Approximately inner and centrally trivial automorphisms

Written by Claude Opus 5.5 (Anthropic), September 2026, revised October 2026. The September text was spot-checked by Claude Opus 5.5 in a separate session; the October revisions are self-checked by the writing AI. Public domain (CC0).

Introduction

Let $N$ be a II₁ factor. Two subgroups of its automorphism group $\operatorname{Aut}N$ measure how far an automorphism is from being inner.

The approximately inner automorphisms form the closure $\overline{\operatorname{Int}N}$ of the group of inner automorphisms. An automorphism $\theta$ is approximately inner when there are unitaries $u_n$ with $u_nxu_n^*\to\theta(x)$ for every $x$.
The centrally trivial automorphisms form the group $\operatorname{Ct}N$ of automorphisms that leave every central sequence asymptotically fixed.

Every inner automorphism belongs to both groups, and each group is a way of saying "inner at infinity". They are nevertheless very different. On the hyperfinite factor every automorphism is approximately inner, but many are not centrally trivial.

This lesson proves two main results.

A characterization of approximate innerness (Theorem 3.2). Let $N$ have separable predual and act on $L^2(N)$. Then $\theta$ is approximately inner if and only if the map $a\mapsto\theta(a)$, $b\mapsto b$ ($a\in N$, $b\in N'$) extends to an automorphism of the C*-algebra generated by $N$ and $N'$. This is also equivalent to the existence of unit vectors $\xi\in L^2(N)$ with $\theta(u)\xi\approx\xi u$ for finitely many unitaries $u$. The hard step turns such vectors, which may be unbounded operators concentrated on tiny projections, into bounded almost intertwiners. This is done by cutting the vectors into pieces with the polar-decomposition estimates of the lesson "Trace inequalities for finite von Neumann algebras", and then gluing the pieces together by a maximality argument. Consequences: every automorphism of a semidiscrete II₁ factor is approximately inner, and a tensor product $\theta_1\otimes\theta_2$ is approximately inner exactly when both $\theta_1$ and $\theta_2$ are.
A numerical invariant for central triviality. For $\theta\in\operatorname{Aut}N$ let $c(\theta)\in[0,2]$ be the largest amount by which $\theta$ can move almost central unit vectors of $L^2(N)$ (Definition 7.1). We prove that $c(\theta)=\|V_\theta-1\|$, where $V_\theta$ is the unitary by which $\theta$ acts on $L^2$ of the central sequence algebra (Theorem 8.2). So $c(\theta)=0$ exactly when $\theta$ is centrally trivial. Together with the structure of the asymptotic period this gives $c(\theta)=\max\{|z-1|:z^p=1\}$, where $p$ is the asymptotic period of $\theta$ (Theorem 8.3). Because $c(\theta\otimes\mathrm{id})\le c(\theta)$, the tensor product of two centrally trivial automorphisms is again centrally trivial (Corollary 8.5). This fact is not obvious, since a central sequence of $N_1\bar\otimes N_2$ need not be built from central sequences of the two factors.

These results are used in the lesson "Uniqueness of the injective II₁ factor".

What is assumed. The lessons "Ultraproducts and the asymptotic centralizer" and "Full factors" of the course on type III factors are prerequisites. From them we use the ultrapower $N^\omega$, the central sequence algebra $N_\omega$, the topology of $\operatorname{Aut}N$, and two facts about central sequences in II₁ factors. From this course we use the lesson "Trace inequalities for finite von Neumann algebras", and, for comparison only, the lesson "Property $\Gamma$ and the algebra generated by a factor and its commutant". Section 1 states exactly what is used from these lessons, with their numbering. Standard operator algebra is listed in "Results used from other lessons", each fact with the lesson that proves it.

Basic references are [Connes 1976], [Connes 1975], [Connes 1994] and [Anantharaman–Popa].

Conventions. Throughout, $N$ is a II₁ factor with its unique normal tracial state $\tau$. We write $\|x\|_p=\tau(|x|^p)^{1/p}$ for $p=1,2$, and $\|x\|$ for the operator norm. $\mathcal U(N)$ is the unitary group. For a nonzero projection $e\in N$, $N_e=eNe$. A free ultrafilter $\omega$ is a nonprincipal ultrafilter on $\mathbb N$. For $p\ge1$ let $\mu_p=\{z\in\mathbb C:z^p=1\}$, and let $\mu_0=\mathbb T$ be the whole unit circle.

Results used from other lessons

(B1) Traces on II₁ factors. The trace $\tau$ is faithful, and every automorphism of $N$ preserves it. Every $x\in N$ is a linear combination of four unitaries with coefficients of modulus at most $\|x\|$. We have $\|x\|_1\le\|x\|_2\le\|x\|$, $\|x^*\|_p=\|x\|_p$, $\|axb\|_p\le\|a\|\,\|x\|_p\,\|b\|$, $|\tau(xy)|\le\|x\|_2\|y\|_2$, $\|xy\|_1\le\|x\|_2\|y\|_2$ and $\|x\|_2^2\le\|x\|\,\|x\|_1$. Also $\|x\|_1=\sup\{|\tau(xy)|:\|y\|\le1\}$. The functionals $\tau(h\,\cdot\,)$, $h\in N$, are norm dense in $N_*$. On bounded sets the strong operator topology is given by $\|\cdot\|_2$. The unit ball of $N$ is complete for $\|\cdot\|_2$, and it is $\|\cdot\|_2$-separable when $N_*$ is separable. If $e_k$ is an increasing sequence of projections with supremum $e$, then $\|e-e_k\|_2\to0$.

Two projections $e,f$ of $N$ are equivalent if and only if $\tau(e)=\tau(f)$, and then there is a unitary $w$ with $wew^*=f$. Every projection $e$ has subprojections of every trace in $[0,\tau(e)]$. Each corner $N_e$ is a II₁ factor with trace $\tau_e=\tau(e)^{-1}\tau|_{N_e}$. Every $x\in N$ can be written $x=w|x|$ with $w$ unitary.

Let $P$ be a finite von Neumann algebra with a faithful normal tracial state. By the comparison theorem, for projections $p,q\in P$ there is a central projection $z$ with $zp\precsim zq$ and $(1-z)q\precsim(1-z)p$. If $p\sim q$ then $1-p\sim1-q$.

These facts are proved in the course Traces and noncommutative integration and in Foundations of von Neumann algebras. On a factor, two normal traces are proportional (Integration for a trace, Corollary 7.4), so $\tau\circ\theta=\tau$ for every automorphism $\theta$; and $\tau$ is faithful because its support is a nonzero central projection (Traces on von Neumann algebras, Proposition 3.2). The four unitaries are Lemma 1.4 there; its proof writes $x=\operatorname{Re}x+i\operatorname{Im}x$ and each self-adjoint part $h$ as $\tfrac{\|h\|}2(u+u^*)$ with $u$ unitary, which gives the bound on the coefficients. The trace norm satisfies $|\tau(yx)|\le\|y\|\,\|x\|_1$ and $\|x^*\|_1=\|x\|_1$ (Proposition 7.1); the duality $\|x\|_1=\sup\{|\tau(xy)|:\|y\|\le1\}$, with the functionals $\tau(h\,\cdot\,)$ norm dense in $N_*$, is Integration for a trace, Theorem 1.1 and Section 1. The other inequalities follow: $\|x\|_2\le\|x\|$ as $\tau(x^*x)\le\|x\|^2$; $|\tau(xy)|\le\|x^*\|_2\|y\|_2=\|x\|_2\|y\|_2$ is the Cauchy–Schwarz inequality for the inner product $\tau(b^*a)$, with $\tau(xx^*)=\tau(x^*x)$; for $x=u|x|$ it gives $\|x\|_1=\tau(u^*x)\le\|x\|_2$; $\|axb\|_2\le\|a\|\,\|x\|_2\|b\|$ because $\tau(b^*x^*a^*axb)\le\|a\|^2\tau(xbb^*x^*)\le\|a\|^2\|b\|^2\tau(xx^*)$; $\|xy\|_1=\tau(v^*xy)\le\|x^*v\|_2\|y\|_2\le\|x\|_2\|y\|_2$ for $xy=v|xy|$; and $\|x\|_2^2=\tau(|x|\,|x|)\le\|x\|\,\|x\|_1$. Every normal positive functional $\varphi$ is within $\varepsilon$ in norm of some $\tau(h\,\cdot\,)$, so $\varphi(x^*x)\le\|h\|\,\|x\|_2^2+\varepsilon\|x\|^2$; hence on bounded sets convergence in $\|\cdot\|_2$ implies convergence for every seminorm $\varphi(x^*x)^{1/2}$, which is the strong topology there (Compact and trace-class operators, Lemma 8.5), and conversely because $\tau$ is normal. In the standard representation (B2), a $\|\cdot\|_2$-Cauchy sequence $(x_n)$ in the unit ball is strongly Cauchy on the dense vectors $\hat y$, since $\|(x_n-x_m)\hat y\|\le\|x_n-x_m\|_2\|y\|$; it converges strongly to some $x$ in the unit ball of $N$, and $\|x_n-x\|_2=\|(x_n-x)\hat1\|\to0$. If $N_*$ is separable, the unit ball is σ-weakly compact and metrizable, so it has a countable σ-weakly dense subset $D$; σ-weak convergence in the ball gives weak convergence of the vectors $\hat x$, and in a Hilbert space a convex set has the same weak and norm closures, so the rational convex combinations of $D$ are $\|\cdot\|_2$-dense in the ball. For increasing projections $e_k$ with supremum $e$, $\|e-e_k\|_2^2=\tau(e-e_k)\to0$ by normality. Projections are equivalent exactly when their traces are equal (Traces on von Neumann algebras, Corollary 5.4); every projection has subprojections of every trace in $[0,\tau(e)]$ (Measurable operators for a trace, Lemma 4.3); and a corner $N_e$ has centre $\mathbb Ce$ (Traces on von Neumann algebras, Lemma 1.5), so it is a II₁ factor with the normal tracial state $\tau_e$. In a finite algebra the comparison theorem holds (Projections and types of von Neumann algebras, Theorem 5.5), and equivalent projections have equivalent complements and are unitarily equivalent (Proposition 14.2). With the polar decomposition $x=v|x|$ in $N$ (The double commutant theorem, Proposition 7.2) this gives a partial isometry $v'$ from $1-v^*v$ to $1-vv^*$, and $w=v+v'$ is a unitary with $x=w|x|$.

(B2) The standard form. $L^2(N)=L^2(N,\tau)$ is the completion of $N$ for $\|\cdot\|_2$, with $x\mapsto\hat x$. Its elements are identified with the closed densely defined operators $\xi$ affiliated with $N$ and measurable for $\tau$, with $\tau(\xi^*\xi)<\infty$. For $a,b\in N$ and $\xi\in L^2(N)$, the vectors $a\xi$ and $\xi b$ are operator products. Left multiplication is the standard representation of $N$, and $J\xi=\xi^*$ is the modular conjugation, so $J b^*J\xi=\xi b$. The commutant $N'$ is the set of right multiplications $JNJ$. Each $\xi\in L^2(N)$ has a polar decomposition $\xi=u|\xi|$ with $u$ a partial isometry of $N$ whose initial projection is the support of $|\xi|$, and the spectral projections of $|\xi|$ lie in $N$. An automorphism $\theta$ extends to the measurable operators, respecting sums, products, adjoints, polar decompositions and functional calculus. Its restriction to $L^2(N)$ is the unitary $U_\theta\hat x=\widehat{\theta(x)}$. For a projection $e\ne0$, the map $\xi\mapsto\tau(e)^{1/2}\xi$ identifies $eL^2(N)e$ with $L^2(N_e,\tau_e)$, compatibly with left and right multiplication by $N_e$. The Hilbert space $L^2(N,\tau)$, the two actions, the conjugation $J$ and the commutation theorem $N'=JNJ$ are proved in Integration for a trace, Sections 2–3 (Theorem 3.1). The identification of $L^2(N)$ with the square-integrable measurable operators, with left and right multiplication and $J\xi=\xi^*$, is Measurable operators for a trace, Fact 2.6, proved in Trace densities and noncommutative integration; polar decompositions of measurable operators are Fact 2.3(g) there. The extension of an automorphism, with $U_\theta$, is Theorem 14.1 there, and the corner identification is Proposition 14.2 there.

(B3) Tensor products. If $N_1,N_2$ are II₁ factors, $N_1\bar\otimes N_2$ is a II₁ factor with trace $\tau_1\otimes\tau_2$, with separable predual if both have one. $L^2(N_1\bar\otimes N_2)=L^2(N_1)\otimes L^2(N_2)$ with $\widehat{x\otimes y}=\hat x\otimes\hat y$ and $J=J_1\otimes J_2$, so $(N_1\bar\otimes N_2)'$ contains $N_1'\otimes1$. The algebraic tensor product is $\|\cdot\|_2$-dense, and $\|x\otimes y\|_2=\|x\|_2\|y\|_2$. For $\theta_j\in\operatorname{Aut}N_j$ there is a unique automorphism $\theta_1\otimes\theta_2$, and $U_{\theta_1\otimes\theta_2}=U_{\theta_1}\otimes U_{\theta_2}$. If $M$ is a finite von Neumann algebra with faithful normal tracial state, so is $M_2(M)=M\otimes M_2(\mathbb C)$, with trace $\tau\otimes\mathrm{tr}$. The product functional $\tau_1\otimes\tau_2$ is a faithful normal finite trace (Integration for a trace, Proposition 6.1), and the centre of $N_1\bar\otimes N_2$ is $Z(N_1)\bar\otimes Z(N_2)=\mathbb C1$ (Spatial tensor products of von Neumann algebras, Corollary 11.5(3)), so $N_1\bar\otimes N_2$ is a II₁ factor; the same corollary and Theorem 11.4 give the commutant. Since $\|x\otimes y\|_2^2=\tau_1(x^*x)\tau_2(y^*y)$, the map $\hat x\otimes\hat y\mapsto\widehat{x\otimes y}$ is isometric, and its range is dense by Kaplansky's density theorem and (B1); it carries $J_1\otimes J_2$ to $J$ on elementary tensors. The automorphism $\theta_1\otimes\theta_2$ is $\operatorname{Ad}(U_{\theta_1}\otimes U_{\theta_2})$, unique by density. For $M_2(M)$ see Traces on von Neumann algebras, Proposition 6.5.

(B4) Minimal tensor norm. For C*-algebras $A,B$ the minimal (spatial) C*-norm $\|\cdot\|_{\min}$ on the algebraic tensor product $A\odot B$ does not depend on the faithful representations used to compute it. Hence $\alpha\odot\mathrm{id}$ is isometric for every $*$-automorphism $\alpha$ of $A$. It is the smallest C*-norm on $A\odot B$. If $M\subset B(H)$ is a factor, the product map $M\odot M'\to B(H)$, $a\otimes b\mapsto ab$, is injective. The first two statements are Tensor products of C*-algebras and the minimal norm, Theorem 2.2 and Corollary 2.3, the minimality is Theorem 4.4 there, and the injectivity of the product map is Theorem 5.1 there.

(B5) Semidiscreteness. A von Neumann algebra $M$ is semidiscrete if there are completely positive contractions $M\xrightarrow{S_i}M_{n_i}\xrightarrow{T_i}M$, with $S_i$ normal, such that $T_iS_i(x)\to x$ σ-weakly for every $x\in M$. (Semidiscreteness was introduced in [Effros–Lance 1977] with normal completely positive maps of finite rank.) If $M\subset B(H)$ is semidiscrete, then $\|\sum_ia_ib_i\|\le\|\sum_ia_i\otimes b_i\|_{\min}$ for $a_i\in M$, $b_i\in M'$. The factor $R\bar\otimes B(\ell^2)$ is semidiscrete. The inequality is proved in Finite models of a von Neumann algebra, Theorem 2.1, where semidiscreteness is defined as above. And $R\bar\otimes B(\ell^2)$ is semidiscrete: the projections $1\otimes p_n$, with $p_n$ of rank $n$ increasing to $1$, cut it to corners $R\otimes M_n$, and a corner condition of this kind suffices (Hypertraces and finite injectivity, Lemma 6.2); $R\otimes M_n$ is the closure of an increasing union of matrix algebras $A_k$, and the trace-preserving conditional expectations onto $A_k$ (Integration for a trace, Theorem 9.1), followed by the inclusions, converge to the identity in $\|\cdot\|_2$ on the dense union, hence pointwise σ-weakly. See also [Effros–Lance 1977].

