Uniqueness of the injective II₁ factor

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

Introduction

The hyperfinite II₁ factor $R$ is the von Neumann algebra obtained by completing the increasing union of the matrix algebras $M_2\subset M_2\otimes M_2\subset M_2\otimes M_2\otimes M_2\subset\cdots$ with respect to the trace. It is the smallest infinite-dimensional factor in a precise sense. This lesson proves the theorem that characterizes it among all factors of type II₁ with separable predual:

For a II₁ factor with separable predual, being isomorphic to $R$, being injective, being semidiscrete, having a hypertrace and satisfying a Følner-type condition all amount to the same thing.

The full list of equivalent conditions is Theorem 1.5. Two consequences stand out.

Every injective II₁ factor on a separable Hilbert space is isomorphic to $R$ (Theorem 1.6).
Every subfactor of $R$ is either finite-dimensional or isomorphic to $R$ (Corollary 1.7).

Why is this hard? Each of the easy conditions (injectivity, hypertrace, Følner) is an averaging property: it gives states or vectors that are almost invariant. Being isomorphic to $R$ is a structural property: it asks for finite-dimensional subalgebras that approximate everything. The bridge between the two is built in four steps.

From averages to norm inequalities. A hypertrace gives, by an argument that parallels the proof of the classical Følner theorem for amenable groups, finite-rank projections that almost commute with the factor (Section 4). These give norm inequalities for the algebra generated by the factor $N$ and its commutant $N'$ (Section 5).
The flip. The norm inequalities pass to $N\bar\otimes N$ and show that the flip $x\otimes y\mapsto y\otimes x$ is a limit of inner automorphisms (Section 6).
Matrix models. The Følner projections compress $N$ into matrix algebras, which embeds $N$ into an ultrapower of $R$ (Section 7).
Central sequences. The flip forces $N$ to have many noncommuting central sequences, and McDuff's theorem then splits off a copy of $R$: $N\cong N\bar\otimes R$ (Section 8). Together with the matrix models and the flip this shows that $N$ is approximately finite-dimensional, hence isomorphic to $R$ (Section 9).

Section 10 relates the hypertrace to amenability in the sense of vanishing derivations, and Section 11 has exercises. Other proofs of the hard implication, avoiding some of these steps, were given later in [Haagerup 1985] and [Popa 1986].

What is assumed. The Course 1 lessons "Ultraproducts and the asymptotic centralizer" and "Full factors" are used for ultrapowers, central sequences and the relation between property Γ and fullness. From this course we use one result from "Property Γ and the algebra generated by a factor and its commutant", two from "Approximately inner and centrally trivial automorphisms" and one from "Injective von Neumann algebras". The lesson "Trace inequalities for finite von Neumann algebras" proves in a general semifinite setting the two inequalities we need on Hilbert–Schmidt operators; we give short proofs of the special cases used here, so this lesson does not depend on it. The section "Results used from other lessons" restates precisely what is used. Standard operator algebra theory is listed in "Standard results and where they are proved", each fact with the lesson that proves it.

Basic references are [Connes 1976], [Connes 1994], [Haagerup 1985], [Popa 1986] and [Anantharaman–Popa, Chapters 10–11].

Conventions. Hilbert spaces are complex; inner products are linear in the first variable. $B(K)$ is the algebra of bounded operators on $K$, $\operatorname{Tr}$ its usual trace, $\|T\|_{\mathrm{HS}}=\operatorname{Tr}(T^*T)^{1/2}$ the Hilbert–Schmidt norm and $\|T\|_{\mathrm{Tr}}=\operatorname{Tr}|T|$ the trace norm. The Hilbert–Schmidt operators form a Hilbert space $\mathrm{HS}(K)$ with $\langle S,T\rangle_{\mathrm{HS}}=\operatorname{Tr}(T^*S)$. $\mathcal U(M)$ is the unitary group of $M$, $\operatorname{Ad}u(x)=uxu^*$, and $[x,y]=xy-yx$. A II₁ factor is always assumed to have separable predual unless said otherwise; $\tau$ is its unique normal tracial state and $\|x\|_2=\tau(x^*x)^{1/2}$. We fix a free ultrafilter $\omega$ on $\mathbb N$ whenever ultrapowers appear.

Standard results and where they are proved

(B1) II₁ factors. A II₁ factor $N$ has a unique normal tracial state $\tau$, and $\tau$ is faithful. For every projection $q$ and $s\in[0,\tau(q)]$ there is a projection $q'\le q$ with $\tau(q')=s$; two projections with the same trace are equivalent. Every element of a finite von Neumann algebra has a polar decomposition $x=w|x|$ with $w$ unitary. For a nonzero projection $e\in N$, $eNe$ is a II₁ factor. On bounded subsets of $N$ the strong* operator topology (in any faithful normal representation) is given by the norm $\|\cdot\|_2$, and the closed unit ball is complete for $\|\cdot\|_2$. One has $\|xy\|_2\le\|x\|\|y\|_2$, $\|xy\|_2\le\|x\|_2\|y\|$, $\|x^*\|_2=\|x\|_2$ and $|\tau(x)|\le\|x\|_2$. A von Neumann algebra with separable predual has a norm-dense sequence in its predual and a $\|\cdot\|_2$-dense sequence in its unit ball, and it is generated by countably many unitaries. A II₁ factor is a simple C*-algebra. Except for simplicity, these facts are proved, with the lessons that contain them, in Approximately inner and centrally trivial automorphisms, (B1); the norm estimates for $\|\cdot\|_2$ are shown there, and the strong* topology agrees with the strong topology on bounded sets of $N$ because $\|x^*\|_2=\|x\|_2$. A countable $\|\cdot\|_2$-dense subset of the unit ball also gives a countable dense subset of $N_*$, through $h\mapsto\tau(h\,\cdot\,)$ and the estimate $\|\tau(h\,\cdot\,)\|\le\|h\|_2$; replacing each element by the unitaries in the decomposition into four unitaries gives countably many unitaries that generate $N$. Simplicity is Property Γ and the algebra generated by a factor and its commutant, Lemma 6.3.

(B2) Dixmier approximation theorem. In a factor $N$, for every $x\in N$ the norm-closed convex hull of $\{uxu^*:u\in\mathcal U(N)\}$ meets the scalars. Consequently a II₁ factor has exactly one tracial state, normal or not: if $t$ is a tracial state, $t$ is constant on that convex hull and continuous, so $t(x)=\tau(x)$. The approximation theorem and this consequence are Traces on von Neumann algebras, Theorem 8.2 and Corollary 8.3. The consequence also follows without the approximation theorem from Theorem 5.5 there: every finite trace $\sigma$, normal or not, satisfies $\sigma(x)=\sigma(T(x))$, and for a factor $T(x)=\tau(x)1$. The σ-weakly closed version of the approximation theorem is Lemma 5.1 there.

(B3) Standard form. Let $N$ have a faithful normal tracial state $\tau$. The GNS space $L^2(N,\tau)$, with $x\mapsto\hat x$ and trace vector $\hat1$, carries the standard form of $N$: $\hat1$ is cyclic and separating, $J\hat x=\widehat{x^*}$ extends to an antiunitary involution, $JNJ=N'$, and $Jx^*J\hat y=\widehat{yx}$ for $x,y\in N$. For finite von Neumann algebras $N_1,N_2$ with traces, $L^2(N_1\bar\otimes N_2)=L^2(N_1)\otimes L^2(N_2)$ with $J=J_1\otimes J_2$, the product trace $\tau_1\otimes\tau_2$ is the trace of $N_1\bar\otimes N_2$, and $(N_1\bar\otimes N_2)'=N_1'\bar\otimes N_2'$ (commutation theorem). If $N_1,N_2$ are II₁ factors, so is $N_1\bar\otimes N_2$. Every faithful normal state of a von Neumann algebra in standard form is a vector state of a cyclic and separating vector. Two faithful normal tracial states preserved by an isomorphism give unitarily equivalent standard forms. The standard form of a trace is Integration for a trace, Sections 2–3 (Theorem 3.1); the tensor product statements are Integration for a trace, Proposition 6.1 and Spatial tensor products of von Neumann algebras, Theorem 11.4 and Corollary 11.5, with the identification of $L^2$ as in Approximately inner and centrally trivial automorphisms, (B3). A faithful normal state $\omega$ is the vector state of a vector $\xi$ of the positive cone (The positive cone of a standard representation, §SF-10); $\xi$ is separating for $N$ because $\omega$ is faithful, and $J\xi=\xi$ makes it separating for $N'=JNJ$, that is, cyclic for $N$. For the last statement, if $\theta$ carries $\tau_1$ to $\tau_2$, then $\hat x\mapsto\widehat{\theta(x)}$ is a unitary that implements $\theta$ and commutes with $J$.

(B4) Conditional expectations. Let $B$ be a von Neumann subalgebra of a von Neumann algebra $N$ with faithful normal tracial state $\tau$. There is a unique $\tau$-preserving conditional expectation $E_B\colon N\to B$; it is normal, unital, completely positive and $B$-bimodular, and $\widehat{E_B(x)}$ is the orthogonal projection of $\hat x$ onto the closure of $\hat B$ in $L^2(N)$. In particular, if $\hat x$ lies in that closure then $x\in B$. Proved in Integration for a trace, Theorem 9.1; complete positivity follows from the bimodule property as in Injective von Neumann algebras, (B3), and if $\hat x$ lies in the closure of $\hat B$, then $\widehat{E_B(x)}=\hat x$, so $x=E_B(x)\in B$ because $\hat1$ is separating.

(B5) Kaplansky density theorem. If a unital $*$-algebra $A_0$ is weakly dense in a von Neumann algebra $M$, every element of $M$ is the strong* limit of a net in $A_0$ bounded by its norm; when $M$ has separable predual a sequence suffices. Proved in Kaplansky's density theorem and its consequences, Theorem 7.1; for a sequence, the strong* topology is metrizable on bounded sets when the Hilbert space is separable (Compact and trace-class operators, Proposition 10.1(a)); this applies to the standard representations on the separable spaces $L^2(N,\tau)$ used in this lesson.

(B6) Slice maps. For a normal functional $\psi$ on $B(K_1)$ the slice map $\psi\otimes\mathrm{id}\colon B(K_1\otimes K_2)\to B(K_2)$, defined by $\chi((\psi\otimes\mathrm{id})(T))=(\psi\otimes\chi)(T)$ for normal $\chi$, is normal with norm at most $\|\psi\|$, satisfies $(\psi\otimes\mathrm{id})((1\otimes a)T(1\otimes b))=a\,(\psi\otimes\mathrm{id})(T)\,b$ and $(\psi\otimes\mathrm{id})((u\otimes1)T(u^*\otimes1))= ((\psi\circ\operatorname{Ad}u)\otimes\mathrm{id})(T)$, and is completely positive when $\psi\ge0$. The normal functionals $\psi\otimes\chi$ separate the points of $B(K_1\otimes K_2)$. Proved in Spatial tensor products of von Neumann algebras, Section 9 and Theorem 10.1.

(B7) Tensor products of C*-algebras. For C*-algebras $A\subset B(K_1)$, $C\subset B(K_2)$, the minimal (spatial) norm $\|\cdot\|_{\min}$ on the algebraic tensor product $A\odot C$ is the operator norm on $K_1\otimes K_2$; it does not depend on the faithful representations used, so an isometric $*$-isomorphism $\theta$ of $A$ gives $\|(\theta\otimes\mathrm{id})(X)\|_{\min}=\|X\|_{\min}$, and $A\otimes_{\min}C\subset B\otimes_{\min}C$ isometrically when $A\subset B$. If $\alpha\colon A\to B$ is unital completely positive, then $\alpha\otimes\mathrm{id}$ extends to a contraction $A\otimes_{\min}C\to B\otimes_{\min}C$. If $A$ and $C$ are simple, $A\otimes_{\min}C$ is simple (Takesaki's theorem). If $D$ is finite-dimensional, $D\odot C$ is already a C*-algebra with a unique C*-norm. Independence and the induced maps: Tensor products of C*-algebras and the minimal norm, Theorem 2.2 and Corollary 2.3. Completely positive maps: Completely positive finite models, Lemma 4.1. Simplicity: Tensor products of C*-algebras and the minimal norm, Theorem 5.2. The finite-dimensional case: Tensor products of C*-algebras and the minimal norm, Example 6.2 and Completely positive finite models, Lemma 4.2, and $D\odot C$, a finite sum of matrix algebras over $C$, is complete.

