Ultraproducts and the asymptotic centralizer

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. The arguments added to (B4) and (B9), and the example in the remark after Proposition 8.7, were drafted by GPT-6 Astra (OpenAI) in ChatGPT web, Pro mode, and checked and adapted by Claude Opus 5.5, October 2026. Public domain (CC0).

Introduction

A von Neumann algebra is an infinite object, and many of its finer invariants are only visible "at infinity": they are carried by bounded sequences $(x_n)$ that asymptotically commute with everything in sight. This lesson builds the algebras that collect such sequences.

For a finite von Neumann algebra $N$ with a faithful normal tracial state $\tau$, the right object is the ultrapower $N^\omega$: bounded sequences modulo those that tend to zero in the trace norm along a free ultrafilter $\omega$. More generally we form the ultraproduct $\prod_\omega N_k$ of a sequence of finite von Neumann algebras, prove it is again a finite von Neumann algebra, and prove it is a factor when the $N_k$ are factors. The ultraproduct does not act on the Hilbert space ultraproduct of the spaces $L^2(N_k)$; it acts, in standard form, on the subspace of uniformly integrable sequences.

For a general countably decomposable von Neumann algebra $M$ with a faithful normal state $\varphi$, the trace norm is no longer available. The key observation is that the sequences which asymptotically commute with every normal functional still form a finite von Neumann algebra after one divides out the sequences tending to zero $*$-strongly. This is the asymptotic centralizer $M_\omega$. The heart of the matter is a quantitative estimate: an element that almost commutes with $\varphi$ is almost fixed by the modular group of $\varphi$, $\|\sigma_t^\varphi(x)-x\|_\varphi^\sharp\le 16\sqrt2\,(1+|t|)\,\|x\|^{1/2}\,\|x\varphi-\varphi x\|^{1/2}.$ From it we deduce that $M_\omega$ does not depend on $\varphi$, that every automorphism of $M$ acts on $M_\omega$ through its class in $\operatorname{Out}M$, and that the modular automorphisms act trivially.

We then give three applications. First, a spectral gap estimate passes to tensor products and computes $(Q_1\bar\otimes Q_2)_\omega$. Second, we build an explicit II₁ factor with no nontrivial central sequences that has a prescribed $\lambda\in(0,1)$ in its fundamental group, and a II∞ factor with two nonconjugate outer classes scaling the trace by the same $\lambda$. Third, we prove that for a factor of type III₀ the asymptotic centralizer is nontrivial and even has nontrivial centre.

Later lessons use this material: "Full factors", "Property Γ and the algebra generated by a factor and its commutant" and "Uniqueness of the injective II₁ factor". Their definitions of $M^\omega$, $M_\omega$ and $\prod_\omega N_k$ are those of Definitions 3.1, 3.4, 8.1 and 8.6 below.

What is assumed. Basic von Neumann algebra theory, including the lessons "The double commutant theorem", "Projections and types of von Neumann algebras", "Traces on von Neumann algebras", "Completely positive maps" and "Spatial tensor products of von Neumann algebras"; Tomita–Takesaki theory, including "The modular group and its analytic algebra" and "The KMS boundary condition determines the modular group"; and, for Section 4, the lesson "Measurable operators for a trace: examples, convergence and the commutant". These are lessons of this programme. We also use the Connes cocycle derivative, and, only in Section 11, the discrete decomposition of factors of type III₀. The precise facts are listed below.

Basic references are [Connes 1974] and [Ando–Haagerup 2014].

Results used from other lessons

Throughout, "von Neumann algebra" means a concrete von Neumann algebra, but every notion used (normal functionals, $\sigma$-weak and $*$-strong topologies on bounded sets) depends only on the abstract algebra.

(B1) Normal functionals. If $M\subset B(H)$ is a von Neumann algebra and $\psi\in M_*^+$, there are vectors $\eta_k\in H$ with $\sum_k\|\eta_k\|^2=\psi(1)$ and $\psi(x)=\sum_k\langle x\eta_k,\eta_k\rangle$. A bounded net $x_i$ tends to $0$ strongly if and only if $\psi(x_i^*x_i)\to0$ for every $\psi\in M_*^+$. The first statement is The double commutant theorem, Theorem 10.1; the second holds because on bounded sets the strong and σ-strong topologies agree (Compact and trace-class operators, Lemma 8.5) and the σ-strong topology is given by the seminorms $\psi(x^*x)^{1/2}$.

(B2) Kaplansky density theorem. If $A\subset B(H)$ is a $*$-algebra, the closed unit ball of $A$ is strongly (indeed $*$-strongly) dense in the closed unit ball of its $\sigma$-weak closure. Proved in Kaplansky's density theorem and its consequences, Theorem 7.1.

(B3) Standard form. Fix a faithful normal state $\varphi$ on $M$; its GNS representation is $(H_\varphi,\xi_\varphi)$. Then $\xi_\varphi$ is cyclic and separating, and with the modular conjugation $J$ and the natural cone $\mathcal P$ the following hold: every $\psi\in M_*^+$ is $\omega_\eta=\langle\,\cdot\,\eta,\eta\rangle$ for a unique $\eta=\xi_\psi\in\mathcal P$; and for $\eta_1,\eta_2\in\mathcal P$ $\|\eta_1-\eta_2\|^2\le\|\omega_{\eta_1}-\omega_{\eta_2}\|.$ This last inequality is the Powers–Størmer inequality in the form proved for general von Neumann algebras. Proved in The positive cone of a standard representation: the inequality in §SF-07, existence and uniqueness of the cone vector in §§SF-10 and SF-07. See also [Haagerup 1975].

(B4) Relative modular operator and cocycle. Let $\varphi,\psi$ be faithful normal states on $M$ and $\xi_\psi\in\mathcal P$ the representing vector of $\psi$ as in (B3). The map $x\xi_\varphi\mapsto x^*\xi_\psi$ is closable, its closure has polar decomposition $J\Delta_{\psi,\varphi}^{1/2}$ with $\Delta_{\psi,\varphi}$ positive self-adjoint and nonsingular, and the Connes cocycle derivative is $(D\psi:D\varphi)_t=\Delta_{\psi,\varphi}^{it}\Delta_\varphi^{-it}$. If $v\in M$ is unitary and $\varphi_v=\varphi(v\,\cdot\,v^*)$, then $(D\varphi_v:D\varphi)_t=v^*\sigma_t^\varphi(v)$. The relative map, its closability and nonsingularity, and the identification of $\Delta_{\psi,\varphi}$ with the spatial derivative $d\psi/d\varphi^{\mathrm{opp}}$, where $\varphi^{\mathrm{opp}}=\langle\,\cdot\,\xi_\varphi,\xi_\varphi\rangle$ on $M'$, are proved in Spatial energy as a corner of a modular operator, §§SI-05, SI-07 and SI-12; the cocycle is defined by the balanced $2\times2$ matrix weight in §SI-14 there; the unitary case is in Comparing supported weights and transporting their cuts, §WC-03; see also Changing reference for a weight cocycle. These lessons do not identify the polar factor with the conjugation $J$ of the standard form. Here is that step, after [Araki 1974, Theorem 1].

A criterion for the modular conjugation. Let $R\subseteq B(K)$ have a cyclic and separating vector $\Omega$ with Tomita operator $S_\Omega$, and let $C$ be an antiunitary involution with $CRC=R'$, $C\Omega=\Omega$ and $\langle\Omega,aCaC\Omega\rangle\ge0$ for $a\in R$. Then $C=J_\Omega$. Indeed, $T_0=CS_\Omega$ on $R\Omega$ satisfies $\langle T_0a\Omega,a\Omega\rangle=\langle\Omega,aCaC\Omega\rangle\ge0$, so it is symmetric; its closure is $T=CS_\Omega$, with $T^*=S_\Omega^*C$. Since $R'\Omega$ is a core for $S_\Omega^*$ (§SI-02 of the same lesson) and $C(R\Omega)=R'\Omega$, the space $R\Omega$ is a core for $T^*$, on which $T^*$ agrees with $T_0$. So $T^*=T\ge0$, $S_\Omega=CT$ is the polar decomposition, and $C=J_\Omega$.

The polar factor is $J$. The vector $\xi_\psi$ is separating for $M$, because $\psi$ is faithful, and for $M'=JMJ$, because $J\xi_\psi=\xi_\psi$; so it is cyclic and separating, as is $\xi_\varphi$. Let $R=M_2(M)$ act on $M_2(H)$ by left multiplication, put $\Omega=\operatorname{diag}(\xi_\varphi,\xi_\psi)$, which is cyclic and separating, and $(\mathcal J\eta)_{ij}=J\eta_{ji}$. Then $\mathcal JR\mathcal J=R'$, $\mathcal J\Omega=\Omega$, and, with $\theta_1=\varphi$, $\theta_2=\psi$, $\langle\Omega,a\mathcal Ja\mathcal J\Omega\rangle=\sum_{i,k}\langle\xi_{\theta_i},a_{ik}Ja_{ik}J\xi_{\theta_k}\rangle\ge0,$ because $aJaJ$ maps $\mathcal P$ into $\mathcal P$ and $\mathcal P$ is self-dual (B3). So $\mathcal J=J_\Omega$. The Tomita map of $\Omega$ sends $a_{21}\xi_\varphi$ in the entry $(2,1)$ to $a_{21}^*\xi_\psi$ in the entry $(1,2)$; restricting $S_\Omega=\mathcal J\Delta_\Omega^{1/2}$ to these entries gives $S_{\psi,\varphi}=J\Delta_{\psi,\varphi}^{1/2}$, with $\Delta_{\psi,\varphi}$ the restriction of $\Delta_\Omega$ to the entry $(2,1)$. Since $\Delta_\Omega$ restricts to $\Delta_\varphi$ on the entry $(1,1)$, and left multiplication by the matrix unit $E_{21}$ carries that entry to the entry $(2,1)$, the balanced-matrix definition $(D\psi:D\varphi)_t\otimes E_{21}=\sigma_t^{\varphi\oplus\psi}(1\otimes E_{21})=\Delta_\Omega^{it}(1\otimes E_{21})\Delta_\Omega^{-it}$ gives $(D\psi:D\varphi)_t=\Delta_{\psi,\varphi}^{it} \Delta_\varphi^{-it}$.

(B5) Analytic elements. If $y\in M$ is entire analytic for $\sigma^\varphi$, then $y\xi_\varphi=J\sigma^\varphi_{-i/2}(y^*)J\,\xi_\varphi$, so $y\xi_\varphi\in M'\xi_\varphi$. The entire analytic elements form a $\sigma$-weakly dense $*$-subalgebra of $M$. Proved in The modular group and its analytic algebra, §§MF-07–MF-09; see also Analytic elements and strip arguments.

(B6) Smoothing and the centralizer. For $f\in L^1(\mathbb R)$ and $x\in M$ the $\sigma$-weak integral $\sigma^\varphi_f(x)=\int f(t)\sigma^\varphi_t(x)\,dt$ lies in $M$, and $\sigma^\varphi_f(x)\xi_\varphi=\int f(t)\Delta_\varphi^{it}x\xi_\varphi\,dt$. The centralizer $M_\varphi=\{x:\sigma_t^\varphi(x)=x\ \forall t\}$ equals $\{x:\varphi(xy)=\varphi(yx)\ \forall y\in M\}$, and the restriction of $\varphi$ to $M_\varphi$ is a faithful normal tracial state. There is a unique $\varphi$-preserving normal conditional expectation $E_\varphi$ of $M$ onto $M_\varphi$; $E_\varphi(x)$ is the unique $y\in M_\varphi$ with $\varphi(zy)=\varphi(zx)$ for all $z\in M_\varphi$. The characterization of the centralizer is Fixed elements and changes of density, §CZ-05; the expectation exists by Conditional expectations from modular invariance, §ME-01, since $M_\varphi$ is invariant under $\sigma^\varphi$, and it is unique by §ME-08 there; the restriction of $\varphi$ to $M_\varphi$ is tracial by the characterization. The smoothing formula is the integral of the covariance $\sigma^\varphi_t(x)\xi_\varphi=\Delta_\varphi^{it}x\xi_\varphi$.

(B7) Tensor products. For von Neumann algebras $Q_1,Q_2$ with faithful normal positive functionals $\psi_1,\psi_2$, the product functional $\psi_1\otimes\psi_2$ on $Q_1\bar\otimes Q_2$ is faithful and normal, and $\|\alpha\otimes\beta\|\le\|\alpha\|\|\beta\|$ for $\alpha\in Q_{1*}$, $\beta\in Q_{2*}$. If $\psi_1$ is a state, the slice map $\mathrm{id}\otimes\psi_1\colon Q_2\bar\otimes Q_1\to Q_2$ (defined by $a\otimes x\mapsto\psi_1(x)a$) is a normal unital completely positive $Q_2$-bimodule map. Tensor products of isomorphisms are isomorphisms, and for normal semifinite traces $(\tau_1\otimes\tau_2)\circ(\gamma_1\otimes\gamma_2)=(\tau_1\circ\gamma_1)\otimes(\tau_2\circ\gamma_2)$. Product functionals and their norms: Spatial tensor products of von Neumann algebras, Section 10; slice maps: Section 9 there; tensor products of maps: Proposition 8.1 there.

(B8) Factors. A II₁ factor has a unique normal tracial state. In a countably decomposable factor, any two properly infinite projections are equivalent; in particular every nonzero projection of a countably decomposable type III factor is equivalent to $1$, so $eMe\cong M$. A finite factor has type I$_n$ for some $n$, or type II₁; and in a finite factor with trace $\tau$, $\tau(p)\le\tau(q)$ implies that $p$ is equivalent to a subprojection of $q$. Uniqueness of the trace: Integration for a trace, Corollary 7.4. Properly infinite countably decomposable projections of a factor majorize each other (Projections and types of von Neumann algebras, Proposition 15.2(2)), so they are equivalent (Proposition 5.1), and equivalent projections have isomorphic corners (Lemma 8.5). Finite factors: Corollaries 7.3 and 10.4 there. Comparison through the trace: Traces on von Neumann algebras, Corollary 5.4.

(B9) Crossed products by a state-preserving action. Let $G$ be a countable group acting on a von Neumann algebra $B$ by automorphisms $\alpha_g$ preserving a faithful normal state $\omega$, and let $P=B\rtimes_\alpha G$, generated by $B$ and unitaries $u_g$ with $u_gbu_g^*=\alpha_g(b)$. There is a faithful normal conditional expectation $E\colon P\to B$ with $E(bu_g)=0$ for $g\ne e$; the state $\varphi=\omega\circ E$ is faithful and normal; with $x_g=E(xu_g^*)$ the map $x\xi_\varphi\mapsto(x_g\xi_\omega)_{g\in G}$ extends to a unitary $L^2(P,\varphi)\to\ell^2(G,L^2(B,\omega))$; and $\sigma^\varphi_t(b)=\sigma^\omega_t(b)$, $\sigma_t^\varphi(u_g)=u_g$. We use the regular crossed product: with $K=L^2(B,\omega)$, it acts on $\ell^2(G,K)$ by $(\pi(b)\eta)_s=\alpha_{s^{-1}}(b)\eta_s$ and $(u_g\eta)_s=\eta_{g^{-1}s}$ (Changing the Hilbert space of a regular crossed product), and another faithful normal representation of $B$ gives an isomorphic algebra, by an isomorphism that fixes $B$ and the $u_g$ (the comparison theorem there). We identify $\pi(b)$ with $b$.

The expectation. Let $V_s:K\to\ell^2(G,K)$ be the inclusion as the coordinate $s$, and $E(x)=V_e^*xV_e$. On finite sums $V_e^*(\sum_gb_gu_g)V_e=b_e$; by normality, and because $B$ is ultraweakly closed, $E$ maps $P$ onto $B$. It is a normal unital completely positive $B$-bimodule map with $E(bu_g)=0$ for $g\neq e$. Moreover $V_s^*xV_s=\alpha_{s^{-1}}(E(x))$ for $x\in P$, first for finite sums and then by normality. So if $x\ge0$ and $E(x)=0$, then $x^{1/2}V_s=0$ for all $s$, and $x=0$. Hence $E$ is faithful, and so is $\varphi=\omega\circ E$.

The unitary. Put $\Omega=V_e\xi_\omega$. Then $\langle x\Omega,\Omega\rangle=\varphi(x)$, and $bu_g\Omega=V_g(\alpha_{g^{-1}}(b)\xi_\omega)$, so $\Omega$ is cyclic and $x\xi_\varphi\mapsto x\Omega$ is a unitary $W:L^2(P,\varphi)\to\ell^2(G,K)$. Both sides of $V_g^*xV_e=\alpha_{g^{-1}}(E(xu_g^*))$ are normal in $x$ and agree on the elements $bu_h$, so this holds for all $x\in P$. As $\omega\circ\alpha_g=\omega$, the formula $U_g(b\xi_\omega)=\alpha_g(b)\xi_\omega$ defines unitaries of $K$, and the coordinatewise unitary $(D\eta)_g=U_g\eta_g$ gives $(DW(x\xi_\varphi))_g=U_gV_g^*xV_e\xi_\omega=x_g\xi_\omega$. This is the asserted unitary.

