Trace-preserving polynomial flows

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

Throughout this lesson, \((A_1,\ldots,A_n,C)\) is a freely generating Haar \((n+1)\)-tuple in a von Neumann algebra \(M\) with faithful normal tracial state \(\tau\), \(n\ge1\), and the notation of Section 3 of Free independence and Haar tuples is in force: \(\Gamma=\mathbb F_n\), \(A_g\) is the value of \(g\in\Gamma\) at \(A=(A_1,\ldots,A_n)\), \(S=\arg(C)\), \(S_g=A_gSA_g^*\), and \(s(k)=\sum_gk(g)S_g\). Write \(\mathbb R^{(\Gamma)}\) for the finitely supported real functions on \(\Gamma\), with the left translation \((\lambda(g)k)(b)=k(g^{-1}b)\), and \(\delta_e\) for the indicator function of \(e\).

Fix \(h_1,\ldots,h_n\in\mathbb R^{(\Gamma)}\). This lesson constructs trace-preserving automorphisms \(\beta_t\) of \(M\), \(t\in\mathbb R\), that fix \(S\) and move the generators along the differential equation \[ \begin{gathered} \frac d{dt}A_j(t)=i\,s_t(h_j)A_j(t),\\ s_t(k)=\sum_{g}k(g)A_g(t)SA_g(t)^* , \end{gathered} \] where \(A_g(t)\) is the value of \(g\) at \(A(t)=(A_1(t),\ldots,A_n(t))\). The main result, Theorem 6.2, adds two estimates: \(\beta_t\) moves \(A_j\) by at most \(3\pi|t|\|h_j\|_{\ell^2}\) in operator norm, and it moves the value \(A_w\) of a word \(w\) to within \(3\pi|t|\|D_w-\delta_e\|_{\ell^2}\) of \(e^{itS}A_w\), where \(D\) is the cocycle of Section 1.

The difficulty is that the moving tuple is not known in advance to have the same joint distribution as the original one. The proof shows this in two steps. Section 3 proves, by a computation at time zero that uses only the freeness of Lesson 1, that every derivative of every moment vanishes at \(t=0\). Section 5 shows that the moments are real-analytic in \(t\), so they are constant.

We use: from Free independence and Haar tuples, Lemmas 1.3, 1.4, 2.1, 3.1, 3.2 and Corollary 4.2; Riemann integrals of continuous functions with values in a Banach space and the fundamental theorem of calculus for them, Lemma 0.1 of the lesson on Cauchy's theorem; the Borel functional calculus of a self-adjoint operator, Theorem 3.1 of the spectral theorem lesson; and Taylor's theorem with the Lagrange remainder, proved in the core course Real Analysis I, whose text is [Lebl RA], Theorem 4.3.2.

1. Cocycles on the free group

Lemma 1.1. There is exactly one map \(D\colon\Gamma\to\mathbb R^{(\Gamma)}\), \(g\mapsto D_g\), with \[ \begin{gathered} D_{x_j}=h_j\quad(1\le j\le n),\\ D_{gb}=D_g+\lambda(g)D_b\quad(g,b\in\Gamma). \end{gathered} \] It satisfies \(D_e=0\), \(D_{g^{-1}}=-\lambda(g^{-1})D_g\), and \(D_{y_1y_2\cdots y_r}=\sum_{k=1}^r\lambda(y_1\cdots y_{k-1})D_{y_k}\) for all \(y_1,\ldots,y_r\in\Gamma\).

Proof. On \(G=\mathbb R^{(\Gamma)}\times\Gamma\) define \((k,g)(k',g')=(k+\lambda(g)k',gg')\). This is a group: both ways of bracketing a triple product give \((k+\lambda(g)k'+\lambda(gg')k'',gg'g'')\), the unit is \((0,e)\), and \((k,g)^{-1}=(-\lambda(g^{-1})k,g^{-1})\). By Lemma 2.1 of the first lesson there is a unique homomorphism \(\theta\colon\Gamma\to G\) with \(\theta(x_j)=(h_j,x_j)\). Its second component is a homomorphism \(\Gamma\to\Gamma\) fixing every \(x_j\), hence the identity, so \(\theta(g)=(D_g,g)\) for a map \(D\), and the homomorphism property of \(\theta\) is the stated identity. Conversely, any \(D\) with these properties makes \(g\mapsto(D_g,g)\) a homomorphism with the same values on the \(x_j\), so \(D\) is unique. The remaining formulas express \(\theta(e)=(0,e)\), \(\theta(g^{-1})=\theta(g)^{-1}\) and \(\theta(y_1\cdots y_r)=\theta(y_1)\cdots\theta(y_r)\). \(\square\)

We call \(D\) the cocycle with generator values \(h_1,\ldots,h_n\).