(B8) Completely positive maps. Arveson's extension theorem: a unital completely positive (ucp) map from a unital C*-subalgebra $A\subset C$ into $B(K)$ extends to a ucp map $C\to B(K)$. Tomiyama's theorem: a projection of norm one $P$ from a C*-algebra $C$ onto a C*-subalgebra $B$ is ucp and satisfies $P(b_1cb_2)=b_1P(c)b_2$. Multiplicative domain: if $\Phi$ is ucp and $\Phi(b^*b)=\Phi(b)^*\Phi(b)$, $\Phi(bb^*)=\Phi(b)\Phi(b)^*$, then $\Phi(bc)=\Phi(b)\Phi(c)$ and $\Phi(cb)=\Phi(c)\Phi(b)$ for all $c$. Norm at the unit: a 2-positive linear map $\Gamma$ between unital C*-algebras has $\|\Gamma\|=\|\Gamma(1)\|$. A bounded net of linear maps from a Banach space into a von Neumann algebra has a subnet converging pointwise $\sigma$-weakly. Arveson's theorem: Completely positive finite models, Theorem 2.3. Tomiyama's theorem: Injective von Neumann algebras, (B3). Multiplicative domain: Completely positive maps, Theorem 4.1(3), which also gives $\|\Gamma\|=\|\Gamma(1)\|$ when $\Gamma$ is 2-positive (Theorem 4.1(2)). (For a merely positive map the same identity is the Russo–Dye theorem; it is not used.) The last statement holds because the balls of the von Neumann algebra are σ-weakly compact (Compact and trace-class operators, Corollary 9.7(a)), so the product of balls is compact (Weak topologies, Theorem 2.3).

(B9) Functionals. A state on a C*-subalgebra extends to a state of the whole algebra (Hahn–Banach); a linear functional $\varphi$ on a unital C*-algebra with $\|\varphi\|=\varphi(1)=1$ is a state. The normal states of $B(K)$ are weak* dense in its state space, and they are the functionals $\operatorname{Tr}(\rho\,\cdot\,)$ with $\rho\ge0$, $\operatorname{Tr}\rho=1$, with $\|\operatorname{Tr}(\rho\,\cdot)-\operatorname{Tr}(\sigma\,\cdot)\|= \|\rho-\sigma\|_{\mathrm{Tr}}$. Every self-adjoint bounded functional $h$ on a C*-algebra has a unique decomposition $h=h_+-h_-$ with $h_\pm\ge0$ and $\|h\|=\|h_+\|+\|h_-\|$ (Jordan decomposition). In a Banach space, the weak and norm closures of a convex set coincide (Mazur). The bounded trilinear forms on a product $E_1\times E_2\times E_3$ of Banach spaces form the dual of the projective tensor product. Extension of states: Hahn–Banach, Baire and the basic theorems on Banach spaces, Corollary 2.3 and the positivity criterion The universal enveloping von Neumann algebra of a C*-algebra, and W*-algebras, Lemma 8.1. Normal states of $B(K)$ and the trace norm: Compact and trace-class operators, Theorem 6.2 and Proposition 6.3. They are weak* dense in the state space: otherwise the separation theorem (Hahn–Banach, Baire and the basic theorems on Banach spaces, Corollary 6.4) gives a self-adjoint $x$ and a state $\varphi$ with $\varphi(x)>\sup_\omega\omega(x)$ over the normal states, while $\sup_\omega\omega(x)\ge\sup_{\|\xi\|=1}\langle x\xi,\xi\rangle=\max\sigma(x)$ (The spectral theorem for bounded self-adjoint operators, Proposition 8.2) and $\varphi(x)\le\max\sigma(x)$. Jordan decomposition: The universal enveloping von Neumann algebra, Proposition 3.4. Mazur: Weak topologies, Theorem 4.1. Projective tensor products: Tensor products of Banach and Hilbert spaces, Theorem 4.1, applied twice.

(B10) Compact operators. A positive Hilbert–Schmidt operator $h$ on $K$ has an orthonormal basis of eigenvectors $(\xi_i)$, $h\xi_i=\lambda_i\xi_i$ with $\lambda_i\ge0$, $\sum\lambda_i^2=\|h\|_{\mathrm{HS}}^2$, and its spectral projections for $(t,\infty)$, $t>0$, have finite rank. For $S,T\in\mathrm{HS}(K)$ and $A,B\in B(K)$: $\|ASB\|_{\mathrm{HS}}\le\|A\|\|S\|_{\mathrm{HS}}\|B\|$, $|\operatorname{Tr}(ST)|\le\|S\|_{\mathrm{HS}} \|T\|_{\mathrm{HS}}$, $ST$ is trace class, $\operatorname{Tr}(ST)=\operatorname{Tr}(TS)$, and $\|T\|_{\mathrm{HS}}^2=\sum_{i,j}|\langle T\eta_j,\xi_i\rangle|^2$ for orthonormal bases $(\xi_i),(\eta_j)$. Proved in Compact and trace-class operators, Sections 2–4 (Schmidt decomposition, Hilbert–Schmidt and trace-class operators, Lemma 3.1, Proposition 4.9, Theorem 4.6).

(B11) The hyperfinite factor and its uniqueness. Let $A_\infty=\bigcup_kM_2^{\otimes k}$ (algebraic tensor powers, $x\mapsto x\otimes1$) with its unique tracial state $\mathrm{tr}$. The hyperfinite II₁ factor $R$ is the weak closure of $A_\infty$ in the GNS representation of $\mathrm{tr}$; it is a II₁ factor with separable predual, and $A_\infty$ is weakly dense in it. Murray–von Neumann theorem: let $N$ be a II₁ factor with separable predual such that for every finite set $x_1,\dots,x_n\in N$ and every $\varepsilon>0$ there are a unital subfactor $Q\subset N$ isomorphic to a matrix algebra $M_{2^m}$ and elements $q_j\in Q$ with $\|x_j-q_j\|_2\le \varepsilon$. Then $N\cong R$. The construction of $R$ is Infinite tensor products, Corollary 5.3, and the theorem is Hyperfinite finite factors, Theorem 5.1. See also [Anantharaman–Popa, §11.2].

(B12) Amenability of approximately finite-dimensional algebras. If a von Neumann algebra has finite-dimensional $*$-subalgebras $D_1\subset D_2\subset\cdots$ with weakly dense union, it is amenable in the sense of Definition 10.1. This is the Johnson–Kadison–Ringrose theorem. [Johnson–Kadison–Ringrose 1972]

Results used from other lessons

From "Ultraproducts and the asymptotic centralizer" (Course 1). Let $(N,\tau)$ be a von Neumann algebra with a faithful normal tracial state. The ultrapower $N^\omega$ is the quotient of the bounded sequences in $N$ by those with $\lim_{k\to\omega}\|x_k\|_2=0$, with trace $\tau_\omega(x)=\lim_{k\to\omega}\tau(x_k)$ (Definitions 3.1 and 3.4 there); an element is written as the class of a representing sequence. By Theorem 3.2 there, $N^\omega$ is a finite von Neumann algebra and $\tau_\omega$ is a faithful normal tracial state on it; the proof shows that an element of norm less than $1$ has a representing sequence bounded by $1$. The constant sequences embed $N$ in $N^\omega$, and $N_\omega=N'\cap N^\omega$ is the algebra of classes of central sequences: bounded sequences with $\lim_{k\to\omega}\|[x_k,y]\|_2=0$ for all $y\in N$ (Definition 3.4 and Proposition 8.7(2) there). The remark following Definition 3.4 there shows that a unital $*$-homomorphism between von Neumann algebras with faithful normal tracial states that preserves the traces is injective and normal.

From "Full factors" (Course 1). A von Neumann algebra with separable predual is full if $\operatorname{Int}M$ is closed in $\operatorname{Aut}M$ for the $u$-topology, in which $\alpha_i\to\alpha$ means $\|\varphi\circ\alpha_i-\varphi\circ\alpha\|\to0$ for all $\varphi\in M_*$ (Definition 2.1 and Definition 4.3 there); $\operatorname{Aut}M$ is then a metrizable topological group (Proposition 2.3 there). A II₁ factor $N$ has property Γ if for all $y_1,\dots,y_k\in N$ and $\varepsilon>0$ there is a unitary $u$ with $\tau(u)=0$ and $\|[u,y_j]\|_2<\varepsilon$ (Definition 6.1 there). We use:

Theorem 6.5 there: a II₁ factor with separable predual is full exactly when it does not have property Γ.
Lemma 6.3 there (asymptotic independence): if $(y_k)$ is a bounded sequence with $\|[y_k,y]\|_2\to0$ along $\omega$ for all $y\in N$, then $\tau(xy_k)-\tau(x)\tau(y_k)\to0$ along $\omega$ for every $x\in N$.

From "Property Γ and the algebra generated by a factor and its commutant" (this course). Let $N$ be a II₁ factor acting standardly on $H=L^2(N,\tau)$, and let $C^*(N,N')$ be the C*-algebra generated by $N$ and $N'$. By Theorem 2.5 there ((a)⇔(d)), $N$ has property Γ if and only if $C^*(N,N')$ contains no nonzero compact operator. No separability of the predual is needed there. We use only the implication from the absence of compact operators to property Γ.

From "Approximately inner and centrally trivial automorphisms" (this course). Let $M$ be a II₁ factor acting on $L^2(M)$. There an automorphism is called approximately inner if it lies in the closure of $\operatorname{Int}M$ for the $u$-topology (Definition 3.1 there); for separable predual this is the set $\overline{\operatorname{Int}}M$ of Section 1.4 below (Lemma 1.4 below, or Lemma 2.1 there). An automorphism $\theta$ is centrally trivial if $\|\theta(x_k)-x_k\|_2\to0$ for every central sequence $(x_k)$, that is, every bounded sequence with $\|[x_k,y]\|_2\to0$ for all $y\in M$ (Definition 6.1 there); $\operatorname{Ct}M$ is the set of centrally trivial automorphisms. On bounded sequences, convergence to $0$ in $\|\cdot\|_2$ is the same as strong convergence on $L^2(M)$. We use two results, both proved there only for separable predual.

(L1) Criterion for approximate innerness (Theorem 3.2 there, (b)⇒(a)). Let $M$ have separable predual and $\theta\in\operatorname{Aut}M$. If $\|\sum_i\theta(a_i)b_i\|=\|\sum_ia_ib_i\|$ for all finite families $a_i\in M$, $b_i\in M'$, then $\theta\in\overline{\operatorname{Int}}M$. The converse, (a)⇒(b) there (Proposition 3.5 there), is Lemma 5.5 below, which we prove.
(L2) Centrally trivial automorphisms and tensor products (Corollary 8.5 there). Let $M_1,M_2$ be II₁ factors with separable preduals and $\theta_j\in\operatorname{Aut}M_j$. Then $\theta_1\otimes\theta_2\in \operatorname{Ct}(M_1\bar\otimes M_2)$ if and only if $\theta_1\in\operatorname{Ct}M_1$ and $\theta_2\in\operatorname{Ct}M_2$. We use the "if" direction with $\theta_2=\mathrm{id}$: if $\theta\in\operatorname{Ct}M_1$, then $\theta\otimes\mathrm{id}\in\operatorname{Ct}(M_1\bar\otimes M_2)$. There it is proved from Corollary 8.4(i), Proposition 7.4(b) and Proposition 7.3, without the structure theorem for the asymptotic period.