(B6) Infinite tensor products of matrix algebras. Let $I$ be $\mathbb N$ or $\mathbb Z$, and let $R_I=\bar\bigotimes_{n\in I}(M_2(\mathbb C),\mathrm{tr})$. Then $R_I$ is a II₁ factor with separable predual, and the $*$-algebra $A_{\mathrm{fin}}$ of finite tensor products is $\|\cdot\|_2$-dense in it. Every $\tau$-preserving $*$-automorphism of $A_{\mathrm{fin}}$ extends uniquely to an automorphism of $R_I$. We write $R=R_{\mathbb N}$. The II∞ factor $R\bar\otimes B(\ell^2)$ has, for every $\lambda>0$, an automorphism multiplying its trace by $\lambda$. The infinite tensor product and its trace are constructed in Infinite tensor products, Section 5 (Corollary 5.3). Density of $A_{\mathrm{fin}}$ in $\|\cdot\|_2$ follows from Kaplansky's density theorem and (B1), and a $\tau$-preserving automorphism of $A_{\mathrm{fin}}$ is a unitary on $L^2$ that extends by continuity and normalizes the generated algebra. The trace-scaling automorphisms rest on the fact that every nonzero corner of $R\bar\otimes M_n$ is isomorphic to $R$ (Hyperfinite finite factors, Corollary 5.2); see also [Anantharaman–Popa, §11.2].

(B7) Unitary operators. For a unitary $V$, $\|V-1\|=\max\{|\lambda-1|:\lambda\in\operatorname{Sp}V\}$. Every point of $\operatorname{Sp}V$ is an approximate eigenvalue. If $V^p=1$ with $p\ge1$, then $\operatorname{Sp}V\subset\mu_p$. The first statement is the spectral radius formula for the normal operator $V-1$ (C*-algebras: continuous functional calculus, automatic continuity, positive cones, approximate identities and quotients); the second is The spectral theorem for bounded self-adjoint operators, Proposition 8.2; the third is the spectral mapping theorem for the polynomial $z^p$ (Banach algebras, spectrum, holomorphic functional calculus and Gelfand theory, Theorem 6.10).

(B8) Central sequences and the asymptotic period. Let $N$ have separable predual, let $\theta\in\operatorname{Aut}N$, and let $\omega$ be a free ultrafilter; $\theta_\omega$ is the automorphism of $N_\omega$ induced by $\theta$ (Section 1).

(a) If $\theta\notin\operatorname{Ct}N$, then for every nonzero projection $e\in N_\omega$ and every $\varepsilon>0$ there is a nonzero projection $f\le e$ in $N_\omega$ with $\|f\theta_\omega(f)\|\le\varepsilon$. [Connes 1975, Proposition 2.1.2 and Theorem 1.2.1]
(b) If no power $\theta^k$ with $k\ge1$ lies in $\operatorname{Ct}N$, then for every unitary $u\in N_\omega$ there is a unitary $v\in N_\omega$ with $\theta_\omega(v)=uv$. [Connes 1975, Theorem 2.1.3]

These two facts enter only through Proposition 6.6.

1. What is used from other lessons

From "Ultraproducts and the asymptotic centralizer". Let $\omega$ be a free ultrafilter. The ultrapower $N^\omega$ is the C*-algebra of bounded sequences $(x_k)$ in $N$ modulo those with $\lim_{k\to\omega}\|x_k\|_2=0$ (Definitions 3.1 and 3.4 there). By Theorem 3.2 there, $N^\omega$ is a finite von Neumann algebra, and $\tau_\omega((x_k))=\lim_{k\to\omega}\tau(x_k)$ is a faithful normal tracial state on it. Constant sequences embed $N$ in $N^\omega$. The central sequence algebra is $N_\omega=N'\cap N^\omega$ (Definition 3.4 there). A bounded sequence $(x_k)$ is central if $\|[x_k,y]\|_2\to0$ for every $y\in N$. Each automorphism $\theta$ preserves $\tau$, hence $\|\cdot\|_2$, so it induces a $\tau_\omega$-preserving $*$-automorphism $\theta^\omega[(x_k)]=[(\theta(x_k))]$ of $N^\omega$. Since $\theta^\omega(N)=N$, it maps $N_\omega$ onto itself, and we put $\theta_\omega=\theta^\omega|_{N_\omega}$. By Proposition 8.7 there, $N_\omega$ is the asymptotic centralizer of $N$, and $\theta_\omega$ is the automorphism of Theorem 8.3 there. The ultraproduct of Hilbert spaces $H_\omega$ consists of the bounded sequences $(\xi_k)$ in a Hilbert space $H$ modulo those with $\lim_{k\to\omega}\|\xi_k\|=0$, with inner product $\lim_{k\to\omega}\langle\xi_k,\eta_k\rangle$. It is a Hilbert space (Section 4 there).

From "Full factors". The $u$-topology on $\operatorname{Aut}M$ is the topology in which $\alpha_i\to\alpha$ if and only if $\|\varphi\circ\alpha_i-\varphi\circ\alpha\|\to0$ for every $\varphi\in M_*$ (Definition 2.1 there). If $M$ has separable predual, $\operatorname{Aut}M$ is a Polish group in this topology (Proposition 2.3 there). In particular it is a metrizable topological group, so closures are sequential closures. For a II₁ factor $M$ with separable predual:

Lemma 6.3 there (asymptotic independence). If $(y_k)$ is a bounded central sequence and $x\in M$, then $\tau(xy_k)-\tau(x)\tau(y_k)\to0$.
Lemma 6.4 there. Every projection of $M_\omega$ is the class of a sequence of projections of $M$.
Corollary 5.2 there. $M$ is full (that is, $\operatorname{Int}M$ is closed in $\operatorname{Aut}M$) if and only if $M_\omega=\mathbb C$.

From "Trace inequalities for finite von Neumann algebras". That lesson works with a faithful normal semifinite trace; we use only the finite case. Let $P$ be a finite von Neumann algebra with a faithful normal tracial state $\tau$. For $\xi\in L^2(P)$ (a $\tau$-measurable operator, possibly unbounded) with polar decomposition $\xi=u|\xi|$ and for $a>0$, put $u_a(\xi)=u\,\chi_{]a,\infty[}(|\xi|),$ the truncated polar part of $\xi$ (Notation 1.2 there), a partial isometry of $P$ with initial projection $\chi_{]a,\infty[}(|\xi|)$.

(T1) Powers–Størmer inequality (Theorem 3.1 there). For positive $h,k\in L^2(P)$, in particular for $h,k\in P_+$, $\|h-k\|_2^2\le\|h^2-k^2\|_1$.
(T2) A common level for truncated polar parts (Theorem 7.1 and Corollary 7.2 there). For $\delta\in\,]0,1[$ and $n\ge1$ put $\varepsilon_T(\delta,n)=(\delta/6n)^8$. Let $\xi_1,\dots,\xi_n\in L^2(P)$ with $\xi_1\ne0$ and $\|\xi_j-\xi_1\|_2\le\varepsilon_T(\delta,n)\|\xi_1\|_2$ for all $j$ (for instance, if the set $\{\xi_1,\dots,\xi_n\}$ has diameter less than $\varepsilon_T(\delta,n)\|\xi_1\|_2$). Then there is $a>0$ with $\|u_a(\xi_j)-u_a(\xi_1)\|_2\le\delta\|u_a(\xi_1)\|_2$ for all $j$, and $\|\xi_1-u_a(\xi_1)|\xi_1|\|_2\le\delta\|\xi_1\|_2$. The vectors $\xi_j$ are arbitrary elements of $L^2(P)$, not only bounded ones.
(T3) Conjugating close projections (Theorem 9.4 and Corollary 9.5 there). If $e,f\in P$ are equivalent projections, there is a unitary $W\in P$ with $WeW^*=f$ and $|W-1|\le\sqrt2\,|e-f|$. In particular $\|W-1\|_1\le\sqrt2\,\|e-f\|_1$.

From "Property $\Gamma$ and the algebra generated by a factor and its commutant" (used only in Remark 5.5). For a II₁ factor $N$ acting on $L^2(N)$, $N$ has property $\Gamma$ if and only if the C*-algebra $C^*(N,N')$ generated by $N$ and $N'$ contains no nonzero compact operator (Theorem 2.5 there). A semidiscrete II₁ factor has property $\Gamma$ (Corollary 6.5 there).

2. Tools for automorphisms of a II₁ factor

For $\theta\in\operatorname{Aut}N$ and $\xi\in L^2(N)$ we write $\theta(\xi)=U_\theta\xi$ (B2). Then $\theta(a\xi b)=\theta(a)\theta(\xi)\theta(b)$ and $\theta(J\xi)=J\theta(\xi)$. For $x\in N$ put $[x,\xi]=x\xi-\xi x=(x-Jx^*J)\xi$. Right multiplication by a unitary is isometric on $L^2(N)$, so $\|u\xi-\xi u\|=\|u\xi u^*-\xi\|$.

2.1 The topology of the automorphism group

Lemma 2.1. Let $\theta_i$ be a net and $\theta$ an element of $\operatorname{Aut}N$. Then $\theta_i\to\theta$ in the $u$-topology if and only if $\|\theta_i(x)-\theta(x)\|_2\to0$ for every $x\in N$, and then also $\theta_i^{-1}\to\theta^{-1}$. Consequently, if $N$ has separable predual, $\theta\in\overline{\operatorname{Int}N}$ if and only if there are unitaries $u_n$ with $\|u_nx-\theta(x)u_n\|_2\to0$ for every $x\in N$.

Proof. For $h\in N$ let $\varphi_h=\tau(h\,\cdot\,)$. Since automorphisms preserve $\tau$, $\varphi_h(\theta_i(x))=\tau(\theta_i^{-1}(h)x)$. By (B1), $\|\varphi_h\circ\theta_i-\varphi_h\circ\theta\|=\|\theta_i^{-1}(h)-\theta^{-1}(h)\|_1 .$ Since $\|\varphi\circ\alpha\|=\|\varphi\|$ for every automorphism $\alpha$, and the $\varphi_h$ are dense in $N_*$, convergence in the $u$-topology is equivalent to $\|\theta_i^{-1}(h)-\theta^{-1}(h)\|_1\to0$ for all $h\in N$. The elements $\theta_i^{-1}(h)-\theta^{-1}(h)$ have norm at most $2\|h\|$, and $\|y\|_1\le\|y\|_2\le(\|y\|\,\|y\|_1)^{1/2}$, so this is equivalent to $\|\theta_i^{-1}(h)-\theta^{-1}(h)\|_2\to0$ for all $h$. Put $x=\theta^{-1}(h)$. Since $\theta_i$ preserves $\|\cdot\|_2$, $\|\theta_i^{-1}(h)-\theta^{-1}(h)\|_2=\|h-\theta_i(\theta^{-1}(h))\|_2=\|\theta(x)-\theta_i(x)\|_2 ,$ and $x$ runs through $N$ when $h$ does. So $u$-convergence of $\theta_i$ and of $\theta_i^{-1}$ are both equivalent to pointwise $\|\cdot\|_2$-convergence of $\theta_i$. The last assertion follows because $\operatorname{Aut}N$ is metrizable (Section 1) and $\|u_nxu_n^*-\theta(x)\|_2=\|u_nx-\theta(x)u_n\|_2$. $\square$

2.2 Matrix units and almost central vectors

Lemma 2.2. Let $\{e_{ij}\}$ and $\{f_{ij}\}$, $1\le i,j\le q$, be two systems of matrix units in $N$ with $\sum_ie_{ii}=\sum_if_{ii}=1$. There is a unitary $w\in N$ with $we_{ij}w^*=f_{ij}$ for all $i,j$.

Proof. Both $e_{11}$ and $f_{11}$ have trace $1/q$, so there is a partial isometry $v$ with $v^*v=e_{11}$ and $vv^*=f_{11}$ (B1). Put $w=\sum_if_{i1}ve_{1i}$. Then $w^*w=\sum_{i,k}e_{k1}v^*f_{1k}f_{i1}ve_{1i}=\sum_ie_{i1}v^*f_{11}ve_{1i}=\sum_ie_{ii}=1$, and in the same way $ww^*=1$. Moreover $we_{ij}=f_{i1}ve_{1j}$, so $we_{ij}w^*=f_{i1}ve_{11}v^*f_{1j}=f_{ij}$. $\square$

Lemma 2.3. (i) Let $e\in N$ be a projection with $0<\tau(e)<1$, let $q\ge1$ and $r=\lfloor q\tau(e)\rfloor$. There are matrix units $\{g_{ij}\}_{1\le i,j\le q}$ in $N$ with $\sum_ig_{ii}=1$ and $P_-=\sum_{i\le r}g_{ii}\ \le\ e\ \le\ P_+=\sum_{i\le r+1}g_{ii}.$ (ii) Let $\{g_{ij}\}_{1\le i,j\le q}$ be matrix units in $N$ with $\sum_ig_{ii}=1$, and let $\zeta\in L^2(N)$ and $\kappa\ge0$ satisfy $\|[g_{ij},\zeta]\|\le\kappa\|\zeta\|$ for all $i,j$. Then $\big|\|g_{ii}\zeta\|^2-\|\zeta\|^2/q\big|\le8\kappa\|\zeta\|^2$ for every $i$. Hence $\big|\|P\zeta\|^2-\tfrac{|I|}{q}\|\zeta\|^2\big|\le8q\kappa\|\zeta\|^2$ for $P=\sum_{i\in I}g_{ii}$.

In words: a vector that almost commutes with a matrix subalgebra spreads its mass evenly over the diagonal.

Proof. (i) Since $r/q\le\tau(e)$, we can choose orthogonal projections $g_1,\dots,g_r\le e$ of trace $1/q$ (B1). The remainder $e'=e-\sum_{i\le r}g_i$ has trace $\tau(e)-r/q\in[0,1/q[$. As $\tau(e)<1$ we have $r\le q-1$, so $\tau(1-e)=1-\tau(e)\ge1/q-\tau(e')$. Choose $h\le1-e$ with $\tau(h)=1/q-\tau(e')$, and put $g_{r+1}=e'+h$. Split $1-\sum_{i\le r+1}g_i$, which has trace $(q-r-1)/q$, into $q-r-1$ projections of trace $1/q$. All the $g_i$ have trace $1/q$, so there are partial isometries $v_i$ with $v_i^*v_i=g_1$ and $v_iv_i^*=g_i$ (with $v_1=g_1$). Then $g_{ij}=v_iv_j^*$ are matrix units with $g_{ii}=g_i$. By construction $P_-\le e=P_-+e'\le P_-+g_{r+1}=P_+$.

(ii) We may assume $\|\zeta\|=1$. Since $g_{1i}^*g_{1i}=g_{ii}$, we have $\|g_{1i}\zeta\|=\|g_{ii}\zeta\|$. Right multiplication by $g_{1i}$ satisfies $\|\zeta g_{1i}\|^2=\langle\zeta g_{1i}g_{1i}^*,\zeta\rangle=\|\zeta g_{11}\|^2$. Hence $\big|\|g_{ii}\zeta\|-\|g_{11}\zeta\|\big|\le\big|\|g_{1i}\zeta\|-\|\zeta g_{1i}\|\big|+\big|\|\zeta g_{11}\|-\|g_{11}\zeta\|\big|\le2\kappa .$ Put $a_i=\|g_{ii}\zeta\|$. Then $\sum_ia_i^2=\|\zeta\|^2=1$ and $|a_i-a_j|\le4\kappa$. If $m=\min a_i$ and $M=\max a_i$, then $qm^2\le1\le qM^2$, so $m\le q^{-1/2}\le M\le m+4\kappa$ and $|a_i-q^{-1/2}|\le4\kappa$ for every $i$. As $a_i+q^{-1/2}\le2$, we get $|a_i^2-1/q|\le8\kappa$. The last statement follows by summing over $i\in I$. $\square$

2.3 Restricting an automorphism to a corner

If $\alpha\in\operatorname{Aut}N$ and $\alpha(e)=e$, we write $\alpha^e$ for the restriction of $\alpha$ to $N_e$, an automorphism of $N_e$.

Definition 2.4. Automorphisms $\alpha$ of a II₁ factor $P$ and $\beta$ of a II₁ factor $P_1$ are outer conjugate if there are an isomorphism $\sigma:P\to P_1$ and a unitary $v\in P_1$ with $\beta=\operatorname{Ad}v\circ\sigma\alpha\sigma^{-1}$.