The modular group. $E(u_gxu_g^*)=\alpha_g(E(x))$ (check it on the elements $bu_h$ and use normality), and $\omega\circ\alpha_g=\omega$, so $\varphi\circ\operatorname{Ad}u_g=\varphi$. Hence $\varphi(u_gy)=\varphi(u_g(yu_g)u_g^*) =\varphi(yu_g)$ for all $y\in P$, and $u_g$ lies in the centralizer of $\varphi$ by Fixed elements and changes of density, §CZ-05. Since $\varphi|_B=\omega$ and $\varphi\circ E=\varphi$, the setting of Conditional expectations from modular invariance, §ME-01 applies; $\sigma^\varphi_t(b)=\sigma^\omega_t(b)$ is (ME.42) in §ME-09 there, and $E\circ\sigma_t^\varphi=\sigma_t^\omega\circ E$ is §ME-10.

(B10) Infinite tensor products of matrix algebras. Let $S$ be a countable set, $0<\lambda\le1$, and $\omega_\lambda=\mathrm{Tr}(\rho\,\cdot\,)$ on $M_2(\mathbb C)$ with $\rho=\operatorname{diag}(\frac1{1+\lambda},\frac\lambda{1+\lambda})$. The infinite tensor product $B=\bar\bigotimes_{s\in S}(M_2(\mathbb C),\omega_\lambda)$ carries a faithful normal product state $\omega$; its modular group acts sitewise by $\operatorname{Ad}\rho^{it}$; every permutation of $S$ induces a $\omega$-preserving automorphism permuting the tensor factors; and, with $K=L^2(M_2,\omega_\lambda)\ominus\mathbb C\xi_{\omega_\lambda}$, $L^2(B,\omega)\ominus\mathbb C\xi_\omega=\bigoplus_{F}\ \bigotimes_{s\in F}K_s ,$ the sum over finite nonempty $F\subset S$, where the automorphism induced by a permutation $\pi$ maps the summand of $F$ unitarily onto that of $\pi(F)$. For $\lambda=1$ and $S=\mathbb N$, $B$ is the hyperfinite II₁ factor $R$. The product state, its faithfulness and normality, and the product modular group are proved in Infinite tensor products, Proposition 5.2 and Theorem 6.1, and $R$ in Corollary 5.3 there. The orthogonal decomposition is the expansion of the product vectors that differ from $\xi_\omega$ at finitely many sites, and a permutation of the sites permutes these vectors.

(B11) Measure-theoretic realization. An abelian von Neumann algebra $C$ whose predual is separable, equipped with a faithful normal state, is $L^\infty(X,\mu)$ for a standard Borel space $X$ and a probability measure $\mu$; every automorphism of $C$ is $f\mapsto f\circ T^{-1}$ for a Borel bijection $T$ with $\mu\circ T$ and $\mu$ equivalent. The realization comes from Abelian operator algebras, Theorems 8.2 and 8.4, as explained below. The automorphisms. Let $\alpha$ be a $*$-automorphism of $L^\infty(X,\mu)$, with $X$ standard and $\mu$ $\sigma$-finite. Choose Borel sets $B_1,B_2,\ldots$ that separate the points of $X$ and generate its Borel sets (preimages of a countable base under a Borel isomorphism with a Polish space), and Borel sets $C_n$ with $\alpha([1_{B_n}])=[1_{C_n}]$: $\alpha$ maps projections to projections, and these are the classes of indicators. Put $S(x)=(1_{B_n}(x))_n$ and $T_0(x)=(1_{C_n}(x))_n$ in $\{0,1\}^{\mathbb N}$. $S$ is an injective Borel map, so $S(X)$ is Borel and $S$ is a Borel isomorphism onto it (Polish spaces and standard Borel spaces, Theorem 4.3(5)). The Borel sets $Q\subseteq\{0,1\}^{\mathbb N}$ with $[1_{T_0^{-1}(Q)}]=\alpha([1_{S^{-1}(Q)}])$ contain the sets $\{c:c_n=1\}$ and are closed under complements and countable unions: $\alpha$ preserves complements of projections and suprema of countable families of projections, being an order isomorphism, and in $L^\infty$ the supremum of the $[1_{A_k}]$ is $[1_{\bigcup_kA_k}]$. So they are all the Borel sets. For $Q$ the complement of $S(X)$ this gives $\mu(T_0^{-1}(Q))=0$. Let $\varphi=S^{-1}\circ T_0$ on the conull Borel set $T_0^{-1}(S(X))$, and constant elsewhere. For Borel $B\subseteq X$, taking $Q=S(B)$ gives $\alpha([1_B])=[1_B\circ\varphi]$; by linearity and norm limits ($\alpha$ is isometric) $\alpha([f])=[f\circ\varphi]$ for every bounded Borel $f$, and $\mu(\varphi^{-1}(N))=0$ when $\mu(N)=0$. In the same way $\alpha^{-1}([f])=[f\circ\psi]$ for a Borel $\psi$. Then $[f\circ\psi\circ\varphi]=[f]$ for all $f$; for $f=1_{B_n}$, $n\ge1$, this gives $S\circ\psi\circ\varphi=S$, hence $\psi\circ\varphi=\mathrm{id}$, almost everywhere, and likewise $\varphi\circ\psi=\mathrm{id}$ almost everywhere. Let $N$ be the null set where one of these identities fails, and $X_*$ the complement of $\bigcup_{k\ge0}(\varphi^{-k}(N)\cup\psi^{-k}(N))$, a conull Borel set. If $x\in X_*$, then $\varphi(x)\in X_*$: $\varphi^{k}(\varphi(x))\notin N$, $\varphi(x)\notin N$, and $\psi^k(\varphi(x))=\psi^{k-1}(x)\notin N$ for $k\ge1$, since $\psi\varphi x=x$. Likewise $\psi(x)\in X_*$. So $\varphi$ restricts to a Borel bijection of $X_*$ with inverse $\psi$. Let $T$ be $\psi$ on $X_*$ and the identity on $X\setminus X_*$. Then $T$ is a Borel bijection of $X$, $\mu\circ T$ and $\mu$ have the same null sets, and $\alpha(f)=f\circ T^{-1}$. The realization. The minimal projections of $C$ are mutually orthogonal, so countably many, since the state is faithful; with $q$ their sum, $Cq\cong\ell^\infty(I)$ for a countable set $I$. The unit ball of $C=(C_*)^*$ is $\sigma$-weakly compact and metrizable (Weak topologies, Tychonoff, Banach–Alaoglu, Mazur, bipolars, Krein–Milman and Eberlein–Šmulian, Theorem 3.1 and Proposition 3.2), hence separable, so $C(1-q)$ is countably generated, $\sigma$-finite and without minimal projections, and Theorem 8.4 makes it $0$ or $L^\infty[0,1]$. So $C\cong L^\infty(X,m)$ with $X=[0,1]\sqcup I$ standard and $m$ Lebesgue measure plus counting measure. The faithful normal state is integration against a density $h>0$ in $L^1(m)$ (Measure and Hilbert space tools for Haar integration, Theorem 4.2, which makes $L^1(m)$ a predual of $L^\infty(m)$, the only one by The universal enveloping von Neumann algebra of a C*-algebra, and W*-algebras, Corollary 11.3), and $\mu=h\,m$.

(B12) Discrete decomposition of type III₀ factors. Suppose $M$ is a type III₀ factor whose predual is separable. There are a von Neumann subalgebra $N\subset M$, a faithful normal semifinite trace $\tau$ on $N$, a faithful normal conditional expectation $E\colon M\to N$ and a unitary $U\in M$ such that: (a) $UNU^*=N$, $E(aU^m)=0$ for $a\in N$, $m\ne0$, and the linear span of $\{aU^m:a\in N,\ m\in\mathbb Z\}$ is $\sigma$-weakly dense in $M$; (b) the centre $C$ of $N$ has no minimal projections, and $\theta=\operatorname{Ad}U$ restricts to an ergodic automorphism of $C$; (c) for every projection $e\in N$ with $0<\tau(e)<\infty$, the state $\varphi_e(x)=\tau(E(x))/\tau(e)$ on $eMe$ is faithful and normal, its centralizer is $eNe$, and $1$ is an isolated point of $\operatorname{Sp}\Delta_{\varphi_e}$. Statement (c) is the corner form of the following: $N$ is the centralizer of the faithful normal semifinite weight $\tau\circ E$, and $1$ is isolated in the spectrum of its modular operator. Since $e$ lies in that centralizer, the modular group of $\varphi_e$ is the restriction of $\sigma^{\tau\circ E}$ to $eMe$, whose spectrum is contained in that of $\sigma^{\tau\circ E}$, and the centralizer of $\varphi_e$ is $eNe$. [Connes 1973, Theorem 5.3.1 and its proof]

(B13) Square-integrable operators. For a von Neumann algebra $N$ carrying a faithful normal tracial state $\tau$, $L^2(N,\tau)$ is realized by $\tau$-measurable operators affiliated with $N$. For $\xi\in L^2$ the modulus $|\xi|$ and the polar decomposition $\xi=v|\xi|$ exist, with $v\in N$ and spectral projections of $|\xi|$ in $N$; $\|y\xi\|_2\le\|y\|\|\xi\|_2$ and $\|\xi y\|_2\le\|\xi\|_2\|y\|$ for $y\in N$; $\|\xi\|_2=\||\xi|\|_2$; and $x\mapsto x^*$ extends to an antiunitary involution $J$. The GNS representation of $\tau$ is the standard form of $N$, with trace vector $\hat1$ and conjugation $J$. Proved in Operators recovered from small trace defects, §MT-12 and Trace densities and noncommutative integration, §TI-15, with the standard form in Integration for a trace, Sections 2–3.

(B14) Recurrence. Let $T$ be an ergodic Borel bijection of a standard probability space $(X,\mu)$ without atoms, with $\mu\circ T$ equivalent to $\mu$. Then $T$ is conservative, and for each Borel $A$ with $\mu(A)>0$, almost every $x\in X$ has $T^kx\in A$ for infinitely many $k>0$ and infinitely many $k<0$. Here ergodic means that every Borel set $Y$ with $TY=Y$ is null or conull, and $T^{-1}$ is Borel as well. Proof. A Borel set $W$ is wandering if the sets $T^nW$, $n\in\mathbb Z$, are disjoint. Such a $W$ is null: otherwise, as $\mu$ has no atoms, $W$ contains a Borel $W'$ with $0<\mu(W')<\mu(W)$; then $Y=\bigcup_{n\in\mathbb Z}T^nW'$ satisfies $TY=Y$, $\mu(Y)>0$, and $Y$ misses $W\setminus W'$, which has positive measure, against ergodicity. This is conservativity. Let $A_0$ be the set of $x\in A$ with $T^kx\notin A$ for all $k\ge1$. Then $A_0\cap T^kA_0=\varnothing$ for $k\ne0$, so $A_0$ is wandering and null. If $x\in A$ visits $A$ only finitely often at positive times, and $m\ge0$ is its last visit, then $T^mx\in A_0$; so these $x$ lie in the null set $\bigcup_{m\ge0}T^{-m}A_0$, null because $\mu\circ T$ and $\mu$ have the same null sets. The set of $x\in X$ whose positive orbit meets $A$ infinitely often is invariant under $T$ and contains almost all of $A$, so it is conull by ergodicity. The same argument for $T^{-1}$ gives the negative times.

1. Conventions: ultrafilters, the bimodule of normal functionals, $*$-strong convergence

Ultrafilters. A free ultrafilter on $\mathbb N$ is an ultrafilter containing no finite set; equivalently, a point $\omega$ of $\beta\mathbb N\setminus\mathbb N$, where $\beta\mathbb N$ is the Stone–Čech compactification. Fix one, $\omega$, for the whole lesson. Every bounded sequence $(c_n)$ of complex numbers has a unique limit $\lim_{n\to\omega}c_n$: the number $c$ such that $\{n:|c_n-c|<\varepsilon\}\in\omega$ for all $\varepsilon>0$. It is the value at $\omega$ of the continuous extension of $n\mapsto c_n$ to $\beta\mathbb N$. The map $(c_n)\mapsto\lim_\omega c_n$ is linear, multiplicative and order preserving, and changing a sequence on a set not in $\omega$ does not change its limit. Several statements in Sections 1, 6 and 7 hold for an arbitrary filter on an index set (for example the filter of tails of a directed set), and we say so when it matters.

The bimodule $M_*$. Write $M_*$ for the predual of a von Neumann algebra $M$. For $x\in M$ and $\psi\in M_*$ define $x\psi,\ \psi x,\ [x,\psi]\in M_*$ by $(x\psi)(y)=\psi(yx),\qquad(\psi x)(y)=\psi(xy),\qquad[x,\psi]=x\psi-\psi x .$ One checks at once that $x(y\psi)=(xy)\psi$, $(\psi x)y=\psi(xy)$, $(x\psi)y=x(\psi y)$ and $\|x\psi\|,\|\psi x\|\le\|x\|\|\psi\|$, so $M_*$ is a Banach $M$-bimodule. With $\psi^*(y)=\overline{\psi(y^*)}$ we have $(x\psi)^*=\psi^*x^*$, hence $[x,\psi]^*=-[x^*,\psi^*] .\tag{1.1}$ For $x\in M$ and $\psi\in M_*^+$ put $\|x\|_\psi=\psi(x^*x)^{1/2},\qquad\|x\|_\psi^\sharp=\psi(x^*x+xx^*)^{1/2}=\big(\|x\|_\psi^2+\|x^*\|_\psi^2\big)^{1/2}.$ If $\psi=\varphi$ is a faithful normal state with GNS vector $\xi_\varphi$, then $\|x\|_\varphi=\|x\xi_\varphi\|$. By (B6), $[x,\varphi]=0$ exactly when $x\in M_\varphi$.

Lemma 1.1. Let $\varphi\in M_*^+$ be a state and $x,y\in M$. Then:

(a) $\|[x,\varphi]\|=\|[x^*,\varphi]\|$;

(b) $\|x\varphi\|\le\|x\|_\varphi$ and (c) $\|\varphi x\|\le\|x^*\|_\varphi$; hence $\|[x,\varphi]\|\le\sqrt2\,\|x\|_\varphi^\sharp$;

(d) $[xy,\varphi]=x[y,\varphi]+[x,\varphi]y$, so $\|[xy,\varphi]\|\le\|x\|\|[y,\varphi]\|+\|y\|\|[x,\varphi]\|$, and $\|[x^k,\varphi]\|\le k\|x\|^{k-1}\|[x,\varphi]\|$;

(e) $\varphi(y^*x^*xy)\le\|y\|\|x\|^2\|[y,\varphi]\|+\|y\|^2\|x\|\|x\|_\varphi$;

(f) if $\|x\|,\|y\|\le1$, then $(\|xy\|_\varphi^\sharp)^2\le\|[x,\varphi]\|+\|[y,\varphi]\|+\|x\|_\varphi^\sharp+\|y\|_\varphi^\sharp$.

For a general $\psi\in M_*^+$, (b) and (c) read $\|x\psi\|\le\psi(1)^{1/2}\|x\|_\psi$ and $\|\psi x\|\le\psi(1)^{1/2}\|x^*\|_\psi$.

Proof. (a) follows from (1.1) since $\varphi^*=\varphi$ and $\psi\mapsto\psi^*$ is isometric. (b) By the Cauchy–Schwarz inequality, $|\varphi(zx)|\le\varphi(zz^*)^{1/2}\varphi(x^*x)^{1/2}\le\|z\|\|x\|_\varphi$. (c) Similarly $|\varphi(xz)|\le\varphi(xx^*)^{1/2}\varphi(z^*z)^{1/2}\le\|x^*\|_\varphi\|z\|$. The bound on $[x,\varphi]$ follows since $\|x\|_\varphi+\|x^*\|_\varphi\le\sqrt2\|x\|^\sharp_\varphi$. (d) Using the bimodule rules, $xy\varphi-\varphi xy=x(y\varphi-\varphi y)+(x\varphi-\varphi x)y$; the power bound follows by induction. (e) Write $\varphi(y^*x^*xy)=(y\varphi)(y^*x^*x)=[y,\varphi](y^*x^*x)+\varphi(yy^*x^*x)$. The first term is at most $\|[y,\varphi]\|\|y\|\|x\|^2$. For the second, the Cauchy–Schwarz inequality gives $|\varphi((yy^*x^*)x)|\le\|yy^*x^*\|\,\|x\|_\varphi\le\|y\|^2\|x\|\|x\|_\varphi$. (f) By (e), $\varphi(y^*x^*xy)\le\|[y,\varphi]\|+\|x\|_\varphi$. Applying (e) to the pair $(y^*,x^*)$ in place of $(x,y)$ gives $\varphi(xyy^*x^*)\le\|[x^*,\varphi]\|+\|y^*\|_\varphi=\|[x,\varphi]\|+\|y^*\|_\varphi$. Add, and use $\|x\|_\varphi,\|y^*\|_\varphi\le\|x\|^\sharp_\varphi,\|y\|^\sharp_\varphi$. The statement for general $\psi$ is the same computation. $\square$

Definition 1.2. Consider a von Neumann algebra $M$, a filter $\mathcal F$ on a set $I$, and a bounded family $(x_i)_{i\in I}$ in $M$. We say $x_i\to0$ strongly along $\mathcal F$ if $\lim_{\mathcal F}\psi(x_i^*x_i)=0$ for every $\psi\in M_*^+$, and $*$-strongly along $\mathcal F$ if moreover $x_i^*\to0$ strongly along $\mathcal F$. For $\mathcal F=\omega$ we write $x_n\to0$ ($*$-)strongly as $n\to\omega$.