From "Injective von Neumann algebras" (this course). There a von Neumann algebra is injective if it is injective as a C*-algebra (Definition 1.2 there). By Theorem 2.1 there, $M\subset B(K)$ is injective if and only if there is a projection of norm one from $B(K)$ onto $M$; this is the form in which we use the notion. By Corollary 2.2(a) there, the existence of such a projection depends only on the isomorphism class of $M$, not on the faithful normal representation.

1. The setting and the main theorem

A reference for this section is [Connes 1976, §4.2].

1.1 Standard form, conjugate spaces and the minimal norm

Throughout, $N$ is a II₁ factor with separable predual and trace $\tau$, acting standardly on $H=L^2(N,\tau)$ with trace vector $\hat1$ and conjugation $J$ (B3). Thus $N'=JNJ$, and for $b\in N$ $JbJ\,\hat1=\widehat{b^*},\qquad \langle a\,JbJ\,\hat1,\hat1\rangle=\tau(ab^*)\qquad(a,b\in N). \tag{1.1}$

For a Hilbert space $K$ let $\overline K$ be the conjugate space: the same additive group, with vectors written $\bar\xi$, scalar multiplication $\lambda\bar\xi=\overline{\bar\lambda\xi}$ and inner product $\langle\bar\xi,\bar\eta\rangle=\langle\eta,\xi\rangle$. For $T\in B(K)$ the operator $\bar T\bar\xi=\overline{T\xi}$ is bounded on $\overline K$, and $T\mapsto\bar T$ is a conjugate-linear, multiplicative, $*$-preserving isometry. For a von Neumann algebra $N\subset B(K)$, $\overline N=\{\bar x:x\in N\}$ is a von Neumann algebra on $\overline K$, the conjugate algebra.

On $H=L^2(N)$ the map $V\colon\overline H\to H$, $V\bar\xi=J\xi$, is a linear unitary, and $V\bar bV^*=JbJ$ for $b\in B(H)$. Hence, for $a_j,b_j\in N$, $\Big\|\sum_ja_j\otimes\bar b_j\Big\|_{B(H\otimes\overline H)}=\Big\|\sum_ja_j\otimes Jb_jJ\Big\|_{B(H\otimes H)}. \tag{1.2}$

The minimal norm on the algebraic tensor product $N\odot N'$ is the norm of $B(H\otimes H)$ (B7); since the minimal norm does not depend on the faithful representations, the left side of (1.2) is also the minimal norm of $\sum a_j\otimes\bar b_j\in N\odot\overline N$, whatever faithful normal representation of $N$ is used. The product map is the $*$-homomorphism $\mu\colon N\odot N'\to B(H),\qquad \mu\Big(\sum_jx_j\otimes y_j\Big)=\sum_jx_jy_j .$ Its range is dense in the C*-algebra $C^*(N,N')$ generated by $N$ and $N'$.

The unitary $\mathrm{HS}(K)\to K\otimes\overline K$ sending the rank-one operator $\xi\otimes\eta^*\colon\zeta\mapsto \langle\zeta,\eta\rangle\xi$ to $\xi\otimes\bar\eta$ carries left multiplication by $a$ and right multiplication by $b^*$, $T\mapsto aTb^*$, to $a\otimes\bar b$. In particular, for $u$ unitary, $T\mapsto uTu^*$ corresponds to $u\otimes\bar u$. (Indeed $a(\xi\otimes\eta^*)b^*=(a\xi)\otimes(b\eta)^*$, and inner products match: $\operatorname{Tr}((\xi'\otimes\eta'^*)^*(\xi\otimes\eta^*))=\langle\xi,\xi'\rangle\langle\eta',\eta\rangle= \langle\xi\otimes\bar\eta,\xi'\otimes\bar\eta'\rangle$.)

1.2 Hypertraces, Følner projections and semidiscreteness

Definition 1.1. Let $M\subset B(K)$ be a von Neumann algebra. A hypertrace for $M$ (on $K$) is a state $\varphi$ on $B(K)$ such that $\varphi(xT)=\varphi(Tx)$ for all $x\in M$ and $T\in B(K)$; equivalently $\varphi(uTu^*)=\varphi(T)$ for all $u\in\mathcal U(M)$, that is, $M$ lies in the centralizer $\{x:\varphi(xT)=\varphi(Tx)\ \forall T\}$ of $\varphi$.

The two forms are equivalent because the unitaries span $M$, and $\varphi(u(Tu)u^*)=\varphi(Tu)$. If $M=N$ is a II₁ factor, the restriction of a hypertrace to $N$ is a tracial state, hence equals $\tau$ (B2).

Definition 1.2. A II₁ factor $N\subset B(K)$ satisfies the Følner condition (on $K$) if for all $x_1,\dots,x_n\in N$ and $\varepsilon>0$ there is a nonzero finite-rank projection $e\in B(K)$ with $\|[x_j,e]\|_{\mathrm{HS}}\le\varepsilon\|e\|_{\mathrm{HS}},\qquad \Big|\tau(x_j)-\frac{\operatorname{Tr}(ex_je)}{\operatorname{Tr}(e)}\Big|\le\varepsilon\qquad(j=1,\dots,n). \tag{1.3}$

Note $\|e\|_{\mathrm{HS}}^2=\operatorname{Tr}(e)$ is the rank of $e$, and $\operatorname{Tr}(exe)/\operatorname{Tr}(e)= \langle xe,e\rangle_{\mathrm{HS}}/\langle e,e\rangle_{\mathrm{HS}}$. The name comes from amenable groups: see Example 4.4.

Definition 1.3. A von Neumann algebra $M$ is semidiscrete if there are a net of finite-dimensional C*-algebras $D_i$ and normal ucp maps $\alpha_i\colon M\to D_i$, $\beta_i\colon D_i\to M$ with $\beta_i(\alpha_i(x))\to x$ $\sigma$-weakly for every $x\in M$.

1.3 The hyperfinite factor

We use the model $R$ of (B11). Inside $R$ let $A_k=M_2^{\otimes k}$, a unital subfactor isomorphic to $M_{2^k}$, so that $A_1\subset A_2\subset\cdots$ and $\bigcup_kA_k$ is weakly dense. A II₁ factor is called hyperfinite if it is isomorphic to $R$.

1.4 Approximately inner and centrally trivial automorphisms

For a II₁ factor $M$, an automorphism $\theta$ is approximately inner if there are unitaries $u_n\in M$ with $\|u_nxu_n^*-\theta(x)\|_2\to0$ for all $x\in M$. Then also $\|u_n^*xu_n-\theta^{-1}(x)\|_2=\|x-u_n\theta^{-1}(x) u_n^*\|_2\to0$. We write $\overline{\operatorname{Int}}M$ for the set of approximately inner automorphisms. The next lemma shows that this agrees with the closure of $\operatorname{Int}M$ for the $u$-topology, so that the results of "Full factors" apply.

Lemma 1.4. Let $M$ be a II₁ factor with separable predual. A net $\alpha_i$ in $\operatorname{Aut}M$ converges to $\alpha$ in the $u$-topology if and only if $\|\alpha_i(x)-\alpha(x)\|_2\to0$ and $\|\alpha_i^{-1}(x)-\alpha^{-1}(x)\|_2\to0$ for all $x\in M$. Hence the $u$-closure of $\operatorname{Int}M$ is $\overline{\operatorname{Int}}M$.

Proof. Every automorphism preserves $\tau$ (B1), hence $\|\cdot\|_2$. Suppose $\alpha_i\to\alpha$ in the $u$-topology. For $x\in M$ put $\varphi=\tau(\alpha(x)^*\,\cdot\,)\in M_*$. Then $\|\alpha_i(x)-\alpha(x)\|_2^2=2\tau(x^*x)-2\operatorname{Re}(\varphi\circ\alpha_i)(x)\to2\tau(x^*x)- 2\operatorname{Re}(\varphi\circ\alpha)(x)=0 .$ Since $\operatorname{Aut}M$ is a topological group for the $u$-topology, also $\alpha_i^{-1}\to\alpha^{-1}$, and the same computation applies. Conversely, suppose the two $\|\cdot\|_2$-convergences hold. For $y\in M$ and $\varphi=\tau(y\,\cdot\,)$, trace invariance gives $\varphi(\alpha_i(x))=\tau(\alpha_i^{-1}(y)x)$, so $\|\varphi\circ\alpha_i-\varphi\circ\alpha\|=\sup_{\|x\|\le1}|\tau((\alpha_i^{-1}(y)-\alpha^{-1}(y))x)|\le \|\alpha_i^{-1}(y)-\alpha^{-1}(y)\|_2\to0 .$ The functionals $\tau(y\,\cdot\,)$, $y\in M$, are norm dense in $M_*$ (the predual of $M$ is $L^1(M,\tau)$, in which $M$ is dense), and $\|\chi\circ\alpha_i\|=\|\chi\|$, so $\|\varphi\circ\alpha_i-\varphi\circ\alpha\|\to0$ for all $\varphi\in M_*$. For the last statement, $\operatorname{Aut}M$ is metrizable, so the $u$-closure of $\operatorname{Int}M$ consists of limits of sequences $\operatorname{Ad}u_n$, and by the first part and the remark before the lemma these are exactly the approximately inner automorphisms. $\square$

1.5 The main theorem

The flip of $N\bar\otimes N$ is the automorphism $\sigma$ with $\sigma(x\otimes y)=y\otimes x$.

Theorem 1.5 (main theorem). Let $N$ be a II₁ factor with separable predual, acting standardly on $H=L^2(N,\tau)$. The following conditions are equivalent.

(a) $N$ is hyperfinite: $N\cong R$.
(b) $N$ is injective.
(c) $N$ is semidiscrete.
(d) $N$ has a hypertrace on $H$.
(e) $N$ satisfies the Følner condition on $H$.
(f) For all $a_1,\dots,a_n,b_1,\dots,b_n\in N$, $\Big|\tau\Big(\sum_ja_jb_j^*\Big)\Big|\le\Big\|\sum_ja_j\otimes\bar b_j\Big\|_{B(H\otimes\overline H)} .$
(g) For all $x_1,\dots,x_n\in N$ and $y_1,\dots,y_n\in N'$, $\Big\|\sum_jx_jy_j\Big\|_{B(H)}=\Big\|\sum_jx_j\otimes y_j\Big\|_{B(H\otimes H)} .$ That is, the product map extends to an isometric $*$-isomorphism of $N\otimes_{\min}N'$ onto $C^*(N,N')$.
(h) The flip of $N\bar\otimes N$ is approximately inner.
(i) $N\cong N\bar\otimes R$, and for all $x_1,\dots,x_n\in N$ and $\varepsilon>0$ there are $z_1,\dots,z_n\in R$ and a unitary $X\in N\bar\otimes R$ with $\|x_j\otimes1-X(1\otimes z_j)X^*\|_2\le\varepsilon\qquad(j=1,\dots,n).$

Reference: [Connes 1976, Part II and Theorem 4.2.1]; see also [Anantharaman–Popa, Chapters 10–11].

In (d) and (e), $H$ may be replaced by any Hilbert space on which $N$ acts faithfully and normally (Remark 5.2). The implications are proved in the following order: $\text{(a)}\Rightarrow\text{(c)}\Rightarrow\text{(b)}\Rightarrow\text{(d)}\Rightarrow\text{(e)}\Rightarrow \text{(f)}\Rightarrow\text{(g)}\Rightarrow\text{(b)},\qquad \text{(g)}\Rightarrow\text{(h)}\Rightarrow\text{(g)},\qquad \text{(e)}+\text{(g)}+\text{(h)}\Rightarrow\text{(i)}\Rightarrow\text{(a)}.$ The first chain shows that (b)–(g) are equivalent and follow from (a); the second adds (h); the third closes the circle. The assembly is at the start of Section 9. The two applications are these.

