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Infinitesimal normalizers of type III factors

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

A unitary \(v\) normalizes a von Neumann algebra \(P\subseteq B(H)\) when \(vPv^*=P\). Products \(ab'\) of unitaries \(a\in P\) and \(b'\in P'\) are normalizers of a simple kind. This lesson proves three rigidity statements about normalizers of a type III factor, each with constants that do not depend on the factor. A self-adjoint operator whose commutators with \(P\) lie close to \(P\) is close to \(P+P'\) (Lemma 2.1, the uniform infinitesimal estimate). A norm-continuous path of unitaries that nearly normalize \(P\) stays close to the products \(ab'\) (Lemma 3.1). An exact normalizer close to the identity is such a product (Lemma 4.1). The first statement is proved by contradiction in a norm ultraproduct of factors, where Sakai's theorem on derivations of simple C*-algebras provides an implementing element. These statements are used to align the modular actions of a prepared pair [OpenAI-289, Section 7].

We use: Close operator algebras (Proposition 3.1 and Lemma 4.2); the inner derivation theorem for von Neumann algebras (every derivation \(\delta\) of a von Neumann algebra is inner, and if \(\delta(x^*)=\delta(x)^*\) it is implemented by \(ik\) with \(k=k^*\) in the algebra and \(\|k\|\le\|\delta\|\)); the facts on projections of countably decomposable type III factors, (B8) of Ultraproducts and the asymptotic centralizer (every nonzero projection is equivalent to \(1\)); the Haar probability measure of a compact group; and one theorem used without proof, Sakai's theorem below.

Theorem 0.1 (Sakai [Sakai]). Every derivation of a simple unital C*-algebra is inner: if \(A\) is simple and unital and \(\delta:A\to A\) is a derivation, there is \(a\in A\) with \(\delta(x)=ax-xa\) for all \(x\in A\).

Throughout, a separably acting algebra acts on a separable Hilbert space; a separably acting type III factor is countably decomposable, and every nonzero projection in it is equivalent to \(1\). For a subset \(P\subseteq B(H)\) and \(h\in B(H)\), \(\|\operatorname{ad}_h|_P\|=\sup_{x\in P_1}\|[h,x]\|\).

1. Distance to the commutant of a properly infinite algebra

Lemma 1.1. Let \(P\subseteq B(H)\) be a properly infinite von Neumann algebra and \(h\in B(H)\). Then \[ \operatorname{dist}(h,P')\le\tfrac32\,\|\operatorname{ad}_h|_P\| . \]

Proof. Put \(d=\|\operatorname{ad}_h|_P\|\). As in Section 4 of Close operator algebras, \(P\) contains an infinite family of isometries \((s_i)\); put \(e_{ij}=s_is_j^*\) and let \(S\) be the von Neumann algebra generated by these matrix units. The finite-dimensional unital algebras \[ \begin{aligned} S_n&=\operatorname{span}\{e_{ij}:i,j\le n\}\\ &\quad +\mathbb C\Bigl(1-\textstyle\sum_{i\le n}e_{ii}\Bigr) \end{aligned} \] increase and generate \(S\). Let \(h_n=\int_{\mathcal U(S_n)}uhu^*\,du\). For a unitary \(u\in P\), \(\|uhu^*-h\|=\|[u,h]\|\le d\), and \([uhu^*,x]=u[h,u^*xu]u^*\) has norm at most \(d\|x\|\) for \(x\in P\). So \(\|h_n-h\|\le d\), \(\|[h_n,x]\|\le d\|x\|\) for \(x\in P\), and \(h_n\) commutes with \(S_n\). A weak operator cluster point \(h_0\) of \((h_n)\) inherits the two bounds, since multiplication by a fixed operator is weakly continuous and norm balls are weakly closed, and it commutes with every \(S_n\), hence with \(S\).

The isometries \(t_1,\dots,t_k\) of Lemma 4.2 of Close operator algebras are strong sums of matrix units \(e_{ij}\), so they lie in \(S\), and the unitary \(V_k\) there satisfies \(V_kM_k(P)V_k^*=P\) and \(V_kh_0^{(k)}V_k^*=h_0\). Hence, for \(X\in M_k(P)\), \(\|[h_0^{(k)},X]\|=\|[h_0,V_kXV_k^*]\|\le d\|X\|\), so \(\|\operatorname{ad}_{h_0}|_P\|_{\rm cb}\le d\). Proposition 3.1 of that lesson gives \(\operatorname{dist}(h_0,P')\le d/2\), and \(\operatorname{dist}(h,P')\le\|h-h_0\|+d/2\le\frac32d\). \(\square\)

2. The uniform infinitesimal estimate

Lemma 2.1 (uniform infinitesimal estimate). There is an absolute constant \(K\) such that, for every separably acting type III factor \(P\subseteq B(H)\) and every self-adjoint \(h\in B(H)\), \[ \operatorname{dist}(h,P+P')\le K\sup_{x\in P_1}\operatorname{dist}([h,x],P). \] Here \(P+P'\) is the linear span of \(P\cup P'\); no closedness of this subspace is needed.