This is an equivalence relation. It is the same as conjugacy of the classes of $\alpha$ and $\beta$ in the outer automorphism groups.

Lemma 2.5. Let $Q$ be a II₁ factor, $\Phi\in\operatorname{Aut}Q$ and $p\in Q$ a nonzero projection.

(i) There is a unitary $W\in Q$ with $W\Phi(p)W^*=p$. If $p'$ is a projection with $\tau(p')=\tau(p)$ and $W'$ is a unitary with $W'\Phi(p')W'^*=p'$, then $(\operatorname{Ad}W\circ\Phi)^p$ and $(\operatorname{Ad}W'\circ\Phi)^{p'}$ are outer conjugate.

(ii) Let $v\in Q$ be a partial isometry with $vv^*=p$ and $v^*v=\Phi(p)$. Then ${}_v\Phi(y)=v\Phi(y)v^*$ defines an automorphism of $Q_p$, and ${}_v\Phi=\operatorname{Ad}s\circ(\operatorname{Ad}W\circ\Phi)^p$, where $s=vW^*$ is a unitary of $Q_p$.

Proof. (i) $\Phi(p)$ and $p$ have the same trace, so $W$ exists (B1). Choose a unitary $w$ with $wpw^*=p'$ and put $\sigma=\operatorname{Ad}w|_{Q_p}:Q_p\to Q_{p'}$. The automorphism $\beta=\operatorname{Ad}w\circ\operatorname{Ad}W\circ\Phi\circ\operatorname{Ad}w^*$ of $Q$ fixes $p'$, and its restriction to $Q_{p'}$ is $\sigma\circ(\operatorname{Ad}W\circ\Phi)^p\circ\sigma^{-1}$. Let $\gamma=\operatorname{Ad}W'\circ\Phi$. Since $\Phi\circ\operatorname{Ad}w^*\circ\Phi^{-1}=\operatorname{Ad}\Phi(w^*)$, we get $\beta\circ\gamma^{-1}=\operatorname{Ad}v_0$ with $v_0=wW\Phi(w^*)W'^*$. Now $v_0p'v_0^*=\beta(\gamma^{-1}(p'))=\beta(p')=p'$, so $v_0$ commutes with $p'$, and $v=v_0p'$ is a unitary of $Q_{p'}$. On $Q_{p'}$ we have $\beta=\operatorname{Ad}v\circ\gamma$, which is the claim.

(ii) For $y\in Q_p$, $\Phi(y)\in Q_{\Phi(p)}$, so ${}_v\Phi(y)\in Q_p$. Multiplicativity follows from $v^*v\Phi(y')=\Phi(y')$, and $y\mapsto\Phi^{-1}(v^*yv)$ is an inverse. Next, $ss^*=vv^*=p$ and $s^*s=Wv^*vW^*=W\Phi(p)W^*=p$, and $s=ps=sp$. So $s$ is a unitary of $Q_p$, and $v\Phi(y)v^*=s\,(W\Phi(y)W^*)\,s^*$. $\square$

2.4 An exhaustion principle

The two hardest proofs below glue together small pieces by a maximality argument. The abstract part is the following.

Lemma 2.6. Let $(\mathcal R,\le)$ be a partially ordered set and $t:\mathcal R\to[0,1]$ a map with $t(r)<t(r')$ whenever $r<r'$. Suppose that every increasing sequence $r_1\le r_2\le\cdots$ in $\mathcal R$ has an upper bound in $\mathcal R$. Then every $r_0\in\mathcal R$ lies below a maximal element of $\mathcal R$.

Proof. We apply Zorn's lemma to $\{r:r\ge r_0\}$. Let $C$ be a nonempty chain and $s=\sup t(C)$. For $c,c'\in C$, $t(c)<t(c')$ forces $c<c'$, since $c'\le c$ would give $t(c')\le t(c)$. If $s=t(c_0)$ with $c_0\in C$, then no $c\in C$ satisfies $c_0<c$, so $c_0$ is an upper bound of $C$. Otherwise choose $c_k\in C$ with $t(c_{k+1})>\max(t(c_k),s-1/(k+1))$. The $c_k$ increase, so they have an upper bound $b$. Every $c\in C$ has $t(c)<s$, hence $t(c)<t(c_k)$ for some $k$, hence $c<c_k\le b$. A maximal element above $r_0$ is maximal in $\mathcal R$. $\square$

2.5 Lifting unitaries

Lemma 2.7. Let $\omega$ be a free ultrafilter. Every unitary of $N^\omega$ is the class of a sequence of unitaries of $N$.

Proof. Let $(y_k)$ represent the unitary $u$. Write $y_k=w_k|y_k|$ with $w_k$ unitary (B1). For $t\ge0$, $|t-1|\le|t^2-1|$, so by functional calculus $\|y_k-w_k\|_2=\|w_k(|y_k|-1)\|_2\le\|y_k^*y_k-1\|_2 ,$ which tends to $0$ along $\omega$ because $u^*u=1$. So $(w_k)$ represents $u$. $\square$

2.6 Almost invariant projections

The next lemma turns almost invariant projections into a single almost invariant projection that almost contains them. It is the main use of (T1) and (T2). First, three properties of the maps $u_a$ of Section 1. Let $\xi\in L^2(N)$, $a>0$, $v,w\in\mathcal U(N)$ and $\alpha\in\operatorname{Aut}N$. Then $u_a(v\xi w)=v\,u_a(\xi)\,w,\qquad u_a(\alpha(\xi))=\alpha(u_a(\xi)),\qquad u_a(h)=\chi_{]a,\infty[}(h)\ \text{if }h\ge0. \tag{2.1}$ Indeed, if $\xi=u|\xi|$, then $v\xi w=(vuw)(w^*|\xi|w)$ is the polar decomposition of $v\xi w$, so $u_a(v\xi w)=vuw\,w^*\chi_{]a,\infty[}(|\xi|)w$. The second identity holds because $\alpha$ respects polar decompositions and functional calculus (B2). For the third, the partial isometry of $h\ge0$ is its support projection, which dominates $\chi_{]a,\infty[}(h)$. (The first two identities are also in Lemma 6.1 of "Trace inequalities for finite von Neumann algebras".)

Lemma 2.8. Let $\alpha_1,\dots,\alpha_m\in\operatorname{Aut}N$ and $\delta\in\,]0,1[$. Put $\eta(\delta,m)=\big(\varepsilon_T(\delta/2,m+1)/4\big)^2$. Let $e_1,e_2$ be projections with $\tau(e_1)=\tau(e_2)>0$ and $\|\alpha_k(e_s)-e_s\|_2\le\eta(\delta,m)\|e_s\|_2$ for all $k$ and $s=1,2$. Then there is a projection $e\le e_1\vee e_2$ with $\|ee_s-e_s\|_2\le\delta\|e_s\|_2\ (s=1,2),\qquad\|\alpha_k(e)-e\|_2\le\delta\|e\|_2\ (k=1,\dots,m).$ In particular $e\ne0$.

For inner $\alpha_k$ this is Corollary 8.3 of "Trace inequalities for finite von Neumann algebras", with other constants. Lemma 8.1 below needs it for an outer automorphism as well.

Proof. Write $\eta=\eta(\delta,m)$. For a projection $f$ and an automorphism $\alpha$, the operator $\alpha(f)-f$ equals $(\alpha(f)-f)(\alpha(f)\vee f)$, and $\tau(\alpha(f)\vee f)\le2\tau(f)$. So by (B1), $\|\alpha_k(e_s)-e_s\|_1\le\|\alpha_k(e_s)-e_s\|_2\sqrt2\|e_s\|_2\le\sqrt2\,\eta\,\tau(e_s)$.

Put $x_0=(e_1+e_2)^{1/2}$ and $x_k=\alpha_k(x_0)=(\alpha_k(e_1)+\alpha_k(e_2))^{1/2}$. Then $\|x_0\|_2^2=2\tau(e_1)$, and by (T1) $\|x_k-x_0\|_2^2\le\|\alpha_k(e_1+e_2)-(e_1+e_2)\|_1\le2\sqrt2\,\eta\,\tau(e_1)=\sqrt2\,\eta\,\|x_0\|_2^2 .$ So the set $\{x_0,\dots,x_m\}$ has diameter at most $2\cdot2^{1/4}\eta^{1/2}\|x_0\|_2<3\eta^{1/2}\|x_0\|_2$, and $3\eta^{1/2}<\varepsilon_T(\delta/2,m+1)$. By (T2) there is $a>0$ such that, with $e=u_a(x_0)$, $\|u_a(x_k)-e\|_2\le\tfrac\delta2\|e\|_2,\qquad\|x_0-ex_0\|_2\le\tfrac\delta2\|x_0\|_2 .$ By (2.1), $e=\chi_{]a,\infty[}(x_0)$ is a projection and $u_a(x_k)=\alpha_k(e)$, which gives the second estimate. Since $e$ is a spectral projection of $x_0$, $e\le\operatorname{supp}x_0=e_1\vee e_2$, and $1-e$ commutes with $x_0$. As $e_1\le x_0^2$, $\|ee_1-e_1\|_2^2=\tau((1-e)e_1(1-e))\le\tau((1-e)x_0^2(1-e))=\|(1-e)x_0\|_2^2\le\tfrac{\delta^2}4\cdot2\tau(e_1)\le\delta^2\|e_1\|_2^2 ,$ and in the same way for $e_2$. Finally $\|ee_1\|_2\ge(1-\delta)\|e_1\|_2>0$, so $e\ne0$. $\square$

3. Approximately inner automorphisms: the characterization

Let $N$ act on $H=L^2(N)$, so $N'=JNJ$. Let $C^*(N,N')$ be the C*-algebra generated by $N$ and $N'$.

Definition 3.1. An automorphism $\theta$ of $N$ is approximately inner if $\theta\in\overline{\operatorname{Int}N}$, the closure in the $u$-topology.

Theorem 3.2. Let $N$ be a II₁ factor with separable predual, and $\theta\in\operatorname{Aut}N$. The following are equivalent.

(a) $\theta$ is approximately inner.
(b) For all $a_1,\dots,a_k\in N$ and $b_1,\dots,b_k\in N'$, $\big\|\sum_i\theta(a_i)b_i\big\|=\big\|\sum_ia_ib_i\big\|$.
(c) For all unitaries $u_1,\dots,u_n\in N$ and $\varepsilon>0$ there is a unit vector $\xi\in L^2(N)$ with $\|\theta(u_j)\xi-\xi u_j\|<\varepsilon$ for all $j$.
(d) There is a bounded sequence $(x_k)$ in $N$ with $\|x_k\|_2\not\to0$ and $\|x_ka-\theta(a)x_k\|_2\to0$ for every $a\in N$.

Reference: [Connes 1976, Theorem 3.9.2].

The proof occupies this section and the next: (a)⇒(b)⇒(c) in Propositions 3.5 and 3.6, (c)⇒(d) in Proposition 4.3, and (d)⇒(a) in Proposition 4.4.

Remarks 3.3.

Condition (b) says exactly that there is an automorphism of $C^*(N,N')$ that equals $\theta$ on $N$ and the identity on $N'$. Indeed, an automorphism of a C*-algebra is isometric. Conversely, assume (b). Since $N$ and $N'$ commute, the elements $\sum_ia_ib_i$ form a $*$-algebra $\mathcal A_0$ dense in $C^*(N,N')$. By (b) the formula $\sum a_ib_i\mapsto\sum\theta(a_i)b_i$ is well defined on $\mathcal A_0$, and it is an isometric $*$-homomorphism. It extends to an isometric $*$-endomorphism of $C^*(N,N')$ whose range is closed and contains $N$ and $N'$, so it is an automorphism.
In (c), $\xi u_j=Ju_j^*J\xi$, so (c) says $\|\theta(u_j)Ju_jJ\xi-\xi\|<\varepsilon$. Since $\hat N$ is dense in $L^2(N)$, one may take $\xi=\hat x$ with $x\in N$ and $\|x\|_2=1$. The estimate then reads $\|\theta(u_j)x-xu_j\|_2<\varepsilon$. By (B1), (c) implies the same statement for arbitrary finite subsets of $N$ in place of unitaries.
In (d) one may take the $x_k$ unitary. Conversely, unitaries $u_k$ with $\|u_ka-\theta(a)u_k\|_2\to0$ for all $a$ give $\operatorname{Ad}u_k\to\theta$ (Lemma 2.1). So "(d) with unitaries" is just (a).
The implications (a)⇒(b)⇒(c) hold for every II₁ factor, with the same proofs.
The difficulty lies in (c)⇒(d). The vectors in (c) may be unbounded operators whose mass sits on projections of tiny trace, and truncating them does not preserve the approximate intertwining. Section 4 cuts them into pieces with Lemma 2.8 and glues the pieces with Lemma 2.6.

Example 3.4. If $\theta=\operatorname{Ad}w$ is inner, then (c) holds exactly with $\xi=\hat w$, since $\theta(u)w=wu$. If $N$ is full, then $\operatorname{Int}N$ is closed, and Theorem 3.2 characterizes the inner automorphisms of $N$ among all automorphisms by condition (b) or (c).

Proposition 3.5. (a)⇒(b), for every II₁ factor.

Proof. Let $\operatorname{Ad}u_i\to\theta$ (a net). By Lemma 2.1, $\|u_iau_i^*-\theta(a)\|_2\to0$ for $a\in N$. For $y\in N$, right multiplication commutes with left multiplication, so $(u_iau_i^*-\theta(a))\hat y=Jy^*J(u_iau_i^*-\theta(a))\hat1\to0$. The operators $u_iau_i^*-\theta(a)$ are bounded by $2\|a\|$, and $\hat N$ is dense, so $u_iau_i^*\to\theta(a)$ strongly on $H$. Multiplying by fixed $b_j\in N'$ and summing, $u_i(\sum_ja_jb_j)u_i^*=\sum_ju_ia_ju_i^*b_j\to\sum_j\theta(a_j)b_j$ strongly. The norm is lower semicontinuous for strong limits, so $\Big\|\sum_j\theta(a_j)b_j\Big\|\le\liminf_i\Big\|u_i\Big(\sum_ja_jb_j\Big)u_i^*\Big\|=\Big\|\sum_ja_jb_j\Big\| .$ By Lemma 2.1, $\operatorname{Ad}u_i^*\to\theta^{-1}$. The same argument applied to $\theta^{-1}$ and the elements $\theta(a_j)$ gives the reverse inequality. $\square$

Proposition 3.6. (b)⇒(c), for every II₁ factor.

Proof. Let $u_1,\dots,u_n$ be unitaries and $\varepsilon>0$. Consider $T=1+\sum_{j=1}^nu_jJu_jJ,\qquad S=1+\sum_{j=1}^n\theta(u_j)Ju_jJ .$ Here $Ju_jJ\in N'$, so (b) gives $\|S\|=\|T\|$. Each term of $T$ is unitary, so $\|T\|\le n+1$. Also $u_jJu_jJ\hat1=u_j\widehat{u_j^*}=\hat1$, so $T\hat1=(n+1)\hat1$ and $\|T\|=n+1$. Let $\eta>0$ and choose a unit vector $\xi$ with $\|S\xi\|\ge n+1-\eta$. Put $\zeta_0=\xi$ and $\zeta_j=\theta(u_j)Ju_jJ\xi$, all unit vectors. For $n+1$ unit vectors, $\sum_{j,k=0}^n\|\zeta_j-\zeta_k\|^2=2(n+1)^2-2\Big\|\sum_j\zeta_j\Big\|^2\le2(n+1)^2-2(n+1-\eta)^2\le4(n+1)\eta .$ The pairs $(j,0)$ and $(0,j)$ occur in the sum, so $\|\zeta_j-\xi\|^2\le2(n+1)\eta$. Since $\zeta_j-\xi=\theta(u_j)\xi u_j^*-\xi$ and right multiplication by $u_j$ is isometric, $\|\theta(u_j)\xi-\xi u_j\|\le(2(n+1)\eta)^{1/2}$, which is less than $\varepsilon$ for small $\eta$. $\square$

4. From almost intertwining vectors to approximate innerness

4.1 Local intertwiners

The first lemma extracts from an almost intertwining vector a bounded partial intertwiner living on a projection that almost commutes with the given unitaries.