Theorem 1.6. Every injective II₁ factor acting on a separable Hilbert space is isomorphic to $R$.

Reference: [Connes 1976, Theorem 4.2.1].

Corollary 1.7. Every subfactor of $R$ is either finite-dimensional or isomorphic to $R$.

Reference: [Connes 1976, Corollary 4.2.2].

Here a subfactor of $R$ is a von Neumann subalgebra containing $1$ which is a factor; Exercise 3 treats subfactors with a smaller unit. So among infinite-dimensional factors $R$ is the smallest: it embeds unitally in every II₁ factor (every II₁ factor contains increasing unital copies of $M_{2^k}$, and Lemma 2.1 below extends the resulting trace-preserving map from $\bigcup_kA_k$), and by Corollary 1.7 it has no other infinite-dimensional subfactors.

2. Tools

This section collects facts used several times: extension of trace-preserving homomorphisms, lifting of unitaries from ultrapowers, two inequalities on Hilbert–Schmidt operators, and two averaging arguments.

2.1 Trace-preserving homomorphisms

Lemma 2.1. Let $M$ and $P$ be von Neumann algebras with faithful normal tracial states $\tau_M,\tau_P$, let $A_0\subset M$ be a weakly dense unital $*$-subalgebra and $\pi_0\colon A_0\to P$ a unital $*$-homomorphism with $\tau_P\circ\pi_0=\tau_M$ on $A_0$. Then $\pi_0$ extends uniquely to a unital normal injective $*$-homomorphism $\pi\colon M\to P$ with $\tau_P\circ\pi=\tau_M$. Its range is the von Neumann algebra generated by $\pi_0(A_0)$.

Proof. For $a\in A_0$, $\|\pi_0(a)\|_2^2=\tau_P(\pi_0(a^*a))=\|a\|_2^2$, and $\|\pi_0(a)\|=\|a\|$: for a faithful normal tracial state the operator norm is $\|y\|=\lim_{p\to\infty}\tau((y^*y)^p)^{1/2p}$ (spectral theorem), and $\tau_P((\pi_0(a)^*\pi_0(a))^p)=\tau_M((a^*a)^p)$. Let $x\in M$. By (B5) there is a net $a_i\in A_0$ with $\|a_i\|\le\|x\|$ and $a_i\to x$ strong*, so $\|a_i-x\|_2\to0$ (B1). Then $(\pi_0(a_i))$ is $\|\cdot\|_2$-Cauchy and bounded by $\|x\|$; the ball of radius $\|x\|$ of $P$ is complete for $\|\cdot\|_2$ (B1), so $\pi_0(a_i)\to\pi(x)$ in $\|\cdot\|_2$. The limit does not depend on the net, since two nets can be merged. If $a_i\to x$, $b_i\to y$ in this way, then $a_ib_i\to xy$ in $\|\cdot\|_2$ (as $\|a_ib_i-xy\|_2\le\|a_i\|\|b_i-y\|_2+\|a_i-x\|_2\|y\|$) with $\|a_ib_i\|\le\|x\|\|y\|$, so $\pi(xy)=\pi(x)\pi(y)$; similarly $\pi$ is linear, $*$-preserving, unital and trace-preserving. By the result from "Ultraproducts and the asymptotic centralizer" recalled above, $\pi$ is injective and normal. The range of a normal $*$-homomorphism is a von Neumann algebra; it contains $\pi_0(A_0)$, and each $\pi(x)$ is a strong limit of the bounded net $\pi_0(a_i)$, so the range is the von Neumann algebra generated by $\pi_0(A_0)$. $\square$

Lemma 2.2. (a) If $Q_1\subset Q_2\subset\cdots$ are unital subalgebras of a II₁ factor $M$ with $Q_k\cong M_2^{\otimes k}$ compatibly (that is, $Q_{k+1}$ is generated by $Q_k$ and a copy of $M_2$ commuting with it), then the von Neumann algebra generated by $\bigcup_kQ_k$ is isomorphic to $R$.

(b) $R\bar\otimes R\cong R$. For each $k$, $R$ is generated by the commuting subalgebras $A_k\cong M_{2^k}$ and $R_{>k}$, the von Neumann algebra generated by the tensor factors of index greater than $k$; and $R_{>k}\cong R$, $R\cong A_k\bar\otimes R_{>k}$.

Proof. (a) The algebraic union $\bigcup_kQ_k$ is isomorphic, as a $*$-algebra with compatible inclusions, to $A_\infty$, and the isomorphism preserves traces because a matrix algebra has only one tracial state. Lemma 2.1, applied to the inverse isomorphism $A_\infty\to\bigcup Q_k\subset M$, gives a normal injective $*$-homomorphism of $R$ onto the von Neumann algebra generated by $\bigcup Q_k$.

(b) In $R\bar\otimes R$ let $B_{2k}=A_k\otimes A_k$ and $B_{2k-1}=A_k\otimes A_{k-1}$ (with $A_0=\mathbb C$). Each $B_{l+1}$ is generated by $B_l$ and one copy of $M_2$ commuting with it, $B_l\cong M_2^{\otimes l}$, and $\bigcup_lB_l=\bigcup_kA_k\otimes A_k$ is weakly dense in $R\bar\otimes R$. So (a) gives $R\bar\otimes R\cong R$. For fixed $k$, let $C_l$ ($l\ge1$) be the subalgebra of $R$ generated by the tensor factors of index $k+1,\dots,k+l$; then $C_l\cong M_2^{\otimes l}$ increase in the same way, and (a) gives $R_{>k}\cong R$. Finally $A_k\odot\bigcup_lC_l=\bigcup_mA_m$ is a weakly dense unital $*$-subalgebra of $A_k\bar\otimes R_{>k}$ (viewed abstractly), the map $a\otimes c\mapsto ac$ into $R$ is a $*$-homomorphism preserving the traces (the trace of $R$ on $A_{k+l}$ is the product of the traces of $A_k$ and $C_l$, by uniqueness of the trace on a matrix algebra), and Lemma 2.1 gives $A_k\bar\otimes R_{>k}\cong R$, the image being generated by $\bigcup_mA_m$. $\square$

2.2 Ultrapowers and unitaries

Lemma 2.3. Let $P$ be a II₁ factor and $W$ a unitary of $P^\omega$. Then $W$ has a representing sequence of unitaries of $P$.

Proof. Let $(w_k)$ be a bounded representing sequence. Then $\lim_\omega\|w_k^*w_k-1\|_2=0$. Write $w_k=s_k|w_k|$ with $s_k$ unitary (B1). For $t\ge0$, $|t-1|\le|t^2-1|$, so by functional calculus $\|s_k-w_k\|_2=\||w_k|-1\|_2\le\|w_k^*w_k-1\|_2$, which tends to $0$ along $\omega$. $\square$

2.3 Two inequalities for Hilbert–Schmidt operators

The following inequality is due to Powers and Størmer; the lesson "Trace inequalities for finite von Neumann algebras" proves it for arbitrary semifinite von Neumann algebras. We need only the case of $B(K)$.

Proposition 2.4 (Powers–Størmer inequality). Let $\rho,\sigma$ be positive trace-class operators on $K$. Then $\|\rho^{1/2}-\sigma^{1/2}\|_{\mathrm{HS}}^2\le\|\rho-\sigma\|_{\mathrm{Tr}} .$

Reference: [Powers–Størmer 1970].

Proof. Put $A=\rho^{1/2}$, $B=\sigma^{1/2}$ (Hilbert–Schmidt), $X=A-B$, $Y=A+B$, and let $P$ be the spectral projection of $X$ for $[0,\infty)$, so that $X(2P-1)=(2P-1)X=|X|$. Since $XY+YX=2(A^2-B^2)$, the cyclicity of the trace (B10) gives $\operatorname{Tr}\big((A^2-B^2)(2P-1)\big)=\tfrac12\operatorname{Tr}\big(XY(2P-1)\big)+\tfrac12\operatorname{Tr}\big(YX(2P-1)\big) =\operatorname{Tr}(Y|X|).$ The left side is at most $\|A^2-B^2\|_{\mathrm{Tr}}=\|\rho-\sigma\|_{\mathrm{Tr}}$ in absolute value. On the other hand $Y-X=2B\ge0$ and $Y+X=2A\ge0$. As $|X|^{1/2}$ commutes with $P$, $\operatorname{Tr}(Y|X|)=\operatorname{Tr}\big(|X|^{1/2}PYP|X|^{1/2}\big)+\operatorname{Tr}\big(|X|^{1/2}(1-P)Y(1-P)|X|^{1/2}\big),$ and $PYP\ge PXP=P|X|$, $(1-P)Y(1-P)\ge-(1-P)X(1-P)=(1-P)|X|$. Hence $\operatorname{Tr}(Y|X|)\ge \operatorname{Tr}(|X|^2P)+\operatorname{Tr}(|X|^2(1-P))=\|X\|_{\mathrm{HS}}^2$. $\square$

For a positive Hilbert–Schmidt operator $h$ and $t>0$ let $E_t(h)=\chi_{(t,\infty)}(h)$, a finite-rank projection (B10). The next lemma is the joint distribution argument: it controls spectral projections of two positive operators, on average over the level $t$, by the distance of the operators.

Lemma 2.5. Let $h,k$ be positive Hilbert–Schmidt operators on $K$. Then $s\mapsto\|E_{\sqrt s}(h)-E_{\sqrt s} (k)\|_{\mathrm{HS}}^2$ is measurable on $(0,\infty)$ and $\int_0^\infty\|E_{\sqrt s}(h)\|_{\mathrm{HS}}^2\,ds=\|h\|_{\mathrm{HS}}^2,\qquad \int_0^\infty\|E_{\sqrt s}(h)-E_{\sqrt s}(k)\|_{\mathrm{HS}}^2\,ds\le\|h-k\|_{\mathrm{HS}}\,\|h+k\|_{\mathrm{HS}}.$

Proof. Take orthonormal eigenbases $h\xi_i=\lambda_i\xi_i$, $k\eta_j=\mu_j\eta_j$ (B10), and put $c_{ij}=|\langle\eta_j,\xi_i\rangle|^2$. Since $\langle(E_t(h)-E_t(k))\eta_j,\xi_i\rangle=(\mathbf 1_{\lambda_i>t}- \mathbf 1_{\mu_j>t})\langle\eta_j,\xi_i\rangle$, $\|E_t(h)-E_t(k)\|_{\mathrm{HS}}^2=\sum_{i,j}c_{ij}\,|\mathbf 1_{\lambda_i>t}-\mathbf 1_{\mu_j>t}| ,$ a countable sum of measurable functions of $t$. For $\lambda,\mu\ge0$ the function $s\mapsto|\mathbf 1_{\lambda>\sqrt s}-\mathbf 1_{\mu>\sqrt s}|$ is the indicator of an interval of length $|\lambda^2-\mu^2|$. By Tonelli's theorem and the Cauchy–Schwarz inequality, $\int_0^\infty\|E_{\sqrt s}(h)-E_{\sqrt s}(k)\|_{\mathrm{HS}}^2ds=\sum_{i,j}c_{ij}|\lambda_i-\mu_j|(\lambda_i+\mu_j)\le \Big(\sum_{i,j}c_{ij}(\lambda_i-\mu_j)^2\Big)^{1/2}\Big(\sum_{i,j}c_{ij}(\lambda_i+\mu_j)^2\Big)^{1/2}.$ Since $\langle(h\mp k)\eta_j,\xi_i\rangle=(\lambda_i\mp\mu_j)\langle\eta_j,\xi_i\rangle$, the two sums are $\|h-k\|_{\mathrm{HS}}^2$ and $\|h+k\|_{\mathrm{HS}}^2$ (B10). The first identity is the case $k=0$. $\square$