By (B1), for a filter of tails this is ordinary strong convergence of a bounded net in any normal faithful representation. The next lemma says that one faithful state suffices to test it.

Lemma 1.3. Fix a faithful normal state $\varphi$ on $M$ together with a bounded family $(x_i)$. Then $x_i\to0$ strongly along $\mathcal F$ if and only if $\lim_{\mathcal F}\|x_i\|_\varphi=0$; and $x_i\to0$ $*$-strongly along $\mathcal F$ if and only if $\lim_{\mathcal F}\|x_i\|^\sharp_\varphi=0$.

Proof. One direction is trivial. Conversely, work in the GNS space of $\varphi$, where $\xi=\xi_\varphi$ is separating for $M$, hence cyclic for $M'$. Let $\|x_i\|\le c$ and $\psi\in M_*^+$; by (B1) $\psi(z)=\sum_k\langle z\eta_k,\eta_k\rangle$. Given $\varepsilon>0$, pick $K$ with $\sum_{k>K}\|\eta_k\|^2<\varepsilon$, and $y_k'\in M'$ with $\|\eta_k-y_k'\xi\|<\varepsilon^{1/2}/K^{1/2}$ for $k\le K$. Then $\|x_i\eta_k\|\le c\|\eta_k-y'_k\xi\|+\|y'_k\|\|x_i\xi\|$, so $\psi(x_i^*x_i)\le c^2\varepsilon+\sum_{k\le K}\big(c\,\varepsilon^{1/2}K^{-1/2}+\|y_k'\|\,\|x_i\|_\varphi\big)^2 .$ Taking the limit along $\mathcal F$ gives $\lim_{\mathcal F}\psi(x_i^*x_i)\le2c^2\varepsilon$. The $*$-version follows by applying this to $(x_i^*)$. $\square$

Lemma 1.4. Fix a faithful normal state $\varphi$ on $M$. Then the closed unit ball of $M$ is complete for the metric $d(x,y)=\|x-y\|^\sharp_\varphi$, and this metric defines the $*$-strong topology on it.

Proof. The second statement is Lemma 1.3 for the filter of neighbourhoods. Let $(x_k)$ be $d$-Cauchy in the ball. Then $(x_k\xi)$ and $(x_k^*\xi)$ are Cauchy in $H_\varphi$. For $y'\in M'$, $\|x_ky'\xi-x_ly'\xi\|\le\|y'\|\|(x_k-x_l)\xi\|$, so $x_k\eta$ converges for $\eta$ in the dense subspace $M'\xi$; since $\|x_k\|\le1$, it converges for every $\eta$, to $T\eta$ with $\|T\|\le1$. Likewise $x_k^*\to S$ strongly, and $S=T^*$ by comparing weak limits. Thus $x_k\to T$ $*$-strongly, $T$ lies in the strongly closed ball of $M$, and $\|x_k-T\|^\sharp_\varphi\to0$. $\square$

2. A completeness criterion for finite von Neumann algebras

All the algebras in this lesson are first built as quotient C*-algebras. The following criterion shows that they are von Neumann algebras. For a tracial state $\tau$ write $\|x\|_2=\tau(x^*x)^{1/2}$.

Proposition 2.1. Let $A$ be a unital C*-algebra with a faithful tracial state $\tau$. Suppose the closed unit ball of $A$ is complete for the metric $\|x-y\|_2$. Then the GNS representation $\pi_\tau$ maps $A$ onto a von Neumann algebra. So $A$ is itself a finite von Neumann algebra, and $\tau$ is a faithful normal tracial state on it.

Proof. Let $(H,\pi,\xi)$ be the GNS triple of $\tau$. Since $\tau(x^*x)=\|\pi(x)\xi\|^2$ and $\tau$ is faithful, $\pi$ is injective, hence isometric. For $b\in A$ we have $\|xb\|_2^2=\tau(xbb^*x^*)\le\|b\|^2\tau(xx^*)=\|b\|^2\|x\|_2^2$, using the trace property twice. So $R_b\colon\pi(x)\xi\mapsto\pi(xb)\xi$ extends to a bounded operator, which commutes with $\pi(A)$.

Let $T\in\pi(A)''$ with $\|T\|\le1$. By (B2) there is a net $(x_i)$ in the unit ball of $A$ with $\pi(x_i)\to T$ strongly. Then $\pi(x_i)\xi\to T\xi$, so $(x_i)$ is a Cauchy net for $\|\cdot\|_2$. In a complete metric space Cauchy nets converge, so $x_i\to x$ in $\|\cdot\|_2$ for some $x$ in the unit ball, and $\pi(x)\xi=T\xi$. For $b\in A$, $T\pi(b)\xi=TR_b\xi=R_bT\xi=R_b\pi(x)\xi=\pi(x)\pi(b)\xi$. Hence $T=\pi(x)$ on a dense subspace, so $T=\pi(x)$. Thus the ball of $\pi(A)''$ lies in $\pi(A)$, and $\pi(A)=\pi(A)''$. The trace $\tau=\langle\pi(\cdot)\xi,\xi\rangle$ is a vector state, hence normal. $\square$

Lemma 2.2. In the setting of Proposition 2.1 it suffices to check that every $\|\cdot\|_2$-Cauchy sequence $(x^{(p)})$ with $\sup_p\|x^{(p)}\|<1$ has a $\|\cdot\|_2$-limit of norm at most $1$.

Proof. Let $(x^{(p)})$ be Cauchy in the closed unit ball and $0<r<1$. Then $(rx^{(p)})$ has a limit $y_r$ with $\|y_r\|\le1$, so $x^{(p)}\to y_r/r$. Since $\|\cdot\|_2$ is a norm ($\tau$ is faithful), the limit $x=y_r/r$ does not depend on $r$, and $\|x\|\le1/r$ for every $r<1$, so $\|x\|\le1$. $\square$

3. Ultraproducts of finite von Neumann algebras

Definition 3.1. Let $(N_k,\tau_k)_{k\in\mathbb N}$ be von Neumann algebras with faithful normal tracial states. Let $\ell^\infty=\prod_kN_k$ be the C*-algebra of bounded sequences $x=(x_k)$, and $\mathcal I_\omega=\{x\in\ell^\infty:\ \lim_{k\to\omega}\|x_k\|_{2}=0\}.$ The ultraproduct is $\prod_\omega N_k=\ell^\infty/\mathcal I_\omega$, with $\tau_\omega(x)=\lim_{k\to\omega}\tau_k(x_k)$. A bounded sequence mapping to $x$ is a representing sequence of $x$.

Since $\|yx\|_2\le\|y\|\|x\|_2$, $\|xy\|_2\le\|x\|_2\|y\|$ and $\|x^*\|_2=\|x\|_2$ for a trace, $\mathcal I_\omega$ is a norm-closed two-sided self-adjoint ideal, so $\prod_\omega N_k$ is a C*-algebra. By Lemma 1.3 (with $\varphi=\tau_k$ in each coordinate, and $\|x\|^\sharp=\sqrt2\|x\|_2$ for a trace), $\mathcal I_\omega$ consists of the sequences tending to $0$ $*$-strongly along $\omega$ in the sense that $\lim_\omega\|x_k\|_{\tau_k}^\sharp=0$.

Theorem 3.2. $\prod_\omega N_k$ is again a finite von Neumann algebra, and $\tau_\omega$ is a faithful normal tracial state on it.

Proof. $\tau_\omega$ is well defined since $|\tau_k(x_k)|\le\|x_k\|_2$; it is tracial since each $\tau_k$ is; and $\tau_\omega(x^*x)=\lim_\omega\|x_k\|_2^2$ vanishes only on $\mathcal I_\omega$. We check Lemma 2.2. Let $(x^{(p)})$ be $\|\cdot\|_2$-Cauchy with $\|x^{(p)}\|<1$; passing to a subsequence we may assume $\|x^{(p+1)}-x^{(p)}\|_2<2^{-p}$ (a Cauchy sequence converges once a subsequence does). By the definition of the quotient norm, each $x^{(p)}$ has a representing sequence $(y^{(p)}_k)$ with $\sup_k\|y_k^{(p)}\|<1$. Put $x_k^{(1)}=y^{(1)}_k$. Given $(x^{(p)}_k)$, the set $W_p=\{k:\|y^{(p+1)}_k-x^{(p)}_k\|_2<2^{-p}\}$ belongs to $\omega$; let $x^{(p+1)}_k=y^{(p+1)}_k$ for $k\in W_p$ and $x^{(p+1)}_k=x^{(p)}_k$ otherwise. This is still a representing sequence of $x^{(p+1)}$, and now $\|x^{(p+1)}_k-x^{(p)}_k\|_2<2^{-p}$ for all $k$. For fixed $k$, $(x_k^{(p)})_p$ is Cauchy in the unit ball of $N_k$, which is complete by Lemma 1.4; let $x_k$ be its limit. Then $\|x_k\|\le1$ and $\|x_k-x_k^{(p)}\|_2\le2^{1-p}$ for all $k$, so the class $x$ of $(x_k)$ satisfies $\|x\|\le1$ and $\|x-x^{(p)}\|_2\le2^{1-p}$. Proposition 2.1 finishes the proof. $\square$

Theorem 3.3. If every $N_k$ is a factor, then $\prod_\omega N_k$ is a factor.

Proof. Write $Q=\prod_\omega N_k$. First, projections lift to projections: let $z\in Q$ be a projection and $(y_k)$ a representing sequence, which we may take self-adjoint (replace $y_k$ by $(y_k+y_k^*)/2$). Then $\lim_\omega\|y_k^2-y_k\|_2=\|z^2-z\|_2=0$. Let $p_k=\chi_{[1/2,\infty)}(y_k)$. For real $s$ one has $|\chi_{[1/2,\infty)}(s)-s|\le2|s^2-s|$ (for $s\ge\frac12$ because $2s\ge1$, for $s<\frac12$ because $2|s-1|>1$). By functional calculus and positivity of the trace, $\|p_k-y_k\|_2\le2\|y_k^2-y_k\|_2$, so $(p_k)$ represents $z$.

Now let $z$ be a central projection of $Q$, with lift $(p_k)$, and put $t=\tau_\omega(z)=\lim_\omega\tau_k(p_k)$. Suppose $0<t<1$. In the finite factor $N_k$, one of $p_k$, $1-p_k$ is equivalent to a subprojection of the other (B8); so $N_k$ contains a partial isometry $v_k$ with $v_k^*v_k\le p_k$, $v_kv_k^*\le1-p_k$ and $\tau_k(v_k^*v_k)=\min(\tau_k(p_k),1-\tau_k(p_k))$. Let $v\in Q$ be the class of $(v_k)$. Then $v^*v\le z$ and $vv^*\le1-z$, so $v=vz$ and $v=(1-z)v$. As $z$ is central, $v=(1-z)vz=(1-z)zv=0$. But $\tau_\omega(v^*v)=\min(t,1-t)>0$, a contradiction. Hence $\tau_\omega(z)\in\{0,1\}$, and faithfulness gives $z\in\{0,1\}$. The centre of a von Neumann algebra is generated by its projections, so it is $\mathbb C$. $\square$

Definition 3.4. Fix a von Neumann algebra $N$ with a faithful normal tracial state $\tau$. The ultrapower is $N^\omega=\prod_\omega N$ (all $N_k=N$). The map sending $x\in N$ to the class of the constant sequence $(x,x,\dots)$ is an injective normal $*$-homomorphism, by which we regard $N\subset N^\omega$. The central sequence algebra is the relative commutant $N_\omega=N'\cap N^\omega$. A bounded sequence $(x_n)$ is central if $\lim_\omega\|[x_n,y]\|_2=0$ for all $y\in N$, that is, if its class lies in $N_\omega$.

Normality of the embedding $\iota$: it is trace preserving, $\tau_\omega\circ\iota=\tau$. If $x_i\uparrow x$ is a bounded increasing net in $N$, then $\iota(x_i)\uparrow y$ for some $y\le\iota(x)$, and $\tau_\omega(\iota(x)-y)\le\tau_\omega(\iota(x-x_i))=\tau(x-x_i)\to0$ because $\tau$ is normal; as $\tau_\omega$ is faithful, $y=\iota(x)$. The same argument shows that every trace-preserving (or trace-scaling) unital $*$-homomorphism between finite von Neumann algebras with faithful normal traces is injective and normal. In Section 8 we show that $N_\omega$ is the asymptotic centralizer of $N$ (Proposition 8.7), so the notation is consistent.

Example 3.5. Let $R=\bar\bigotimes_{n\ge1}(M_2(\mathbb C),\mathrm{tr})$ be the hyperfinite II₁ factor (B10), and let $s_n$ be $\operatorname{diag}(1,-1)$ placed in the $n$-th tensor factor. Each $s_n$ is a self-adjoint unitary with trace $0$, and $s_n$ commutes with the algebraic tensor product of the first $n-1$ factors. Since that union is $\|\cdot\|_2$-dense and $\|[s_n,y]\|_2\le2\|y-y_0\|_2$ whenever $[s_n,y_0]=0$, the sequence $(s_n)$ is central. Its class $s\in R_\omega$ satisfies $s^2=1$ and $\tau_\omega(s)=0$, so $s\notin\mathbb C$. Thus $R_\omega\ne\mathbb C$: the hyperfinite factor has nontrivial central sequences. By contrast, Theorem 10.3 below gives a II₁ factor with $N_\omega=\mathbb C$.

4. The Hilbert space of uniformly integrable sequences

Keep the setting of Definition 3.1, let $H_k=L^2(N_k,\tau_k)$ (B13), and let $H_\omega$ denote the Hilbert space ultraproduct of the $H_k$: bounded sequences $(\xi_k)$, $\xi_k\in H_k$, modulo those with $\lim_\omega\|\xi_k\|=0$, with $\langle(\xi_k),(\eta_k)\rangle=\lim_\omega\langle\xi_k,\eta_k\rangle$. It is a Hilbert space (completeness is proved exactly like Theorem 3.2, choosing representatives coordinatewise). Each bounded sequence of operators $A_k\in B(H_k)$ gives an operator $(\xi_k)\mapsto(A_k\xi_k)$ on $H_\omega$, which vanishes exactly when $\lim_\omega\|A_k\|=0$.

So bounded sequences in $\prod_kN_k$ act on $H_\omega$, but the action does not factor through $\prod_\omega N_k$: the ideal $\mathcal I_\omega$ is defined by the $2$-norm, not the operator norm.

Example 4.1. Let $N_k=L^\infty[0,1]$ with Lebesgue measure. The projections $x_k=\chi_{[0,1/k]}$ satisfy $\|x_k\|_2=k^{-1/2}\to0$, so they represent $0$ in $\prod_\omega N_k$. The unit vectors $\xi_k=k^{1/2}\chi_{[0,1/k]}$ satisfy $x_k\xi_k=\xi_k$, so $(x_k)$ acts on $H_\omega$ as a nonzero operator. The vector $(\xi_k)$ has its mass escaping to large values: $\|\chi_{(a,\infty)}(|\xi_k|)\,|\xi_k|\|_2=1$ as soon as $k>a^2$. This is exactly the behaviour excluded below.

For $\xi\in L^2(N,\tau)$ and $a>0$ let $E_a(|\xi|)=\chi_{(a,\infty)}(|\xi|)\in N$, the spectral projection of $|\xi|$ for the interval $(a,\infty)$.

Theorem 4.2. Let $\mathcal H\subset H_\omega$ be the set of classes $\xi=(\xi_k)$ satisfying the uniform integrability condition $\forall\varepsilon>0\ \exists a>0:\qquad\lim_{k\to\omega}\big\|E_a(|\xi_k|)\,|\xi_k|\big\|_2<\varepsilon .\tag{4.1}$ Then:

(i) condition (4.1) holds for one representing sequence of $\xi$ if and only if it holds for all of them, and $\mathcal H$ is the closure of the set $\mathcal H_0$ of classes of sequences $(x_k)$ with $x_k\in N_k$ and $\sup_k\|x_k\|<\infty$; in particular $\mathcal H$ is a closed subspace;

(ii) every bounded sequence $(x_k)\in\prod_kN_k$ maps $\mathcal H$ into itself, and its restriction to $\mathcal H$ depends only on the class of $(x_k)$ in $\prod_\omega N_k$; this defines a faithful normal representation $\pi$ of $\prod_\omega N_k$ on $\mathcal H$;

(iii) $\pi$ is the standard form: the vector $\mathbf 1=(1)_k$ is a trace vector, cyclic and separating, and the antiunitary $J(\xi_k)=(J_k\xi_k)$ restricts to the modular conjugation of $\tau_\omega$ on $\mathcal H$.

Reference: due to Connes.

Proof. (i) We show that for a bounded sequence $(\xi_k)$, (4.1) is equivalent to

(4.2) for every $\varepsilon>0$ there is a bounded sequence $(x_k)$, $x_k\in N_k$, with $\lim_\omega\|\xi_k-x_k\|_2<\varepsilon$.