Lemma 4.1. Let $N$ be a II₁ factor and let $\theta\in\operatorname{Aut}N$ satisfy (c) of Theorem 3.2. Let $u_1,\dots,u_n\in\mathcal U(N)$ and $\varepsilon>0$. There are a nonzero projection $e\in N$ and $x\in N$ with $\|x\|\le1,\quad xe=x,\quad\theta(e)x=x,\quad\|x\|_1\ge\tfrac14\tau(e),$ $\|[u_j,e]\|_1\le\varepsilon\tau(e),\qquad\|xu_j-\theta(u_j)x\|_1\le\varepsilon\tau(e)\qquad(j=1,\dots,n).$

Proof. Put $\delta_1=\min(1/8,\varepsilon/10)$, $\delta_2=\min(\delta_1,\eta(\delta_1,n)/2)$ with $\eta$ as in Lemma 2.8, and $\delta=\varepsilon_T(\delta_2,n+1)/4$.

Step 2: two almost invariant projections. Let $e_1=w^*w$ and $e_2=\theta^{-1}(ww^*)$. Both have trace $\|w\|_2^2$. Put $w_j'=\theta(u_j)wu_j^*$, so $\|w_j'-w\|_2\le\delta_2\|w\|_2$. Then $w_j'^*w_j'=u_je_1u_j^*$ and $w_j'w_j'^*=\theta(u_je_2u_j^*)$. Since $\|w\|,\|w'_j\|\le1$, $\|w_j'^*w_j'-w^*w\|_2\le\|w_j'^*(w_j'-w)\|_2+\|(w_j'-w)^*w\|_2\le2\delta_2\|w\|_2 ,$ and likewise for $w_j'w_j'^*-ww^*$. As $\theta$ preserves $\|\cdot\|_2$, we get $\|u_je_su_j^*-e_s\|_2\le2\delta_2\|e_s\|_2\le\eta(\delta_1,n)\|e_s\|_2$ for $s=1,2$.

Step 3: a common projection. By Lemma 2.8 with $\alpha_j=\operatorname{Ad}u_j$, there is a projection $e\le e_1\vee e_2$ with $\|ee_s-e_s\|_2\le\delta_1\|e_s\|_2$ and $\|u_jeu_j^*-e\|_2\le\delta_1\|e\|_2$. Note that $\tau(e)\le\tau(e_1\vee e_2)\le2\tau(e_1)$. Also $(1-\delta_1)\|e_1\|_2\le\|ee_1\|_2\le\|e\|_2$, so $\|w\|_2\le\frac43\|e\|_2$.

Step 4: the intertwiner. Put $x=\theta(e)we$. Then $\|x\|\le1$, $xe=x$ and $\theta(e)x=x$. Using $w=we_1=\theta(e_2)w$, $x-w=\theta(e)we_1(e-1)+(\theta(e)-1)\theta(e_2)w ,$ so $\|x-w\|_2\le\|e_1(e-1)\|_2+\|(e-1)e_2\|_2\le2\delta_1\|w\|_2$. Since $x=xe$ and $w=we_1$, the operator $x-w$ vanishes on the complement of $e\vee e_1\le e_1\vee e_2$, whose trace is at most $2\|w\|_2^2$. By (B1), $\|x-w\|_1\le2\delta_1\|w\|_2\cdot\sqrt2\|w\|_2=2\sqrt2\delta_1\tau(e_1)$. Hence, with $\|w\|_1=\tau(e_1)$ and $2\sqrt2\delta_1\le\frac12$, $\|x\|_1\ge(1-2\sqrt2\delta_1)\tau(e_1)\ge\tfrac12\tau(e_1)\ge\tfrac14\tau(e) .$ Next, $\|wu_j-\theta(u_j)w\|_2=\|w-\theta(u_j)wu_j^*\|_2\le\delta_2\|w\|_2$, so $\|xu_j-\theta(u_j)x\|_2\le2\|x-w\|_2+\delta_2\|w\|_2\le5\delta_1\|w\|_2\le\tfrac{20}3\delta_1\|e\|_2 .$ Since $xu_j=xu_j(u_j^*eu_j)$ and $\theta(u_j)x=\theta(u_j)xe$, the operator $xu_j-\theta(u_j)x$ vanishes on the complement of $e\vee u_j^*eu_j$, whose trace is at most $2\tau(e)$. So $\|xu_j-\theta(u_j)x\|_1\le\frac{20}3\sqrt2\,\delta_1\tau(e)<10\delta_1\tau(e)\le\varepsilon\tau(e)$. Finally $[u_j,e]=(u_jeu_j^*-e)u_j$, and $u_jeu_j^*-e$ vanishes on the complement of $e\vee u_jeu_j^*$, so $\|[u_j,e]\|_1\le\delta_1\|e\|_2\sqrt2\|e\|_2\le\varepsilon\tau(e)$. $\square$

Lemma 4.2. Let $N$ be a II₁ factor and let $\theta\in\operatorname{Aut}N$ satisfy (c) of Theorem 3.2. Let $f\in N$ be a nonzero projection and $v\in N$ a partial isometry with $vv^*=f$ and $v^*v=\theta(f)$. Then the automorphism ${}_v\theta:y\mapsto v\theta(y)v^*$ of $N_f$ satisfies (c) relative to the II₁ factor $N_f$.

Proof. Condition (c) is unchanged when $L^2$-norms are multiplied by a constant, so we may use $\tau$ on $N_f$ in place of $\tau_f$ and $fL^2(N)f$ in place of $L^2(N_f)$ (B2).

Step 1: inner perturbations. If $\alpha\in\operatorname{Aut}P$ satisfies (c) on a II₁ factor $P$ and $w\in\mathcal U(P)$, so does $\operatorname{Ad}w\circ\alpha$: if $\|\alpha(u)x-xu\|_2$ is small, then $\|w\alpha(u)w^*(wx)-(wx)u\|_2=\|\alpha(u)x-xu\|_2$ is small too, and $\|wx\|_2=\|x\|_2$.

Step 2: reduction. By Lemma 2.5(ii), ${}_v\theta=\operatorname{Ad}s\circ(\operatorname{Ad}W\circ\theta)^f$ for any unitary $W$ of $N$ with $W\theta(f)W^*=f$, and $s$ a unitary of $N_f$. By Step 1 (in $N_f$) it suffices to find one such $W$ for which $(\operatorname{Ad}W\circ\theta)^f$ satisfies (c). If $f=1$ this is Step 1 (in $N$). So assume $\tau(f)<1$.

Step 3: a convenient perturbation. Choose an integer $m$ with $1/m\le\tau(f)$, and let $\{g_{ij}\}_{i,j\le m}$ be matrix units as in Lemma 2.3(i) for $e=f$ and $q=m$, with $r=\lfloor m\tau(f)\rfloor\ge1$. Then $f=P_-+f'$ with $g_{11}\le P_-$ and $f'=f-P_-\le g_{r+1,r+1}$. By Lemma 2.2 there is a unitary $W_1$ with $W_1\theta(g_{ij})W_1^*=g_{ij}$, and $\theta_1=\operatorname{Ad}W_1\circ\theta$ fixes all $g_{ij}$. Then $\theta_1(f')\le g_{r+1,r+1}$ has the same trace as $f'$. In the II₁ factor $g_{r+1,r+1}Ng_{r+1,r+1}$ there is a unitary $W_2$ with $W_2\theta_1(f')W_2^*=f'$ (if $f'=0$ take $W_2=g_{r+1,r+1}$). Then $W_3=\sum_ig_{i,r+1}W_2g_{r+1,i}$ is a unitary of $N$ commuting with all $g_{kl}$, since $W_3g_{kl}=g_{k,r+1}W_2g_{r+1,l}=g_{kl}W_3$. It also satisfies $W_3\theta_1(f')W_3^*=W_2\theta_1(f')W_2^*=f'$. So $\theta_2=\operatorname{Ad}(W_3W_1)\circ\theta$ fixes every $g_{ij}$ and $f'$, hence $f$. By Step 1, $\theta_2$ satisfies (c) on $N$.

Step 4: (c) for $\theta_2^f$. Let $u_1,\dots,u_n$ be unitaries of $N_f$ and $\kappa\in\,]0,1/(64m^2)]$. Put $\bar u_j=u_j+1-f\in\mathcal U(N)$. By Remark 3.3(2) applied to the finite set $\{\bar u_j\}\cup\{g_{ij}\}$, there is $x\in N$ with $\|x\|_2=1$, $\|\theta_2(\bar u_j)x-x\bar u_j\|_2\le\kappa$ and $\|[g_{ij},x]\|_2\le\kappa$, because $\theta_2(g_{ij})=g_{ij}$. By Lemma 2.3(ii), $\|g_{11}x\|_2^2\ge1/m-8\kappa\ge1/(2m)$. Now $g_{11}x(1-g_{11})=-g_{11}[x,g_{11}](1-g_{11})$ has norm at most $\kappa$. Put $y=fxf\in N_f$. Since $g_{11}\le f$, $\|y\|_2\ge\|g_{11}yg_{11}\|_2=\|g_{11}xg_{11}\|_2\ge\|g_{11}x\|_2-\kappa\ge(2m)^{-1/2}-\kappa\ge(4m)^{-1/2} .$ Since $\theta_2(f)=f$, we have $f\bar u_j=\bar u_jf=u_j$ and $f\theta_2(\bar u_j)=\theta_2(\bar u_j)f=\theta_2(u_j)$. Hence $yu_j-\theta_2(u_j)y=f(x\bar u_j-\theta_2(\bar u_j)x)f$, and $\|yu_j-\theta_2(u_j)y\|_2\le\kappa\le2m^{1/2}\kappa\,\|y\|_2$. As $\kappa$ is arbitrary, $y/\|y\|_2$ gives (c) for $\theta_2^f$. $\square$

4.2 Gluing the local intertwiners

Proposition 4.3. (c)⇒(d) in Theorem 3.2.

Proof. Reduction. We show: for all $u_1,\dots,u_n\in\mathcal U(N)$ and $\delta>0$ there is $y\in N$ with $\|y\|\le1,\qquad\|y\|_1\ge\tfrac14,\qquad\|yu_j-\theta(u_j)y\|_1\le3\delta\quad(j\le n). \tag{4.1}$ This gives (d). Let $(w_m)$ be a sequence of unitaries, $\|\cdot\|_2$-dense in $\mathcal U(N)$ (B1). Let $y_m$ satisfy (4.1) for $w_1,\dots,w_m$ and $\delta=1/m$. Then $\|y_m\|_2\ge\|y_m\|_1\ge1/4$. Also $\|y_mw_j-\theta(w_j)y_m\|_2^2\le2\cdot3/m$ by (B1), so this tends to $0$ for each $j$. For an arbitrary unitary $u$ and any $j$, $\|y_mu-\theta(u)y_m\|_2\le2\|u-w_j\|_2+\|y_mw_j-\theta(w_j)y_m\|_2 ,$ so $\|y_mu-\theta(u)y_m\|_2\to0$. By (B1) the same holds for every $a\in N$.

The ordered set. Fix $u_1,\dots,u_n$ and $\delta$. For a projection $E$ and a partial isometry $V$ with $VV^*=E$ and $V^*V=\theta(E)$, write ${}_V\theta(y)=V\theta(y)V^*$ for $y\in N_E$. Let $\mathcal R$ be the set of tuples $r=(E,U_1,\dots,U_n,V,X)$ of elements of $N$ such that:

(R1) $E$ is a projection, and the $U_j$ are unitaries commuting with $E$;
(R2) $\|U_j-u_j\|_1\le\delta\tau(E)$;
(R3) $V$ is a partial isometry with $VV^*=E$ and $V^*V=\theta(E)$;
(R4) $X\in N_E$, $\|X\|\le1$, $\|X\|_1\ge\frac14\tau(E)$;
(R5) $\|XU_j-{}_V\theta(U_jE)X\|_1\le\delta\tau(E)$ for all $j$.

Write $r\le r'$ when $E\le E',\qquad\|U_j'-U_j\|_1\le\delta\tau(E'-E),\qquad EV'=V'\theta(E)=V,\qquad X'E=EX'=X .$ This relation is reflexive ($EV=V=V\theta(E)$ by (R3)). It is transitive: for instance, if $r\le r'\le r''$, then $EV''=EE'V''=EV'=V$ and $V''\theta(E)=V''\theta(E')\theta(E)=V'\theta(E)=V$. If $r\le r'$ and $E=E'$, then $U_j'=U_j$, $V'=EV'=V$ and $X'=X'E=X$. So $\le$ is a partial order, and $t(r)=\tau(E)$ is strictly increasing, because $\tau$ is faithful. The tuple $r_0=(0,u_1,\dots,u_n,0,0)$ belongs to $\mathcal R$.

Increasing sequences are bounded. Let $r^1\le r^2\le\cdots$ in $\mathcal R$, and let $E$ be the supremum of the $E^k$, so $\|E-E^k\|_2\to0$. For $k\le l$, $\|U_j^l-U_j^k\|_1\le\delta\tau(E^l-E^k)$, and $\|z\|_2^2\le2\|z\|_1$ for $\|z\|\le2$. So $(U_j^k)_k$ is $\|\cdot\|_2$-Cauchy and bounded, and it converges in $\|\cdot\|_2$ to some $U_j$ (B1). Products and adjoints of bounded $\|\cdot\|_2$-convergent sequences converge in $\|\cdot\|_2$, because $\|ab\|_2\le\|a\|\,\|b\|_2$ and $\|ab\|_2\le\|a\|_2\|b\|$. Hence $U_j$ is unitary and commutes with $E$. Next, $V^l-V^k=(E^l-E^k)V^l$ and $X^l-X^k=(E^l-E^k)X^l$, so both sequences are $\|\cdot\|_2$-Cauchy, with limits $V$ and $X$. Passing to the limit, $VV^*=E$, $V^*V=\theta(E)$, $X=EXE$, $\|X\|\le1$, and $E^kV=V\theta(E^k)=V^k,\qquad XE^k=E^kX=X^k .$ Then $\|X\|_1\ge\|E^kXE^k\|_1=\|X^k\|_1\ge\frac14\tau(E^k)$ for all $k$, so (R4) holds. Since $\|\cdot\|_1\le \|\cdot\|_2$, the quantities in (R2) and (R5) pass to the limit, as do the bounds $\|U_j^k-U_j\|_1\le\delta\tau(E-E^k)$. So $r=(E,U,V,X)\in\mathcal R$, and $r^k\le r$ for all $k$.

A maximal element is complete. By Lemma 2.6 there is a maximal $r=(E,U_1,\dots,U_n,V,X)$ in $\mathcal R$. Suppose $F=1-E\ne0$. Since $\tau(\theta(F))=\tau(F)$, there is a partial isometry $Y$ with $YY^*=F$ and $Y^*Y=\theta(F)$. The automorphism $\psi={}_Y\theta$ of $N_F$ satisfies (c) by Lemma 4.2. The elements $v_j=U_jF$ are unitaries of $N_F$ by (R1). Lemma 4.1, applied in the II₁ factor $N_F$ with $\varepsilon=\delta/9$, gives a nonzero projection $e\le F$ and $x\in N_F$ with $\|x\|\le1,\ \ xe=x,\ \ \psi(e)x=x,\ \ \|x\|_1\ge\tfrac14\tau(e),\ \ \|[v_j,e]\|_1\le\tfrac\delta9\tau(e),\ \ \|xv_j-\psi(v_j)x\|_1\le\tfrac\delta9\tau(e).$ All these conditions are homogeneous in the trace, so it does not matter whether they are read with $\tau$ or $\tau_F$. By (T3) in $N_F$ there are unitaries $W_j$ of $N_F$ with $W_jv_jev_j^*W_j^*=e$ and $\|W_j-F\|_1\le\sqrt2\,\|v_jev_j^*-e\|_1=\sqrt2\,\|[v_j,e]\|_1\le\frac\delta3\tau(e)$. The unitaries $v_j'=W_jv_j$ of $N_F$ commute with $e$ and satisfy $\|v_j'-v_j\|_1\le\frac\delta3\tau(e)$. Since $\psi(e)$ and $e$ have the same trace, choose a partial isometry $Y'\in N_F$ with $Y'^*Y'=\psi(e)$ and $Y'Y'^*=e$. Put $E'=E+e,\quad U_j'=U_jE+v_j',\quad V'=V+Y'Y,\quad X'=X+Y'x .$ We check that $r'=(E',U',V',X')\in\mathcal R$ and $r\le r'$. Since $e\ne0$, this contradicts maximality.