2.4 Two averaging arguments

Lemma 2.6. Let $\varphi$ be a state on $B(K)$, $u_1,\dots,u_p\in B(K)$ unitaries with $\varphi\circ\operatorname{Ad}u_k=\varphi$, $T_1,\dots,T_q\in B(K)$, and $\eta>0$. There is a normal state $\psi$ on $B(K)$ with $\sum_{k=1}^p\|\psi\circ\operatorname{Ad}u_k-\psi\|\le\eta,\qquad|\psi(T_i)-\varphi(T_i)|\le\eta\quad(i=1,\dots,q).$

Proof. Consider the Banach space $Z=B(K)_*^p\times\mathbb C^q$ with norm $\sum_k\|\chi_k\|+\sum_i|c_i|$; its dual is $B(K)^p\times\mathbb C^q$. The set $W$ of the points $\big((\psi\circ\operatorname{Ad}u_k-\psi)_k, (\psi(T_i))_i\big)$, $\psi$ a normal state, is convex. By (B9) some net of normal states $\psi_\alpha$ converges weak* to $\varphi$. Then $\psi_\alpha\circ\operatorname{Ad}u_k-\psi_\alpha\to\varphi\circ\operatorname{Ad}u_k-\varphi=0$ for the weak topology of $B(K)_*$ (whose dual is $B(K)$), and $\psi_\alpha(T_i)\to\varphi(T_i)$. So the point $(0,(\varphi(T_i))_i)$ is in the weak closure of $W$, hence in its norm closure (B9). $\square$

Lemma 2.7. Let $N$ be a II₁ factor, $x_1,\dots,x_n\in N$ and $\varepsilon>0$. There are unitaries $u_1,\dots,u_p\in N$ and $\delta>0$ such that every state $\psi$ of $N$ (normal or not) with $\|\psi\circ\operatorname{Ad}u_k-\psi\|\le\delta$ for $k\le p$ satisfies $|\psi(x_j)-\tau(x_j)|\le\varepsilon$ for $j\le n$.

Proof. Suppose not. Let $\Lambda$ be the set of pairs $\lambda=(F,\delta)$, $F\subset\mathcal U(N)$ finite and $\delta>0$, directed by $(F,\delta)\le(F',\delta')$ if $F\subset F'$ and $\delta'\le\delta$. For each $\lambda$ choose a state $\psi_\lambda$ with $\|\psi_\lambda\circ\operatorname{Ad}u-\psi_\lambda\|\le\delta$ for $u\in F$ and an index $j(\lambda)$ with $|\psi_\lambda(x_{j(\lambda)})-\tau(x_{j(\lambda)})|>\varepsilon$. Some $j$ has $\Lambda_j=\{\lambda:j(\lambda)=j\}$ cofinal (if each $\Lambda_j$ had an element $\lambda_j$ with nothing of $\Lambda_j$ above it, an upper bound of $\lambda_1,\dots,\lambda_n$ would lie in no $\Lambda_j$). The state space is weak* compact, so the net $(\psi_\lambda)_{\lambda\in\Lambda_j}$ has a limit point $\psi$. For each unitary $u$ and $y\in N$, $|\psi_\lambda(uyu^*)-\psi_\lambda(y)|\le\delta\|y\|$ once $u\in F$, so $\psi\circ\operatorname{Ad}u=\psi$. Thus $\psi(uy)=\psi(u(yu)u^*)=\psi(yu)$, and by linearity $\psi$ is tracial, so $\psi=\tau$ by (B2). But $|\psi(x_j)-\tau(x_j)|\ge\varepsilon$, a contradiction. $\square$

Lemma 2.8 (testing a hypertrace). Let $N\subset B(K)$ be a II₁ factor, $A_0\subset N$ a weakly dense unital $*$-subalgebra and $\varphi$ a state on $B(K)$ with $\varphi|_N=\tau$ and $\varphi(aT)=\varphi(Ta)$ for $a\in A_0$, $T\in B(K)$. Then $\varphi$ is a hypertrace for $N$.

Proof. Let $x\in N$ and $T\in B(K)$. By (B5) choose $a_i\in A_0$ bounded with $\|a_i-x\|_2\to0$. By the Cauchy–Schwarz inequality for the state $\varphi$, $|\varphi((x-a_i)T)|\le\varphi((x-a_i)(x-a_i)^*)^{1/2}\varphi(T^*T)^{1/2}=\|(x-a_i)^*\|_2\,\varphi(T^*T)^{1/2}\to0,$ and likewise $|\varphi(T(x-a_i))|\le\varphi(TT^*)^{1/2}\|x-a_i\|_2\to0$. Hence $\varphi(xT)=\lim\varphi(a_iT)= \lim\varphi(Ta_i)=\varphi(Tx)$. $\square$

The hypothesis $\varphi|_N=\tau$ cannot be dropped from Lemma 2.8: without it, the restriction of $\varphi$ to $N$ might be a non-normal state, and the $\|\cdot\|_2$-approximation above would give no control.

3. The hyperfinite factor has all the properties

All conditions of Theorem 1.5 are invariant under isomorphism: an isomorphism of II₁ factors preserves the traces (B1), hence is implemented by a unitary between the standard forms (B3), which carries hypertraces, Følner projections and the norms in (f) and (g) to the corresponding objects; and injectivity does not depend on the representation (Corollary 2.2(a) of "Injective von Neumann algebras"). So for (a)⇒(c) and (a)⇒(h) we may assume $N=R$.

Proposition 3.1 ((a)⇒(c)). $R$ is semidiscrete.

Proof. Let $E_k\colon R\to A_k$ be the trace-preserving conditional expectation (B4); it is normal and ucp. Take $D_k=A_k$, $\alpha_k=E_k$ and $\beta_k$ the inclusion. For $x\in R$ and $a\in A_k$, $\|x-E_k(x)\|_2\le\|x-a\|_2$ because $\widehat{E_k(x)}$ is the orthogonal projection of $\hat x$ onto $\widehat{A_k}$. Since $\bigcup A_k$ is $\|\cdot\|_2$-dense in $R$ (B5), $\|x-E_k(x)\|_2\to0$. The net $E_k(x)$ is bounded by $\|x\|$, so it converges strongly on $L^2(R)$, hence $\sigma$-weakly, to $x$. $\square$

Proposition 3.2 ((c)⇒(b)). A semidiscrete von Neumann algebra $M\subset B(K)$ is injective.

Proof. Let $\alpha_i,\beta_i,D_i$ be as in Definition 1.3. Represent $D_i$ faithfully as a block-diagonal subalgebra of some $M_{n_i}=B(\mathbb C^{n_i})$, and let $F_i\colon M_{n_i}\to D_i$ be the compression to the diagonal blocks, a ucp projection. By Arveson's extension theorem (B8), $\alpha_i$ extends to a ucp map $\bar\alpha_i\colon B(K)\to M_{n_i}$. The maps $\Phi_i=\beta_i\circ F_i\circ\bar\alpha_i\colon B(K)\to M$ are ucp and bounded by $1$, so a subnet converges pointwise $\sigma$-weakly to a map $\Phi\colon B(K)\to M$ with $\|\Phi\|\le1$ (B8). For $x\in M$, $\Phi_i(x)=\beta_i(\alpha_i(x))\to x$. So $\Phi$ is a projection of norm one onto $M$. $\square$

Proposition 3.3 ((b)⇒(d)). If $N$ is injective, then $N$ has a hypertrace on $H$; more generally on every Hilbert space on which it acts faithfully and normally.

Proof. By Theorem 2.1 and Corollary 2.2(a) of "Injective von Neumann algebras", there is a projection of norm one $P\colon B(K)\to N$. By Tomiyama's theorem (B8), $P(xT)=xP(T)$ and $P(Tx)=P(T)x$ for $x\in N$. Then $\varphi=\tau\circ P$ is a state ($P$ is ucp) and $\varphi(xT)=\tau(xP(T))=\tau(P(T)x)=\varphi(Tx)$. $\square$

Proposition 3.4 ((a)⇒(h)). The flip of $R\bar\otimes R$ is approximately inner.

Proof. Let $(\varepsilon_{cd})_{c,d\le 2^k}$ be matrix units of $A_k$ and $S_k=\sum_{c,d}\varepsilon_{cd}\otimes\varepsilon_{dc}\in A_k\otimes A_k .$ A direct computation on matrix units gives $S_k^*=S_k$, $S_k^2=\sum_{c,d}\varepsilon_{cc}\otimes\varepsilon_{dd}=1$ and $S_k(\varepsilon_{ij}\otimes\varepsilon_{lm})S_k=\sum\varepsilon_{cd}\varepsilon_{ij}\varepsilon_{c'd'}\otimes \varepsilon_{dc}\varepsilon_{lm}\varepsilon_{d'c'}=\varepsilon_{lm}\otimes\varepsilon_{ij}$ (only $c=l$, $d=i$, $c'=j$, $d'=m$ contribute). So $\operatorname{Ad}S_k=\sigma$ on $A_k\otimes A_k$, and hence on $A_m\otimes A_m$ for $m\le k$. The union of the $A_m\otimes A_m$ is $\|\cdot\|_2$-dense in $R\bar\otimes R$ (B5). For $X\in R\bar\otimes R$ and $X_0\in A_m\otimes A_m$, since $\operatorname{Ad}S_k$ and $\sigma$ preserve $\|\cdot\|_2$, $\|S_kXS_k^*-\sigma(X)\|_2\le2\|X-X_0\|_2$ for $k\ge m$. So $\operatorname{Ad}S_k\to\sigma$ pointwise in $\|\cdot\|_2$. $\square$

The flip is not inner, however (Exercise 1). This is the typical behaviour of the conditions of Theorem 1.5: they are approximate versions of properties that finite-dimensional algebras have exactly.

4. Hypertraces and Følner projections

A reference for this section is [Connes 1976, §2.3–§2.6]; see also [Anantharaman–Popa, §10.3].

4.1 From a hypertrace to Følner projections

The proof of the next theorem follows the pattern of the classical proof of Følner's theorem for amenable groups: an invariant mean is approximated by almost invariant normal states (Lemma 2.6); their densities give almost invariant Hilbert–Schmidt vectors (Proposition 2.4); and a suitable level set of such a vector is an almost invariant finite-rank projection (Lemma 2.5).

Theorem 4.1 ((d)⇒(e)). Let $N\subset B(K)$ be a II₁ factor acting faithfully and normally, with a hypertrace $\varphi$ on $K$. Then $N$ satisfies the Følner condition on $K$. Moreover, for every finite set of unitaries $u_1,\dots,u_p\in N$ and $\delta>0$ there is a nonzero finite-rank projection $e$ with $\|[u_k,e]\|_{\mathrm{HS}}\le\delta\|e\|_{\mathrm{HS}}$ for all $k$.