Condition (4.2) depends only on the class of $(\xi_k)$ and says that the class lies in $\overline{\mathcal H_0}$, so this proves (i).

(4.1) ⇒ (4.2). Given $\varepsilon$, choose $a$ as in (4.1). Let $\xi_k=v_k|\xi_k|$ be the polar decomposition and $x_k=\xi_k\,(1-E_a(|\xi_k|))=v_k\,|\xi_k|\chi_{[0,a]}(|\xi_k|)$. Then $x_k\in N_k$, $\|x_k\|\le a$, and $\xi_k-x_k=v_k\,|\xi_k|E_a(|\xi_k|)$. Since $v_k^*v_k$ is the support of $|\xi_k|$, $\|\xi_k-x_k\|_2=\||\xi_k|E_a(|\xi_k|)\|_2$, whose limit along $\omega$ is less than $\varepsilon$.

(4.2) ⇒ (4.1). We use an elementary estimate: if $\xi\in L^2(N,\tau)$, $x\in N$, $\|x\|\le a$ and $\|\xi-x\|_2\le\delta$, then $\|E_{2a}(|\xi|)\,|\xi|\|_2\le2\delta$. Indeed, put $E=E_{2a}(|\xi|)$. Since $|\xi|E\ge2aE$, $\|\xi E\|_2^2=\tau(E|\xi|^2E)\ge4a^2\tau(E)=4a^2\|E\|_2^2$. On the other hand $\|\xi E\|_2\le\|xE\|_2+\|(\xi-x)E\|_2\le a\|E\|_2+\delta\le\frac12\|\xi E\|_2+\delta$. So $\||\xi|E\|_2=\|\xi E\|_2\le2\delta$. Now, given $\varepsilon$, take $(x_k)$ as in (4.2) for $\varepsilon/2$, and $a>0$ with $\|x_k\|\le a$ for all $k$. On the set $\{k:\|\xi_k-x_k\|_2<\varepsilon/2\}\in\omega$ the estimate gives $\|E_{2a}(|\xi_k|)|\xi_k|\|_2\le\varepsilon$, so (4.1) holds with $2a$ (with $\le\varepsilon$, which is enough since $\varepsilon$ is arbitrary).

(ii) Let $x=(x_k)$ be bounded. It maps $\mathcal H_0$ into $\mathcal H_0$ and is bounded on $H_\omega$, so it maps $\mathcal H=\overline{\mathcal H_0}$ into itself. If $x\in\mathcal I_\omega$ and $(y_k)$ is bounded, then $\|x_ky_k\|_2=\|y_k^*x_k^*\|_2\le\|y_k\|\|x_k\|_2\to0$ along $\omega$; so $x$ vanishes on $\mathcal H_0$, hence on $\mathcal H$. This defines a $*$-representation $\pi$ of $\prod_\omega N_k$ on $\mathcal H$.

(iii) Let $Q=\prod_\omega N_k$ and $(L^2(Q,\tau_\omega),\hat1)$ the GNS space of $\tau_\omega$. The map $\hat x\mapsto(x_k)\in\mathcal H_0$ is well defined and isometric, since $\|\hat x\|^2=\tau_\omega(x^*x)=\lim_\omega\|x_k\|_2^2$; its range is $\mathcal H_0$, so it extends to a unitary $W\colon L^2(Q,\tau_\omega)\to\mathcal H$ with $W\hat1=\mathbf1$. On $\mathcal H_0$, $W$ intertwines left multiplication by $x$ with $\pi(x)$, hence $\pi$ is unitarily equivalent to the GNS representation of $\tau_\omega$. By Theorem 3.2 and Proposition 2.1 that representation is faithful and normal and is the standard form of $Q$, with trace vector $\hat1$ (cyclic and separating) and conjugation $\hat x\mapsto\widehat{x^*}$ (B13). Finally, each $J_k$ is antiunitary, so $(\xi_k)\mapsto(J_k\xi_k)$ is a well-defined antiunitary on $H_\omega$; on $\mathcal H_0$ it sends $(x_k)$ to $(x_k^*)=W\widehat{x^*}$. Hence it maps $\mathcal H$ onto itself and equals $W\hat JW^*$ there. $\square$

Remark. In Example 4.1 the class of $(\xi_k)$ is orthogonal to $\mathcal H$: for bounded $(y_k)$, $|\langle\xi_k,y_k\rangle|\le\|y_k\|\,\|\xi_k\|_1=\|y_k\|k^{-1/2}\to0$.

5. Sequences asymptotically commuting with a state

A reference for Sections 5 to 9 is [Connes 1974].

From now on the von Neumann algebra $M$ is countably decomposable, and $\varphi$ is a faithful normal state on it. Write $\ell^\infty(M)$ for the C*-algebra of bounded sequences in $M$, and put $\mathcal I_\omega=\{(x_n)\in\ell^\infty(M):\ x_n\to0\ *\text{-strongly as }n\to\omega\}.$ By Lemma 1.3, $(x_n)\in\mathcal I_\omega$ exactly when $\lim_\omega\|x_n\|_\varphi^\sharp=0$, whichever faithful normal state $\varphi$ is used. Note that $\mathcal I_\omega$ is a norm-closed self-adjoint subspace, but in general not an ideal of $\ell^\infty(M)$ (Exercise 12.2).

Theorem 5.1. Let $A_{\varphi,\omega}=\{(x_n)\in\ell^\infty(M):\ \lim_{n\to\omega}\|[x_n,\varphi]\|=0\}.$ (1) $A_{\varphi,\omega}$ is a $*$-subalgebra of $\ell^\infty(M)$, closed in norm, that contains $\mathcal I_\omega$.

(2) If $(x_n),(y_n)\in\ell^\infty(M)$ and $x_n-y_n\to0$ $*$-strongly as $n\to\omega$, then $(x_n)\in A_{\varphi,\omega}$ if and only if $(y_n)\in A_{\varphi,\omega}$.

(3) $\varphi_\omega((x_n))=\lim_\omega\varphi(x_n)$ is a tracial state on $A_{\varphi,\omega}$.

(4) For $X=(x_n)\in A_{\varphi,\omega}$: $\varphi_\omega(X^*X)=0$ if and only if $X\in\mathcal I_\omega$.

(5) $\mathcal I_\omega$ is a two-sided ideal of $A_{\varphi,\omega}$, and the quotient $M_{\varphi,\omega}=A_{\varphi,\omega}/\mathcal I_\omega$ is again a finite von Neumann algebra, on which $\varphi_\omega$ induces a faithful normal tracial state $\tau_{\varphi,\omega}$.

(2) follows from the same inequality applied to $z_n=x_n-y_n$.

(3) For $X=(x_n)$, $Y=(y_n)$ in $A_{\varphi,\omega}$, $|\varphi(x_ny_n)-\varphi(y_nx_n)|=|[y_n,\varphi](x_n)|\le\|x_n\|\|[y_n,\varphi]\|\to0$ along $\omega$.

(4) $\varphi_\omega(X^*X)=\lim_\omega\|x_n\|^2_\varphi$, and by (3) this equals $\varphi_\omega(XX^*)=\lim_\omega\|x_n^*\|_\varphi^2$. So $\varphi_\omega(X^*X)=0$ iff $\lim_\omega\|x_n\|^\sharp_\varphi=0$ iff $X\in\mathcal I_\omega$ (Lemma 1.3).

(5) Let $X=(x_n)\in A_{\varphi,\omega}$ and $Z=(z_n)\in\mathcal I_\omega$. Then $\varphi(z_n^*x_n^*x_nz_n)\le\|x_n\|^2\|z_n\|^2_\varphi\to0$, and, by Lemma 1.1(e) applied to the pair $(z_n^*,x_n^*)$, $\varphi(x_nz_nz_n^*x_n^*)\le\|x_n\|\|z_n\|^2\|[x_n^*,\varphi]\|+\|x_n\|^2\|z_n\|\|z_n^*\|_\varphi\to0 .$ So $XZ\in\mathcal I_\omega$. Since $\mathcal I_\omega$ and $A_{\varphi,\omega}$ are self-adjoint, $ZX=(X^*Z^*)^*\in\mathcal I_\omega$ as well. Hence $M_{\varphi,\omega}$ is a C*-algebra, and by (3),(4) the functional $\varphi_\omega$ induces a faithful tracial state $\tau=\tau_{\varphi,\omega}$, with $\|x\|_2^2=\lim_\omega\|x_n\|_\varphi^2=\lim_\omega\|x_n^*\|^2_\varphi$ for any representing sequence.

We check Lemma 2.2. Let $(x^{(p)})$ be $\|\cdot\|_2$-Cauchy with $\|x^{(p)}\|<1$; as before we may assume $\|x^{(p+1)}-x^{(p)}\|_2<2^{-(p+1)}$. Choose representing sequences $(y^{(p)}_n)\in A_{\varphi,\omega}$ with $\sup_n\|y_n^{(p)}\|<1$ (quotient norm). Put $x^{(1)}_n=y^{(1)}_n$. Given $(x^{(p)}_n)$, we have $\lim_\omega\|y_n^{(p+1)}-x_n^{(p)}\|_\varphi^\sharp=\sqrt2\,\|x^{(p+1)}-x^{(p)}\|_2<\sqrt2\cdot2^{-(p+1)}<2^{-p}$, so $W_p=\{n:\|y^{(p+1)}_n-x^{(p)}_n\|^\sharp_\varphi<2^{-p}\}\in\omega$. Let $x^{(p+1)}_n=y^{(p+1)}_n$ on $W_p$ and $x^{(p)}_n$ off $W_p$. This changes $(y^{(p+1)}_n)$ only off a set in $\omega$, so it is still a representing sequence in $A_{\varphi,\omega}$, and $\|x^{(p+1)}_n-x^{(p)}_n\|^\sharp_\varphi<2^{-p}$ for all $n$. By Lemma 1.4, for each $n$ the sequence $(x^{(p)}_n)_p$ converges $*$-strongly to some $x_n$ with $\|x_n\|\le1$ and $\|x_n-x_n^{(p)}\|^\sharp_\varphi\le2^{1-p}$ for all $n$ and $p$. Then, by Lemma 1.1(b),(c), $\lim_\omega\|[x_n,\varphi]\|\le\lim_\omega\|[x_n^{(p)},\varphi]\|+\sqrt2\cdot2^{1-p}=\sqrt2\cdot2^{1-p}\quad\text{for all }p,$ so $(x_n)\in A_{\varphi,\omega}$. Its class $x$ has $\|x\|\le1$ and $\|x-x^{(p)}\|_2\le2^{1-p}\to0$. Proposition 2.1 finishes the proof. $\square$

Example 5.2. If $M$ is finite and $\varphi=\tau$ is a faithful normal tracial state, then $[x,\tau]=0$ for all $x$, so $A_{\tau,\omega}=\ell^\infty(M)$, $\mathcal I_\omega$ is the ideal of Definition 3.1, and $M_{\tau,\omega}=M^\omega$. If instead $\varphi$ is not tracial, $M_{\varphi,\omega}$ is usually much smaller than any reasonable ultrapower; for instance, if $\varphi$ has $1$ isolated in $\operatorname{Sp}\Delta_\varphi$, Theorem 6.6 below shows that every element of $M_{\varphi,\omega}$ is represented by a sequence in the centralizer $M_\varphi$, so that $M_{\varphi,\omega}=(M_\varphi)^\omega$ (with the trace $\varphi|_{M_\varphi}$).

6. Almost commuting with $\varphi$ means almost fixed by the modular group

This section proves the key estimates. The route is spectral: everything reduces to two elementary inequalities for functions of one positive variable.

Lemma 6.1. For $t\in\mathbb R$ and $\lambda>0$ put $g_t(\lambda)=1-\lambda^{it}+it(\lambda-1)$ and $C_t=8t^2+5|t|+8$. Then $|g_t(\lambda)|\le C_t\,(\lambda^{1/2}-1)^2 .\tag{6.1}$

Proof. Put $u=\lambda^{1/2}$ and $s=\log\lambda=2\log u$. If $u\le\frac12$, then $|g_t(\lambda)|\le2+|t|$ and $(u-1)^2\ge\frac14$, so $|g_t|\le4(2+|t|)(u-1)^2\le C_t(u-1)^2$. Let $u\ge\frac12$. Write $g_t(\lambda)=\big(1-e^{its}+its\big)+it\,h(u),\qquad h(u)=u^2-1-2\log u\ge0 .$ We use $|e^{iy}-1-iy|\le y^2/2$ for real $y$, so the first term is at most $t^2s^2/2$. Next, $|\log u|\le2|u-1|$ for $u\ge\frac12$ (for $u\ge1$, $\log u\le u-1$; for $\frac12\le u\le1$, $\log\frac1u\le\frac1u-1=\frac{1-u}u\le2(1-u)$), so $s^2\le16(u-1)^2$. Finally $h(u)\le5(u-1)^2$: for $u\ge1$, the function $k(u)=2(u-1)^2-h(u)=u^2-4u+3+2\log u$ has $k(1)=0$ and $k'(u)=2(u-1)^2/u\ge0$, so $h\le2(u-1)^2$; for $\frac12\le u\le1$, $h(u)\le u^2-1+2(\frac1u-1)=\frac{(1-u)^2(2+u)}u\le5(1-u)^2$. Altogether $|g_t|\le(8t^2+5|t|)(u-1)^2\le C_t(u-1)^2$. $\square$

Theorem 6.2. Let $\varphi,\psi$ be faithful normal states on a von Neumann algebra $P$ and $t\in\mathbb R$. Then $|1-\varphi((D\psi:D\varphi)_t)|\le C_t\,\|\psi-\varphi\| .$ Proof. Work in the GNS space of $\varphi$ with the notation of (B3), (B4), and write $\Delta=\Delta_{\psi,\varphi}$. The closure of $x\xi_\varphi\mapsto x^*\xi_\psi$ sends $\xi_\varphi$ to $\xi_\psi$, so $\xi_\varphi$ lies in the domain of $\Delta^{1/2}$ and $J\Delta^{1/2}\xi_\varphi=\xi_\psi$; as $J$ fixes the natural cone, $\Delta^{1/2}\xi_\varphi=J\xi_\psi=\xi_\psi$. Let $\mu$ be the spectral measure of $\Delta$ in the vector $\xi_\varphi$; it lives on $(0,\infty)$ since $\Delta$ is nonsingular. Then $\int d\mu=\|\xi_\varphi\|^2=1,\quad\int\lambda\,d\mu=\|\xi_\psi\|^2=\psi(1)=1,\quad \int(\lambda^{1/2}-1)^2d\mu=\|\xi_\psi-\xi_\varphi\|^2 .$ Since $\Delta_\varphi^{-it}\xi_\varphi=\xi_\varphi$, (B4) gives $\varphi((D\psi:D\varphi)_t)=\langle\Delta^{it}\xi_\varphi,\xi_\varphi\rangle=\int\lambda^{it}d\mu$. As $\int(\lambda-1)d\mu=0$, $1-\varphi((D\psi:D\varphi)_t)=\int g_t(\lambda)\,d\mu(\lambda),$ and by (6.1) and (B3) its modulus is at most $C_t\|\xi_\psi-\xi_\varphi\|^2\le C_t\|\psi-\varphi\|$. $\square$

The point of adding $it(\lambda-1)$, which integrates to zero, is that it removes the first-order term: without it one only gets $\|\psi-\varphi\|^{1/2}$ on the right.

Corollary 6.3. For a faithful normal state $\varphi$ on $M$, every unitary $v\in M$ and every $t\in\mathbb R$, one has $\big(\|\sigma_t^\varphi(v)-v\|_\varphi^\sharp\big)^2\le4C_t\,\|[v,\varphi]\| .$ Proof. Let $\varphi_v=\varphi(v\,\cdot\,v^*)$. In bimodule notation $\varphi_v=v^*\varphi v$, so $\|\varphi_v-\varphi\|=\|v^*(\varphi v-v\varphi)\|=\|[v,\varphi]\|$ (left multiplication by a unitary is isometric on $M_*$). Since $\sigma^\varphi_t(v)$ is unitary and $(D\varphi_v:D\varphi)_t=v^*\sigma_t^\varphi(v)$ by (B4), $\|(\sigma_t^\varphi(v)-v)\xi_\varphi\|^2=2\operatorname{Re}\big(1-\varphi(v^*\sigma_t^\varphi(v))\big)\le2C_t\|[v,\varphi]\|$ by Theorem 6.2. Applying this to $v^*$ and using $\|[v^*,\varphi]\|=\|[v,\varphi]\|$ bounds $\|(\sigma^\varphi_t(v)-v)^*\xi_\varphi\|^2$ in the same way. Add. $\square$

To pass from unitaries to arbitrary elements we write a self-adjoint element as the real part of a unitary, and we need to control the commutator of the "imaginary part".

Lemma 6.4. Let $c=c^*\in M$ with $\|c\|\le\frac12$, and $\psi\in M_*$. Then $\|[(1-c^2)^{1/2},\psi]\|\le\tfrac1{\sqrt3}\,\|[c,\psi]\| .$ In particular this holds for $0\le c\le\frac12$, with the constant $\frac1{\sqrt3}<\frac23$.