2. A formal algebra and a derivation

Let \(\mathcal P\) be the unital complex \(*\)-algebra generated by symbols \(a_1,\ldots,a_n,z\) subject only to the relations \(a_j^*a_j=a_ja_j^*=1\) and \(z^*=z\). Concretely, \(\mathcal P\) is the free unital algebra on the symbols \(a_j\), \(a_j^*\), \(z\) (spanned by the finite products of symbols, called monomials) modulo the two-sided ideal generated by the elements \(a_j^*a_j-1\) and \(a_ja_j^*-1\); its involution is induced by the conjugate-linear map that reverses monomials, exchanges \(a_j\) and \(a_j^*\) and fixes \(z\). For every unital \(*\)-algebra \(\mathcal Q\), unitaries \(u_1,\ldots,u_n\in\mathcal Q\) and self-adjoint \(y\in\mathcal Q\), there is exactly one unital \(*\)-homomorphism \(\mathcal P\to\mathcal Q\) with \(a_j\mapsto u_j\) and \(z\mapsto y\).

The degree of a monomial is its number of symbols. Its label is the element of \(\Gamma\) obtained by multiplying its group symbols in order, with \(a_j\mapsto x_j\) and \(a_j^*\mapsto x_j^{-1}\), and ignoring every \(z\).

The \(a_j\) are unitaries of \(\mathcal P\); let \(a_g\) be the value of \(g\in\Gamma\) at \((a_1,\ldots,a_n)\) (Lemma 2.1 of the first lesson), represented by the reduced word of \(g\). Put \[ \begin{gathered} z_g=a_gza_g^*,\\ \sigma(k)=\sum_gk(g)z_g\quad(k\in\mathbb R^{(\Gamma)}). \end{gathered} \] Each \(\sigma(k)\) is self-adjoint, and \(a_g\sigma(k)a_g^*=\sum_bk(b)z_{gb}=\sigma(\lambda(g)k)\).

Lemma 2.1. There is exactly one linear map \(\delta\colon\mathcal P\to\mathcal P\) with \(\delta(pq)=\delta(p)q+p\,\delta(q)\) and \[ \begin{gathered} \delta z=0,\qquad \delta a_j=i\sigma(h_j)a_j,\\ \delta a_j^*=-ia_j^*\sigma(h_j). \end{gathered} \] It satisfies \(\delta(p^*)=\delta(p)^*\), and for every \(g\in\Gamma\) \[ \delta a_g=i\sigma(D_g)a_g,\qquad \delta z_g=i[\sigma(D_g),z_g]. \] For a monomial \(m\), \(\delta m\) is a linear combination of monomials with the same label as \(m\).

Proof. Let \(\mathcal P[\varepsilon]=\mathcal P\oplus\varepsilon\mathcal P\) with \(\varepsilon\) central and \(\varepsilon^2=0\). Put \(d_j=i\sigma(h_j)a_j\) and \(d_j^\sharp=-ia_j^*\sigma(h_j)\). The unital homomorphism from the free algebra to \(\mathcal P[\varepsilon]\) with \(a_j\mapsto a_j+\varepsilon d_j\), \(a_j^*\mapsto a_j^*+\varepsilon d_j^\sharp\) and \(z\mapsto z\) vanishes on the defining relations: \[ \begin{aligned} &(a_j^*+\varepsilon d_j^\sharp)(a_j+\varepsilon d_j)=1\\ &\quad +\varepsilon\big(ia_j^*\sigma(h_j)a_j-ia_j^*\sigma(h_j)a_j\big)\\ &\quad =1, \end{aligned} \] and similarly \[ \begin{aligned} &(a_j+\varepsilon d_j)(a_j^*+\varepsilon d_j^\sharp)\\ &\quad=1+\varepsilon\big(i\sigma(h_j)-i\sigma(h_j)\big)=1. \end{aligned} \] So it induces a unital homomorphism \(\Theta\colon\mathcal P\to\mathcal P[\varepsilon]\). Write \(\Theta(p)=p_0+\varepsilon\,\delta(p)\). The map \(p\mapsto p_0\) is a unital homomorphism \(\mathcal P\to\mathcal P\) fixing the generators, hence the identity, and multiplicativity of \(\Theta\) is the product rule for \(\delta\). A map with the product rule is determined by its values on the generators, which gives uniqueness.