Proof. Step 1: almost commuting projections. Let $u_1,\dots,u_p\in\mathcal U(N)$ and $\delta>0$. Put $\eta=(\delta^2/2p)^2$. By Lemma 2.6 (with $\varphi\circ\operatorname{Ad}u_k^*=\varphi$) there is a normal state $\psi=\operatorname{Tr}(\rho\,\cdot\,)$ with $\|\psi\circ\operatorname{Ad}u_k^*-\psi\|\le\eta$ for all $k$. Since $\psi(u_k^*Tu_k)=\operatorname{Tr}(u_k\rho u_k^*T)$, this says $\|u_k\rho u_k^*-\rho\|_{\mathrm{Tr}}\le\eta$ (B9). Let $h=\rho^{1/2}$, a positive Hilbert–Schmidt operator with $\|h\|_{\mathrm{HS}}=1$. As $(u_k\rho u_k^*)^{1/2}=u_khu_k^*$, Proposition 2.4 gives $\|u_khu_k^*-h\|_{\mathrm{HS}}\le\eta^{1/2}$. Now $E_t(u_khu_k^*)=u_kE_t(h)u_k^*$, and $\|h+u_khu_k^*\|_{\mathrm{HS}}\le2$, so Lemma 2.5 gives $\int_0^\infty\sum_{k=1}^p\|u_kE_{\sqrt s}(h)u_k^*-E_{\sqrt s}(h)\|_{\mathrm{HS}}^2\,ds\le2p\,\eta^{1/2}=\delta^2= \delta^2\int_0^\infty\|E_{\sqrt s}(h)\|_{\mathrm{HS}}^2\,ds .$ Let $f(s)=\sum_k\|u_kE_{\sqrt s}(h)u_k^*-E_{\sqrt s}(h)\|_{\mathrm{HS}}^2-\delta^2\|E_{\sqrt s}(h)\|_{\mathrm{HS}}^2$, so $\int f\le0$. If $f(s)>0$ for every $s$ with $E_{\sqrt s}(h)\neq0$, then, as $f=0$ where $E_{\sqrt s}(h)=0$ and $E_{\sqrt s}(h)\ne0$ for all $s<\|h\|^2$, we would get $\int f>0$. So there is $s>0$ with $e=E_{\sqrt s}(h)\ne0$ and $\sum_k\|u_keu_k^*-e\|_{\mathrm{HS}}^2\le\delta^2\|e\|_{\mathrm{HS}}^2$. Since $\|[u_k,e]\|_{\mathrm{HS}}= \|u_keu_k^*-e\|_{\mathrm{HS}}$, each $\|[u_k,e]\|_{\mathrm{HS}}\le\delta\|e\|_{\mathrm{HS}}$, and $e$ has finite rank (B10). This proves the last statement.

Step 2: almost commuting projections give almost tracial states. Let $e\ne0$ be a finite-rank projection and $u$ a unitary of $N$ with $\|[u,e]\|_{\mathrm{HS}}\le\delta\|e\|_{\mathrm{HS}}$. Put $\psi_e(y)=\operatorname{Tr}(eye)/\operatorname{Tr}(e)$ for $y\in N$, a normal state of $N$. With $f=u^*eu$, cyclicity gives $\operatorname{Tr}(euyu^*e)=\operatorname{Tr}(fyf)=\langle yf,f\rangle_{\mathrm{HS}}$, and $\|f-e\|_{\mathrm{HS}}=\|eu-ue\|_{\mathrm{HS}}\le\delta\|e\|_{\mathrm{HS}}$, $\|f\|_{\mathrm{HS}}=\|e\|_{\mathrm{HS}}$. Therefore $|\psi_e(uyu^*)-\psi_e(y)|=\frac{|\langle yf,f\rangle_{\mathrm{HS}}-\langle ye,e\rangle_{\mathrm{HS}}|}{\|e\|_{\mathrm{HS}}^2} \le\frac{|\langle y(f-e),f\rangle|+|\langle ye,f-e\rangle|}{\|e\|_{\mathrm{HS}}^2}\le2\delta\|y\|,$ that is, $\|\psi_e\circ\operatorname{Ad}u-\psi_e\|\le2\delta$.

Step 3: the Følner condition. Let $x_1,\dots,x_n\in N$ and $0<\varepsilon\le1$; put $c=1+\max_j\|x_j\|$. Each $x_j$ is a combination $\sum_{i=1}^4\lambda_{ji}w_{ji}$ of unitaries with $\sum_i|\lambda_{ji}|\le2\|x_j\|$ (write $x_j=a+ib$ with $a,b$ self-adjoint of norm at most $\|x_j\|$, and $a=\frac{\|a\|}2(w+w^*)$ with $w=a/\|a\|+i(1-a^2/\|a\|^2)^{1/2}$). Let $u'_1,\dots,u'_q$ and $\delta_0$ be given by Lemma 2.7 for $x_1,\dots,x_n$ and $\varepsilon$. Apply Step 1 to the unitaries $u'_l$ and $w_{ji}$, with $\delta=\min(\delta_0/2,\varepsilon/2c)$. The resulting projection $e$ satisfies $\|[x_j,e]\|_{\mathrm{HS}}\le2\|x_j\|\delta\|e\|_{\mathrm{HS}}\le\varepsilon\|e\|_{\mathrm{HS}}$, and by Step 2 the state $\psi_e$ satisfies $\|\psi_e\circ\operatorname{Ad}u'_l-\psi_e\|\le\delta_0$, so $|\psi_e(x_j)-\tau(x_j)|\le\varepsilon$ by Lemma 2.7. This is (1.3). $\square$

Remark 4.2. Steps 2 and 3 show that the second inequality in (1.3) follows from the first: if for every finite set of unitaries and every $\delta>0$ there is a nonzero finite-rank projection almost commuting with them in the sense of Step 1, then the Følner condition holds. Conversely the Følner condition gives such projections. So the Følner condition is equivalent to its commutator half alone.

4.2 Reformulations by representations

Let $\mathcal U=\mathcal U(N)$ with the discrete topology, $\pi$ its identity representation on $K$ and $\bar\pi(u)=\bar u$ the conjugate representation on $\overline K$. A unitary representation of a discrete group weakly contains the trivial representation if for every finite subset $F$ of the group and every $\varepsilon>0$ there is a unit vector $\zeta$ with $\|g\zeta-\zeta\|\le\varepsilon$ for $g\in F$.

Proposition 4.3. Let $N\subset B(K)$ be a II₁ factor acting faithfully and normally. The representation $u\mapsto(T\mapsto uTu^*)$ of $\mathcal U$ on $\mathrm{HS}(K)$ is unitarily equivalent to $\pi\otimes\bar\pi$. The following are equivalent:

(1) $N$ has a hypertrace on $K$;
(2) $\pi\otimes\bar\pi$ weakly contains the trivial representation of $\mathcal U$;
(3) $\big\|\sum_{j=1}^nu_j\otimes\bar u_j\big\|_{B(K\otimes\overline K)}=n$ for all $n$ and all unitaries $u_1,\dots,u_n\in N$.

By Theorem 1.5 (with Remark 5.2 for a general $K$) these conditions are also equivalent to $N\cong R$.

Proof. The unitary equivalence is the identification $\mathrm{HS}(K)\cong K\otimes\overline K$ of Section 1.1.

(1)⇒(2). Let $u_1,\dots,u_p\in\mathcal U$ and $\varepsilon>0$. As in Step 1 of Theorem 4.1, there is a normal state $\operatorname{Tr}(\rho\,\cdot\,)$ with $\|u_k\rho u_k^*-\rho\|_{\mathrm{Tr}}\le\varepsilon^2$, and $h=\rho^{1/2}$ is a unit vector of $\mathrm{HS}(K)$ with $\|u_khu_k^*-h\|_{\mathrm{HS}}\le\varepsilon$.

(2)⇒(1). For a unit vector $\zeta\in\mathrm{HS}(K)$ let $\psi_\zeta(T)=\operatorname{Tr}(\zeta^*T\zeta)$, a normal state. By cyclicity $\psi_\zeta(uTu^*)=\psi_{u^*\zeta u}(T)$, and $|\psi_{\zeta'}(T)-\psi_\zeta(T)|\le 2\|T\|\|\zeta'-\zeta\|_{\mathrm{HS}}$ for unit vectors. Choose, for each finite $F\subset\mathcal U$ and $\varepsilon>0$, a unit vector $\zeta_{F,\varepsilon}$ with $\|u^*\zeta u-\zeta\|_{\mathrm{HS}}\le\varepsilon$ for $u\in F$ (apply (2) to $F^{-1}$). A weak* limit point of the states $\psi_{\zeta_{F,\varepsilon}}$ is invariant under every $\operatorname{Ad}u$, $u\in\mathcal U$: it is a hypertrace.

(3)⇒(2). Let $u_2,\dots,u_n\in\mathcal U$ and $u_1=1$; put $U_j=u_j\otimes\bar u_j$. For unit vectors $a_j$, $\sum_{i,j}\|a_i-a_j\|^2=2n^2-2\|\sum_ja_j\|^2$. By (3) there are unit vectors $\zeta_m$ with $\|\sum_jU_j\zeta_m\|\to n$, so $\|U_j\zeta_m-\zeta_m\|^2=\|U_j\zeta_m-U_1\zeta_m\|^2\le2n^2-2\|\sum_iU_i\zeta_m\|^2\to0$. $\square$

Example 4.4 (amenable groups). Let $G$ be a countable discrete group with infinite conjugacy classes, so that its von Neumann algebra $L(G)\subset B(\ell^2(G))$, generated by the left translations $\lambda_g$, is a II₁ factor in standard form, with trace $\tau(x)=\langle x\delta_1,\delta_1\rangle$. Suppose $G$ is amenable, so that it has Følner sets: for every finite $S\subset G$ and $\varepsilon>0$ there is a finite nonempty $F\subset G$ with $|gF\,\triangle\,F|\le\varepsilon|F|$ for $g\in S$. Let $e_F$ be the projection onto $\ell^2(F)$. Then $\lambda_ge_F\lambda_g^*=e_{gF}$, so $\|[\lambda_g,e_F]\|_{\mathrm{HS}}^2=\|e_{gF}-e_F\|_{\mathrm{HS}}^2=|gF\,\triangle\,F|\le\varepsilon\,\|e_F\|_{\mathrm{HS}}^2 ,$ and $\operatorname{Tr}(e_F\lambda_ge_F)/\operatorname{Tr}(e_F)=|\{h\in F:gh=h\}|/|F|=\tau(\lambda_g)$. Moreover, for every $y\in L(G)$, $\operatorname{Tr}(e_Fye_F)/|F|=|F|^{-1}\sum_{h\in F}\langle y\delta_h,\delta_h\rangle=\tau(y)$, because $\delta_h=\rho_{h}\delta_1$ for a unitary $\rho_h$ of the right regular representation, which commutes with $y$. Let $\varphi$ be a weak* limit point of the states $T\mapsto\operatorname{Tr}(e_FTe_F)/|F|$ along Følner sets for larger and larger $S$ and smaller $\varepsilon$. The computation in Step 2 of Theorem 4.1 holds verbatim for every $y\in B(\ell^2(G))$, with $\delta=\varepsilon^{1/2}$; so $\varphi$ is invariant under each $\operatorname{Ad}\lambda_g$, and $\varphi|_{L(G)}=\tau$. The group algebra is weakly dense, so Lemma 2.8 shows that $\varphi$ is a hypertrace. By Theorem 1.5, $L(G)\cong R$: the von Neumann algebra of an amenable group with infinite conjugacy classes is the hyperfinite factor.

Remark 4.5 (the analogy with amenable groups). Example 4.4 explains the names. For a discrete group, a hypertrace of $L(G)$ plays the role of an invariant mean on $G$, and the Følner condition (1.3) the role of Følner sets: a finite set $F$ is replaced by a finite-rank projection $e$, its cardinality by $\operatorname{Tr}(e)= \|e\|_{\mathrm{HS}}^2$, and the symmetric difference $gF\triangle F$ by the commutator $[u,e]$. The proof of Theorem 4.1 is Namioka's argument for Følner's theorem [Namioka 1964], transposed: in the group case one approximates an invariant mean by almost invariant densities in $\ell^1(G)$, passes to $\ell^2$ by a square root, and takes a level set; here the densities are trace-class operators, the square root is controlled by the Powers–Størmer inequality, and the level sets are spectral projections, controlled by Lemma 2.5.

Proposition 4.6. If a factor $M$ acting on a Hilbert space $K$ has a hypertrace on $K$, then $M$ is finite. In particular a factor with a hypertrace is either a matrix algebra or a II₁ factor, and in the second case (with separable predual) it is hyperfinite.