(R1): $E'$ is a projection since $e\le F$. $U_jE$ is a unitary of $N_E$ and $v_j'$ one of $N_F$, so $U_j'$ is unitary. It commutes with $E$ and with $e$.

(R2) and the order condition on $U$: $U_j'-U_j=v_j'-v_j$, so $\|U_j'-U_j\|_1\le\frac\delta3\tau(e)\le\delta\tau(E'-E)$ and $\|U_j'-u_j\|_1\le\delta\tau(E)+\delta\tau(e)=\delta\tau(E')$.

(R3): $(Y'Y)^*(Y'Y)=Y^*\psi(e)Y=Y^*Y\theta(e)Y^*Y=\theta(e)$, because $e\le F$ gives $\theta(e)\le\theta(F)$. Also $(Y'Y)(Y'Y)^*=Y'FY'^*=e$. The partial isometries $V$ and $Y'Y$ have orthogonal initial projections $\theta(E),\theta(e)$ and orthogonal final projections $E,e$, so $V'$ is a partial isometry with $V'V'^*=E'$ and $V'^*V'=\theta(E')$. Moreover $EV'=V+EeY'Y=V$ and $V'\theta(E)=V+Y'Y\theta(e)\theta(E)=V$.

(R4) and the order condition on $X$: $x=xe$ and $x=\psi(e)x=Y'^*Y'x$ give $Y'x=eY'xe\in N_e$. So $X'$ is the sum of the orthogonal blocks $X\in N_E$ and $Y'x\in N_e$. Hence $\|X'\|\le1$, $X'E=EX'=X$, and $\|X'\|_1=\|X\|_1+\|Y'x\|_1$. Since $|Y'x|^2=x^*Y'^*Y'x=x^*x$, we get $\|Y'x\|_1=\|x\|_1\ge\frac14\tau(e)$, so $\|X'\|_1\ge\frac14\tau(E')$.

(R5): We have $U_j'E'=U_jE+v_j'e$. First, $V'\theta(E)=V$ gives $V'\theta(U_jE)V'^*=V\theta(U_jE)V^*$. Since $V^*=V^*E$ and $EY'=0$, we have $V^*X'=V^*X$. Second, $V'\theta(e)=Y'Y$, so $V'\theta(v_j'e)V'^*=Y'Y\theta(v_j'e)Y^*Y'^*$. Since $Y'^*X=Y'^*eX=0$ and $Y'^*Y'x=x$, this gives $Y'Y\theta(v_j'e)Y^*Y'^*X'=Y'\psi(v_j'e)x=Y'\psi(v_j')x$. Together, ${}_{V'}\theta(U_j'E')X'={}_V\theta(U_jE)X+Y'\psi(v_j')x .$ On the other side, $Xv_j'=0$ and $xU_jE=xFU_jE=xU_jFE=0$, so $X'U_j'=XU_j+Y'xv_j'$. Therefore $X'U_j'-{}_{V'}\theta(U_j'E')X'=\big(XU_j-{}_V\theta(U_jE)X\big)+Y'\big(xv_j'-\psi(v_j')x\big).$ The first term has $\|\cdot\|_1\le\delta\tau(E)$ by (R5) for $r$, because $XU_j=XU_jE$. For the second, note that $\|\psi(z)\|_1=\|Y\theta(z)Y^*\|_1=\|z\|_1$ for $z\in N_F$. Hence $\|xv_j'-\psi(v_j')x\|_1\le\|x(v_j'-v_j)\|_1+\|xv_j-\psi(v_j)x\|_1+\|\psi(v_j-v_j')x\|_1\le\big(\tfrac13+\tfrac19+\tfrac13\big)\delta\tau(e) .$ So (R5) holds for $r'$.

Conclusion. A maximal element $r\ge r_0$ therefore has $E=1$. Then $V$ is a unitary, $\|X\|\le1$, $\|X\|_1\ge\frac14$, $\|U_j-u_j\|_1\le\delta$ and $\|XU_j-V\theta(U_j)V^*X\|_1\le\delta$. Put $y=V^*X$. Then $\|y\|\le1$ and $\|y\|_1=\|X\|_1\ge\frac14$. Also $yU_j-\theta(U_j)y=V^*(XU_j-V\theta(U_j)V^*X)$, and $\|yu_j-\theta(u_j)y\|_1\le\|y(u_j-U_j)\|_1+\|yU_j-\theta(U_j)y\|_1+\|\theta(U_j-u_j)y\|_1\le3\delta .$ This is (4.1). $\square$

4.3 From intertwining sequences to unitaries

Proposition 4.4. (d)⇒(a) in Theorem 3.2.

Proof. Fix a free ultrafilter $\omega$. Let $\mathcal Y=\{y\in N^\omega:\ yx=\theta(x)y\ \text{for all }x\in N\},$ the set of intertwiners. Here $N\subset N^\omega$ by constant sequences, and $\theta^\omega(N)=N$, so $\theta^\omega$ maps $N_\omega=N'\cap N^\omega$ onto itself. It also maps the centre $Z(N_\omega)$ onto itself and preserves $\tau_\omega$.

Step 1: intertwiners reach every corner. We claim: for every nonzero projection $a\in N_\omega$ there is $y\in\mathcal Y$ with $\theta^\omega(a)ya\ne0$.

Let $(x_k)$ be as in (d). Passing to a subsequence we may assume $\|x_k\|_2^2\ge\beta>0$ for all $k$; the subsequence still satisfies (d). Let $\alpha=\tau_\omega(a)>0$, and let $(a_n)$ be projections of $N$ representing $a$ (Lemma 6.4 of "Full factors"). The set $S=\{n:\tau(a_n)\ge\alpha/2\}$ belongs to $\omega$. The sequence $h_k=x_k^*x_k$ is bounded and central, since $[h_k,x]=x_k^*(x_kx-\theta(x)x_k)+(\theta(x^*)x_k-x_kx^*)^*x_k$ and both terms tend to $0$ in $\|\cdot\|_2$. Fix $n\in S$. By asymptotic independence (Lemma 6.3 of "Full factors"), $\tau(a_nh_k)-\tau(a_n)\tau(h_k)\to0$ as $k\to\infty$, so $\liminf_k\|x_ka_n\|_2^2=\liminf_k\tau(a_nh_k)\ge\alpha\beta/2$. Moreover $\|\theta(a_n)x_ka_n-x_ka_n\|_2=\|(\theta(a_n)x_k-x_ka_n)a_n\|_2\to0\qquad(k\to\infty).$ Choose $k_n\ge n$ with $\|\theta(a_n)x_{k_n}a_n\|_2\ge\frac12(\alpha\beta/2)^{1/2}$ when $n\in S$, and $k_n=n$ otherwise. As $k_n\to\infty$, $\|x_{k_n}x-\theta(x)x_{k_n}\|_2\to0$ for every $x\in N$. So the class $y$ of $(x_{k_n})_n$ lies in $\mathcal Y$, and $\|\theta^\omega(a)ya\|_2\ge\frac12(\alpha\beta/2)^{1/2}>0$.

Step 2: a two-by-two algebra. Let $Q=M_2(N^\omega)$, a finite von Neumann algebra with faithful normal tracial state $\tau_Q=\tau_\omega\otimes\mathrm{tr}$ (B3), with matrix units $e_{ij}$. Let $\sigma(x)=x\otimes e_{11}+\theta(x)\otimes e_{22}$ for $x\in N$, and let $P=\sigma(N)'\cap Q$, a von Neumann subalgebra of $Q$. A direct computation shows that a matrix $\big(\begin{smallmatrix}c&b\\y&d\end{smallmatrix}\big)$ lies in $P$ exactly when $c,d\in N_\omega$, $y\in\mathcal Y$ and $b^*\in\mathcal Y$. Here we use $\theta(N)'\cap N^\omega=N_\omega$. In particular $p_1=1\otimes e_{11}$ and $p_2=1\otimes e_{22}$ lie in $P$, $p_1Pp_1=N_\omega\otimes e_{11}$, $p_2Pp_2=N_\omega\otimes e_{22}$, and $y\otimes e_{21}\in P$ for $y\in\mathcal Y$.

Step 3: the centre of $P$. Let $z$ be a central projection of $P$. It commutes with $p_1$ and $p_2$, so $z=a\otimes e_{11}+b\otimes e_{22}$ with $a,b$ projections of $N_\omega$. Because $z$ commutes with $p_1Pp_1$ and $p_2Pp_2$, both $a$ and $b$ lie in $Z(N_\omega)$. Because $z$ commutes with $y\otimes e_{21}$, $by=ya\qquad(y\in\mathcal Y). \tag{4.2}$ We claim $b=\theta^\omega(a)$. The projections $a$ and $(\theta^\omega)^{-1}(1-b)$ are central in $N_\omega$, so $a'=a\,(\theta^\omega)^{-1}(1-b)$ is a projection with $a'\le a$ and $\theta^\omega(a')=\theta^\omega(a)(1-b)$. For $y\in\mathcal Y$, (4.2) gives $ya'=yaa'=bya'$, hence $\theta^\omega(a')ya'=\theta^\omega(a)(1-b)\,b\,ya'=0$. By Step 1, $a'=0$, that is, $\theta^\omega(a)\le b$. Applying this to the central projection $1-z$, whose corners are $1-a$ and $1-b$, we get $\theta^\omega(1-a)\le1-b$. Hence $b=\theta^\omega(a)$.

Step 4: equivalence of the corners. By the comparison theorem (B1) there is a central projection $z$ of $P$ with $zp_1\precsim zp_2$ and $(1-z)p_2\precsim(1-z)p_1$. By Step 3, $z=a\otimes e_{11}+\theta^\omega(a)\otimes e_{22}$, so $\tau_Q(zp_1)=\tfrac12\tau_\omega(a)=\tfrac12\tau_\omega(\theta^\omega(a))=\tau_Q(zp_2).$ If $zp_1\sim q\le zp_2$, then $\tau_Q(zp_2-q)=0$, so $q=zp_2$ by faithfulness. Thus $zp_1\sim zp_2$, and in the same way $(1-z)p_1\sim(1-z)p_2$. Adding the two partial isometries gives $p_1\sim p_2$ in $P$. A partial isometry $W\in P$ with $W^*W=p_1$ and $WW^*=p_2$ has the form $u\otimes e_{21}$ with $u\in\mathcal Y$, $u^*u=1$ and $uu^*=1$. So $\mathcal Y$ contains a unitary $u$.

Step 5: conclusion. By Lemma 2.7, $u$ is represented by unitaries $u_k\in N$, and $\lim_{k\to\omega}\|u_kx-\theta(x)u_k\|_2=0$ for every $x\in N$. Let $(x_i)$ be $\|\cdot\|_2$-dense in the unit ball of $N$. For each $m$, the set of $k$ with $\|u_kx_i-\theta(x_i)u_k\|_2<1/m$ for all $i\le m$ belongs to $\omega$, so it is nonempty. Choose $k_m$ in it. For $x$ in the unit ball and any $i$, $\|u_{k_m}x-\theta(x)u_{k_m}\|_2\le2\|x-x_i\|_2+\|u_{k_m}x_i-\theta(x_i)u_{k_m}\|_2$. Hence $\|u_{k_m}x-\theta(x)u_{k_m}\|_2\to0$ for every $x\in N$. By Lemma 2.1, $\theta\in\overline{\operatorname{Int}N}$. $\square$

Proof of Theorem 3.2. Propositions 3.5, 3.6, 4.3 and 4.4. $\square$

5. Consequences

Corollary 5.1. Every automorphism of a semidiscrete II₁ factor $N$ with separable predual is approximately inner.

Reference: [Connes 1976, §4.2].

Proof. Let $\theta\in\operatorname{Aut}N$, $a_i\in N$ and $b_i\in N'$. By (B4) the operator norm is a C*-norm on $N\odot N'$ (the product map is injective), so it dominates the minimal norm. By (B5) it is also dominated by the minimal norm. Hence $\|\sum a_ib_i\|=\|\sum a_i\otimes b_i\|_{\min}$ for all such sums. By (B4), $\theta\odot\mathrm{id}$ is isometric for the minimal norm, so $\Big\|\sum_i\theta(a_i)b_i\Big\|=\Big\|\sum_i\theta(a_i)\otimes b_i\Big\|_{\min}=\Big\|\sum_ia_i\otimes b_i\Big\|_{\min}=\Big\|\sum_ia_ib_i\Big\| .$ This is condition (b), and Theorem 3.2 applies. $\square$

For factors that are not finite, (a) and (b) are no longer equivalent.

Proposition 5.2. Let $M=R\bar\otimes B(\ell^2)$, a semidiscrete II∞ factor with trace $\mathrm{Tr}$, acting on a Hilbert space $H$. Every automorphism $\theta$ of $M$ satisfies $\|\sum\theta(a_i)b_i\|=\|\sum a_ib_i\|$ for $a_i\in M$, $b_i\in M'$. But an automorphism with $\mathrm{Tr}\circ\theta=\lambda\mathrm{Tr}$, $\lambda\ne1$, which exists by (B6), is not in the $u$-closure of $\operatorname{Int}M$.

Proof. The first statement is the computation in the proof of Corollary 5.1, which uses only (B4) and (B5). For the second, let $\alpha_i\to\alpha$ in the $u$-topology with $\mathrm{Tr}\circ\alpha_i=\mathrm{Tr}$ (for example $\alpha_i$ inner), and let $\mathrm{Tr}\circ\alpha=\lambda\mathrm{Tr}$. This is the only possibility, since $\mathrm{Tr}\circ\alpha$ is a trace and traces on a II∞ factor are proportional. Choose a projection $f$ with $\mathrm{Tr}(f)=1$, let $\psi=\mathrm{Tr}(f\,\cdot\,f)\in M_*^+$, and let $g=\alpha^{-1}(f)$, so $\mathrm{Tr}(g)=1/\lambda$. For $y\ge0$, $\mathrm{Tr}(fyf)\le\mathrm{Tr}(y)$, so $\psi(\alpha_i(g))\le\mathrm{Tr}(\alpha_i(g))=1/\lambda$. As $i$ grows, $\psi(\alpha_i(g))\to\psi(\alpha(g))= \mathrm{Tr}(f)=1$. Hence $1\le1/\lambda$. Since $\operatorname{Aut}M$ is a topological group (Section 1), $\alpha_i^{-1}\to\alpha^{-1}$. Since $\mathrm{Tr}\circ\alpha^{-1}=\lambda^{-1}\mathrm{Tr}$, the same argument gives $1\le\lambda$. So $\lambda=1$. $\square$

Example 5.3. For $R_I$ as in (B6) there is a direct proof that every automorphism is approximately inner. Let $A_1\subset A_2\subset\cdots$ be the matrix algebras of finite tensor products over increasing finite sets exhausting $I$. Then $\bigcup_mA_m=A_{\mathrm{fin}}$ is $\|\cdot\|_2$-dense. Let $\theta\in\operatorname{Aut}R_I$. The images under $\theta$ of the matrix units of $A_m$ form a system of matrix units with sum $1$, so by Lemma 2.2 there is a unitary $w_m$ with $\theta(a)=w_maw_m^*$ for all $a\in A_m$. For $x\in R_I$ and $a\in A_l$, $l\le m$, $\|w_mxw_m^*-\theta(x)\|_2\le\|w_m(x-a)w_m^*\|_2+\|\theta(a-x)\|_2=2\|x-a\|_2 ,$ so $\operatorname{Ad}w_m\to\theta$ by Lemma 2.1. The hyperfinite factor is semidiscrete, so this is a special case of Corollary 5.1.

Corollary 5.4. Let $N_1,N_2$ be II₁ factors with separable preduals and $\theta_j\in\operatorname{Aut}N_j$. Then $\theta_1\otimes\theta_2$ is approximately inner exactly when both $\theta_1$ and $\theta_2$ are.

Reference: [Connes 1976, §3.9.2], the corollary stated after Theorem 3.9.2.