The set of \(p\) with \(\delta(p^*)=\delta(p)^*\) is closed under sums, products and adjoints, since \(\delta((pq)^*)=\delta(q^*)p^*+q^*\delta(p^*)\). It contains the generators: \(\delta(a_j)^*=-ia_j^*\sigma(h_j)=\delta(a_j^*)\), and \(\delta z=0\). So it is \(\mathcal P\).

The set of \(g\in\Gamma\) with \(\delta a_g=i\sigma(D_g)a_g\) contains \(e\) (\(\delta1=0\) by the product rule) and every \(x_j\). It contains \(x_j^{-1}\): \(\delta a_j^*=-ia_j^*\sigma(h_j)a_ja_j^*\), which is \(-i\sigma(\lambda(x_j^{-1})h_j)a_j^*=i\sigma(D_{x_j^{-1}})a_{x_j^{-1}}\). It is closed under products: if it contains \(g\) and \(b\), then \[ \begin{aligned} &\delta(a_ga_b)\\ &\quad =i\sigma(D_g)a_ga_b+ia_g\sigma(D_b)a_g^*a_ga_b\\ &\quad =i\sigma\big(D_g+\lambda(g)D_b\big)a_{gb}=i\sigma(D_{gb})a_{gb}. \end{aligned} \] So it is \(\Gamma\). Then \(\delta(a_g^*)=\delta(a_g)^*=-ia_g^*\sigma(D_g)\), and the product rule gives \(\delta z_g=i\sigma(D_g)z_g-iz_g\sigma(D_g)\).

Finally, \(\delta m\) is the sum, over the symbols of \(m\), of \(m\) with that symbol replaced by its image. The image of \(a_j\) is \(i\sum_bh_j(b)\,a_bza_b^*a_j\), a combination of monomials of label \(bb^{-1}x_j=x_j\); the image of \(a_j^*\) is a combination of monomials of label \(x_j^{-1}\); the image of \(z\) is \(0\). So every term has the label of \(m\). \(\square\)

3. The trace identity at time zero

Let \(\mathrm{ev}_0\colon\mathcal P\to M\) be the unital \(*\)-homomorphism with \(a_j\mapsto A_j\) and \(z\mapsto S\). Then \(\mathrm{ev}_0(a_g)=A_g\), \(\mathrm{ev}_0(z_g)=S_g\) and \(\mathrm{ev}_0(\sigma(k))=s(k)\), and \(\mathrm{ev}_0\) maps a monomial to a product of letters \(A_j,A_j^*,S\) with the same label. We make no assumption on the kernel of \(\mathrm{ev}_0\).

Lemma 3.1. \(\tau(\mathrm{ev}_0(\delta p))=0\) for every \(p\in\mathcal P\). Consequently \(\tau(\mathrm{ev}_0(\delta^rp))=0\) for every \(p\in\mathcal P\) and every \(r\ge1\).

Proof. By linearity we may take \(p\) to be a monomial \(m\). If its label is not \(e\), then by Lemma 2.1 \(\delta m\) is a combination of monomials with that label, and their images have trace \(0\) by Lemma 3.2 of the first lesson.

Let the label of \(m\) be \(e\). Write \(m=w_0zw_1z\cdots zw_q\) with words \(w_l\) in the group symbols, and let \(g_l\in\Gamma\) be the value of \(w_l\), so \(w_l=a_{g_l}\) in \(\mathcal P\). Since \(a_gz=a_gza_g^*a_g=z_ga_g\) in \(\mathcal P\), moving the group symbols to the right gives \[ \begin{aligned} m&=z_{g_0}z_{g_0g_1}\cdots z_{g_0\cdots g_{q-1}}\,a_{g_0\cdots g_q}\\ &=z_{g_0}z_{g_0g_1}\cdots z_{g_0\cdots g_{q-1}}, \end{aligned} \] because \(g_0\cdots g_q\) is the label \(e\). So \(m=F(z_b:b\in I)\) for a finite set \(I\subseteq\Gamma\) and a noncommutative monomial \(F\) in variables \(Z_b\), \(b\in I\). By the product rule, \(\delta m\) is the sum, over the occurrences of a variable \(Z_b\) in \(F\), of \(F(z)\) with that occurrence replaced by \(\delta z_b=i[\sigma(D_b),z_b]\). Applying \(\mathrm{ev}_0\), the summand \(D_b(b)S_b\) of \(s(D_b)\) commutes with \(S_b\), so \[ \begin{gathered} \mathrm{ev}_0(\delta z_b)=i[Y_b,S_b],\\ Y_b=\sum_{c\ne b}D_b(c)S_c\in W^*(S_c:c\ne b). \end{gathered} \] Fix \(b\in I\) and put \(U_u=e^{iuY_b}\) for \(u\in\mathbb R\), a unitary of \(W^*(S_c:c\ne b)\). Let \(f_b(u)\) be the trace of \(F(S)\) with every occurrence of \(S_b\) replaced by \(U_uS_bU_u^*\) and the other variables \(S_c\) left alone. By Lemma 3.1 of the first lesson, \(W^*(S_b)\) and \(W^*(S_c:c\ne b)\) are free, so Lemma 1.3 of that lesson shows that \(f_b\) is constant. The map \(u\mapsto U_uS_bU_u^*\) is norm-differentiable with derivative \(i[Y_b,S_b]\) at \(u=0\), so by the product rule \(0=f_b'(0)\) is the trace of the sum over the occurrences of \(Z_b\) of \(F(S)\) with that occurrence replaced by \(i[Y_b,S_b]\). Summing over \(b\in I\) gives \(\tau(\mathrm{ev}_0(\delta m))=\sum_bf_b'(0)=0\).