Proof. The restriction $t$ of a hypertrace to $M$ is a tracial state. If $M$ were not finite, $1$ would be the sum of two orthogonal projections $p,q$, each equivalent to $1$ (in an infinite factor the identity is properly infinite and can be halved); with $v^*v=1$, $vv^*=p$ we get $t(p)=t(vv^*)=t(v^*v)=1$, likewise $t(q)=1$, and $1=t(1)=t(p)+t(q)=2$, a contradiction. A finite factor is a matrix algebra or a II₁ factor. For a II₁ factor with separable predual, Theorem 1.5 and Remark 5.2 give $N\cong R$. $\square$

This explains the word "hypertrace": the state extends the trace of $N$ to all of $B(K)$ while keeping the trace property with respect to $N$.

5. From Følner projections to norm inequalities

A reference for this section is [Connes 1976, §2.7–§2.8]; the argument of Proposition 5.3 goes back to [Effros–Lance 1977].

Proposition 5.1 ((e)⇒(f)). Let $N\subset B(K)$ be a II₁ factor acting faithfully and normally and satisfying the Følner condition on $K$. Then for all $a_j,b_j\in N$ $\Big|\tau\Big(\sum_ja_jb_j^*\Big)\Big|\le\Big\|\sum_ja_j\otimes\bar b_j\Big\|_{\min}.$

Proof. By homogeneity we may assume $\|a_j\|\le1$. Let $\varepsilon>0$ and choose, by (1.3), a nonzero finite-rank projection $e$ with $\|[b_j^*,e]\|_{\mathrm{HS}}\le\varepsilon\|e\|_{\mathrm{HS}}$ for all $j$ and $|\tau(c)-\operatorname{Tr}(ece)/\operatorname{Tr}(e)|\le\varepsilon$ for $c=\sum_ja_jb_j^*$. Let $m=\operatorname{Tr}(e)$, $L=eK$, $Q=eB(K)e\cong M_m$ with normalized trace $\mathrm{tr}(y)=\operatorname{Tr}(y)/m$, and $a'_j=ea_je$, $b'_j=eb_je$.

First estimate. $ea_jb_j^*e-a'_jb'^*_j=ea_j(1-e)b_j^*e$, and $(1-e)b_j^*e=(1-e)(b_j^*e-eb_j^*)$, so by (B10) $|\operatorname{Tr}(ea_j(1-e)b_j^*e)|\le\|ea_j(1-e)\|_{\mathrm{HS}}\|(1-e)b_j^*e\|_{\mathrm{HS}}\le \|e\|_{\mathrm{HS}}\cdot\varepsilon\|e\|_{\mathrm{HS}}=\varepsilon m .$ Hence $|\mathrm{tr}(\sum_ja'_jb'^*_j)-\tau(\sum_ja_jb_j^*)|\le(n+1)\varepsilon$.

Second estimate. Let $\xi_1,\dots,\xi_m$ be an orthonormal basis of $L$ and $\zeta=m^{-1/2}\sum_i\xi_i\otimes \bar\xi_i$, a unit vector in $L\otimes\overline L\subset K\otimes\overline K$. For $x,y\in Q$, $\langle(x\otimes\bar y)\zeta,\zeta\rangle=\frac1m\sum_{i,l}\langle x\xi_i,\xi_l\rangle\langle\bar y\bar\xi_i,\bar\xi_l\rangle =\frac1m\sum_{i,l}\langle x\xi_i,\xi_l\rangle\overline{\langle y\xi_i,\xi_l\rangle}=\mathrm{tr}(xy^*).$ Since $(e\otimes\bar e)\zeta=\zeta$ and $(e\otimes\bar e)(a\otimes\bar b)(e\otimes\bar e)=eae\otimes\overline{ebe}$, $\mathrm{tr}\Big(\sum_ja'_jb'^*_j\Big)=\Big\langle\Big(\sum_ja_j\otimes\bar b_j\Big)\zeta,\zeta\Big\rangle ,$ which has modulus at most $\|\sum_ja_j\otimes\bar b_j\|$. With the first estimate and $\varepsilon\to0$, the claim follows. The norm on $K\otimes\overline K$ is the minimal norm of $N\odot\overline N$ (B7). $\square$

Remark 5.2. The right side of (f) is the minimal norm on $N\odot\overline N$, which depends only on $N$. So Theorem 4.1 and Proposition 5.1 show: if $N$ has a hypertrace on some Hilbert space on which it acts faithfully and normally, then (f) holds, and hence (by the rest of the proof) $N\cong R$. This is the sense in which $H$ may be replaced by any such space in (d) and (e).

Proposition 5.3 ((f)⇒(g)). Let $N$ satisfy (f). Then the product map $\mu$ extends to an isometric $*$-isomorphism of $N\otimes_{\min}N'$ onto $C^*(N,N')$.

Proof. Every element of $N'$ is $JbJ$ with $b\in N$. By (1.1) and (1.2), condition (f) says exactly that the linear functional $w(A)=\langle\mu(A)\hat1,\hat1\rangle$ on $N\odot N'$ satisfies $|w(A)|\le\|A\|_{\min}$. So $w$ extends to a functional on $N\otimes_{\min}N'$ with $\|w\|\le1=w(1)$, a state (B9). For $A,B\in N\odot N'$, $B^*A^*AB\le\|A\|_{\min}^2B^*B$ in $N\otimes_{\min}N'$, so $w(B^*A^*AB)\le\|A\|_{\min}^2w(B^*B)$. As $\mu$ is a $*$-homomorphism on $N\odot N'$, $w(C^*C)=\|\mu(C)\hat1\|^2$, so $\|\mu(A)\mu(B)\hat1\|\le\|A\|_{\min}\|\mu(B)\hat1\|\qquad(A,B\in N\odot N').$ The vectors $\mu(B)\hat1$ include $N\hat1$, which is dense in $H$; hence $\|\mu(A)\|\le\|A\|_{\min}$. Thus $\mu$ extends to a $*$-homomorphism of $N\otimes_{\min}N'$ onto $C^*(N,N')$. The C*-algebras $N$ and $N'$ are simple (B1), so $N\otimes_{\min}N'$ is simple (B7); the kernel of $\mu$ is a closed ideal not containing $1$, hence zero, and an injective $*$-homomorphism of C*-algebras is isometric. $\square$

Proposition 5.4 ((g)⇒(b)). Let $M\subset B(K)$ be any von Neumann algebra such that $\|\sum x_jy_j\|\le\|\sum x_j\otimes y_j\|_{\min}$ for $x_j\in M$, $y_j\in M'$. Then $M$ is injective.

Proof. The hypothesis gives a unital $*$-homomorphism $\mu\colon M\otimes_{\min}M'\to B(K)$ with $\mu(x\otimes y)=xy$. Since $M\otimes_{\min}M'\subset B(K)\otimes_{\min}M'$ (B7), Arveson's extension theorem gives a ucp map $\Phi\colon B(K)\otimes_{\min}M'\to B(K)$ extending $\mu$. On $1\otimes M'$, $\Phi$ is the $*$-homomorphism $1\otimes y\mapsto y$, so $1\otimes M'$ lies in the multiplicative domain of $\Phi$ (B8). Put $P(T)=\Phi(T\otimes1)$ for $T\in B(K)$. For $y\in M'$, $yP(T)=\Phi((1\otimes y)(T\otimes1))=\Phi(T\otimes y)=\Phi((T\otimes1)(1\otimes y))=P(T)y ,$ so $P(T)\in M''=M$. Also $P(x)=\mu(x\otimes1)=x$ for $x\in M$ and $\|P\|\le1$. So $P$ is a projection of norm one onto $M$. $\square$

At this point (b), (d), (e), (f) and (g) are equivalent, by Propositions 3.3, 5.1, 5.3, 5.4 and Theorem 4.1.

Lemma 5.5. Let $M$ be a II₁ factor acting on $L^2(M)$ and $\theta\in\overline{\operatorname{Int}}M$. Then $\|\sum_i\theta(a_i)b_i\|=\|\sum_ia_ib_i\|$ for all $a_i\in M$, $b_i\in M'$.

Proof. Let $u_n$ be unitaries with $\|u_nau_n^*-\theta(a)\|_2\to0$ for all $a$. For $y'\in M'$, $\|(u_nau_n^*-\theta(a))y'\hat1\|\le\|y'\|\|u_nau_n^*-\theta(a)\|_2$, and $M'\hat1$ is dense, so the bounded sequence $u_nau_n^*$ tends strongly to $\theta(a)$. Since $u_n$ commutes with $b_i$, $\sum_iu_na_iu_n^*b_i=u_n(\sum_ia_ib_i)u_n^*$, which has norm $\|\sum a_ib_i\|$. For a vector $\xi$, $\sum_i\theta(a_i)b_i\xi=\lim_n\sum_iu_na_iu_n^*b_i\xi$, so $\|\sum\theta(a_i)b_i\|\le\|\sum a_ib_i\|$. Applying this to $\theta^{-1}\in\overline{\operatorname{Int}}M$ and the elements $\theta(a_i)$ gives the reverse inequality. $\square$

Lemma 5.5 is the easy half of Theorem 3.2 of "Approximately inner and centrally trivial automorphisms" (Proposition 3.5 there); the other half is the criterion (L1).

Proposition 5.6 ((h)⇒(g)). If the flip of $N\bar\otimes N$ is approximately inner, then (g) holds.

Proof. Put $M=N\bar\otimes N$ on $H\otimes H=L^2(M)$ (B3); $M'=N'\bar\otimes N'$. For $x_j\in N$, $y_j\in N'$ take $a_j=x_j\otimes1\in M$, $b_j=1\otimes y_j\in M'$. Then $\sum a_jb_j=\sum x_j\otimes y_j$ and $\sum\sigma(a_j)b_j=1\otimes\sum x_jy_j$. Lemma 5.5 gives $\|\sum x_j\otimes y_j\|=\|\sum x_jy_j\|$. $\square$

5.1 Semidiscreteness and purifications

Proposition 5.7 holds for arbitrary von Neumann algebras, Proposition 5.8 for $\sigma$-finite ones, that is, those with a faithful normal state. Neither is needed for Theorem 1.5.

Proposition 5.7. If a von Neumann algebra $M\subset B(K)$ is semidiscrete, then $\|\sum x_jy_j\|\le\|\sum x_j\otimes y_j\|_{\min}$ for all $x_j\in M$, $y_j\in M'$.