Proof. The binomial series $(1-z)^{1/2}=1-\sum_{n\ge1}\beta_nz^n$ has $\beta_n\ge0$ and $\sum\beta_n=1$. Since $\|c^2\|\le\frac14$, $(1-c^2)^{1/2}=1-\sum_n\beta_nc^{2n}$ with norm convergence, and $x\mapsto[x,\psi]$ is bounded and linear. By Lemma 1.1(d) (which holds for any $\psi\in M_*$), $\|[c^{2n},\psi]\|\le2n\,2^{1-2n}\|[c,\psi]\|$. Hence the left side is at most $\sum_n\beta_n\,n\,4^{1-n}\,\|[c,\psi]\|=\Big[\tfrac{d}{dz}\big(1-(1-z)^{1/2}\big)\Big]_{z=1/4}\|[c,\psi]\| =\tfrac12\big(\tfrac34\big)^{-1/2}\|[c,\psi]\|=\tfrac1{\sqrt3}\|[c,\psi]\| .\qquad\square$

Theorem 6.5. For a faithful normal state $\varphi$ on $M$, every $x\in M$ and every $t\in\mathbb R$, we have $\|\sigma_t^\varphi(x)-x\|_\varphi^\sharp\le8\,C_t^{1/2}\,\|x\|^{1/2}\,\|[x,\varphi]\|^{1/2}\le16\sqrt2\,(1+|t|)\,\|x\|^{1/2}\,\|[x,\varphi]\|^{1/2}.$ Proof. For $x=0$ there is nothing to prove, and both sides are homogeneous of degree $1$ in $x$, so we may assume $\|x\|=1$; put $\varepsilon=\|[x,\varphi]\|$. Let $a=(x+x^*)/2$ and $b=(x-x^*)/2i$. They are self-adjoint of norm at most $1$, and $\|[a,\varphi]\|,\|[b,\varphi]\|\le\varepsilon$ by Lemma 1.1(a). Let $c=a/2$ and $u=c+i(1-c^2)^{1/2}$. Then $u$ is unitary, $c=(u+u^*)/2$, and by Lemma 6.4 $\|[u,\varphi]\|\le\big(1+\tfrac1{\sqrt3}\big)\tfrac\varepsilon2\le\varepsilon .$ Since $\|y^*\|^\sharp_\varphi=\|y\|^\sharp_\varphi$, Corollary 6.3 gives $\|\sigma_t^\varphi(c)-c\|^\sharp_\varphi\le\|\sigma^\varphi_t(u)-u\|^\sharp_\varphi\le2C_t^{1/2}\varepsilon^{1/2}$, hence $\|\sigma_t^\varphi(a)-a\|^\sharp_\varphi\le4C_t^{1/2}\varepsilon^{1/2}$. The same holds for $b$, and $x=a+ib$ gives the first inequality. The second follows from $C_t\le8(1+|t|)^2$. $\square$

The constant grows linearly in $|t|$. This is the growth one expects: $\sigma_{t+s}(x)-x=\sigma_t(\sigma_s(x)-x)+(\sigma_t(x)-x)$, $\sigma_t$ is isometric for $\|\cdot\|^\sharp_\varphi$, and $\|[\sigma_s(x),\varphi]\|=\|[x,\varphi]\|$, so the best constant is subadditive in $t$.

Theorem 6.6. Fix a faithful normal state $\varphi$, a filter $\mathcal F$ on a set $I$, and a bounded family $(x_i)_{i\in I}$ in $M$ with $\lim_{\mathcal F}\|[x_i,\varphi]\|=0$.

(1) $\lim_{\mathcal F}\sup_{|t|\le T}\|\sigma^\varphi_t(x_i)-x_i\|^\sharp_\varphi=0$ for every $T>0$.

(2) Suppose $1$ is an isolated point of $\operatorname{Sp}\Delta_\varphi$, and let $E_\varphi$ be the $\varphi$-preserving conditional expectation onto the centralizer $M_\varphi$. Then $\lim_{\mathcal F}\|x_i-E_\varphi(x_i)\|^\sharp_\varphi=0$. More precisely, there is a constant $K_\varphi$ with $\|x-E_\varphi(x)\|^\sharp_\varphi\le K_\varphi\,\|x\|^{1/2}\,\|[x,\varphi]\|^{1/2}\qquad(x\in M).$

Proof. (1) is immediate from Theorem 6.5.

(2) Choose $\delta>0$ with $\operatorname{Sp}\Delta_\varphi\cap(e^{-\delta},e^\delta)=\{1\}$. Choose a smooth function $\hat f$ supported in $(-\delta,\delta)$ with $\hat f(0)=1$, and let $f$ be the Schwartz function with $\hat f(s)=\int f(t)e^{its}\,dt$. Then $\int f=1$ and $\int(1+|t|)|f(t)|\,dt<\infty$. By the spectral theorem, $\int f(t)\Delta_\varphi^{it}\,dt=\hat f(\log\Delta_\varphi)=P_0$, the projection onto $\ker(\Delta_\varphi-1)$, because $\log\Delta_\varphi$ has spectrum in $(-\infty,-\delta]\cup\{0\}\cup[\delta,\infty)$. Put $y=\sigma^\varphi_f(x)\in M$ (B6). Then $y\xi_\varphi=P_0x\xi_\varphi$.

$y$ is $E_\varphi(x)$. For $s\in\mathbb R$, $\sigma_s^\varphi(y)\xi_\varphi=\Delta_\varphi^{is}P_0x\xi_\varphi=y\xi_\varphi$, so $y\in M_\varphi$ because $\xi_\varphi$ is separating. If $z\in M_\varphi$, then $\Delta^{is}_\varphi z^*\xi_\varphi=z^*\xi_\varphi$, so $P_0z^*\xi_\varphi=z^*\xi_\varphi$ and $\varphi(zy)=\langle P_0x\xi_\varphi,z^*\xi_\varphi\rangle=\varphi(zx)$. An element $y\in M_\varphi$ with $\varphi(zy)=\varphi(zx)$ for all $z\in M_\varphi$ is unique (take $z=(y-y')^*$ for two such), and it is $E_\varphi(x)$.

The estimate. Since $\int f=1$, $(y-x)\xi_\varphi=\int f(t)(\sigma^\varphi_t(x)-x)\xi_\varphi\,dt$ and $(y-x)^*\xi_\varphi=\int\overline{f(t)}(\sigma^\varphi_t(x)-x)^*\xi_\varphi\,dt$. Minkowski's inequality for the $H_\varphi\oplus H_\varphi$-valued integral and Theorem 6.5 give $\|y-x\|^\sharp_\varphi\le\int|f(t)|\,\|\sigma_t^\varphi(x)-x\|^\sharp_\varphi\,dt \le16\sqrt2\int(1+|t|)|f(t)|\,dt\ \|x\|^{1/2}\|[x,\varphi]\|^{1/2}.\qquad\square$

Example 6.7. The spectral gap in (2) cannot be dropped. Let $M=B(\ell^2(\mathbb N))$ and $\varphi=\mathrm{Tr}(\rho\,\cdot\,)$ with $\rho=\sum_j\mu_je_{jj}$, where $\mu_j>0$ are distinct, $\sum\mu_j=1$, and $\mu_{j+1}/\mu_j\to1$ (for instance $\mu_j$ proportional to $(j+1)^{-2}$). The eigenvalues of $\Delta_\varphi$ are the ratios $\mu_i/\mu_j$, which accumulate at $1$, and $M_\varphi$ is the diagonal algebra. For $x_n=e_{n+1,n}$ one computes $x_n\varphi=\mathrm{Tr}(x_n\rho\,\cdot\,)=\mu_n\mathrm{Tr}(e_{n+1,n}\,\cdot\,)$ and $\varphi x_n=\mu_{n+1}\mathrm{Tr}(e_{n+1,n}\,\cdot\,)$, so $\|[x_n,\varphi]\|=\mu_n-\mu_{n+1}$ (in absolute value), while $E_\varphi(x_n)=0$ and $\|x_n\|_\varphi^\sharp=(\mu_n+\mu_{n+1})^{1/2}$. The ratio $\|x_n-E_\varphi(x_n)\|^\sharp_\varphi/\|[x_n,\varphi]\|^{1/2}=\big((\mu_n+\mu_{n+1})/|\mu_n-\mu_{n+1}|\big)^{1/2}$ tends to infinity, so no constant $K_\varphi$ exists. Taking instead $\mu_{j+1}/\mu_j\le q<1$ for all $j$ (so that $1$ is isolated), the same ratio stays bounded, as Theorem 6.6 predicts.

7. Asymptotically central families

The next theorem says that asymptotic commutation with all normal functionals is a robust notion: it can be tested on a total set of functionals, or on one faithful state together with a dense set of operators, and it implies asymptotic commutation with every operator.

Theorem 7.1. Let $M$ be countably decomposable, $\mathcal F$ a filter on a set $I$, and $(x_i)_{i\in I}$ a bounded family in $M$. The following are equivalent.

(α) For some faithful $\varphi\in M_*^+$ and some $\sigma$-weakly dense subset $\mathcal S\subset M$, one has $\lim_{\mathcal F}\|[x_i,\varphi]\|=0$, and $[x_i,y]\to0$ strongly along $\mathcal F$ for every $y\in\mathcal S$.

(β) There is a total subset $\mathcal D\subset M_*$ (its linear span is norm dense) with $\lim_{\mathcal F}\|[x_i,\psi]\|=0$ for all $\psi\in\mathcal D$.

(γ) $\lim_{\mathcal F}\|[x_i,\psi]\|=0$ for all $\psi\in M_*$, and $[x_i,y]\to0$ $*$-strongly along $\mathcal F$ for every $y\in M$.

Proof. (γ)⇒(α) is clear, since $M$ has a faithful normal state.

(α)⇒(β). Let $\mathcal D=\{y\varphi:y\in\mathcal S\}$. From the bimodule rules, $[x,y\varphi]=[x,y]\varphi+y[x,\varphi]$, so by Lemma 1.1(b) (for the positive functional $\varphi$) $\|[x_i,y\varphi]\|\le\varphi(1)^{1/2}\|[x_i,y]\|_\varphi+\|y\|\,\|[x_i,\varphi]\|\to0 .$ The set $\mathcal D$ is total: if $a\in M=(M_*)^*$ annihilates $\mathcal D$, then $\varphi(ay)=0$ for all $y\in\mathcal S$; as $y\mapsto\varphi(ay)$ is $\sigma$-weakly continuous, $\varphi(ay)=0$ for all $y\in M$, and $y=a^*$ gives $a=0$. By the Hahn–Banach theorem the span of $\mathcal D$ is dense.

(β)⇒(γ). Let $c=\sup\|x_i\|$. Since $\|[x,\psi]-[x,\psi']\|\le2c\|\psi-\psi'\|$, the set of $\psi$ with $\lim_{\mathcal F}\|[x_i,\psi]\|=0$ is a closed subspace, so it is all of $M_*$. Now fix $a\in M$ and $\psi\in M_*^+$, and let $d=[a,x]=ax-xa$ for some $x\in M$. Then $\psi(d^*d)=\psi(d^*ax)-\psi(d^*xa)=(x\psi)(d^*a)-(x(a\psi))(d^*).$ Replacing $x\psi$ by $\psi x$ costs at most $\|[x,\psi]\|\|d\|\|a\|$, and replacing $x(a\psi)$ by $(a\psi)x$ costs at most $\|[x,a\psi]\|\|d\|$. After both replacements the two terms are $\psi(xd^*a)$ and $(a\psi)(xd^*)=\psi(xd^*a)$, which cancel. Hence $\psi([a,x]^*[a,x])\le2\|a\|\|x\|\big(\|a\|\,\|[x,\psi]\|+\|[x,a\psi]\|\big).\tag{7.1}$ Similarly, $[a,x][a,x]^*=[a^*,x^*]^*[a^*,x^*]$, so (7.1) applied to $a^*,x^*$ together with (1.1) bounds $\psi([a,x][a,x]^*)$ by $2\|a\|\|x\|(\|a\|\|[x^*,\psi]\|+\|[x^*,a^*\psi]\|)=2\|a\|\|x\|(\|a\|\|[x,\psi]\|+\|[x,\psi a]\|)$, since $\psi^*=\psi$ and $(a^*\psi)^*=\psi a$. Apply these to $x=x_i$: the right sides tend to $0$ along $\mathcal F$, because $\psi,a\psi,\psi a\in M_*$. Hence $[x_i,a]\to0$ $*$-strongly along $\mathcal F$. $\square$

Remark. The proof of (β)⇒(γ) shows more: commutation with functionals alone implies commutation with operators. In (α) one only needs strong (not $*$-strong) convergence of the commutators, and $\mathcal S$ need not be an algebra.

8. The asymptotic centralizer

Definition 8.1. Assume that the von Neumann algebra $M$ is countably decomposable, and put $A_\omega(M)=\{(x_n)\in\ell^\infty(M):\ \lim_{n\to\omega}\|[x_n,\psi]\|=0\ \text{for every }\psi\in M_*\}.$ The asymptotic centralizer of $M$ at $\omega$ is $M_\omega=A_\omega(M)/\mathcal I_\omega$. A bounded sequence in $A_\omega(M)$ is called $\omega$-central.

By Theorem 7.1, a bounded sequence is $\omega$-central if and only if it asymptotically commutes with one faithful normal state and strongly with a $\sigma$-weakly dense set of operators; and then $[x_n,y]\to0$ $*$-strongly for every $y\in M$. Clearly $A_\omega(M)=\bigcap_\varphi A_{\varphi,\omega}$ over all faithful normal states, and $\mathcal I_\omega\subset A_\omega(M)$ (the proof of Theorem 5.1(1) works for every $\psi\in M_*^+$, and $M_*$ is spanned by $M_*^+$). Nothing in the definition depends on a choice of state.

Theorem 8.2. $M_\omega$ is always a finite von Neumann algebra. For every faithful normal state $\varphi$, $M_\omega$ is a von Neumann subalgebra of $M_{\varphi,\omega}$, namely the set of classes of sequences $(x_n)\in A_{\varphi,\omega}$ such that $[x_n,y]\to0$ strongly as $n\to\omega$ for all $y\in M$; and $\tau_{\varphi,\omega}$ restricts to a faithful normal tracial state on $M_\omega$.

Reference: [Connes 1974, Theorem 2.9].

Proof. Fix $\varphi$. Since $A_\omega(M)\subset A_{\varphi,\omega}$ and the ideal is $\mathcal I_\omega$ in both cases, $M_\omega$ is a C*-subalgebra of $M_{\varphi,\omega}$, and the description of it follows from Theorem 7.1 ((α) with $\mathcal S=M$, and (γ)). By Proposition 2.1 and Lemma 2.2 it suffices to show that $M_\omega$ has a $\|\cdot\|_2$-closed unit ball in $M_{\varphi,\omega}$, whose own unit ball is complete by Theorem 5.1.

Let $x^{(p)}$ be in the unit ball of $M_\omega$ and $x^{(p)}\to x\in M_{\varphi,\omega}$ in $\|\cdot\|_2$. Choose representing sequences $(x^{(p)}_n)\in A_\omega(M)$ and $(x_n)\in A_{\varphi,\omega}$. Let $\mathcal S=\{y\in M:\ y\xi_\varphi\in M'\xi_\varphi\}$; by (B5) it contains the entire analytic elements, so it is $\sigma$-weakly dense. For $y\in\mathcal S$ with $y\xi_\varphi=y'\xi_\varphi$, $y'\in M'$, and any $z\in M$, $\|zy\|_\varphi=\|zy'\xi_\varphi\|=\|y'z\xi_\varphi\|\le\|y'\|\,\|z\|_\varphi .$ With $z_n=x_n-x^{(p)}_n$, which represents $x-x^{(p)}$, we get $\lim_\omega\|[x_n,y]\|_\varphi\le\lim_\omega\|[x^{(p)}_n,y]\|_\varphi+\lim_\omega\big(\|z_ny\|_\varphi+\|yz_n\|_\varphi\big) \le0+(\|y'\|+\|y\|)\,\|x-x^{(p)}\|_2 ,$ using Theorem 7.1(γ) for $(x^{(p)}_n)$. Letting $p\to\infty$, $[x_n,y]\to0$ strongly as $n\to\omega$ for all $y\in\mathcal S$. Together with $\lim_\omega\|[x_n,\varphi]\|=0$, Theorem 7.1 ((α)⇒(γ), for the ultrafilter $\omega$) shows $(x_n)\in A_\omega(M)$, so $x\in M_\omega$. $\square$

Theorem 8.3. (1) Every $*$-isomorphism $\alpha\colon M\to M_1$ induces a $*$-isomorphism $\alpha_\omega\colon M_\omega\to M_{1,\omega}$, $\alpha_\omega[(x_n)]=[(\alpha(x_n))]$, and $(\beta\alpha)_\omega=\beta_\omega\alpha_\omega$.

(2) If $u\in M$ is unitary, $(\operatorname{Ad}u)_\omega=\mathrm{id}$. Hence $\theta\mapsto\theta_\omega$ is a homomorphism $\operatorname{Aut}M\to\operatorname{Aut}M_\omega$ that factors through $\operatorname{Out}M=\operatorname{Aut}M/\operatorname{Int}M$.