For the second statement apply the first to \(\delta^{r-1}p\). \(\square\)

The conjugations by \(U_u\) are used only to compute a derivative at the initial tuple: different variables \(S_b\) are conjugated separately, and no simultaneous motion of the tuple is involved.

4. Solutions of the differential equation

Lemma 4.1. Let \(E\) be a real Banach space and \(V\colon E\to E\) a map that is Lipschitz on every bounded set.

  1. Let \(x_0\in E\), \(t_0\in\mathbb R\), and suppose \(\|V(x)\|\le K\) and \(\|V(x)-V(y)\|\le L\|x-y\|\) for \(x,y\) in the closed ball \(\bar B(x_0,1)\). For \(0<\delta\le\min(1/K,1/(2L))\) there is a continuously differentiable \(X\colon[t_0-\delta,t_0+\delta]\to E\) with \(X'=V(X)\) and \(X(t_0)=x_0\).
  2. Two continuously differentiable solutions of \(X'=V(X)\) on an interval that agree at one point agree on the whole interval.

Proof. (1) The continuous maps \(X\colon[t_0-\delta,t_0+\delta]\to\bar B(x_0,1)\) form a complete metric space for the supremum distance. The map \(\Phi(X)(t)=x_0+\int_{t_0}^tV(X(s))\,ds\) sends it to itself, since \(\|\Phi(X)(t)-x_0\|\le\delta K\le1\), and it is a contraction with constant \(\delta L\le\frac12\). Its fixed point is continuously differentiable with \(X'=V(X)\), by the fundamental theorem of calculus (Lemma 0.1 of the lesson on Cauchy's theorem).

(2) Let \(X,Y\) be solutions on an interval \(I\) with \(X(t_1)=Y(t_1)\). The set where they agree is closed in \(I\). Let \(X(t_2)=Y(t_2)\). On a closed interval \(J\ni t_2\) of length at most \(1/(2L')\), where \(L'\) is a Lipschitz constant of \(V\) on a bounded set containing \(X(J)\cup Y(J)\), the difference \(X(t)-Y(t)\) is \(\int_{t_2}^t(V(X(s))-V(Y(s)))\,ds\), so \(\sup_J\|X-Y\|\le\frac12\sup_J\|X-Y\|\), and \(X=Y\) on \(J\). So the set is also open in \(I\), and it is \(I\). \(\square\)

Let \(B\subseteq M\) be a norm-closed unital \(*\)-subalgebra containing \(S\). For \(X=(X_1,\ldots,X_n)\in B^n\) let \(X_g\) be the value at \(X\) of the reduced word of \(g\), with \(x_j^{-1}\) evaluated as \(X_j^*\); this is a polynomial in the \(X_j\) and \(X_j^*\), and it agrees with the group evaluation when the \(X_j\) are unitary. Put \[ \begin{gathered} H_j(X)=\sum_gh_j(g)X_gSX_g^*,\\ V_j(X)=iH_j(X)X_j . \end{gathered} \] Each \(H_j(X)\) is self-adjoint, for every \(X\). We regard \(B^n\), with the maximum of the coordinate norms, as a real Banach space. \(V=(V_1,\ldots,V_n)\) is a finite sum of products of coordinates, their adjoints and \(S\), so it is Lipschitz on every bounded set.

Lemma 4.2. For every tuple \(X^0\in B^n\) of unitaries there is exactly one continuously differentiable \(X\colon\mathbb R\to B^n\) with \(X'=V(X)\) and \(X(0)=X^0\), and every \(X_j(t)\) is unitary.

