4} + 2k} - \sqrt{2k} \leq \frac{1}{2}$| may be rearranged to |$\frac{1+\sqrt{1+8k}}{2} \leq \sqrt{2k} + 1$|⁠. So, for any |$k \geq 0$|⁠, there is an integer |$m$| such that |$\frac{m(m-1)}{2} \geq k$| and |$m \leq \sqrt{2k} + 2$|⁠. Now choose |$K> 0$| such that |$\sum _{i=1}^{n-1} (\sqrt{\frac{2}{K^{i}}} + \frac{2}{K^{i}}) < \varepsilon $| and set |$N = K^{n-1}$|⁠.

By the conditions on |$u_{i}$|⁠, we may choose, for each |$i = 1, \cdots , n-1$|⁠, pairwise orthogonal projections |$F^{(i)}_{1}, \cdots , F^{(i)}_{K^{i}}$| in |$W^{*}(u_{i})$| all of trace |$\frac{1}{K^{i}}$|⁠. Fix unital embeddings |${\mathbb{M}}_{K}({\mathbb{C}}) \subset{\mathbb{M}}_{K^{2}}({\mathbb{C}}) \subset \cdots \subset{\mathbb{M}}_{K^{n-1}}({\mathbb{C}}) \subset R$| with the embeddings being the standard diagonal ones. Let |$\{E^{(i)}_{\alpha \beta }\}_{1 \leq \alpha , \beta \leq K^{i}}$| be the standard system of matrix units for |${\mathbb{M}}_{K^{i}}({\mathbb{C}})$|⁠. Note that the embeddings between the matrix algebras are standard diagonal embeddings, so |$E^{(i)}_{jj} = \sum _{k=(j-1)K+1}^{jK} E^{(i+1)}_{kk}$|⁠.

If |$M$| is generated by a countable set of sequentially commuting Haar unitaries, then |$M$| is singly generated.

Let |$\{u_{0}, u_{1}, \cdots \}$| be an enumeration of a countable generating set of sequentially commuting Haar unitaries in which every unitary appears infinitely many times. By Lemma 3.8, we may fix an irreducible |$R \subset M$| s.t. |$W^{*}(R, u_{0})$| is generated by a unitary |$v$| and a self-adjoint |$T_{0}$| whose support is majorized by a projection |$p_{0} \in R$| with |$\tau (p_{0}) < \frac{1}{2}$|⁠.

Now, for each |$k> 0$|⁠, let |$n_{k}$| be such that |$u_{0} \sim _{n_{k}}^{\mathcal{U}} u_{k}$|⁠. Let |$\varepsilon _{k} = \frac{1}{2^{k+1}}$|⁠. Choose |$N = N_{k}> 0$| that satisfies the conclusions of Lemma 3.9 with |$\varepsilon = \varepsilon _{k}$| and |$n = n_{k}$|⁠. Now, using Lemma 3.3 and the proof of Proposition 3.5, we see that there exists finite-dimensional unitaries |$(x^{(k)}_{i})_{i=1}^{n_{k}}$| and a Haar unitary |$u^{\prime}_{k}$| s.t. |$[u_{0}, x^{(k)}_{1}] = [x^{(k)}_{i}, x^{(k)}_{i+1}] = [x^{(k)}_{n_{k}}, u^{\prime}_{k}] = 0$| for all |$i=1, \cdots , n_{k}-1$|⁠; |$\|u^{\prime}_{k} - u_{k}\|_{2} < \varepsilon _{k}$|⁠; and all atoms of |$x^{(k)}_{i}$| have the same trace and the number of atoms are multiples of |$N_{k}$|⁠, for all |$i = 1, \cdots , n_{k}$|⁠. By Lemma 3.9, then, |$W^{*}(R, u_{0}, x^{(k)}_{1}, \cdots , x^{(k)}_{n_{k}}, u^{\prime}_{k})$| is generated by |$R$|⁠, |$u_{0}$|⁠, and a self-adjoint |$T_{k}$| whose support is majorized by a projection |$p_{k} \in R$| with |$\tau (p_{k}) < \varepsilon _{k} = \frac{1}{2^{k+1}}$|⁠. As |$\tau (p_{0}) + \sum _{k=1}^\infty \tau (p_{k}) < 1$|⁠, we may, after conjugating |$T_{k}$| by unitaries in |$R$|⁠, assume all |$p_{k}$| (as well as |$p_{0}$|⁠) are orthogonal to each other. For |$k> 0$|⁠, by multiplying by appropriate scalars and adding appropriate scalar multiples, we may assume the nonzero parts of the spectra of |$T_{k}$| are disjoint from each other and are all contained in a fixed compact interval disjoint from the spectrum of |$T_{0}$|⁠. Then |$T = T_{0} + \sum _{k=1}^\infty T_{k}$| is a well-defined self-adjoint, and by functional calculus, we see that |$W^{*}(R, u_{0}) \subset W^{*}(v, T)$|⁠. Therefore, we also have |$u^{\prime}_{k} \in W^{*}(v, T)$| for all |$k> 0$|⁠. Since |$\{u_{0}, u_{1}, \cdots \}$| generates |$M$|⁠, all generators appear infinitely many times, and |$\|u^{\prime}_{k} - u_{k}\|_{2} < \varepsilon _{k} \to 0$|⁠, we have |$\{u_{0}, u^{\prime}_{1}, u^{\prime}_{2}, \cdots \}$| generates |$M$|⁠, so |$M$| is generated by a unitary |$v$| and a self-adjoint |$T$|⁠, whence it is singly generated (by |$T + \ln (v)$|⁠, say).

We note that a refinement of the above proof shows the following: Fix |$\varepsilon> 0$|⁠. Then, for |$N> 0$| large enough, there exists a matrix subalgebra |${\mathbb{M}}_{N}({\mathbb{C}}) \subset M$| s.t. |$M$| is generated by the right shift unitary |$v = \sum _{i=1}^{N-1} e_{i(i+1)} + e_{N1}$| in |${\mathbb{M}}_{N}({\mathbb{C}})$| and a self-adjoint |$T$| whose support is majorized by a diagonal projection |$p \in{\mathbb{M}}_{N}({\mathbb{C}})$| with trace less than |$\varepsilon $|⁠. This implies |${\mathcal{G}}(M) = 0$| in the sense of [20]. In particular, by Theorem 5.5 of [20], if |$M$| is generated by a sequence of subfactors |$M_{k}$|⁠, each one generated by a countable set of sequentially commuting Haar unitaries, and furthermore |$M_{k} \cap M_{k+1}$| is diffuse for all |$k$|⁠, then |$M$| is singly generated.