Proof. Suppose $\operatorname{Ad}u_n\to\theta_1$ and $\operatorname{Ad}v_n\to\theta_2$. For $a\in N_1$ and $b\in N_2$, using $\|x\otimes y\|_2=\|x\|_2\|y\|_2\le\|x\|_2\|y\|$, $\|u_nau_n^*\otimes v_nbv_n^*-\theta_1(a)\otimes\theta_2(b)\|_2\le\|u_nau_n^*-\theta_1(a)\|_2\|b\|+\|a\|\,\|v_nbv_n^*-\theta_2(b)\|_2\to0 .$ By linearity $\operatorname{Ad}(u_n\otimes v_n)(x)\to(\theta_1\otimes\theta_2)(x)$ in $\|\cdot\|_2$ for $x$ in the algebraic tensor product. All the maps involved preserve $\|\cdot\|_2$, and the algebraic tensor product is dense (B3), so the convergence holds for all $x$. By Lemma 2.1, $\theta_1\otimes\theta_2$ is approximately inner.

Conversely, let $\theta_1\otimes\theta_2$ be approximately inner. $N_1\bar\otimes N_2$ acts standardly on $L^2(N_1)\otimes L^2(N_2)$, and $b\otimes1$ lies in its commutant for $b\in N_1'$ (B3). For $a_i\in N_1$ and $b_i\in N_1'$, condition (b) for $\theta_1\otimes\theta_2$ (Proposition 3.5) gives $\Big\|\Big(\sum_i\theta_1(a_i)b_i\Big)\otimes1\Big\|=\Big\|\sum_i(\theta_1\otimes\theta_2)(a_i\otimes1)(b_i\otimes1)\Big\|=\Big\|\Big(\sum_ia_ib_i\Big)\otimes1\Big\| .$ So $\theta_1$ satisfies (b), and Theorem 3.2 shows it is approximately inner. The same argument applies to $\theta_2$. $\square$

Remark 5.5. The identity $\|\sum a_ib_i\|=\|\sum a_i\otimes b_i\|_{\min}$ of Corollary 5.1 is also the heart of the proof, in the lesson "Property $\Gamma$ and the algebra generated by a factor and its commutant", that semidiscrete II₁ factors have property $\Gamma$. The opposite case is that of a full II₁ factor $N$ with separable predual. Then $\operatorname{Int}N$ is closed, so Theorem 3.2 says that an automorphism of $N$ is inner if and only if it satisfies (b). By Theorem 6.5 of "Full factors", $N$ does not have property $\Gamma$, so by the lesson just named $C^*(N,N')$ contains nonzero compact operators. Both facts express the rigidity of full factors.

6. Centrally trivial automorphisms

Definition 6.1. An automorphism $\theta$ of $N$ is centrally trivial if $\|\theta(y_k)-y_k\|_2\to0$ for every bounded central sequence $(y_k)$. Their set is denoted $\operatorname{Ct}N$.

Lemma 6.2. (i) $\operatorname{Ct}N$ is a normal subgroup of $\operatorname{Aut}N$ containing $\operatorname{Int}N$.

(ii) Let $\omega$ be a free ultrafilter. $\theta_\omega$ preserves $\tau_\omega$, so it defines a unitary $V_\theta$ of $L^2(N_\omega,\tau_\omega)$ with $V_\theta\hat y=\widehat{\theta_\omega(y)}$, and $V_{\theta\theta'}=V_\theta V_{\theta'}$. Moreover $\theta_\omega=\mathrm{id}$ if and only if $V_\theta=1$. If $V_\theta=1$, then $\theta\in\operatorname{Ct}N$. If $N$ has separable predual, the converse holds.

Proof. (i) If $\theta,\theta'\in\operatorname{Ct}N$ and $(y_k)$ is central, then $(\theta'(y_k))$ is $\|\cdot\|_2$-close to $(y_k)$, hence central, and $\theta\theta'(y_k)-y_k\to0$. Inverses: $\|\theta^{-1}(y_k)-y_k\|_2= \|y_k-\theta(y_k)\|_2$. Normality: an automorphism $\sigma$ maps central sequences to central sequences, and $\|\sigma\theta\sigma^{-1}(y_k)-y_k\|_2=\|\theta(\sigma^{-1}(y_k))-\sigma^{-1}(y_k)\|_2$. Inner automorphisms: $\|uy_ku^*-y_k\|_2=\|[u,y_k]\|_2\to0$.

(ii) The first assertions are clear, and $\theta_\omega=\mathrm{id}$ if and only if $V_\theta=1$ because $N_\omega$ is dense in $L^2(N_\omega)$. If $\theta\notin\operatorname{Ct}N$, there is a bounded central sequence $(y_k)$ with $\|\theta(y_k)-y_k\|_2\ge\beta>0$ along a subsequence. The subsequence is central, and its class $y\in N_\omega$ has $\|\theta_\omega(y)-y\|_2\ge\beta$, so $V_\theta\ne1$.

Now let $N$ have separable predual and $\theta\in\operatorname{Ct}N$. Suppose some $y\in N_\omega$, with representing sequence $(y_k)$, has $\beta=\lim_{k\to\omega}\|\theta(y_k)-y_k\|_2>0$. Let $(x_i)$ be $\|\cdot\|_2$-dense in the unit ball of $N$ (B1). For each $m$, the set of $k$ with $\|[y_k,x_i]\|_2<1/m$ for $i\le m$ and $\|\theta(y_k)-y_k\|_2>\beta/2$ belongs to $\omega$, so it is infinite. Choose $k_1<k_2<\cdots$ with $k_m$ in the $m$-th set. Since $\|[y,x]\|_2\le\|[y,x_i]\|_2+2\|y\|\,\|x-x_i\|_2$, the sequence $(y_{k_m})_m$ is central, and $\theta$ does not fix it asymptotically. This contradicts $\theta\in\operatorname{Ct}N$. $\square$

Definition 6.3. The asymptotic period $p_a(\theta)$ of $\theta\in\operatorname{Aut}N$ is the smallest integer $p\ge1$ with $\theta^p\in\operatorname{Ct}N$, or $0$ if there is none. It is the order of the class of $\theta$ in $\operatorname{Aut}N/\operatorname{Ct}N$, with order $0$ for infinite order.

If $N$ has separable predual and $p=p_a(\theta)\ge1$, then $V_\theta^p=V_{\theta^p}=1$ by Lemma 6.2(ii), so $\operatorname{Sp}V_\theta\subset\mu_p$ (B7). The converse inclusion rests on the two results on the asymptotic period recalled in (B8); it is Proposition 6.6 below. In the next two examples it can be checked by hand.

Example 6.4 (product type automorphisms). Let $p\ge2$, let $\zeta=e^{2\pi i/p}$, and let $d=\operatorname{diag}(1,\zeta)\in M_2(\mathbb C)$. On $R=\bar\bigotimes_{n\ge1}(M_2(\mathbb C),\mathrm{tr})$ let $\theta_\zeta$ be the automorphism acting as $\operatorname{Ad}d$ on every tensor factor. It exists by (B6), and it is approximately inner, as is every automorphism of $R$ (Example 5.3). Let $E_{12}^{(n)}$ be the matrix unit $E_{12}$ placed in the $n$-th factor. For $k\ge1$ put $y_n^{(k)}=2^{k/2}E_{12}^{(n)}E_{12}^{(n+1)}\cdots E_{12}^{(n+k-1)} .$ Then $\|y_n^{(k)}\|_2=1$. The sequence $(y_n^{(k)})_n$ is central, since it commutes with each finite tensor product for large $n$ and these are dense. Also $\theta_\zeta(y_n^{(k)})=\bar\zeta^ky_n^{(k)}$, since $dE_{12}d^*=\bar\zeta E_{12}$. So each $\bar\zeta^k$ is an eigenvalue of $V_{\theta_\zeta}$, with an eigenvector of norm one in $N_\omega$. Since $\theta_\zeta^p=\mathrm{id}$ and $\theta_\zeta^k(y_n^{(1)})-y_n^{(1)}=(\bar\zeta^k-1)y_n^{(1)}\ne0$ for $0<k<p$, we get $p_a(\theta_\zeta)=p$ and $\operatorname{Sp}V_{\theta_\zeta}=\mu_p$.

Example 6.5 (the shift). On $R_{\mathbb Z}$ let $\sigma$ be the shift of tensor factors, $n\mapsto n+1$ (B6). Let $s_n$ be $\operatorname{diag}(1,-1)$ in the $n$-th factor. Then $\tau(s_ns_m)=\delta_{nm}$ and $\sigma(s_n)=s_{n+1}$. Fix $z\in\mathbb T$ and $L\ge1$, and put $w_N=L^{-1/2}\sum_{j=0}^{L-1}z^{-j}s_{N+j}$. Then $\|w_N\|_2=1$, $\|w_N\|\le L^{1/2}$, $(w_N)_N$ is central, and $\sigma(w_N)-zw_N=zL^{-1/2}\big(z^{-L}s_{N+L}-s_N\big),\qquad\|\sigma(w_N)-zw_N\|_2=(2/L)^{1/2}.$ So every $z\in\mathbb T$ is an approximate eigenvalue of $V_\sigma$, and $\operatorname{Sp}V_\sigma=\mathbb T$. In particular no power $\sigma^k$, $k\ne0$, is centrally trivial (otherwise $V_\sigma^k=1$ by Lemma 6.2(ii)), so $p_a(\sigma)=0$.

In general the spectrum of $V_\theta$ is determined by the asymptotic period.

Proposition 6.6 (the spectrum of $V_\theta$). Let $N$ have separable predual, let $\theta\in\operatorname{Aut}N$ have asymptotic period $p$, and let $\omega$ be a free ultrafilter. Then $\operatorname{Sp}V_\theta=\mu_p$. More precisely, every $\lambda\in\mu_p$ is an eigenvalue of $V_\theta$, with an eigenvector in $N_\omega$.

Proof. The inclusion $\operatorname{Sp}V_\theta\subset\mu_p$ was shown above for $p\ge1$, and for $p=0$ it holds because $\mu_0=\mathbb T$. Let $\lambda\in\mu_p$.

Case $p=0$. By (B8)(b) with $u=\lambda1$ there is a unitary $v\in N_\omega$ with $\theta_\omega(v)=\lambda v$. Then $V_\theta\hat v=\lambda\hat v$ and $\hat v\ne0$.

Case $p\ge1$. For $0<q<p$ the automorphism $\theta^q$ is not centrally trivial, by the definition of $p$, and $(\theta^q)_\omega=\theta_\omega^q$. Apply (B8)(a) with $\varepsilon=1/(2p)$ successively: to $\theta$ and $e=1$, which gives a nonzero projection $f_1$; then to $\theta^2$ and $e=f_1$, which gives $f_2\le f_1$; and so on up to $\theta^{p-1}$. Let $f=f_{p-1}$, or $f=1$ if $p=1$. Then $f\ne0$, and for $0<q<p$ we have $f\le f_q$, so $\|f\theta_\omega^q(f)\|=\|f\,f_q\,\theta_\omega^q(f_q)\,\theta_\omega^q(f)\|\le\|f_q\theta_\omega^q(f_q)\|\le\tfrac1{2p}.$ Put $x=\sum_{q=0}^{p-1}\lambda^{-q}\theta_\omega^q(f)\in N_\omega$. Since $\theta^p\in\operatorname{Ct}N$, Lemma 6.2(ii) gives $\theta_\omega^p=\mathrm{id}$, and with $\lambda^p=1$, $\theta_\omega(x)=\sum_{q=0}^{p-1}\lambda^{-q}\theta_\omega^{q+1}(f)=\lambda\sum_{r=1}^{p}\lambda^{-r}\theta_\omega^{r}(f) =\lambda x ,$ because $\lambda^{-p}\theta_\omega^p(f)=f=\lambda^{0}\theta_\omega^{0}(f)$. Moreover $fxf=f+\sum_{q=1}^{p-1}\lambda^{-q}f\theta_\omega^q(f)f$, so $\|fxf-f\|\le\frac{p-1}{2p}<1=\|f\|$, and hence $x\ne0$. So $V_\theta\hat x=\lambda\hat x$ with $\hat x\ne0$, as $\tau_\omega$ is faithful. $\square$

7. The invariant c(θ)

Definition 7.1. For $\theta\in\operatorname{Aut}N$ let $S(\theta)$ be the set of $c\ge0$ with the following property: for every finite set $F\subset N$ and every $\varepsilon>0$ there is a unit vector $\xi\in L^2(N)$ with $\|[x,\xi]\|\le\varepsilon\ \ (x\in F)\qquad\text{and}\qquad\|\theta(\xi)-\xi\|\ge c .$ Put $c(\theta)=\sup S(\theta)$.

Recall $[x,\xi]=(x-Jx^*J)\xi$, so $\xi$ is required to be almost central. Taking $\xi=\hat1$ shows $0\in S(\theta)$.

Proposition 7.2. (i) If $c\in S(\theta)$ and $0\le c'\le c$, then $c'\in S(\theta)$. Hence $[0,c(\theta)[\,\subset S(\theta)\subset[0,c(\theta)]$, and $0\le c(\theta)\le2$.

(ii) The number $c(\theta)$ does not change if, in Definition 7.1, $F$ is restricted to finite sets of unitaries and $\xi$ to vectors $\hat x/\|x\|_2$ with $x\in N$.

Proof. (i) is clear, and $\|\theta(\xi)-\xi\|\le2$. (ii) By (B1) each $x\in F$ is a combination $\sum_{i\le4}\lambda_iw_i$ of unitaries with $|\lambda_i|\le\|x\|$, so $\|[x,\xi]\|\le4\|x\|\max_i\|[w_i,\xi]\|$. If $\|\xi-\hat x\|\le\eta$, then $\|[a,\xi]-[a,\hat x]\|\le2\|a\|\eta$ and $\big|\|\theta(\xi)-\xi\|-\|\theta(\hat x)-\hat x\|\big|\le2\eta$. Since $\hat N$ is dense, a small loss in $c$ is absorbed by (i). $\square$

The vectors $\hat x$ in (ii) come with no bound on $\|x\|$. Bounded almost central elements are the same as elements of $N_\omega$, and this gives a lower bound.

Proposition 7.3. For every free ultrafilter $\omega$, $c(\theta)\ge\|V_\theta-1\|$. In particular, if $\theta\notin\operatorname{Ct}N$ then $c(\theta)>0$.

Proposition 7.4. Let $\theta\in\operatorname{Aut}N$.

(a) If $\theta'$ is outer conjugate to $\theta$, then $c(\theta')=c(\theta)$.

(b) If $M$ is a finite factor (a matrix algebra or a II₁ factor), then $c(\theta\otimes\mathrm{id}_M)\le c(\theta)$. If $M=M_q(\mathbb C)$, then $c(\theta\otimes\mathrm{id}_M)=c(\theta)$.

Reference: Connes; the asymptotic period is [Connes 1976, Definition 3.4.1].

Proof. (a) An isomorphism $\sigma:N\to N_1$ preserves traces, so it induces a unitary $W:L^2(N)\to L^2(N_1)$ with $W(a\xi b)=\sigma(a)W(\xi)\sigma(b)$ and $WU_\theta W^*=U_{\sigma\theta\sigma^{-1}}$. Finite subsets correspond under $\sigma$, so $c(\sigma\theta\sigma^{-1})=c(\theta)$. For $\theta'=\operatorname{Ad}u\circ\theta$ we have $\theta'(\xi)=u\theta(\xi)u^*$, and $\|u\theta(\xi)u^*-\theta(\xi)\|=\|[u,\theta(\xi)]\|=\|[\theta^{-1}(u),\xi]\| .$ If $\xi$ is admissible for $c$ and $F\cup\{\theta^{-1}(u)\}$ with tolerance $\varepsilon$, then $\|\theta'(\xi)-\xi\|\ge c-\varepsilon$. With (i) of Proposition 7.2 this gives $c(\theta')\ge c(\theta)$, and by symmetry equality.