Proof. Choose \(K,L\) as in Lemma 4.1(1) that serve for every ball \(\bar B(x_0,1)\) with \(\|x_0\|\le1\) (all such balls lie in the ball of radius \(2\) about \(0\)), and fix \(\delta\) accordingly. By Lemma 4.1, for every time \(t_0\) and every unitary tuple at \(t_0\), a solution exists on \([t_0-\delta,t_0+\delta]\), and solutions are unique.

Along a solution, the product rule gives \[ \begin{aligned} &\frac d{dt}(X_j^*X_j)=(iH_j(X)X_j)^*X_j\\ &\quad +X_j^*\,iH_j(X)X_j=-iX_j^*H_j(X)X_j\\ &\quad +iX_j^*H_j(X)X_j=0, \end{aligned} \] so a solution that starts at a unitary tuple satisfies \(X_j^*X_j=1\) throughout. Then \(X_jX_j^*\) is a projection and \(\tau(1-X_jX_j^*)=1-\tau(X_j^*X_j)=0\), so \(X_jX_j^*=1\) because \(\tau\) is faithful: \(X_j\) stays unitary.

Let \(T\) be the supremum of the times \(T'>0\) for which a solution exists on \([0,T']\); by uniqueness these solutions fit together. If \(T<\infty\), take \(t_0<T\) with \(T-t_0<\delta/2\) at which a solution is defined; its value there is a unitary tuple, and the solution on \([t_0-\delta,t_0+\delta]\) extends it beyond \(T\), a contradiction. The same argument applies for negative times. \(\square\)

5. Moment invariance

Let \(B=C^*(A_1,\ldots,A_n,S)\) be the norm-closed \(*\)-algebra generated by the \(A_j\) and \(S\), and let \(A(t)\) be the solution of Lemma 4.2 in \(B\) with \(A(0)=A\). For each \(t\), let \(\mathrm{ev}_t\colon\mathcal P\to B\) be the unital \(*\)-homomorphism with \(a_j\mapsto A_j(t)\) and \(z\mapsto S\); it exists because the \(A_j(t)\) are unitary. Then \(\mathrm{ev}_t(a_g)=A_g(t)\), \(\mathrm{ev}_t(\sigma(k))=s_t(k)\), and the differential equation reads \(A_j'(t)=\mathrm{ev}_t(\delta a_j)\).

Lemma 5.1. For every \(p\in\mathcal P\), the function \(t\mapsto\mathrm{ev}_t(p)\) is infinitely differentiable in norm, with \(\frac{d^r}{dt^r}\mathrm{ev}_t(p)=\mathrm{ev}_t(\delta^rp)\), and \[ \tau(\mathrm{ev}_t(p))=\tau(\mathrm{ev}_0(p))\qquad(t\in\mathbb R). \]

Proof. Derivatives. For a monomial \(m\), \(\mathrm{ev}_t(m)\) is a product of the norm-differentiable functions \(A_j(t)\), \(A_j(t)^*\) and the constant \(S\), with derivatives \(\mathrm{ev}_t(\delta a_j)\), \(\mathrm{ev}_t(\delta a_j)^*=\mathrm{ev}_t(\delta a_j^*)\) and \(0=\mathrm{ev}_t(\delta z)\). By the product rule, \(\frac d{dt}\mathrm{ev}_t(m)=\mathrm{ev}_t(\delta m)\). By linearity this holds for all \(p\), and since \(t\mapsto\mathrm{ev}_t(\delta p)\) is again differentiable, induction gives all orders.

Bounds. Let \(\ell\) be the largest length \(|b|\) of an element \(b\) in the support of some \(h_j\) (\(\ell=0\) if all \(h_j\) vanish), \(J=2\ell+1\), \(B_1=\max(1,\max_j\sum_b|h_j(b)|)\) and \(R=\max(1,\|S\|)\). Represent \(p\) by a linear combination of monomials of degree at most \(l\ge1\) whose coefficients have absolute values summing to \(b_0\). The image of \(a_j\) under \(\delta\) is \(i\sum_bh_j(b)a_bza_b^*a_j\), with \(a_b\) and \(a_b^*\) written as reduced words: a combination of monomials of degree at most \(2\ell+2\) with absolute coefficient sum at most \(B_1\); the same holds for \(a_j^*\), and \(\delta z=0\). Hence \(\delta\) maps a combination of monomials of degree at most \(l'\) with absolute coefficient sum \(b'\) to one of degree at most \(l'+J\) with absolute coefficient sum at most \(l'B_1b'\). By induction, \(\delta^rp\) is represented by monomials of degree at most \(l+rJ\) with absolute coefficient sum at most \(b_0B_1^r\prod_{v=0}^{r-1}(l+vJ)\). A monomial of degree \(d\) evaluates at time \(t\) to an operator of norm at most \(R^d\), since the group letters are unitary. Using \(l+vJ\le(l+J)(v+1)\), we get for all \(t\in\mathbb R\) and \(r\ge0\) \[ \begin{gathered} \|\mathrm{ev}_t(\delta^rp)\|\le b_0R^l\,c^r\,r!,\\ c=B_1R^J(l+J). \end{gathered} \]