(3) For every faithful normal state $\varphi$ and $t\in\mathbb R$, $(\sigma^\varphi_t)_\omega=\mathrm{id}$. The same holds for the modular group of any faithful normal semifinite weight, whose automorphisms differ from $\sigma_t^\varphi$ by inner ones; so the modular homomorphism $\delta\colon\mathbb R\to\operatorname{Out}M$ acts trivially on $M_\omega$.

Proof. (1) For $\psi\in M_{1*}$, $[\alpha(x),\psi]=[x,\psi\circ\alpha]\circ\alpha^{-1}$, so $\|[\alpha(x),\psi]\|=\|[x,\psi\circ\alpha]\|$; and $\psi(\alpha(x)^*\alpha(x))=(\psi\circ\alpha)(x^*x)$. Hence $(\alpha(x_n))$ is $\omega$-central iff $(x_n)$ is, and lies in $\mathcal I_\omega$ iff $(x_n)$ does.

(2) For $(x_n)\in A_\omega(M)$, $ux_nu^*-x_n=[u,x_n]u^*$. By Theorem 7.1(γ), $[u,x_n]\to0$ $*$-strongly, and right multiplication by the fixed unitary $u^*$ preserves this (for $\psi\in M_*^+$, $\psi(uz^*zu^*)=(u^*\psi u)(z^*z)$ and $\psi(zu^*uz^*)=\psi(zz^*)$).

(3) If $(x_n)\in A_\omega(M)$, then $\|[x_n,\varphi]\|\to0$, so $\|\sigma^\varphi_t(x_n)-x_n\|^\sharp_\varphi\to0$ by Theorem 6.5, that is, $(\sigma^\varphi_t(x_n)-x_n)\in\mathcal I_\omega$. For another weight, $\sigma^\psi_t=\operatorname{Ad}(D\psi:D\varphi)_t\circ\sigma^\varphi_t$ by the Connes cocycle theorem, and (2) applies. $\square$

Example 8.4. If $M$ is finite-dimensional, $M_\omega=Z(M)$, the centre. Indeed, let $(x_n)$ be $\omega$-central with $\|x_n\|\le c$. The ball of $M$ is compact, so $x=\lim_\omega x_n$ exists in norm, and $[x,y]=\lim_\omega[x_n,y]=0$ for all $y$ (all Hausdorff vector topologies agree in finite dimensions), so $x\in Z(M)$. Then $x_n-x\to0$, so the class of $(x_n)$ is that of the constant sequence $x$; and constant central sequences are $\omega$-central. In particular $M_\omega=\mathbb C$ for a matrix algebra.

Example 8.5. If $M$ is abelian, then $x\psi=\psi x$ for all $x,\psi$, so $A_\omega(M)=\ell^\infty(M)$ and $M_\omega=\ell^\infty(M)/\mathcal I_\omega$, the ultrapower. For $M=L^\infty[0,1]$ with Lebesgue measure this is far from trivial: the Rademacher functions $r_n$ define a self-adjoint unitary class $r$ whose trace is $\lim_\omega\int_0^1r_n=0$, so $r\notin\mathbb C$.

For comparison with Section 3 we record the ultrapower of a general countably decomposable algebra. For finite $M$ it is Definition 3.4; in general it is only needed here as an ambient algebra.

Definition 8.6. Let $M$ be countably decomposable. Let $\mathcal M^\omega(M)=\{x\in\ell^\infty(M):\ x\mathcal I_\omega\subset\mathcal I_\omega,\ \mathcal I_\omega x\subset\mathcal I_\omega\}$, the largest subalgebra of $\ell^\infty(M)$ in which $\mathcal I_\omega$ is a two-sided ideal. It is a norm-closed $*$-subalgebra (as $\mathcal I_\omega$ is closed and self-adjoint). The ultrapower is $M^\omega=\mathcal M^\omega(M)/\mathcal I_\omega$. Constant sequences lie in $\mathcal M^\omega(M)$ (for $(z_n)\in\mathcal I_\omega$ and fixed $y$, $yz_n$ and $z_ny$ tend to $0$ $*$-strongly), which gives an injective $*$-homomorphism $M\to M^\omega$.

Proposition 8.7. (1) $A_\omega(M)\subset\mathcal M^\omega(M)$, and $M_\omega$ is a C*-subalgebra of $M'\cap M^\omega$.

(2) If $M$ is finite with faithful normal tracial state $\tau$, then $\mathcal M^\omega(M)=\ell^\infty(M)$, $M^\omega$ is the ultrapower of Definition 3.4, and $M_\omega=M'\cap M^\omega$.

Proof. (1) The first claim is the computation in Theorem 5.1(5), done with an arbitrary $\psi\in M_*^+$ in place of $\varphi$ (Lemma 1.1(e) holds for positive functionals up to the factor $\psi(1)^{1/2}$). The classes of $\omega$-central sequences commute with constants by Theorem 7.1(γ).

(2) For a trace, $\|xz\|_2\le\|x\|\|z\|_2$ and $\|zx\|_2\le\|z\|_2\|x\|$, so $\mathcal I_\omega$ is an ideal of $\ell^\infty(M)$; by Lemma 1.3 it is the ideal of Definition 3.1. Since $[x,\tau]=0$ for all $x$, Theorem 7.1 ((α) with $\varphi=\tau$, $\mathcal S=M$) says that $(x_n)$ is $\omega$-central exactly when $[x_n,y]\to0$ strongly along $\omega$ for all $y$, i.e. when its class commutes with $M$ in $M^\omega$. $\square$

Remark. For a general countably decomposable $M$, $M^\omega$ is a von Neumann algebra, and $M_\omega$ is the centralizer of the restriction of the ultrapower state to $M'\cap M^\omega$ [Ando–Haagerup 2014, Proposition 4.35]. The inclusion in (1) can be strict. Let $M=R_\lambda=\bigotimes_{n\ge1}(M_2(\mathbb C),\omega_\lambda)$ with $0<\lambda<1$ and $\omega_\lambda=\operatorname{Tr}(\operatorname{diag}((1+\lambda)^{-1},\lambda(1+\lambda)^{-1})\,\cdot\,)$, let $\varphi$ be the product state, and let $u_n$ be the matrix unit $e_{12}$ in the $n$-th tensor factor. Writing an element of $M$ as a $2\times2$ matrix over the other tensor factors, one finds that left and right multiplication by $u_n$ and by $u_n^*$ have norm at most $\lambda^{-1/2}$ for $\|\cdot\|_\varphi^\sharp$, uniformly in $n$; so $(u_n)\in\mathcal M^\omega(M)$. For each $y$ in the union of the finite tensor products, $[u_n,y]=0$ for large $n$, and by the same uniform bounds $[u_n,y]\to0$ $*$-strongly for every $y\in M$; so the class of $(u_n)$ lies in $M'\cap M^\omega$. But $\|[u_n,\varphi]\|=(1-\lambda)/(1+\lambda)$ for every $n$, and changing $(u_n)$ by an element of $\mathcal I_\omega$ does not change $\lim_\omega\|[u_n,\varphi]\|$, because $\|[z,\varphi]\|\le\|z\|_\varphi+\|z^*\|_\varphi$. So the class is not in $M_\omega$ [Ando–Haagerup 2014, Example 5.1]. We do not use these facts.

9. Spectral gap and tensor products

Definition 9.1. A spectral gap datum for a von Neumann algebra $Q$ consists of a faithful normal state $\varphi_0$, elements $b_1,\dots,b_p\in Q$, faithful positive normal functionals $\varphi_1,\dots,\varphi_p$ on $Q$, and a constant $K>0$, such that $\|x-\varphi_0(x)1\|_{\varphi_0}^2\le K^2\sum_{k=1}^p\|[x,b_k]\|_{\varphi_k}^2\qquad(x\in Q).\tag{9.1}$

A hypothesis of the form "$\|[x,b_k]\|_{\varphi_k}<\varepsilon$ for all $k$ implies $\|x-\varphi_0(x)\|_{\varphi_0}\le K\varepsilon$" gives (9.1) with the same $K$, since the maximum of the $\|[x,b_k]\|_{\varphi_k}$ is at most the square root of the sum of their squares. If $Q$ has a spectral gap datum, then $Q'\cap Q\subseteq\{b_k\}'\cap Q=\mathbb C1$: every central element commutes with each $b_k$, and the equality on the right holds by (9.1) and faithfulness of $\varphi_0$. So $Q$ is a factor.

Theorem 9.2. Let $Q_1$ have a spectral gap datum $(\varphi_0,b_k,\varphi_k,K)$, let $Q_2$ be any von Neumann algebra with a faithful normal state $\varphi$, and put $E_0=\mathrm{id}\otimes\varphi_0\colon Q_2\bar\otimes Q_1\to Q_2\otimes1$.

(a) For every $X\in Q_2\bar\otimes Q_1$, $\|X-E_0(X)\|_{\varphi\otimes\varphi_0}^2\le K^2\sum_{k=1}^p\|[X,1\otimes b_k]\|^2_{\varphi\otimes\varphi_k}.$ (b) If $Q_2$ is countably decomposable and $\pi(y)=y\otimes1$, then $\pi_\omega\colon[(y_n)]\mapsto[(y_n\otimes1)]$ is an isomorphism of $Q_{2,\omega}$ onto $(Q_2\bar\otimes Q_1)_\omega$.

Proof. (a) First let $X$ be in the algebraic tensor product, $X=\sum_{j=1}^mc_j\otimes y_j$. Replacing the $c_j$ by an orthonormal basis $a_1,\dots,a_r$ of their span for the inner product $\langle a,a'\rangle=\varphi(a'^*a)$ (positive definite as $\varphi$ is faithful), we may write $X=\sum_ja_j\otimes x_j$ with $\varphi(a_i^*a_j)=\delta_{ij}$. For any positive functional $\psi$ on $Q_1$ and $z_j\in Q_1$, $\|\sum_ja_j\otimes z_j\|^2_{\varphi\otimes\psi}=\sum_{i,j}\varphi(a_i^*a_j)\psi(z_i^*z_j)=\sum_j\|z_j\|_\psi^2$. Since $E_0(X)=\sum_ja_j\varphi_0(x_j)\otimes1$ and $[X,1\otimes b_k]=\sum_ja_j\otimes[x_j,b_k]$, (9.1) gives $\|X-E_0(X)\|^2_{\varphi\otimes\varphi_0}=\sum_j\|x_j-\varphi_0(x_j)\|^2_{\varphi_0}\le K^2\sum_{j,k}\|[x_j,b_k]\|^2_{\varphi_k} =K^2\sum_k\|[X,1\otimes b_k]\|^2_{\varphi\otimes\varphi_k}.$ For general $X$, (B2) gives a bounded net $X_i$ in the algebraic tensor product with $X_i\to X$ $\sigma$-strongly. For a bounded net $Z_i\to0$ strongly, a fixed $Y$ and a normal positive $\psi$ we have $\|Z_iY\|_\psi\to0$ and $\|YZ_i\|_\psi\to0$; so both commutator terms converge. Also $\|E_0(Z)\|_{\varphi\otimes\varphi_0}\le\|Z\|_{\varphi\otimes\varphi_0}$ by the Kadison–Schwarz inequality $E_0(Z)^*E_0(Z)\le E_0(Z^*Z)$ and $(\varphi\otimes\varphi_0)\circ E_0=\varphi\otimes\varphi_0$. So both sides pass to the limit.

(b) Write $Q=Q_2\bar\otimes Q_1$ and $\Phi=\varphi\otimes\varphi_0$, a faithful normal state (B7). If $(y_n)\in A_\omega(Q_2)$, then $\|[y_n\otimes1,\Phi]\|=\|[y_n,\varphi]\otimes\varphi_0\|\le\|[y_n,\varphi]\|\to0$, and for $c\otimes d$ in the algebraic tensor product (a $\sigma$-weakly dense set), $[y_n\otimes1,c\otimes d]=[y_n,c]\otimes d\to0$ strongly along $\omega$ (since $\|([y_n,c]\otimes d)\|_\Phi\le\|d\|\,\|[y_n,c]\|_\varphi$); by linearity the same holds throughout the algebraic tensor product. By Theorem 7.1, $(y_n\otimes1)\in A_\omega(Q)$. Since $\|y\otimes1\|^\sharp_\Phi=\|y\|^\sharp_\varphi$, the sequence $(y_n\otimes1)$ is in $\mathcal I_\omega$ iff $(y_n)$ is. So $\pi_\omega$ is a well-defined injective $*$-homomorphism.

For surjectivity, let $(X_n)\in A_\omega(Q)$. By Theorem 7.1(γ), $[X_n,1\otimes b_k]\to0$ $*$-strongly, so by (a) applied to $X_n$ and to $X_n^*$ (note $E_0(X^*)=E_0(X)^*$), $\|X_n-E_0(X_n)\|^\sharp_\Phi\to0$ along $\omega$. Write $E_0(X_n)=y_n\otimes1$; then $(y_n\otimes1)$ and $(X_n)$ have the same class, so $(y_n\otimes1)\in A_\omega(Q)$. For $\psi\in Q_{2*}$, restricting $[y_n\otimes1,\psi\otimes\varphi_0]$ to $Q_2\otimes1$ gives $[y_n,\psi]$, so $\|[y_n,\psi]\|\le\|[y_n\otimes1,\psi\otimes\varphi_0]\|\to0$. Hence $(y_n)\in A_\omega(Q_2)$ and $\pi_\omega$ is onto. $\square$

Note that $Q_1$ need not be assumed countably decomposable or a factor: both follow from the existence of the datum.

Corollary 9.3. If $Q_1$ and $Q_2$ have spectral gap data $(\varphi_0,b_k,\varphi_k,K)$ and $(\varphi_0',b'_l,\varphi'_l,K')$, then $Q_2\bar\otimes Q_1$ has the spectral gap datum consisting of $\varphi'_0\otimes\varphi_0$, the elements $1\otimes b_k$ with functionals $\varphi'_0\otimes\varphi_k$, the elements $b'_l\otimes1$ with functionals $\varphi'_l\otimes\varphi_0$, and the constant $K''=\sqrt2\max(K,K')$.

Proof. Let $X\in Q_2\bar\otimes Q_1$ and $E_0(X)=y\otimes1$. By Theorem 9.2(a), $\|X-y\otimes1\|^2\le K^2\sum_k\|[X,1\otimes b_k]\|^2_{\varphi'_0\otimes\varphi_k}$ (norm of $\varphi_0'\otimes\varphi_0$). Since $E_0$ is a $Q_2\otimes1$-bimodule map, $[y,b'_l]\otimes1=E_0([X,b'_l\otimes1])$, and $E_0$ is contractive for $\|\cdot\|_{\varphi'_l\otimes\varphi_0}$ (Kadison–Schwarz, and $(\varphi'_l\otimes\varphi_0)\circ E_0=\varphi_l'\otimes\varphi_0$). So $\|y-\varphi_0'(y)\|^2_{\varphi'_0}\le K'^2\sum_l\|[y,b_l']\|_{\varphi'_l}^2\le K'^2\sum_l\|[X,b'_l\otimes1]\|^2_{\varphi'_l\otimes\varphi_0}$. As $(\varphi_0'\otimes\varphi_0)(X)=\varphi_0'(y)$, the triangle inequality and $(s+t)^2\le2s^2+2t^2$ give the claim. $\square$

Example 9.4 (a spectral gap datum for $B(\ell^2(\mathbb Z))$). Let $F=B(\ell^2(\mathbb Z))$ with matrix units $e_{ij}$, and for $x\in F$ write $x_{ij}=\langle x\delta_j,\delta_i\rangle$. For positive weights $\mu=(\mu_j)$ with $\sum\mu_j<\infty$, the functional $\varphi_\mu=\mathrm{Tr}(\rho_\mu\,\cdot\,)$, $\rho_\mu=\sum_j\mu_je_{jj}$, is faithful and normal, and $\|y\|^2_{\varphi_\mu}=\sum_{i,j}\mu_j|y_{ij}|^2$. Take $\mu^{(0)}_j=c_0\,2^{-5|j|}\ \ (c_0=\tfrac{31}{33},\ \text{so }\varphi_0\text{ is a state}),\qquad\mu^{(1)}_j=\mu^{(2)}_j=2^{-|j|},$ the diagonal operator $D=\sum_j\lambda_je_{jj}$ with $\lambda_j=2^{-m(j)}$, $m(j)=2j$ for $j\ge0$ and $m(j)=2|j|-1$ for $j<0$, and the bilateral shift $S\delta_j=\delta_{j+1}$. Then $\|x-\varphi_0(x)\|^2_{\varphi_0}\le4\|[x,D]\|^2_{\varphi_1}+\tfrac{16}{15}\|[x,S]\|^2_{\varphi_2},$ so $(\varphi_0;D,S;\varphi_1,\varphi_2;K=2)$ is a spectral gap datum.

Proof. The map $m$ is a bijection of $\mathbb Z$ onto $\{0,1,2,\dots\}$ with $m(j)\le2|j|$, so the $\lambda_j$ are distinct powers of $2$ and $|\lambda_i-\lambda_j|\ge2^{-m(j)-1}\ge2^{-2|j|-1}$ for $i\ne j$. The matrix of $x-\varphi_0(x)1$ has off-diagonal entries $x_{ij}$ and diagonal entries $d_j-\bar d$, where $d_j=x_{jj}$ and $\bar d=\sum_j\mu^{(0)}_jd_j=\varphi_0(x)$.