Proof. A sequence of counterexamples. Suppose that no \(K\) works. Then for every \(j\) there are a separably acting type III factor \(P_j\subseteq B(H_j)\) and a self-adjoint \(h_j\) with \(\operatorname{dist}(h_j,P_j+P_j')>j\,\varepsilon_j\), where \(\varepsilon_j=\sup_{x\in(P_j)_1}\operatorname{dist}([h_j,x],P_j)\). Rescale so that \(\operatorname{dist}(h_j,P_j+P_j')=1\); then \(\varepsilon_j<1/j\). Choose \(a+b'\) with \(a\in P_j\), \(b'\in P_j'\) and \(\|h_j-(a+b')\|<2\). Replacing \(a+b'\) by its real part, which again lies in \(P_j+P_j'\), does not increase the distance to the self-adjoint \(h_j\); subtracting this self-adjoint element from \(h_j\) changes neither \(\operatorname{dist}(h_j,P_j+P_j')\) nor \(\varepsilon_j\), because its commutators with \(P_j\) lie in \(P_j\). So we may assume \(\|h_j\|\le2\).

The ultraproduct. Fix a free ultrafilter \(\omega\) on \(\mathbb N\). The norm ultraproduct \(E=\prod_\omega B(H_j)\) is the C*-algebra of bounded sequences \((x_j)\), \(x_j\in B(H_j)\), modulo those with \(\lim_\omega\|x_j\|=0\); let \(A=\prod_\omega P_j\subseteq E\) be the C*-subalgebra of classes of sequences with \(x_j\in P_j\). It is unital and simple. Indeed, let \(a\ne0\) be a positive element with a positive representative \((a_j)\), and \(t=\|a\|/2>0\). On a set of indices in \(\omega\), \(\|a_j\|>t\), so the spectral projection \(e_j=1_{[t,\infty)}(a_j)\) is nonzero, hence equivalent to \(1\): choose isometries \(v_j\in P_j\) with \(v_jv_j^*=e_j\), and \(v_j=1\) elsewhere. Then \(v_j^*a_jv_j\ge t\) on that set, so the class of \((v_j^*a_jv_j)\) is invertible in \(A\) and lies in the ideal generated by \(a\). Every nonzero closed ideal of \(A\) contains a nonzero positive element, hence \(1\).

A derivation. Let \(h\in E\) be the class of \((h_j)\). For a bounded sequence \(x_j\in P_j\), choose \(y_j\in P_j\) with \(\|[h_j,x_j]-y_j\|\le(\varepsilon_j+j^{-1})\sup_i\|x_i\|\). The \(y_j\) are bounded, and the class of \((y_j)\) is \([h,x]\) in \(E\). So \(\delta(x)=[h,x]\) is a bounded derivation of \(A\) into \(A\). By Sakai's theorem there is \(a\in A\) with \([h,x]=[a,x]\) for \(x\in A\). Since \(h\) is self-adjoint, \([a^*,x]=-[a,x^*]^*=[a,x]\), so \(a-a^*\) is central in \(A\); the center of a simple unital C*-algebra is \(\mathbb C1\), and replacing \(a\) by \((a+a^*)/2\) changes \(a\) by a scalar. So we may take \(a\) self-adjoint, with a bounded self-adjoint representative \((a_j)\), \(a_j\in P_j\).

The contradiction. We claim \(\lim_\omega\|\operatorname{ad}_{h_j-a_j}|_{P_j}\|=0\). Otherwise there are \(c>0\), a set of indices in \(\omega\) and contractions \(x_j\in P_j\) there with \(\|[h_j-a_j,x_j]\|\ge c\); with \(x_j=0\) elsewhere, the class \(x\) of \((x_j)\) satisfies \([h-a,x]\ne0\), a contradiction. Since \(P_j\) is a type III factor, it is properly infinite, and Lemma 1.1 gives \[ \begin{aligned} 1&=\operatorname{dist}(h_j,P_j+P_j')\\ &\le\operatorname{dist}(h_j-a_j,P_j')\\ &\le\tfrac32\|\operatorname{ad}_{h_j-a_j}|_{P_j}\|, \end{aligned} \] whose limit along \(\omega\) is \(0\). \(\square\)

3. Paths of approximate normalizers

For a von Neumann algebra \(P\subseteq B(H)\) put \[ \begin{gathered} \mathcal G_P=\mathcal U(P)\,\mathcal U(P'),\\ \mathcal N_P=\{v\in\mathcal U(B(H)):vPv^*=P\}. \end{gathered} \] Since the two factors commute, \(\mathcal G_P\) is a subgroup of \(\mathcal N_P\).