Analyticity. Let \(f(t)=\tau(\mathrm{ev}_t(p))\). Since \(\tau\) is a bounded linear functional, \(f\) is infinitely differentiable with \(f^{(r)}(t)=\tau(\mathrm{ev}_t(\delta^rp))\), so \(|f^{(r)}(t)|\le b_0R^lc^rr!\). Let \(t_0\in\mathbb R\) be a point where \(f^{(r)}(t_0)=0\) for all \(r\ge1\). Taylor's theorem, applied to the real and imaginary parts of \(f\) on the interval between \(t_0\) and \(t_0+u\), gives \[ \begin{gathered} |f(t_0+u)-f(t_0)|\le2\,b_0R^l(c|u|)^{r+1}\\ (r\ge0), \end{gathered} \] which tends to \(0\) as \(r\to\infty\) when \(|u|<1/c\). So \(f\) is constant on \((t_0-1/c,t_0+1/c)\), and all its derivatives of order \(\ge1\) vanish there. By Lemma 3.1, \(t_0=0\) is such a point. Moving in steps of \(1/(2c)\) in both directions shows that \(f\) is constant on \(\mathbb R\). \(\square\)

The constant \(c\) depends on \(p\) and on the \(h_j\); no common radius is needed. Nothing about freeness at positive times was used: only the unitarity of the solution and the bound on \(S\).

6. The automorphisms

Lemma 6.1. For every \(b\in M\), \(\|b\|=\lim_{k\to\infty}\tau((b^*b)^k)^{1/(2k)}\).

Proof. Let \(a=b^*b\), so \(\|a\|=\|b\|^2\). Then \(\tau(a^k)\le\|a\|^k\). For \(0<\varepsilon<\|a\|\), let \(e=1_{[\|a\|-\varepsilon,\|a\|]}(a)\). If \(e\) were \(0\), then \(a=a\,1_{[0,\|a\|-\varepsilon)}(a)\) would have norm at most \(\|a\|-\varepsilon\) by the Borel functional calculus; so \(e\ne0\) and \(\tau(e)>0\). Since \(x^k\ge(\|a\|-\varepsilon)^k1_{[\|a\|-\varepsilon,\|a\|]}(x)\) on the spectrum of \(a\), we get \(\tau(a^k)\ge(\|a\|-\varepsilon)^k\tau(e)\), and \(\liminf_k\tau(a^k)^{1/k}\ge\|a\|-\varepsilon\). \(\square\)

Theorem 6.2. Let \(h_1,\ldots,h_n\in\mathbb R^{(\Gamma)}\), let \(D\) be the cocycle with these generator values, and let \(K=3\pi\). There are trace-preserving normal \(*\)-automorphisms \(\beta_t\) of \(M\), \(t\in\mathbb R\), with the following properties.

  1. \(\beta_0=\mathrm{id}\), \(\beta_s\beta_t=\beta_{s+t}\), \(\beta_t(S)=S\) and \(\beta_t(C)=C\).
  2. \(\beta_t\) maps \(B=C^*(A_1,\ldots,A_n,S)\) onto itself, and \(t\mapsto\beta_t(b)\) is norm-continuous for every \(b\in B\).
  3. \(A_j(t)=\beta_t(A_j)\) is norm-differentiable in \(t\), with \(A_j'(t)=i\,s_t(h_j)A_j(t)\), where \(s_t(k)=\sum_gk(g)A_g(t)SA_g(t)^*\), which is \(\beta_t(s(k))\).
  4. For all \(t\in\mathbb R\), \(1\le j\le n\) and \(w\in\Gamma\), \[ \begin{aligned} &\|\beta_t(A_j)-A_j\|\le|t|K\|h_j\|_{\ell^2},\\ &\|\beta_t(A_w)-e^{itS}A_w\|\\ &\quad \le|t|K\|D_w-\delta_e\|_{\ell^2}. \end{aligned} \] In particular \(\|\beta_1(A_w)-CA_w\|\le K\|D_w-\delta_e\|_{\ell^2}\).