(b) Write $K=L^2(M)$, so $L^2(N\bar\otimes M)=L^2(N)\otimes K$, $U_{\theta\otimes\mathrm{id}}=U_\theta\otimes1$, and $[x\otimes1,\xi]=(T_x\otimes1)\xi$ with $T_x=x-Jx^*J$ (B3). We may assume $c(\theta\otimes\mathrm{id})>0$. Let $0<c<c(\theta\otimes\mathrm{id})$, $\eta\in\,]0,1[$, $x_1,\dots,x_n\in N$ and $\varepsilon>0$. Choose $\delta>0$ with $\delta<\varepsilon^2$ and $n\delta<\frac14c^2(1-\eta^2)$. Take a unit vector $\xi\in L^2(N)\otimes K$ with $\|(T_{x_j}\otimes1)\xi\|\le\delta$ and $\|(U_\theta\otimes1)\xi-\xi\|\ge c$. Expand $\xi=\sum_b\xi_b\otimes b$ along an orthonormal basis $\mathcal B$ of $K$, and put $\mu(b)=\|\xi_b\|^2$, a probability on $\mathcal B$. Let $E_j=\{b:\|T_{x_j}\xi_b\|^2\ge\delta\|\xi_b\|^2\},\qquad G=\{b:\|\theta(\xi_b)-\xi_b\|^2\ge\eta^2c^2\|\xi_b\|^2\}.$ Then $\mu(E_j)\le\delta^{-1}\sum_b\|T_{x_j}\xi_b\|^2=\delta^{-1}\|(T_{x_j}\otimes1)\xi\|^2\le\delta$. Also $c^2\le\sum_b\|\theta(\xi_b)-\xi_b\|^2\le4\mu(G)+\eta^2c^2 ,$ so $\mu(G)\ge\frac14c^2(1-\eta^2)>\sum_j\mu(E_j)$. Hence there is $b\in G\setminus\bigcup_jE_j$ with $\xi_b\ne0$. The unit vector $\xi_b/\|\xi_b\|$ satisfies $\|[x_j,\cdot]\|<\delta^{1/2}<\varepsilon$ and $\|\theta(\cdot)-\cdot\|\ge\eta c$. So $\eta c\in S(\theta)$ for all $\eta<1$ and $c<c(\theta\otimes\mathrm{id})$, which proves the inequality.

Let $M=M_q(\mathbb C)$ with normalized trace and matrix units $E_{kl}$. Every element of $N\bar\otimes M$ is $z=\sum_{k,l}a_{kl}\otimes E_{kl}$ with $a_{kl}\in N$. For $\xi\in L^2(N)$, $[z,\xi\otimes\hat1]=\sum_{k,l}[a_{kl},\xi]\otimes\hat E_{kl}$, whose norm is at most $\sum_{k,l}\|[a_{kl},\xi]\|$. Also $(U_\theta\otimes1)(\xi\otimes\hat1)-\xi\otimes\hat1=(\theta(\xi)-\xi)\otimes\hat1$, and $\|\hat1\|=1$. So every $c\in S(\theta)$ belongs to $S(\theta\otimes\mathrm{id}_M)$.

$c(\theta^e)\ge c(\theta)$. We may assume $c(\theta)>0$. Let $0<c<c_1<c(\theta)$. Choose $q>2/\alpha$ with $c_1\big((\alpha-1/q)/(\alpha+1/q)\big)^{1/2}>c$. Let $\{g_{ij}\}$, $P_\pm$ be as in Lemma 2.3(i). Let $x_1,\dots,x_n\in N_e$, $\varepsilon>0$, and $\kappa_0>0$ small, to be fixed. Since $c_1\in S(\theta)$, there is a unit vector $\xi$ with $\|\theta(\xi)-\xi\|\ge c_1$ and $\|[a,\xi]\|\le\kappa_0$ for all $a$ in the finite set $\{g_{ij},\theta^{-1}(g_{ij}),e,x_j,\theta^{-1}(x_j)\}$. Put $\zeta=\theta(\xi)-\xi$, so $c_1\le\|\zeta\|\le2$. Since $[a,\theta(\xi)]=\theta([\theta^{-1}(a),\xi])$ and $\theta^{-1}(e)=e$, we have $\|[g_{ij},\zeta]\|\le2\kappa_0\le (2\kappa_0/c_1)\|\zeta\|$ and $\|[e,\zeta]\|\le2\kappa_0$. Put $\eta=e\xi e$. Then $\theta(\eta)-\eta=e\zeta e$ and $[x_j,\eta]=e[x_j,\xi]e$, so $\|[x_j,\eta]\|\le\kappa_0$. For any $\chi$, $\|e\chi e-e\chi\|=\|e[e,\chi](1-e)\|\le \|[e,\chi]\|$. Since $P_-\le e\le P_+$, we have $\|P_-\chi\|\le\|e\chi\|\le\|P_+\chi\|$. Using Lemma 2.3(ii), $\tau(P_-)\ge\alpha-1/q$ and $\tau(P_+)\le\alpha+1/q$, we obtain $\|\eta\|\le\big(\alpha+\tfrac1q+8q\kappa_0\big)^{1/2},\qquad\|\eta\|\ge\big(\alpha-\tfrac1q-8q\kappa_0\big)^{1/2}-\kappa_0,$ $\|\theta(\eta)-\eta\|\ge\big(\alpha-\tfrac1q-16q\kappa_0/c_1\big)^{1/2}c_1-2\kappa_0 .$ As $\kappa_0\to0$, the ratio $\|\theta(\eta)-\eta\|/\|\eta\|$ has lower limit at least $c_1((\alpha-1/q)/(\alpha+1/q))^{1/2}>c$, while $\|\eta\|$ stays at least $(\alpha/4)^{1/2}$. Choose $\kappa_0$ so that the ratio exceeds $c$, $\|\eta\|\ge(\alpha/4)^{1/2}$ and $\kappa_0\le\varepsilon(\alpha/4)^{1/2}$. By (B2) the vector $\eta/\|\eta\|$, read in $L^2(N_e)$, is a unit vector with $\|[x_j,\cdot]\|\le\varepsilon$ and $\|\theta^e(\cdot)-\cdot\|\ge c$. Hence $c\in S(\theta^e)$, and $c(\theta^e)\ge c(\theta)$.

$c(\theta)\ge c(\theta^e)$. Choose an integer $q\ge1/\alpha$, let $Q=N\bar\otimes M_q(\mathbb C)$, a II₁ factor with normalized trace $\tau_Q$, and let $\Theta=\theta\otimes\mathrm{id}$. The projections $p_1=1\otimes E_{11}$ and $p_2=e\otimes1$ are $\Theta$-invariant, with $\tau_Q(p_1)=1/q\le\alpha=\tau_Q(p_2)$. Under $x\mapsto x\otimes E_{11}$, $\Theta^{p_1}$ is conjugate to $\theta$. Also $Q_{p_2}=N_e\bar\otimes M_q(\mathbb C)$ and $\Theta^{p_2}=\theta^e\otimes\mathrm{id}$. By (a) and (b), $c(\Theta^{p_1})=c(\theta),\qquad c(\Theta^{p_2})=c(\theta^e).$ Choose $p_3\le p_2$ with $\tau_Q(p_3)=1/q$, and a unitary $w$ of $Q_{p_2}$ with $w\Theta(p_3)w^*=p_3$ (B1). By the first half of the proof, applied in $Q_{p_2}$, and by (a), $c\big((\operatorname{Ad}w\circ\Theta^{p_2})^{p_3}\big)\ge c(\operatorname{Ad}w\circ\Theta^{p_2})=c(\theta^e).$ The unitary $W=w+1-p_2$ of $Q$ satisfies $(\operatorname{Ad}W\circ\Theta)^{p_3}=(\operatorname{Ad}w\circ \Theta^{p_2})^{p_3}$. By Lemma 2.5(i), applied to $\Phi=\Theta$ and the equivalent projections $p_1,p_3$, this automorphism is outer conjugate to $\Theta^{p_1}$. By (a), $c(\theta)=c(\Theta^{p_1})\ge c(\theta^e)$. $\square$

8. Computing c(θ)

8.1 Local almost eigenvectors

Lemma 8.1. Let $N$ be a II₁ factor, $\theta\in\operatorname{Aut}N$, $0\le c<c(\theta)$, $u_1,\dots,u_n\in\mathcal U(N)$ and $\varepsilon>0$. There are a nonzero projection $e\in N$ and $x\in N_e$ with:

$\|[u_j,e]\|_1\le\varepsilon\tau(e)$ for all $j$, and $\|\theta(e)-e\|_1\le\varepsilon\tau(e)$;
$\|x\|\le1$ and $\|x\|_2\ge\frac14\|e\|_2$, hence $\|x\|_1\ge\frac1{16}\tau(e)$;
$\|\theta(x)-x\|_2\ge c\|x\|_2$;
$\|[x,u_j]\|_1\le\varepsilon\tau(e)$ for all $j$.

Proof. Step 1: an approximate eigenvalue. Fix $c_1$ with $c<c_1<c(\theta)$, so $c_1\in S(\theta)$. Let $(v_q)_{q\ge1}$ enumerate the countable set $\{\theta^k(u_j):k\in\mathbb Z,\ j\le n\}$, which is invariant under $\theta$ and $\theta^{-1}$. For each $k$ choose a unit vector $\xi_k$ with $\|[v_q,\xi_k]\|\le1/k$ for $q\le k$ and $\|\theta(\xi_k)-\xi_k\|\ge c_1$. Fix a free ultrafilter $\omega$, and in the ultraproduct $H_\omega$ of $L^2(N)$ (Section 1) let $\mathcal K=\{[(\eta_k)]:\ \lim_{k\to\omega}\|[v_q,\eta_k]\|=0\ \text{for all }q\}.$ This is well defined on classes, since $\|[v,\eta]\|\le2\|\eta\|$, and it is a closed subspace. The unitary $U[(\eta_k)]=[(\theta(\eta_k))]$ of $H_\omega$ maps $\mathcal K$ onto itself, because $\|[v_q,\theta(\eta)]\|=\|[\theta^{-1}(v_q),\eta]\|$ and the family $(v_q)$ is $\theta^{\pm1}$-invariant. Let $U_0=U|_{\mathcal K}$. The class of $(\xi_k)$ is a unit vector of $\mathcal K$ moved by at least $c_1$, so $\|U_0-1\|\ge c_1$. By (B7) there is $\lambda\in\operatorname{Sp}U_0$ with $|\lambda-1|\ge c_1$, and $\lambda$ is an approximate eigenvalue of $U_0$.

Step 2: parameters. Put $\delta_1=\min\big(\frac14,\frac\varepsilon{15},\frac{c_1-c}{10}\big)$, $\delta_2=\min\big(\delta_1,\eta(\delta_1,n+1)/2\big)$ and $\delta=\min\big(\frac12,\varepsilon_T(\delta_2,n+2)/16\big)$.

Step 5: an almost invariant projection. Let $e_1=v^*v$ and $e_2=vv^*$, of equal trace $\|v\|_2^2$. As in Step 2 of Lemma 4.1, applied to the partial isometries $u_jvu_j^*$ and $\bar\lambda\theta(v)$, each close to $v$, we get $\|\alpha(e_s)-e_s\|_2\le2\delta_2\|e_s\|_2\le\eta(\delta_1,n+1)\|e_s\|_2$ for $\alpha\in\{\operatorname{Ad}u_1,\dots,\operatorname{Ad}u_n,\theta\}$. Note that $\theta(e_1)=(\bar\lambda\theta(v))^*(\bar\lambda\theta(v))$. By Lemma 2.8 there is a projection $e\le e_1\vee e_2$ with $\|ee_s-e_s\|_2\le\delta_1\|e_s\|_2$, $\|u_jeu_j^*-e\|_2\le\delta_1\|e\|_2$ and $\|\theta(e)-e\|_2\le\delta_1\|e\|_2$. As in Lemma 4.1, $\|u_jeu_j^*-e\|_1\le\sqrt2\delta_1\tau(e)$ and similarly for $\theta$, which gives 1.

Step 6: the element. Put $x=eve\in N_e$, so $\|x\|\le1$. Since $v=ve_1=e_2v$, $x-v=eve_1(e-1)+(e-1)e_2v$, so $\|x-v\|_2\le2\delta_1\|v\|_2$. Hence $\|x\|_2\ge(1-2\delta_1)\|v\|_2\ge\frac12\|v\|_2$. Also $\tau(e)\le\tau(e_1\vee e_2)\le2\|v\|_2^2$, so $\|x\|_2\ge\frac1{2\sqrt2}\|e\|_2$, and $\|x\|_1\ge\|x\|_2^2\ge\frac1{16}\tau(e)$. This gives 2. For 3, $\theta(x)-\lambda x=\theta(x-v)-\lambda(x-v)+(\theta(v)-\lambda v),$ so $\|\theta(x)-\lambda x\|_2\le(4\delta_1+\delta_2)\|v\|_2\le10\delta_1\|x\|_2$, and $\|\theta(x)-x\|_2\ge(|\lambda-1|-10\delta_1)\|x\|_2\ge(c_1-10\delta_1)\|x\|_2\ge c\|x\|_2$. For 4, $\|[x,u_j]\|_2\le2\|x-v\|_2+\|u_jvu_j^*-v\|_2\le5\delta_1\|v\|_2\le\frac{20}3\delta_1\|e\|_2$. Here $\|v\|_2=\|e_1\|_2\le\|ee_1\|_2/(1-\delta_1)\le\frac43\|e\|_2$. The operator $[x,u_j]$ vanishes on the complement of $e\vee u_j^*eu_j$, so $\|[x,u_j]\|_1\le\frac{20}3\sqrt2\,\delta_1\tau(e)<10\delta_1\tau(e)\le\varepsilon\tau(e)$. $\square$

8.2 The main formula

Theorem 8.2. Let $N$ be a II₁ factor with separable predual, $\theta\in\operatorname{Aut}N$, and $\omega$ a free ultrafilter. Then $c(\theta)=\|V_\theta-1\|=\max\{|\lambda-1|:\lambda\in\operatorname{Sp}V_\theta\}.$ In particular $\|V_\theta-1\|$ does not depend on $\omega$, and $c(\theta)=0$ if and only if $\theta\in\operatorname{Ct}N$.

Proof. By Proposition 7.3 and (B7) it remains to show $\|V_\theta-1\|\ge c$ for every $c$ with $0<c<c(\theta)$. The last statement then follows from Lemma 6.2(ii).

Reduction. We show: for $u_1,\dots,u_n\in\mathcal U(N)$ and $\delta>0$ there are unitaries $U_j$ with $\|U_j-u_j\|_1\le\delta$, an element $X$ with $\|X\|\le1$, $\|X\|_1\ge\frac1{16}$ and $\|[X,U_j]\|_1\le\delta$, and a unitary $P$ with $\|P-1\|_1\le\delta$ and $\|P\theta(X)P^*-X\|_2\ge c\|X\|_2$. This suffices. Indeed, then $\|[X,u_j]\|_1\le\delta+2\|U_j-u_j\|_1\le3\delta$. Moreover $\|P\theta(X)P^*-\theta(X)\|_2\le2\|P-1\|_2\le2(2\delta)^{1/2}$ by (B1), and $\|X\|_2\ge\|X\|_1\ge\frac1{16}$, so $\|\theta(X)-X\|_2\ge(c-32(2\delta)^{1/2})\|X\|_2$. Take a $\|\cdot\|_2$-dense sequence $(w_m)$ in $\mathcal U(N)$, and let $X_m$ be obtained for $w_1,\dots,w_m$ and $\delta=1/m^2$. As in the proof of Proposition 4.3, $(X_m)$ is a bounded central sequence. Its class $X\in N_\omega$ satisfies $\|X\|_2\ge\frac1{16}$ and $\|\theta_\omega(X)-X\|_2\ge c\|X\|_2$, so $\|V_\theta-1\|\ge c$.

The ordered set. Fix $u_j$ and $\delta$. Let $\mathcal R$ be the set of tuples $r=(E,U_1,\dots,U_n,X,P)$ of elements of $N$ such that:

(S1) $E$ is a projection, and the $U_j$ are unitaries commuting with $E$;
(S2) $\|U_j-u_j\|_1\le\delta\tau(E)$;
(S3) $X\in N_E$, $\|X\|\le1$, $\|X\|_1\ge\frac1{16}\tau(E)$, $\|[X,U_j]\|_1\le\delta\tau(E)$;
(S4) $P$ is unitary, $\|P-1\|_1\le\delta\tau(E)$, and $P\theta(E)P^*=E$;
(S5) $\|P\theta(X)P^*-X\|_2\ge c\|X\|_2$.