Two elementary estimates are used. For self-adjoint \(h,k\), \(e^{ih}-e^{ik}=\int_0^1e^{ish}\,i(h-k)\,e^{i(1-s)k}\,ds\), so \(\|e^{ih}-e^{ik}\|\le\|h-k\|\). For self-adjoint \(h\) and any \(x\), Taylor's formula with integral remainder for \(s\mapsto e^{ish}xe^{-ish}\), whose second derivative is \(-e^{ish}[h,[h,x]]e^{-ish}\), gives \[ \begin{aligned} &\|e^{ih}xe^{-ih}-x-i[h,x]\|\\ &\quad \le\tfrac12\|[h,[h,x]]\|\\ &\quad \le2\|h\|^2\|x\| .\quad(3.1) \end{aligned} \] A unitary \(U\) with \(\|U-1\|<2\) has spectrum in an arc avoiding \(-1\), and its principal logarithm \(U=e^{ih}\), \(h=h^*\), satisfies \(\|h\|\le\frac\pi2\|U-1\|\le2\|U-1\|\), because \(|\theta|\le\frac\pi2|e^{i\theta}-1|\) for \(|\theta|\le\pi\).

Lemma 3.1 (a path of approximate normalizers). Let \(K\) be the constant of Lemma 2.1. There are absolute constants \(e_0>0\) and \(C_{\rm path}\) with the following property. Let \(P\subseteq B(H)\) be a separably acting type III factor, \(I\subseteq\mathbb R\) an interval containing \(0\), and \(v:I\to\mathcal U(B(H))\) norm continuous with \(v(0)=1\). If \(0\le e<e_0\) and \[ \sup_{t\in I}\ \sup_{x\in P_1}\operatorname{dist}\bigl(v(t)xv(t)^*,P\bigr)\le e, \] then \(\sup_{t\in I}\operatorname{dist}(v(t),\mathcal G_P)\le C_{\rm path}\,e\). The constants do not depend on the length of \(I\).

Proof. Put \(b(t)=\operatorname{dist}(v(t),\mathcal G_P)\), a continuous function with \(b(0)=0\). Suppose \(b(t)<1/2\). For \(\eta>0\) small choose \(p\in\mathcal G_P\) with \(\|p^*v(t)-1\|<b(t)+\eta\), and write \(p^*v(t)=e^{ih}\) with \(\|h\|\le2(b(t)+\eta)\). Since \(p\) normalizes \(P\), conjugation by \(e^{ih}\) moves \(P_1\) to within \(e\) of \(P\), and by (3.1) \[ \sup_{x\in P_1}\operatorname{dist}([h,x],P)\le e+2\|h\|^2 . \] Lemma 2.1 gives \(a\in P\) and \(b'\in P'\), which we may take self-adjoint, with \(\|h-(a+b')\|\le K(e+2\|h\|^2)+\eta\). The unitary \(e^{i(a+b')}=e^{ia}e^{ib'}\) lies in \(\mathcal G_P\), and so does \(pe^{ia}e^{ib'}\). Hence \(b(t)\le\|e^{ih}-e^{i(a+b')}\|\)\(\le K(e+2\|h\|^2)+\eta\), and letting \(\eta\to0\), \[ \begin{gathered} b(t)\le Ke+8K\,b(t)^2\\ \text{whenever }b(t)<1/2 .\quad(3.2) \end{gathered} \] Choose \(r\le1/2\) with \(8Kr\le1/4\), and put \(e_0=r/(2K)\). If \(e<e_0\), the continuous function \(b\) cannot reach the value \(r\) on the interval between \(0\) and any \(t\in I\): at the first time it did, (3.2) would give \(r\le Ke+8Kr^2<r/2+r/4\). So \(b<r\) on \(I\), and then \(8Kb^2\le b/4\) in (3.2) gives \(b\le\frac43Ke\). Take \(C_{\rm path}=2K\). \(\square\)