Proof. Let \(A(t)\) and \(\mathrm{ev}_t\) be as in Section 5, and fix \(t\).

An isometric \(*\)-homomorphism. If \(\mathrm{ev}_0(p)=0\), then by Lemma 5.1 \(\tau(\mathrm{ev}_t(p)^*\mathrm{ev}_t(p))=\tau(\mathrm{ev}_t(p^*p))\), which is \(\tau(\mathrm{ev}_0(p^*p))=0\), so \(\mathrm{ev}_t(p)=0\). Hence \(\varphi_t(\mathrm{ev}_0(p))=\mathrm{ev}_t(p)\) is a well-defined unital \(*\)-homomorphism from the dense \(*\)-subalgebra \(\mathrm{ev}_0(\mathcal P)\) of \(B\) into \(B\). By Lemma 5.1 applied to \((p^*p)^k\) and by Lemma 6.1, \(\|\varphi_t(b)\|=\|b\|\). So \(\varphi_t\) extends to an isometric \(*\)-homomorphism \(\varphi_t\colon B\to B\) with \(\tau\circ\varphi_t=\tau\). Its range is closed and contains the polynomials in \(A_1(t),\ldots,A_n(t),S\), so it is \(B_t=C^*(A_1(t),\ldots,A_n(t),S)\).

Surjectivity. Apply Lemma 4.2 inside \(B_t\) to the initial tuple \(A(t)\). Its solution \(Y\) is also a solution in \(M\), and so is \(u\mapsto A(t+u)\); both start at \(A(t)\), so \(Y(u)=A(t+u)\) by uniqueness (Lemma 4.2 with \(B=M\)). At \(u=-t\) this gives \(A_j=Y_j(-t)\in B_t\). Since also \(S\in B_t\), we get \(B\subseteq B_t\subseteq B\). So \(\varphi_t\) is a \(*\)-automorphism of \(B\) fixing \(S\).

Normal extension. \(W^*(B)\) contains the \(A_j\) and \(C=e^{iS}\), so \(W^*(B)=M\), and \(B\Omega\) is dense in \(L^2(M)\) by Lemma 1.4 of the first lesson. The map \(b\Omega\mapsto\varphi_t(b)\Omega\) (\(b\in B\)) is well defined and isometric, because \(\|\varphi_t(b)\|_2^2=\tau(\varphi_t(b^*b))=\|b\|_2^2\), and its range \(B\Omega\) is dense; it extends to a unitary \(U_t\) of \(L^2(M)\) with \(U_t\Omega=\Omega\). For \(b,c\in B\), \[ \begin{aligned} &U_tL_bU_t^*\,\varphi_t(c)\Omega=U_t(bc)\Omega\\ &\quad =\varphi_t(b)\varphi_t(c)\Omega, \end{aligned} \] so \(U_tL_bU_t^*=L_{\varphi_t(b)}\). Hence \(\operatorname{Ad}U_t\) maps the von Neumann algebra generated by \(L_B\), which is \(L_M\), onto itself, and \(\beta_t=L^{-1}\circ\operatorname{Ad}U_t\circ L\) is a normal \(*\)-automorphism of \(M\) extending \(\varphi_t\). It preserves the trace: \(\tau(\beta_t(x))=\langle U_tL_xU_t^*\Omega,\Omega\rangle\), which is \(\langle L_x\Omega,\Omega\rangle=\tau(x)\). Two normal \(*\)-homomorphisms that agree on \(B\) agree on \(M\), by Lemma 1.4 of the first lesson, so \(\beta_t\) is the only normal extension of \(\varphi_t\). As \(\beta_t\) is isometric and fixes \(S\), it fixes \(C=\sum_k(iS)^k/k!\).

Group law and continuity. Fix \(s\). The path \(t\mapsto\varphi_s(A(t))\) is continuously differentiable, because \(\varphi_s\) is bounded and linear, and it solves \(X'=V(X)\), because \(\varphi_s\) is a \(*\)-homomorphism fixing \(S\). It starts at \(\varphi_s(A)=A(s)\), as does \(t\mapsto A(s+t)\), so \(\varphi_s(A_j(t))=A_j(s+t)\). Thus \(\varphi_s\varphi_t\) and \(\varphi_{s+t}\) agree on the generators of \(B\), hence on \(B\), and \(\beta_s\beta_t=\beta_{s+t}\) by uniqueness of normal extensions; \(\beta_0=\mathrm{id}\) since \(A(0)=A\). For \(b=\mathrm{ev}_0(p)\), \(t\mapsto\beta_t(b)=\mathrm{ev}_t(p)\) is norm-continuous by Lemma 5.1; since the \(\beta_t\) are isometric on \(B\), continuity passes to the closure \(B\). This proves (1) and (2).