Off-diagonal part. $[x,D]_{ij}=x_{ij}(\lambda_j-\lambda_i)$, and $\mu^{(1)}_j|\lambda_j-\lambda_i|^2\ge2^{-|j|}2^{-4|j|-2}\ge\frac14\mu^{(0)}_j$. Hence $\sum_{i\ne j}\mu_j^{(0)}|x_{ij}|^2\le4\|[x,D]\|^2_{\varphi_1}$.

Diagonal part. As $\bar d$ is the $\mu^{(0)}$-mean, $\sum_j\mu^{(0)}_j|d_j-\bar d|^2\le\sum_j\mu_j^{(0)}|d_j-d_0|^2$. With $\Delta_k=d_{k+1}-d_k$ and the Cauchy–Schwarz inequality, $|d_j-d_0|^2\le|j|\sum_k|\Delta_k|^2$, the sum over the $|j|$ indices $k$ between $0$ and $j$ ($0\le k<j$ if $j>0$, $j\le k<0$ if $j<0$). Exchanging sums and using $|j|2^{-5|j|}\le2^{-4|j|}$, $\sum_j\mu^{(0)}_j|d_j-d_0|^2\le\sum_{k\ge0}|\Delta_k|^2\sum_{j>k}2^{-4j}+\sum_{k<0}|\Delta_k|^2\sum_{j\le k}2^{-4|j|} \le\tfrac{16}{15}\sum_k2^{-4|k|}|\Delta_k|^2 .$ Now $(S^*xS)_{ij}=x_{i+1,j+1}$, so $\Delta_k$ is the $(k,k)$ entry of $S^*xS-x=S^*[x,S]$, and $\sum_k2^{-4|k|}|\Delta_k|^2\le\|S^*[x,S]\|^2_{\varphi_2}\le\|[x,S]\|^2_{\varphi_2}$, since $S$ is unitary. $\square$

In particular (Theorem 9.2(b) with $Q_2=\mathbb C$) $B(H)_\omega=\mathbb C$ for separable infinite-dimensional $H$; compare Example 8.4.

10. A II₁ factor without central sequences with $\lambda$ in its fundamental group

For a II₁ factor $N$ with trace $\tau$, the fundamental group $F(N)$ is the set of ratios $\tau(q)/\tau(p)$ over nonzero projections $p,q\in N$ with $pNp\cong qNq$; this is the usual group $\{t>0:N^t\cong N\}$ of amplification constants. A II₁ factor $N$ has property Γ when $N_\omega\ne\mathbb C$. Fix $\lambda\in(0,1)$.

10.1 A ping-pong estimate

Write $\mathbb F_2$ for the free group on $a,b$. For $\epsilon\in\{a,a^{-1},b,b^{-1}\}$ let $W_\epsilon$ be the set of reduced words whose first letter is $\epsilon$, and put $W_A=W_a\cup W_{a^{-1}}$, $W_B=W_b\cup W_{b^{-1}}$.

Lemma 10.1. Let $\mathbb F_2$ act on a set $\Sigma$ by $\rho$, and suppose $\Sigma\supset A_+\sqcup A_-\sqcup B_+\sqcup B_-$ (disjoint), with $A=A_+\cup A_-$, $B=B_+\cup B_-$, such that $\rho_a(B)\subset A_+,\quad\rho_{a^{-1}}(B)\subset A_-,\quad\rho_b(A)\subset B_+,\quad\rho_{b^{-1}}(A)\subset B_- .$ Let $\mathcal K=\bigoplus_{\sigma\in\Sigma}K_\sigma$ be a direct sum of Hilbert spaces and $T_a,T_b$ unitaries on $\mathcal K$ with $T_s(K_\sigma)=K_{\rho_s\sigma}$ for $s\in\{a,b\}$ and all $\sigma$. For $f\in\mathcal K$ put $\delta=\max_{s\in\{a,b\}}\|T_sf-f\|$. Then:

(i) $\sum_{\sigma\in A\cup B}\|f_\sigma\|^2\le8\|f\|\delta$;

(ii) if $\Sigma=A\cup B$, then $\|f\|\le8\delta$;

(iii) if $\Sigma=A\cup B\cup R$ with $\rho_a(R)\subset A_+$, then $\|f\|\le18\delta$.

Proof. Let $\mu(E)=\sum_{\sigma\in E}\|f_\sigma\|^2$ for $E\subset\Sigma$. Since $T_s$ maps $K_\sigma$ isometrically onto $K_{\rho_s\sigma}$, the vector $g=T_sf$ has $\|g_{\rho_s\sigma}\|=\|f_\sigma\|$, so $\sum_{\sigma\in E}\|g_\sigma\|^2=\mu(\rho_s^{-1}E)$. Hence, with $P_E$ the projection onto $\bigoplus_{\sigma\in E}K_\sigma$, $|\mu(\rho_s^{-1}E)-\mu(E)|=\big|\|P_Eg\|^2-\|P_Ef\|^2\big|\le\|g-f\|\big(\|g\|+\|f\|\big)\le2\|f\|\delta ,$ and replacing $E$ by $\rho_sE$, also $|\mu(\rho_sE)-\mu(E)|\le2\|f\|\delta$. Therefore $\mu(B)\le\mu(\rho_aB)+2\|f\|\delta\le\mu(A_+)+2\|f\|\delta$, and likewise $\mu(B)\le\mu(A_-)+2\|f\|\delta$, $\mu(A)\le\mu(B_\pm)+2\|f\|\delta$. Adding the four inequalities, $2\mu(A)+2\mu(B)\le\mu(A)+\mu(B)+8\|f\|\delta$, which is (i). (ii) follows from $\|f\|^2=\mu(A\cup B)$. For (iii), $\mu(R)\le\mu(\rho_aR)+2\|f\|\delta\le\mu(A)+2\|f\|\delta\le10\|f\|\delta$, so $\|f\|^2\le18\|f\|\delta$. $\square$

Lemma 10.2. The hypotheses of Lemma 10.1 hold in the following two cases.

(1) $\Sigma=\mathbb F_2\setminus\{e\}$ with the conjugation action $\rho_s(g)=sgs^{-1}$, and $A_\pm=W_{a^{\pm1}}$, $B_\pm=W_{b^{\pm1}}$; here $\Sigma=A\cup B$.

(2) $\Sigma$ is any set on which $\mathbb F_2$ acts freely, with the sets defined through a choice of a set $R$ of orbit representatives: $A_\pm=\{gr:g\in W_{a^{\pm1}},r\in R\}$, $B_\pm=\{gr:g\in W_{b^{\pm1}},r\in R\}$; here $\Sigma=A\cup B\cup R$ and $\rho_a(R)\subset A_+$.

Proof. (1) Every $g\ne e$ starts with one of the four letters, so $\Sigma=A\cup B$. Let $g\in B$. The product $ag$ is reduced and starts with $a$. If $g$ does not end with $a$, then $aga^{-1}$ is reduced and starts with $a$; if $g=g'a$, then $g'$ is nonempty and starts with $b^{\pm1}$, and $aga^{-1}=ag'$ is reduced and starts with $a$. So $\rho_a(B)\subset A_+$; the other three inclusions are proved in the same way.

(2) Freeness means that $(g,r)\mapsto gr$ is a bijection $\mathbb F_2\times R\to\Sigma$, so the four sets are disjoint and $\Sigma=A\cup B\cup R$. If $g\in W_B$, then $ag\in W_a$, which gives $\rho_a(B)\subset A_+$; similarly for the others, and $\rho_a(r)=ar\in A_+$. $\square$

10.2 The example

Let $B=\bar\bigotimes_{g\in\mathbb F_2}(M_2(\mathbb C),\omega_\lambda)$ with product state $\omega$ (B10), let $\alpha$ be the action of $\mathbb F_2$ by left translation of the tensor factors (the Bernoulli shift), and let $P=B\rtimes_\alpha\mathbb F_2$ with the state $\varphi=\omega\circ E$ (B9). Write $u_g$ for the canonical unitaries and $\xi=\xi_\varphi$.

Theorem 10.3. For every $x\in P$, $\|x-\varphi(x)1\|_\varphi\le20\max\big(\|[x,u_a]\|_\varphi,\|[x,u_b]\|_\varphi\big).\tag{10.1}$ Consequently $P$ is a factor, $u_a,u_b$ lie in the centralizer $N_0=P_\varphi$, and $N_0$ is a II₁ factor with separable predual such that:

(a) $\lambda\in F(N_0)$;

(b) $N_{0,\omega}=\mathbb C$, so $N_0$ does not have property Γ;

Reference: [Connes 1974, Theorem 2.10(a)].

Proof. The estimate. By (B9), $\sigma^\varphi_t(u_g)=u_g$, so $u_g\in P_\varphi$ and $\varphi\circ\operatorname{Ad}u_g=\varphi$ (B6). Hence $\Theta_s(y\xi)=u_syu_s^*\xi$ defines unitaries $\Theta_s$ of $L^2(P,\varphi)$, with $\Theta_s\xi=\xi$, and $\|\Theta_s(x\xi)-x\xi\|=\|(u_sxu_s^*-x)\xi\|=\|[u_s,x]u_s^*\|_\varphi=\|[u_s,x]\|_\varphi .$ Identify $L^2(P,\varphi)$ with $\ell^2(\mathbb F_2,L^2(B,\omega))$ as in (B9), $x\xi\mapsto(x_g\xi_\omega)_g$ with $x_g=E(xu_g^*)$. Using $E(u_syu_s^*)=\alpha_s(E(y))$ and $u_s^*u_g^*u_s=u_{s^{-1}gs}^*$ one finds $(u_sxu_s^*)_g=\alpha_s(x_{s^{-1}gs})$. So, with $V_s$ the unitary $b\xi_\omega\mapsto\alpha_s(b)\xi_\omega$ of $L^2(B,\omega)$, $\Theta_s$ maps the $g$-th summand to the $sgs^{-1}$-th by $V_s$. The decomposition $\ell^2(\mathbb F_2,L^2(B))=\big(L^2_0(B)\,\delta_e\big)\oplus\mathbb C\,\xi\oplus\bigoplus_{g\ne e}L^2(B)\,\delta_g,\qquad L^2_0(B)=L^2(B)\ominus\mathbb C\xi_\omega,$ is invariant under $\Theta_a,\Theta_b$, because $V_s$ fixes $\xi_\omega$.

Let $f=(x-\varphi(x))\xi$. Its component in $\mathbb C\xi$ is $0$, since $\varphi(x)=\omega(x_e)$. Put $\delta=\max_s\|[x,u_s]\|_\varphi=\max_s\|\Theta_sf-f\|$; the components of $\Theta_sf-f$ in the invariant summands also have norm at most $\delta$. On $\bigoplus_{g\ne e}$, $\Theta_s$ is as in Lemma 10.1 for the conjugation action, so by Lemma 10.2(1) the component $f_1$ of $f$ there has $\|f_1\|\le8\delta$. On $L^2_0(B)$, by (B10), $L^2_0(B)=\bigoplus_FK_F$ over finite nonempty $F\subset\mathbb F_2$, and $V_s$ maps $K_F$ onto $K_{sF}$. The group $\mathbb F_2$ acts freely on such sets $F$: if $gF=F$, then $g$ permutes the finite set $F$, so $g^nh=h$ for some $n\ge1$ and $h\in F$, whence $g^n=e$ and $g=e$ since $\mathbb F_2$ is torsion-free. By Lemma 10.2(2) and Lemma 10.1(iii), the component $f_0$ of $f$ there has $\|f_0\|\le18\delta$. Hence $\|f\|^2\le(18^2+8^2)\delta^2\le400\,\delta^2$.

Consequences. If $z$ is central in $P$, or just commutes with $u_a$ and $u_b$, then (10.1) gives $z=\varphi(z)1$. So $P$ is a factor and $(P_\varphi)'\cap P\subset\{u_a,u_b\}'\cap P=\mathbb C$; in particular $N_0=P_\varphi$ is a factor. By (B6), $\varphi|_{N_0}$ is a faithful normal tracial state, so $N_0$ is finite; it is infinite-dimensional (it contains the diagonal matrices of every tensor factor, since $\sigma^\omega$ fixes them), so it is a II₁ factor (B8). $P$ acts on the separable space $L^2(P,\varphi)$, so $N_0$ has separable predual.

(a) Let $v$ be the matrix unit $e_{21}$ in the tensor factor at $g=e$. By (B9), (B10), $\sigma_t^\varphi(v)=\rho^{it}e_{21}\rho^{-it}=(\rho_2/\rho_1)^{it}v=\lambda^{it}v$. Then $p=v^*v$ and $q=vv^*$ are projections in $N_0$, and $\gamma(y)=vyv^*$ maps $pN_0p$ into $qN_0q$, because $\sigma^\varphi_t(vyv^*)=\lambda^{it}v\,y\,\lambda^{-it}v^*=vyv^*$; its inverse is $z\mapsto v^*zv$. So $pN_0p\cong qN_0q$ and $\tau(q)/\tau(p)=\omega_\lambda(e_{22})/\omega_\lambda(e_{11})=\lambda$. Hence $\lambda\in F(N_0)$.

(c) Let $s_n\in R$ be as in Example 3.5. The sequence $(1\otimes s_n)$ in $N_0\bar\otimes R$ commutes with $\tau\otimes\mathrm{tr}$ and, eventually, with each element of the $\sigma$-weakly dense set $N_0\odot R_{\mathrm{alg}}$ (algebraic tensor product with the union of the finite tensor products). By Theorem 7.1 it is $\omega$-central, and its class is a self-adjoint unitary of trace $0$, so $(N_0\bar\otimes R)_\omega\ne\mathbb C$. Since $N_{0,\omega}=\mathbb C$ and asymptotic centralizers of isomorphic algebras are isomorphic (Theorem 8.3(1)), $N_0\not\cong N_0\bar\otimes R$. $\square$

Remark. In the language of the asymptotic ratio set of II₁ factors, (c) says that $1\notin r_\infty(N_0)$. The construction in [Connes 1974] used the centralizer of a periodic state on Pukánszky's factor, the group measure space factor of a group of nonsingular transformations generated by the Bernoulli shift of a free group and by flips of single coordinates, with a spectral estimate for it; the Bernoulli crossed product above makes the spectral estimate elementary.

Theorem 10.4. There is a II∞ factor $N$ with separable predual and two automorphisms $\theta,\theta'$ of $N$, both scaling the trace by $\lambda$, whose classes in $\operatorname{Out}N$ are not conjugate. In particular, at least two conjugacy classes of $\operatorname{Out}N$ consist of elements with module $\lambda$.

Proof. Let $N_0$, $\tau$, $p,q,\gamma$ be as in Theorem 10.3, $F=B(\ell^2(\mathbb Z))$, $Q_1=N_0\bar\otimes F$ with trace $\mathrm{Tr}_1=\tau\otimes\mathrm{Tr}$, and $N=Q_1\bar\otimes R$ with trace $\mathrm{Tr}_1\otimes\mathrm{tr}$.

Spectral gap for $Q_1$. $N_0$ has the datum $(\tau;u_a,u_b;\tau,\tau;K=20)$, by (10.1) and the remark after Definition 9.1, and $F$ has the datum of Example 9.4. By Corollary 9.3, $Q_1$ has a spectral gap datum. By Theorem 9.2(b), applied after the flip isomorphism $Q_1\bar\otimes R\cong R\bar\otimes Q_1$, $x\mapsto1\otimes x$ induces an isomorphism $\pi_\omega\colon R_\omega\to N_\omega$.

An automorphism of $Q_1$ scaling the trace. The projections $p\otimes1$ and $q\otimes1$ of the countably decomposable II∞ factor $Q_1$ are infinite, so by (B8) there are isometries $w_p,w_q\in Q_1$ with $w_pw_p^*=p\otimes1$, $w_qw_q^*=q\otimes1$. The map $\Gamma=\gamma\otimes\mathrm{id}$ carries $(p\otimes1)Q_1(p\otimes1)=pN_0p\bar\otimes F$ isomorphically onto $qN_0q\bar\otimes F$. By uniqueness of the trace on the II₁ factor $pN_0p$ (B8), $\tau\circ\gamma=c\,\tau$ there with $c=\tau(q)/\tau(p)=\lambda$, hence $\mathrm{Tr}_1\circ\Gamma=\lambda\,\mathrm{Tr}_1$ (B7). Put $\theta_1(X)=w_q^*\,\Gamma(w_pXw_p^*)\,w_q$. This is an automorphism of $Q_1$; for $X\ge0$, $\mathrm{Tr}_1(w_pXw_p^*)=\mathrm{Tr}_1(X^{1/2}w_p^*w_pX^{1/2})=\mathrm{Tr}_1(X)$, and similarly for $w_q$, so $\mathrm{Tr}_1\circ\theta_1=\lambda\mathrm{Tr}_1$.