4. Small exact normalizers

Lemma 4.1 (small exact normalizers). Let \(P\subseteq B(H)\) be a factor and \(v\in\mathcal N_P\) with \(\|v-1\|<1/6\). Then \[ \begin{gathered} v=ab',\\ a\in\mathcal U(P),\ b'\in\mathcal U(P'),\\ \|a-1\|+\|b'-1\|\le10\|v-1\| . \end{gathered} \]

Proof. Write \(v=e^{ih}\) with \(h=h^*\) and \(\|h\|\le2\|v-1\|<1/3\). On the Banach space \(B(H)\), the conjugation \(\operatorname{Ad}_v\) equals \(\exp(i\operatorname{ad}_h)\), and \(\|i\operatorname{ad}_h\|\le2\|h\|<\log2\). For a bounded operator \(Y\) on a Banach space with \(\|Y\|<\log2\), \(\|e^Y-1\|<1\) and the logarithm series gives back \(Y\); so \[ \begin{aligned} i\operatorname{ad}_h&=\log(\operatorname{Ad}_v)\\ &=\sum_{n\ge1}\frac{(-1)^{n+1}}n(\operatorname{Ad}_v-\mathrm{id})^n . \end{aligned} \] Every term maps \(P\) into \(P\), because \(\operatorname{Ad}_v(P)=P\), and \(P\) is norm closed; so \(\delta(x)=i[h,x]\) is a derivation of \(P\) into \(P\), with \(\delta(x^*)=\delta(x)^*\) and \(\|\delta\|\le2\|h\|\). The inner derivation theorem gives \(k=k^*\in P\) with \(\|k\|\le\|\delta\|\le2\|h\|\) and \(\delta(x)=i[k,x]\) for \(x\in P\). Then \(b_0'=h-k\) commutes with \(P\), is self-adjoint and has norm at most \(3\|h\|\). Since \(k\) and \(b_0'\) commute, \(v=e^{ik}e^{ib_0'}\), and \[ \begin{gathered} \begin{aligned} &\|e^{ik}-1\|+\|e^{ib_0'}-1\|\\ &\quad \le\|k\|+\|b_0'\|\le5\|h\|\\ &\quad \le10\|v-1\| . \end{aligned}\\ \square \end{gathered} \]

5. Exercises

Exercise 5.1. Show that \(\mathcal N_P\) is closed in operator norm.

Exercise 5.2. Show that, for a factor \(P\), the unitaries in \(\mathcal G_P\) that lie in \(P\) are the unitaries of \(P\): if \(ab'\in P\) with \(a\in\mathcal U(P)\) and \(b'\in\mathcal U(P')\), then \(b'\) is a scalar.

Exercise 5.3. Derive (3.1) from Taylor's formula and \(\|[h,[h,x]]\|\le4\|h\|^2\|x\|\).

Exercise 5.4. Show that the norm ultraproduct \(A=\prod_\omega P_j\) of the proof of Lemma 2.1 has trivial center, directly from simplicity.

Exercise 5.5. Let \(P=B(H)\) with \(H\) infinite-dimensional and separable. Show that every unitary normalizer of \(P\) lies in \(\mathcal G_P\), and that \(\mathcal G_P=\mathcal U(P)\).

6. Solutions

Solution 5.1. If unitaries \(u_j\to u\) in norm and \(u_jPu_j^*=P\), then \(uxu^*=\lim u_jxu_j^*\in P\) for \(x\in P\), and the same for \(u^*\); so \(uPu^*=P\).

Solution 5.2. \(b'=a^*(ab')\in P\cap P'=\mathbb C1\).

Solution 5.3. For \(f(s)=e^{ish}xe^{-ish}\), \(f(1)-f(0)-f'(0)\)\(=\int_0^1(1-s)f''(s)\,ds\), with \(f'(0)=i[h,x]\) and \(\|f''(s)\|=\|[h,[h,x]]\|\).

Solution 5.4. Let \(z\) be central and \(\lambda\) in its spectrum. Since \(z-\lambda\) is central, the closure of \((z-\lambda)A\) is a closed two-sided ideal. It contains no invertible element: invertible elements form an open set, and an invertible element \((z-\lambda)a=a(z-\lambda)\) would make \(z-\lambda\) invertible. By simplicity the ideal is zero, so \(z=\lambda1\).

Solution 5.5. \(P'=\mathbb C1\), so \(\mathcal G_P=\mathcal U(P)\); every unitary of \(B(H)\) normalizes \(B(H)\) and lies in \(\mathcal U(P)\).

References