(3). \(\beta_t(A_j)=\varphi_t(\mathrm{ev}_0(a_j))=A_j(t)\), \(\beta_t(A_g)=A_g(t)\), and \(s_t(k)=\mathrm{ev}_t(\sigma(k))=\beta_t(s(k))\); the differential equation holds by construction.

(4). By (3) and Corollary 4.2 of the first lesson, \(\|s_t(k)\|=\|s(k)\|\le K\|k\|_{\ell^2}\). Hence \(\|A_j'(t)\|\le K\|h_j\|_{\ell^2}\), and integrating (Lemma 0.1 of the lesson on Cauchy's theorem) gives the first bound. For \(w\in\Gamma\), Lemmas 5.1 and 2.1 give \[ \frac d{dt}A_w(t)=\mathrm{ev}_t(\delta a_w)=i\,s_t(D_w)A_w(t). \] The function \(t\mapsto e^{-itS}\) is norm-differentiable with derivative \(-iSe^{-itS}\), since \(\|e^{-iuS}-1+iuS\|\le\sum_{k\ge2}|u|^k\|S\|^k/k!\). Since \(s_t(\delta_e)=S\) commutes with \(e^{-itS}\), \[ \begin{aligned} &\frac d{dt}\big(e^{-itS}A_w(t)\big)\\ &\quad =ie^{-itS}\big(s_t(D_w)-S\big)A_w(t)\\ &\quad =ie^{-itS}\,s_t(D_w-\delta_e)\,A_w(t), \end{aligned} \] of norm at most \(K\|D_w-\delta_e\|_{\ell^2}\). Integrating from \(0\) to \(t\) and multiplying by the unitary \(e^{itS}\) gives the second bound, and \(e^{iS}=C\). \(\square\)

7. Exercises

Exercise 7.1. Let \(n=2\), \(h_1=h\) and \(h_2=0\). Compute \(D_w\) for \(w=x_1x_2x_1^{-1}\).

Exercise 7.2. Show that \(\beta_t=\mathrm{id}\) for all \(t\) when \(h_1=\cdots=h_n=0\).

Exercise 7.3. Without Corollary 4.2 of the first lesson, the differential equation gives \(\|\beta_t(A_j)-A_j\|\le|t|\,\pi\sum_g|h_j(g)|\). For \(h_j\) equal to \(N^{-1/2}\) on \(N\) distinct elements of \(\Gamma\) and \(0\) elsewhere, compare this bound with the bound of Theorem 6.2 as \(N\to\infty\).

Exercise 7.4. Let \(n=1\) and \(h_1=c\,\delta_e\) with \(c\in\mathbb R\). Show that \(\beta_t(A_1)=e^{ictS}A_1\), and check directly, with Lemma 3.2 of the first lesson, that \(\tau\big((e^{ictS}A_1)^k\big)=0\) for \(k\ne0\).

8. Solutions

7.1. By Lemma 1.1, \(D_{x_1}=h\), \(D_{x_2}=0\), \(D_{x_1^{-1}}=-\lambda(x_1^{-1})h\), and \[ \begin{aligned} D_w&=D_{x_1}+\lambda(x_1)D_{x_2}+\lambda(x_1x_2)D_{x_1^{-1}}\\ &=h-\lambda(x_1x_2x_1^{-1})h . \end{aligned} \]

7.2. The constant path \(A(t)=A\) solves the equation, since \(V=0\). So \(\mathrm{ev}_t=\mathrm{ev}_0\), \(\varphi_t=\mathrm{id}_B\), and \(\beta_t=\mathrm{id}\) by uniqueness of normal extensions.

7.3. The first bound is \(|t|\pi N^{1/2}\), which tends to infinity. The bound of Theorem 6.2 is \(3\pi|t|\), because \(\|h_j\|_{\ell^2}=1\).

7.4. \(H_1(X)=cS\) for every \(X\), so the equation is \(X'=icSX\), solved by \(e^{ictS}A_1\) (differentiate as in the proof of Theorem 6.2(4)). By uniqueness, \(A_1(t)=e^{ictS}A_1\). Expanding \(e^{ictS}\) in its power series, \((e^{ictS}A_1)^k\) is a norm limit of combinations of products of letters \(S\) and \(A_1\) with label \(x_1^k\) (and, for \(k<0\), letters \(A_1^*\)), whose traces are \(0\) by Lemma 3.2 of the first lesson when \(k\ne0\).

References