The two automorphisms. Let $\theta=\theta_1\otimes\mathrm{id}_R$. Let $\beta$ be the automorphism of $R$ that exchanges the tensor factors $2n-1$ and $2n$ for every $n$ (B10), and $\theta'=\theta\circ(\mathrm{id}_{Q_1}\otimes\beta)$. Both scale the trace of $N$ by $\lambda$, since $\mathrm{tr}\circ\beta=\mathrm{tr}$. As $\theta\circ\pi=\pi$, Theorem 8.3(1) gives $\theta_\omega\circ\pi_\omega=\pi_\omega$, so $\theta_\omega=\mathrm{id}$ because $\pi_\omega$ is onto. On the other hand $\theta'_\omega\circ\pi_\omega=\pi_\omega\circ\beta_\omega$, and $\beta_\omega\ne\mathrm{id}$: with $s_n$ as in Example 3.5, the sequence $(s_{2n})$ is central, $\beta(s_{2n})=s_{2n-1}$, and $\|s_{2n-1}-s_{2n}\|_2^2=2-2\,\mathrm{tr}(s_{2n-1}s_{2n})=2$. So $\theta'_\omega\ne\mathrm{id}$.

Non-conjugacy. If $\theta'=\alpha\theta\alpha^{-1}\operatorname{Ad}u$ for some $\alpha\in\operatorname{Aut}N$ and unitary $u$, then by Theorem 8.3, $\theta'_\omega=\alpha_\omega\theta_\omega\alpha_\omega^{-1}=\mathrm{id}$, a contradiction. Inner automorphisms preserve the trace, so the module is defined on $\operatorname{Out}N$, and both classes have module $\lambda$. $\square$

11. Factors of type III₀ have an asymptotic centralizer with nontrivial centre

The mechanism is this. A state with a spectral gap at $1$ forces every $\omega$-central sequence into its centralizer (Theorem 6.6). For a factor of type III₀ the centralizer of a suitable such state has a large centre, on which the rest of the factor acts through a single ergodic transformation. A transformation of the integers is never "strongly ergodic": it has asymptotically invariant sets of measure $\frac12$. These produce central elements of $M_\omega$.

Lemma 11.1. Let $(X,\mu)$ be a standard probability space without atoms, $T$ an ergodic Borel bijection of $X$ with $\mu\circ T$ equivalent to $\mu$, and $\nu$ a finite measure absolutely continuous with respect to $\mu$. There are Borel sets $E_n$ with $\nu(E_n)=\frac12\nu(X)$ and $\mu(E_n\,\triangle\,T^{-1}E_n)\to0$.

Proof. Since $\mu$ has no atoms, there are Borel sets $A_n$ with $0<\mu(A_n)\le1/n$. Fix $n$ and write $A=A_n$. By (B14) the set $X_0$ of points whose orbit meets $A$ infinitely often in both directions has full measure, and it is $T$-invariant. For $x\in X_0$ let $k(x)=\min\{k\ge0:T^{-k}x\in A\}$ and $s(x)=T^{-k(x)}x\in A$, the last visit to $A$ up to time $0$. If $Tx\notin A$ then $k(Tx)=k(x)+1$ and $s(Tx)=s(x)$. For Borel $D\subset A$ put $E_D=\{x\in X_0:s(x)\in D\}$. If exactly one of $x,Tx$ lies in $E_D$, then $s(Tx)\ne s(x)$, so $Tx\in A$. Hence $E_D\,\triangle\,T^{-1}E_D\subset T^{-1}A\cup(X\setminus X_0)$, and $\mu(E_D\triangle T^{-1}E_D)\le\mu(T^{-1}A)$.

The function $D\mapsto\nu(E_D)$ is a finite measure on $A$, with total mass $\nu(X_0)=\nu(X)$, and it is absolutely continuous with respect to $\mu$: indeed $E_D=\bigsqcup_{k\ge0}T^k(D\cap A^{(k)})$, where $A^{(k)}$ is the set of $y\in A$ with $T^jy\notin A$ for $1\le j\le k$, and each $T^k$ preserves null sets. Identify $X$ with a Borel subset of $[0,1]$ by a Borel isomorphism. The function $r\mapsto\nu(E_{A\cap[0,r]})$ is nondecreasing from $0$ to $\nu(X)$, and continuous, because $\mu$ has no atoms (so $\mu(A\cap[r,r'])\to0$ as $r'-r\to0$) and a finite measure absolutely continuous with respect to $\mu$ is small on sets of small $\mu$-measure. By the intermediate value theorem some $D=A\cap[0,r]$ has $\nu(E_D)=\frac12\nu(X)$; let $E_n=E_D$. Finally $\mu(T^{-1}A_n)\to0$ as $\mu(A_n)\to0$, since $\mu\circ T^{-1}$ is absolutely continuous with respect to $\mu$. $\square$

Theorem 11.2. Let $M$ be a type III₀ factor whose predual is separable. Then $M_\omega\ne\mathbb C$, and indeed the centre of $M_\omega$ is not $\mathbb C$.

Reference: [Connes 1974, Theorem 2.12].

Proof. Take $N,\tau,E,U,C,\theta$ as in (B12), and a projection $e\in N$ with $0<\tau(e)<\infty$ (it exists as $\tau$ is semifinite). Let $M_e=eMe$ and $\varphi_e$ the state of (B12)(c). Since $M_e\cong M$ (B8), it suffices by Theorem 8.3(1) to show that $Z((M_e)_\omega)\ne\mathbb C$.

Step 1: $(M_e)_\omega$ is represented in $eNe$. By (B12)(c), $1$ is isolated in $\operatorname{Sp}\Delta_{\varphi_e}$ and the centralizer of $\varphi_e$ is $eNe$. If $(x_n)$ is $\omega$-central in $M_e$, then $\|[x_n,\varphi_e]\|\to0$, and Theorem 6.6(2) gives $x_n-E_{\varphi_e}(x_n)\in\mathcal I_\omega$. So every element of $(M_e)_\omega$ has a representing sequence in $eNe$.

Step 2: asymptotically invariant central projections. Choose a faithful normal state $\psi_N$ of $N$ and realize $C=L^\infty(X,\mu)$, with $\mu$ given by $\psi_N|_C$, and $\theta|_C(f)=f\circ T^{-1}$ (B11). As $C$ has no minimal projections, $\mu$ has no atoms, and $T$ is ergodic by (B12)(b). The positive normal functional $c\mapsto\tau(ce)$ on $C$ is a finite measure $\nu$ absolutely continuous with respect to $\mu$, with $\nu(X)=\tau(e)$. Lemma 11.1 gives projections $c_n=\chi_{E_n}\in C$ with $\tau(c_ne)=\frac12\tau(e)$ and $\mu(E_n\triangle T^{-1}E_n)\to0$. Then $\theta(c_n)=\chi_{TE_n}$, and for every $m\in\mathbb Z$, $\mu(T^mE_n\triangle E_n)\to0$: for $m=1$ because $TE_n\triangle E_n=T(E_n\triangle T^{-1}E_n)$ and $\mu\circ T\ll\mu$, and for general $m$ by the triangle inequality for symmetric differences and the same absolute continuity. With the faithful normal state $\psi_0=\psi_N\circ E$ on $M$, $\|\theta^m(c_n)-c_n\|^2_{\psi_0}=\mu(T^mE_n\triangle E_n)\to0$, so $\theta^m(c_n)-c_n\to0$ strongly in $M$ (Lemma 1.3).

Step 3: the sequence $v_n=c_ne$ is $\omega$-central in $M_e$. We verify Theorem 7.1(α) for $M_e$, the faithful state $\varphi_e$ and the $\sigma$-weakly dense subset $\mathcal S=\operatorname{span}\{eaU^me:a\in N,m\in\mathbb Z\}$ (dense by (B12)(a), as $y\mapsto eye$ is $\sigma$-weakly continuous onto $M_e$). First, $v_n$ lies in $eNe$, the centralizer of $\varphi_e$, so $[v_n,\varphi_e]=0$. Next, $c_n$ is central in $N$, so it commutes with $e$ and $a$, and $U^mc_n=\theta^m(c_n)U^m$; hence $[v_n,eaU^me]=ea\,\big(c_n-\theta^m(c_n)\big)\,U^me .$ For $\chi\in(M_e)_*^+$, extended to $M$ by $\chi(e\,\cdot\,e)$, and $d_n=c_n-\theta^m(c_n)$, we get $\chi\big((ead_nU^me)^*(ead_nU^me)\big)\le\|a\|^2\,\chi'(d_n^*d_n)$ with the normal positive functional $\chi'=\chi(eU^{-m}\,\cdot\,U^me)$, which tends to $0$ by Step 2. So $[v_n,y]\to0$ strongly for $y\in\mathcal S$, and $(v_n)$ is $\omega$-central.

Step 4: conclusion. Let $V\in(M_e)_\omega$ be the class of $(v_n)$. It is a projection with $\lim_\omega\varphi_e(v_n)=\tau(c_ne)/\tau(e)=\frac12$, so $V\notin\mathbb C$ (a scalar projection is $0$ or $1$). Let $Y\in(M_e)_\omega$; by Step 1 it has a representing sequence $(y_n)$ in $eNe$. Since $c_n\in C$, $v_n=c_ne$ commutes with every element of $eNe$: $c_ne\cdot eae=ec_nae=eac_ne=eae\cdot c_ne$. So $y_nv_n=v_ny_n$ for all $n$, and $YV=VY$. Hence $V$ is a nonscalar element of the centre of $(M_e)_\omega$. $\square$

Remark. Separability of the predual is used to quote the discrete decomposition (B12) and to realize the centre $C$ and the transformation $\theta$ on a standard measure space. Theorem 11.2 is the reason, developed in the lesson "Full factors", why factors of type III₀ are never full.

12. Exercises

Exercise 12.1. Let $M=M_1\oplus M_2$ with $M_1,M_2$ countably decomposable. Show that $M_\omega\cong M_{1,\omega}\oplus M_{2,\omega}$, and that the centre $Z(M)$, embedded by constant sequences, lies in the centre of $M_\omega$.

Solution. Every $\psi\in M_*$ is $\psi_1\oplus\psi_2$, $\psi(y_1\oplus y_2)=\psi_1(y_1)+\psi_2(y_2)$, with $\|\psi\|=\|\psi_1\|+\|\psi_2\|$. For $x=x_1\oplus x_2$ one has $[x,\psi]=[x_1,\psi_1]\oplus[x_2,\psi_2]$, so $\|[x,\psi]\|=\|[x_1,\psi_1]\|+\|[x_2,\psi_2]\|$. Hence $(x_n)$ is $\omega$-central iff both $(x_{1,n})$ and $(x_{2,n})$ are; similarly $\psi(x_n^*x_n)=\psi_1(x_{1,n}^*x_{1,n})+\psi_2(x_{2,n}^*x_{2,n})$ shows that $\mathcal I_\omega(M)=\mathcal I_\omega(M_1)\oplus\mathcal I_\omega(M_2)$. Passing to quotients gives the isomorphism. If $z\in Z(M)$, then $z\psi=\psi z$ for every $\psi$, so the constant sequence $z$ is $\omega$-central, and it commutes exactly with every representing sequence, so its class is central in $M_\omega$. The embedding is injective because constant sequences lie in $\mathcal I_\omega$ only when they vanish.

Exercise 12.2. Let $M=B(\ell^2(\mathbb N))$ with matrix units $e_{ij}$. Show that $(e_{nn})_n\in\mathcal I_\omega$ but $(e_{1n}e_{nn})_n=(e_{1n})_n\notin\mathcal I_\omega$. Conclude that $\mathcal I_\omega$ is not a left ideal of $\ell^\infty(M)$, and that $(e_{1n})$ is not in $\mathcal M^\omega(M)$ (Definition 8.6).

Solution. For $\eta\in\ell^2$, $e_{nn}\eta=\eta_n\delta_n\to0$, and $e_{nn}$ is self-adjoint, so $e_{nn}\to0$ $*$-strongly (along any filter containing the tails, in particular along $\omega$). But $e_{1n}^*\delta_1=e_{n1}\delta_1=\delta_n$ has norm $1$ for all $n$, so $(e_{1n})$ does not tend to $0$ $*$-strongly: with the vector state $\psi=\omega_{\delta_1}$, $\psi(e_{1n}e_{1n}^*)=1$. Since $(e_{1n})\in\ell^\infty(M)$ and $(e_{nn})\in\mathcal I_\omega$ but their product is not in $\mathcal I_\omega$, the sequence $(e_{1n})$ does not multiply $\mathcal I_\omega$ into itself. By contrast, by Proposition 8.7 every $\omega$-central sequence does.

Exercise 12.3. Let $R$ and $s_n$ be as in Example 3.5, and $s\in R^\omega$ the class of $(s_n)$. Show that $\tau_\omega(sy)=0$ for every $y\in R\subset R^\omega$. Deduce that $s\notin R$, and that $s$ and $R$ together generate a von Neumann subalgebra of $R^\omega$ isomorphic to $R\bar\otimes\mathbb C^2$.

Solution. For $y$ in the algebraic tensor product of the first $m$ factors and $n>m$, $\tau(s_ny)=\mathrm{tr}(s_n)\tau(y)=0$ by the product structure of the trace. By density in $\|\cdot\|_2$ and $|\tau(s_n(y-y'))|\le\|y-y'\|_2$, $\lim_\omega\tau(s_ny)=0$ for every $y\in R$. If $s$ were in $R$, taking $y=s$ would give $1=\tau_\omega(s^2)=0$. Now $s$ commutes with $R$ (it lies in $R_\omega$), $s^2=1$, so $e=\frac12(1+s)$ is a projection in $R'\cap R^\omega$, with $\tau_\omega(ey)=\frac12\tau(y)$ for $y\in R$. The maps $y\mapsto ye$ and $y\mapsto y(1-e)$ are then trace-scaling $*$-homomorphisms of $R$ (by $\frac12$), so they are injective and normal, and $R\vee\{s\}''=Re+R(1-e)\cong R\oplus R=R\bar\otimes\mathbb C^2$.

Exercise 12.4. Take a faithful normal state $\varphi$ and a unitary $v\in M_\varphi$. Show directly that $\sigma_t^\varphi(v)=v$, and check that Corollary 6.3 is consistent with this. Then let $M=M_2(\mathbb C)$, $\varphi=\mathrm{Tr}(\rho\,\cdot\,)$ with $\rho=\operatorname{diag}(p,1-p)$, $p\ne\frac12$, and $v_\varepsilon=\exp(i\varepsilon(e_{12}+e_{21}))$. Show that $\|[v_\varepsilon,\varphi]\|$ is of order $\varepsilon$ while $\|\sigma^\varphi_t(v_\varepsilon)-v_\varepsilon\|^\sharp_\varphi$ is of order $\varepsilon$ (for fixed $t$ with $(p/(1-p))^{it}\ne1$), so the square root in Corollary 6.3 is not attained for small perturbations of a single unitary; its role is to give a bound that is uniform over all $M$.

Solution. If $v\in M_\varphi$, then $\sigma_t^\varphi(v)=v$ by definition (B6) and $[v,\varphi]=0$; both sides of Corollary 6.3 vanish. For the example, $h=e_{12}+e_{21}$ satisfies $\sigma_t(h)=\kappa^{it}e_{12}+\kappa^{-it}e_{21}$ with $\kappa=p/(1-p)$, so $\sigma_t(v_\varepsilon)-v_\varepsilon=i\varepsilon(\sigma_t(h)-h)+O(\varepsilon^2)$ and $\|\sigma_t(v_\varepsilon)-v_\varepsilon\|^\sharp_\varphi=\varepsilon\|\sigma_t(h)-h\|^\sharp_\varphi+O(\varepsilon^2)$, with $\|\sigma_t(h)-h\|_\varphi^\sharp=|\kappa^{it}-1|\big(\|e_{12}\|_\varphi^{\sharp\,2}+\|e_{21}\|_\varphi^{\sharp\,2}\big)^{1/2}=\sqrt2\,|\kappa^{it}-1|\ne0$. Also $[v_\varepsilon,\varphi]=i\varepsilon[h,\varphi]+O(\varepsilon^2)$, and $[h,\varphi]\ne0$ since $h\notin M_\varphi$ (the centralizer is the diagonal algebra because $p\ne\frac12$). So the left side of Corollary 6.3 is of order $\varepsilon^2$ and the right side of order $\varepsilon$.

References

[Connes 1974] A. Connes, Almost periodic states and factors of type III₁, J. Functional Analysis 16 (1974), 415–445. Free at https://doi.org/10.1016/0022-1236(74)90059-7
[Connes 1973] A. Connes, Une classification des facteurs de type III, Ann. Sci. École Norm. Sup. (4) 6 (1973), no. 2, 133–252. https://doi.org/10.24033/asens.1247 (open access: http://www.numdam.org/item?id=ASENS_1973_4_6_2_133_0). Free at https://alainconnes.org/wp-content/uploads/classificationfacteurs.pdf
[Ando–Haagerup 2014] H. Ando and U. Haagerup, Ultraproducts of von Neumann algebras, J. Funct. Anal. 266 (2014), no. 12, 6842–6913. https://doi.org/10.1016/j.jfa.2014.03.013 (open preprint: https://arxiv.org/abs/1212.5457).
[Araki 1974] H. Araki, Some properties of modular conjugation operator of von Neumann algebras and a non-commutative Radon–Nikodym theorem with a chain rule, Pacific J. Math. 50 (1974), 309–354. Free at https://msp.org/pjm/1974/50-2/pjm-v50-n2-p02-s.pdf
[Haagerup 1975] U. Haagerup, The standard form of von Neumann algebras, Math. Scand. 37 (1975), 271–283. https://doi.org/10.7146/math.scand.a-11606