Write $r\le r'$ when $E\le E'$, $\|U_j'-U_j\|_1\le\delta\tau(E'-E)$, $X'E=EX'=X$, $EP'=P'\theta(E)=EP$ and $\|P'-P\|_1\le\delta\tau(E'-E)$. As in Proposition 4.3, this is a partial order: for transitivity, if $r\le r'\le r''$, then $P''\theta(E)=P''\theta(E')\theta(E)=E'P'\theta(E)=E'EP=EP$. The map $t(r)=\tau(E)$ is strictly increasing, and $(0,u_1,\dots,u_n,0,1)\in\mathcal R$. An increasing sequence $(r^k)$ has an upper bound. The limits $E,U_j,X$ are obtained as in Proposition 4.3. The $P^k$ form a $\|\cdot\|_1$-Cauchy, hence $\|\cdot\|_2$-Cauchy, sequence of unitaries with a unitary limit $P$. Then $P\theta(E)P^*=\lim P^k\theta(E^k)P^{k*}=E$ and $E^kP=E^kP^k=P\theta(E^k)$. Conditions (S3) and (S5) pass to the limit, because all quantities converge in $\|\cdot\|_2$.

A maximal element is complete. Let $r$ be maximal (Lemma 2.6) and suppose $F=1-E\ne0$. By (S4), $\operatorname{Ad}P\circ\theta$ fixes $E$ and $F$. Let $\psi=(\operatorname{Ad}P\circ\theta)^F$. By Proposition 7.4(a) and (c), $c(\psi)=c(\theta)>c$. Choose $c'$ with $c<c'<c(\psi)$, and put $\varepsilon=\min\big(\delta/7,(c'^2-c^2)/192\big)$. By Lemma 8.1 in $N_F$, applied to $\psi$, $c'$ and the unitaries $v_j=U_jF$, there are a nonzero projection $e\le F$ and $x\in N_e$ with $\|[v_j,e]\|_1\le\varepsilon\tau(e),\ \ \|\psi(e)-e\|_1\le\varepsilon\tau(e),\ \ \|x\|\le1,\ \ \|x\|_2^2\ge\tfrac1{16}\tau(e),$ $\|x\|_1\ge\tfrac1{16}\tau(e),\ \ \|\psi(x)-x\|_2\ge c'\|x\|_2,\ \ \|[x,v_j]\|_1\le\varepsilon\tau(e).$ These conditions are homogeneous in the trace. By (T3) in $N_F$, there are unitaries $W_j$ of $N_F$ with $W_jv_jev_j^*W_j^*=e$ and $\|W_j-F\|_1\le\sqrt2\,\|[v_j,e]\|_1\le3\varepsilon\tau(e)$, and a unitary $W$ of $N_F$ with $W\psi(e)W^*=e$ and $\|W-F\|_1\le\sqrt2\,\|\psi(e)-e\|_1\le3\varepsilon\tau(e)$. Put $v_j'=W_jv_j$, which commutes with $e$ and satisfies $\|v_j'-v_j\|_1\le3\varepsilon\tau(e)$, and $Q=E+W$. Define $E'=E+e,\quad U_j'=U_jE+v_j',\quad X'=X+x,\quad P'=QP .$ (S1), (S2): as in Proposition 4.3, $U_j'$ is a unitary commuting with $E'$, and $\|U_j'-U_j\|_1=\|v_j'-v_j\|_1\le3\varepsilon\tau(e)\le\delta\tau(E'-E)$.

(S3): $X'\in N_{E'}$, $\|X'\|\le1$ and $\|X'\|_1=\|X\|_1+\|x\|_1\ge\frac1{16}\tau(E')$. Since $X\in N_E$, $x\in N_F$, and $U_j$ commutes with $E$, we get $[X',U_j']=[X,U_j]+[x,v_j']$. Hence $\|[X',U_j']\|_1\le\delta\tau(E)+\|[x,v_j]\|_1+2\|v_j'-v_j\|_1\le\delta\tau(E)+7\varepsilon\tau(e)\le\delta\tau(E').$ Also $X'E=EX'=X$.

(S4): $Q$ is a unitary commuting with $E$, with $QE=E$ and $\|Q-1\|_1=\|W-F\|_1\le3\varepsilon\tau(e)$. So $P'\theta(E')P'^*=Q(E+P\theta(e)P^*)Q^*=E+W\psi(e)W^*=E'$, and $\|P'-1\|_1\le\|P-1\|_1+\|Q-1\|_1\le\delta\tau(E')$. Moreover $EP'=EQP=EP$, $P'\theta(E)=QEP=EP$, and $\|P'-P\|_1\le3\varepsilon\tau(e)\le\delta\tau(E'-E)$.

(S5): The automorphism $\operatorname{Ad}P\circ\theta$ maps $N_E$ and $N_F$ into themselves, and these are orthogonal in $L^2(N)$. So $\|P\theta(X')P^*-X'\|_2^2=\|P\theta(X)P^*-X\|_2^2+\|\psi(x)-x\|_2^2\ge c^2\|X\|_2^2+c'^2\|x\|_2^2\ge c^2\|X'\|_2^2+\tfrac{c'^2-c^2}{16}\tau(e).$ Next compare $P'$ with $P$. For a unitary $A$, $\|A\theta(X')A^*-X'\|_2^2=2\tau(X'^*X')-2\operatorname{Re} \tau(X'^*A\theta(X')A^*)$. With $Y=P\theta(X')P^*$ we have $P'\theta(X')P'^*=QYQ^*$, and $QYQ^*-Y=(Q-1)YQ^*+Y(Q^*-1)$. Hence $|\tau(X'^*(QYQ^*-Y))|\le2\|Q-1\|_1\le6\varepsilon\tau(e)$, and $\|P'\theta(X')P'^*-X'\|_2^2\ge\|P\theta(X')P^*-X'\|_2^2-12\varepsilon\tau(e)\ge c^2\|X'\|_2^2 ,$ by the choice of $\varepsilon$.

So $r'\in\mathcal R$ and $r<r'$, contradicting maximality. Hence $E=1$, and $r$ provides $U_j$, $X$ and $P$ as required. $\square$

8.3 The asymptotic period

Theorem 8.3. Let $N$ be a II₁ factor with separable predual, $\theta\in\operatorname{Aut}N$, and $p=p_a(\theta)$. Then $c(\theta)=\max\{|z-1|:z\in\mu_p\}.$ Explicitly, $c(\theta)=0$ if $p=1$, $c(\theta)=2$ if $p=0$ or $p$ is even, and $c(\theta)=2\cos(\pi/2p)$ if $p\ge3$ is odd.

Reference: Connes; the asymptotic period is [Connes 1976, Definition 3.4.1].

Proof. By Theorem 8.2, $c(\theta)=\max\{|\lambda-1|:\lambda\in\operatorname{Sp}V_\theta\}$, and $\operatorname{Sp}V_\theta=\mu_p$ by Proposition 6.6. For the explicit values, $|e^{2\pi ik/p}-1|= 2|\sin(\pi k/p)|$ is largest when $k/p$ is closest to $\frac12$. This gives $2$ for even $p$, and $2\sin(\pi(p-1)/2p)=2\cos(\pi/2p)$ for odd $p$. $\square$

Only the inequality $c(\theta)\ge\max_{\mu_p}|z-1|$ uses (B8), through Proposition 6.6. Both inequalities can be checked directly in the examples of Section 6. For $\theta_\zeta$ of Example 6.4, $\operatorname{Sp}V_{\theta_\zeta}=\mu_p$, so Theorem 8.2 gives $c(\theta_\zeta)=2$ for even $p$ and $c(\theta_\zeta)=2\cos(\pi/2p)$ for odd $p$. For example $c(\theta_\zeta)=\sqrt3$ when $p=3$. For the shift of Example 6.5, $c(\sigma)=2$.

Corollary 8.4. Let $N$ be a II₁ factor with separable predual and $\theta\in\operatorname{Aut}N$.

(i) $c(\theta)=0$ if and only if $\theta\in\operatorname{Ct}N$.

(ii) $c(\theta)\in\{0\}\cup[\sqrt3,2]$. In particular $c(\theta)<\sqrt3$ implies $\theta\in\operatorname{Ct}N$.

Proof. (i) is part of Theorem 8.2 and does not use (B8). (ii) By Theorem 8.3, $c(\theta)$ is $0$ for $p=1$, $2$ for $p=0$ or even $p$, and $2\cos(\pi/2p)\ge2\cos(\pi/6)=\sqrt3$ for odd $p\ge3$. $\square$

8.4 Tensor products

Corollary 8.5. Let $N_1,N_2$ be II₁ factors with separable preduals and $\theta_j\in\operatorname{Aut}N_j$. Then $\theta_1\otimes\theta_2\in\operatorname{Ct}(N_1\bar\otimes N_2)$ if and only if $\theta_1\in\operatorname{Ct}N_1$ and $\theta_2\in\operatorname{Ct}N_2$.

Reference: [Connes 1976, Theorem 3.9.3].

Proof. Only if. Suppose $\theta_1\notin\operatorname{Ct}N_1$. Then there is a bounded central sequence $(y_k)$ in $N_1$ with $\|\theta_1(y_k)-y_k\|_2\ge\beta>0$. The sequence $(y_k\otimes1)$ is central in $N_1\bar\otimes N_2$: it commutes asymptotically with the algebraic tensor product, which is dense, and $\|[a,z]\|_2\le\|[a,z_0]\|_2+2\|a\|\,\|z-z_0\|_2$. But $(\theta_1\otimes\theta_2)(y_k\otimes1)-y_k\otimes1=(\theta_1(y_k)-y_k)\otimes1$ has $\|\cdot\|_2\ge\beta$. So $\theta_1\otimes\theta_2\notin\operatorname{Ct}$. The same holds with the roles of $N_1,N_2$ exchanged.

If. Let $\theta_1\in\operatorname{Ct}N_1$. By Corollary 8.4(i), $c(\theta_1)=0$. By Proposition 7.4(b), $c(\theta_1\otimes\mathrm{id})=0$, and by Proposition 7.3, $\theta_1\otimes\mathrm{id}\in\operatorname{Ct}(N_1\bar \otimes N_2)$. The flip isomorphism $N_1\bar\otimes N_2\to N_2\bar\otimes N_1$ conjugates $\mathrm{id}\otimes\theta_2$ to $\theta_2\otimes\mathrm{id}$, and $\operatorname{Ct}$ is preserved by isomorphisms. So $\mathrm{id}\otimes\theta_2\in\operatorname{Ct}$ as well. Since $\operatorname{Ct}$ is a group, $\theta_1\otimes\theta_2=(\theta_1\otimes\mathrm{id})(\mathrm{id}\otimes\theta_2)\in\operatorname{Ct}(N_1\bar\otimes N_2)$. $\square$

Only the "if" direction has real content. A bounded central sequence of $N_1\bar\otimes N_2$ need not be close to sums of products of central sequences of $N_1$ and $N_2$. The invariant $c$ avoids this problem. It is defined by almost central vectors, and these can be decomposed along a basis of $L^2(N_2)$, as in the proof of Proposition 7.4(b).

9. Exercises

Exercise 1. Show that $c(\theta^{-1})=c(\theta)$ and $c(\sigma\theta\sigma^{-1})=c(\theta)$ for all $\theta,\sigma\in\operatorname{Aut}N$.

Solution. $U_{\theta^{-1}}=U_\theta^*$, and $\|U_\theta^*\xi-\xi\|=\|\xi-U_\theta\xi\|$. The condition $\|[x,\xi]\|\le\varepsilon$ does not involve $\theta$, so $S(\theta^{-1})=S(\theta)$. The second equality is Proposition 7.4(a).

Exercise 2. Let $\{g_{ij}\}_{i,j\le2}$ be matrix units in $N$ with $g_{11}+g_{22}=1$, and let $\xi=\hat g_{12}$. Show that $[g_{12},\xi]=0$ but $\|[g_{21},\xi]\|=\sqrt2\|\xi\|$. So near commutation with an element does not control near commutation with its adjoint. Show that in Definition 7.1 one may nevertheless restrict $F$ to finite subsets of any self-adjoint set $F_0$ that generates $N$ as a C*-algebra.

Solution. $g_{12}g_{12}=0$, so $[g_{12},\xi]=0$. Next $[g_{21},\xi]=\widehat{g_{21}g_{12}-g_{12}g_{21}}= \widehat{g_{22}-g_{11}}$, whose norm is $1$, while $\|\xi\|=\tau(g_{22})^{1/2}=2^{-1/2}$. Now let $F_0$ be self-adjoint and generate $N$ as a C*-algebra. The identities $[xy,\xi]=x[y,\xi]+[x,\xi]y$ and $\|[x,\xi]\|\le\|[x_0,\xi]\|+2\|x-x_0\|$ show that near commutation with the finitely many elements of $F_0$ that occur in a noncommutative polynomial $x_0$ controls near commutation with $x_0$, and hence with any $x$ close to $x_0$ in norm. Here $[x,\xi]y$ means right multiplication by $y$, which has norm $\|y\|$. Since $F_0$ is self-adjoint, noncommutative polynomials in its elements form a norm dense $*$-subalgebra of $N$.

Exercise 3. Let $N$ be a full II₁ factor with separable predual. Show that $\overline{\operatorname{Int}N}= \operatorname{Int}N$, that $\operatorname{Ct}N=\operatorname{Aut}N$, and that $c(\theta)=0$ for every $\theta$.

Solution. The first statement is the definition of fullness. By Corollary 5.2 of "Full factors", $N_\omega= \mathbb C$ for every free ultrafilter $\omega$, so $\theta_\omega=\mathrm{id}$ and $\theta\in\operatorname{Ct}N$ by Lemma 6.2(ii). Then $c(\theta)=0$ by Theorem 8.2. So on a full factor the two subgroups are as far apart as possible.

Exercise 4. Let $p=3$, $\zeta=e^{2\pi i/3}$, and $\theta_\zeta$ as in Example 6.4. Compute $c$ of $\theta_\zeta\otimes\theta_\zeta^{-1}$ on $R\bar\otimes R$ without using (B8).

Solution. Let $\Theta=\theta_\zeta\otimes\theta_\zeta^{-1}$. Then $\Theta^3=\mathrm{id}$, so $\operatorname{Sp}V_\Theta\subset\mu_3$. With $y_n^{(k)}$ as in Example 6.4, the sequences $(y_n^{(k)}\otimes1)_n$, $k=1,2$, are central in $R\bar\otimes R$ (as in the proof of Corollary 8.5). They satisfy $\Theta(y_n^{(k)}\otimes1)=\bar\zeta^k\,y_n^{(k)}\otimes1$. So $\bar\zeta,\bar\zeta^2\in\operatorname{Sp}V_\Theta$, and $1\in\operatorname{Sp}V_\Theta$ because $V_\Theta\hat1=\hat1$. Hence $\operatorname{Sp}V_\Theta=\mu_3$, and by Theorem 8.2, $c(\Theta)=|\zeta-1|=\sqrt3$. Note that $\Theta$ is centrally nontrivial even though its two tensor factors are mutually inverse.

Exercise 5. Let $N$ have separable predual, let $\theta$ be approximately inner, and let $f\ne0$ be a projection with $\theta(f)=f$. Show that $\theta^f$ is an approximately inner automorphism of $N_f$.

Solution. By Theorem 3.2, $\theta$ satisfies (c). Lemma 4.2 with $v=f$ (here $vv^*=f=\theta(f)=v^*v$) shows that ${}_f\theta=\theta^f$ satisfies (c) relative to $N_f$. $N_f$ is a II₁ factor with separable predual, so Theorem 3.2 applies to it.

References

[Connes 1976] A. Connes, On the classification of von Neumann algebras and their automorphisms, IHÉS preprint IHES/P/76/132, February 1976. Free at https://repo-archives.ihes.fr/FONDS_IHES/I_Prepublications/CONNES/1976-1984/P_76_132/P_76_132_web.pdf
[Connes 1975] A. Connes, Outer conjugacy classes of automorphisms of factors, Ann. Sci. École Norm. Sup. (4) 8 (1975), no. 3, 383–419. https://doi.org/10.24033/asens.1295. Free at https://alainconnes.org/wp-content/uploads/automorphismes.pdf
[Connes 1994] A. Connes, Noncommutative geometry, Academic Press, San Diego, CA, 1994. Free at https://alainconnes.org/wp-content/uploads/book94bigpdf.pdf
[Effros–Lance 1977] E. G. Effros and E. C. Lance, Tensor products of operator algebras, Adv. Math. 25 (1977), no. 1, 1–34. https://doi.org/10.1016/0001-8708(77)90085-8. Free at https://doi.org/10.1016/0001-8708(77)90085-8
[Anantharaman–Popa] C. Anantharaman and S. Popa, An introduction to II₁ factors, book draft. Free at https://www.math.ucla.edu/~popa/Books/IIun.pdf