Proof. Let $\alpha_i,\beta_i,D_i$ be as in Definition 1.3. Fix $i$ and write $D=D_i\cong\bigoplus_rM_{n_r}$, $\alpha=\alpha_i$, $\beta=\beta_i$. Define $\Gamma\colon D\odot M'\to B(K)$ by $\Gamma(d\otimes y)=\beta(d)y$; $D\odot M'$ is a C*-algebra (B7). We check that $\Gamma$ is positive. It suffices to treat one block $M_n$, where $M_n\odot M'=M_n(M')$. A positive element is $Z^*Z$ with $Z=(z_{ij})\in M_n(M')$, whose $(i,j)$ entry is $\sum_lz_{li}^*z_{lj}$. Since $z_{li}\in M'$ commutes with $\beta(e_{ij})\in M$, $\Gamma(Z^*Z)=\sum_{i,j,l}\beta(e_{ij})z_{li}^*z_{lj}=\sum_lR_l^*\big[\beta(e_{ij})\big]_{i,j}R_l, \qquad R_l\xi=(z_{l1}\xi,\dots,z_{ln}\xi)\in K^n ,$ and the matrix $[\beta(e_{ij})]$ is positive because $\beta$ is completely positive. So $\Gamma$ is positive. It is even completely positive. For $k\ge1$, $M_k(D\odot M')=D\odot M_k(M')$, and on this algebra $\Gamma^{(k)}(d\otimes Y)=(\beta(d)\otimes1_k)Y$. Here $M_k(M')$ is the commutant of $M\otimes1_k$ on $K\otimes\mathbb C^k$, and $d\mapsto\beta(d)\otimes1_k$ is completely positive, so the computation above, with $M'$ replaced by $M_k(M')$ and $\beta$ by $\beta\otimes1_k$, shows that $\Gamma^{(k)}$ is positive. A unital completely positive map is contractive (B8). The map $\alpha\otimes\mathrm{id}$ is contractive from $M\otimes_{\min}M'$ to $D\otimes_{\min}M'$ (B7). Therefore $\Big\|\sum_j\beta_i(\alpha_i(x_j))y_j\Big\|=\Big\|\Gamma\Big((\alpha_i\otimes\mathrm{id})\Big(\sum_jx_j\otimes y_j\Big)\Big)\Big\| \le\Big\|\sum_jx_j\otimes y_j\Big\|_{\min}.$ As $i$ runs, $\sum_j\beta_i(\alpha_i(x_j))y_j\to\sum_jx_jy_j$ weakly, and the norm is weakly lower semicontinuous. $\square$

Let $M$ be $\sigma$-finite, act standardly on $K$ with conjugation $J$, and let $\varphi=\langle\,\cdot\,\xi,\xi\rangle$ be a faithful normal state, with $\xi$ cyclic and separating (B3). The map $y\mapsto Jy^*J$ is a linear $*$-anti-isomorphism of $M$ onto $M'$, that is, an isomorphism of the opposite algebra $M^{\mathrm{op}}$ onto $M'$. Following [Powers–Størmer 1970] and [Woronowicz 1972], we call the functional $\tilde\varphi(x\otimes y^{\mathrm{op}})=\langle xJy^*J\xi,\xi\rangle$ on $M\odot M^{\mathrm{op}}$ the purification of $\varphi$; its restriction to $M\otimes1$ is $\varphi$, and it is the vector state of $\xi$ transported from $C^*(M,M')$.

Proposition 5.8 (purifications). Let $M$ be a $\sigma$-finite von Neumann algebra in standard form on $K$. The following are equivalent:

(1) $\|\sum x_jy_j\|\le\|\sum x_j\otimes y_j\|_{\min}$ for $x_j\in M$, $y_j\in M'$;
(2) for some faithful normal state $\varphi$, the purification $\tilde\varphi$ is bounded for the minimal norm, that is, it extends to a state of $M\otimes_{\min}M^{\mathrm{op}}$;
(3) the same holds for every faithful normal state.

They hold when $M$ is semidiscrete, and they imply that $M$ is injective. For factors with separable predual, injectivity implies semidiscreteness (for II₁ factors by Theorem 1.5; for all such factors this is Theorem 11.1 of the lesson "Classification of injective factors", applied in a separable representation), so for such factors (1)–(3) are equivalent to semidiscreteness.

Proof. Under $y^{\mathrm{op}}\mapsto Jy^*J$, which is isometric for the minimal norm (B7), condition (2) says that $A\mapsto\langle\mu(A)\xi,\xi\rangle$ is bounded by $\|A\|_{\min}$ on $M\odot M'$. (1)⇒(3) is then clear, and (3)⇒(2) is trivial. (2)⇒(1) is the proof of Proposition 5.3 with $\hat1$ replaced by $\xi$, which only uses that $M\xi$ is dense. Proposition 5.7 gives (1) for semidiscrete $M$, and Proposition 5.4 gives injectivity from (1). $\square$

Remark. The hypothesis that $M$ be $\sigma$-finite cannot be dropped from Proposition 5.8. Let $I$ be an uncountable set and $M=\ell^\infty(I)$, acting by multiplication on $\ell^2(I)$; this is a standard form, and $M'=M$. For finitely many $x_j,y_j\in M$, $\Big\|\sum_jx_jy_j\Big\|=\sup_i\Big|\sum_jx_j(i)y_j(i)\Big|\le\sup_{i,k}\Big|\sum_jx_j(i)y_j(k)\Big| =\Big\|\sum_jx_j\otimes y_j\Big\|_{\min},$ so (1) holds. But a normal state of $M$ is given by a probability vector in $\ell^1(I)$, whose support is countable, so $M$ has no faithful normal state: (2) fails, and (3) holds only vacuously. Proposition 5.7, which uses no state, is the form that holds for arbitrary $M$.

6. Tensor squares and the flip

In this section we prove (g)⇒(h). The idea: the conditions (b)–(g) pass to $N\bar\otimes N$, and for a factor satisfying (g) every automorphism is approximately inner by the criterion (L1). Since condition (g) is not obviously stable under tensor products (it involves the whole von Neumann tensor product, not only the algebraic one), we pass through the hypertrace.

Lemma 6.1. Let $N_1\subset B(K_1)$, $N_2\subset B(K_2)$ be II₁ factors with hypertraces $\varphi_1,\varphi_2$. Then $N_1\bar\otimes N_2$ has a hypertrace on $K_1\otimes K_2$.

Proof. Let $\tau_1,\tau_2$ be the traces; $\varphi_i|_{N_i}=\tau_i$. By Lemma 2.6 we can find a net of normal states $\psi_\alpha$ on $B(K_1)$ with $\|\psi_\alpha\circ\operatorname{Ad}u-\psi_\alpha\|\to0$ for every $u\in\mathcal U(N_1)$ and $\psi_\alpha(x)\to\tau_1(x)$ for every $x\in N_1$: index the net by triples (finite set of unitaries, finite subset of $N_1$, $\eta$), ordered by inclusion and decreasing $\eta$. Define states on $B(K_1\otimes K_2)$ by $S_\alpha(T)=\varphi_2\big((\psi_\alpha\otimes\mathrm{id})(T)\big),$ using the slice maps of (B6), and let $\Phi$ be a weak* limit point of $(S_\alpha)$.

Invariance. For $v\in\mathcal U(N_2)$, $(\psi_\alpha\otimes\mathrm{id})((1\otimes v)T(1\otimes v^*))= v(\psi_\alpha\otimes\mathrm{id})(T)v^*$, so $S_\alpha$ and $\Phi$ are $\operatorname{Ad}(1\otimes v)$-invariant. For $u\in\mathcal U(N_1)$, (B6) gives $|S_\alpha((u\otimes1)T(u^*\otimes1))-S_\alpha(T)|=\big|\varphi_2\big(((\psi_\alpha\circ\operatorname{Ad}u-\psi_\alpha) \otimes\mathrm{id})(T)\big)\big|\le\|\psi_\alpha\circ\operatorname{Ad}u-\psi_\alpha\|\,\|T\|\to0,$ so $\Phi$ is $\operatorname{Ad}(u\otimes1)$-invariant, hence $\operatorname{Ad}(u\otimes v)$-invariant. The span of the $u\otimes v$ is the algebraic tensor product $N_1\odot N_2$, which is weakly dense in $N_1\bar\otimes N_2$.

Restriction. Let $X\in N_1\bar\otimes N_2$. The slice $(\psi_\alpha\otimes\mathrm{id})(X)$ commutes with $N_2'$ (as $X$ commutes with $1\otimes N_2'$), so it lies in $N_2$, where $\varphi_2=\tau_2$. Hence $S_\alpha(X)=(\psi_\alpha\otimes\tau_2)(X)=\psi_\alpha(E(X))$, where $E(X)=(\mathrm{id}\otimes\tau_2)(X)\in N_1$. So $S_\alpha(X)\to\tau_1(E(X))=(\tau_1\otimes\tau_2)(X)$, and $\Phi|_{N_1\bar\otimes N_2}=\tau_1\otimes\tau_2$, the trace of $N_1\bar\otimes N_2$.

Lemma 2.8, with $A_0=N_1\odot N_2$, shows that $\Phi$ is a hypertrace. $\square$

Theorem 6.2 ((g)⇒(h)). If $N$ satisfies (g), then every automorphism of $N\bar\otimes N$ is approximately inner; in particular the flip is.

Proof. By Proposition 5.4, $N$ is injective, and by Proposition 3.3 it has a hypertrace on $H$. By Lemma 6.1, $M=N\bar\otimes N$ has a hypertrace on $H\otimes H=L^2(M)$ (B3). $M$ is a II₁ factor with separable predual, so Theorem 4.1, Proposition 5.1 and Proposition 5.3 applied to $M$ show that $M$ satisfies (g): the product map is an isometric isomorphism of $M\otimes_{\min}M'$ onto $C^*(M,M')$. Let $\theta\in\operatorname{Aut}M$ and $a_i\in M$, $b_i\in M'$. By (B7), $\|\sum\theta(a_i)\otimes b_i\|_{\min}=\|\sum a_i\otimes b_i\|_{\min}$, hence $\|\sum\theta(a_i)b_i\|=\|\sum a_ib_i\|$. By the criterion (L1), $\theta$ is approximately inner. $\square$

So far we have proved that (b)–(h) are equivalent and follow from (a). It remains to prove (a) from them. This is done in two independent steps, which the last three sections carry out: a copy of $R$ splits off $N$ (Section 8), and $N$ is approximately conjugate into that copy (Sections 7 and 9).

7. Matrix models and embedding in an ultrapower of $R$

A reference for this section is [Connes 1976, §4.2].

Notation 7.1. Let $\mathbb F_\infty$ be the free group on generators $g_1,g_2,\dots$, and $\mathbb F_n\subset \mathbb F_\infty$ the subgroup generated by $g_1,\dots,g_n$. The length of a reduced word $w=g_{i_1}^{s_1}\cdots g_{i_L}^{s_L}$ ($s_l=\pm1$) is $L$. For unitaries $u=(u_1,u_2,\dots)$ of an algebra, $w\mapsto u(w)$ is the homomorphism of $\mathbb F_\infty$ into the unitary group with $u(g_i)=u_i$ (for a finite family $u_1,\dots,u_n$ we set $u_i=1$ for $i>n$).

Lemma 7.2 (matrix models). Let $N\subset B(K)$ be a II₁ factor satisfying the Følner condition on $K$. Let $u_1,\dots,u_n\in\mathcal U(N)$, $k\ge1$ and $\varepsilon>0$. There are $m\ge1$ and unitaries $v_1,\dots,v_n\in M_m$ such that $|\tau(u(w))-\mathrm{tr}_m(v(w))|\le\varepsilon\qquad\text{for every }w\in\mathbb F_n\text{ of length less than }k,$ where $\mathrm{tr}_m$ is the normalized trace of $M_m$.

Proof. Put $\delta=\varepsilon/(2k)$. By (1.3) there is a nonzero finite-rank projection $e$ with $\|[u_i,e]\|_{\mathrm{HS}}\le\delta\|e\|_{\mathrm{HS}}$ for $i\le n$ and $|\tau(u(w))-\operatorname{Tr}(eu(w)e)/ \operatorname{Tr}(e)|\le\delta$ for the finitely many $w$ of length less than $k$. Let $m=\operatorname{Tr}(e)$ and $Q=eB(K)e\cong M_m$, with normalized trace $\mathrm{tr}$ and norm $\|y\|_{2,Q}=\mathrm{tr}(y^*y)^{1/2}= \|y\|_{\mathrm{HS}}/\|e\|_{\mathrm{HS}}$.

Unitaries. Let $T_i=eu_ie\in Q$, and let $v_i$ be a unitary of $Q$ with $T_i=v_i|T_i|$ (B1). Then $e-T_i^*T_i=eu_i^*(1-e)u_ie=|(1-e)u_ie|^2$ and $(1-e)u_ie=(1-e)(u_ie-eu_i)$, so $\mathrm{tr}(e-|T_i|^2)=\|(1-e)u_ie\|_{\mathrm{HS}}^2/m\le\delta^2$. As $0\le|T_i|\le e$, $\mathrm{tr}((e-|T_i|)^2)\le\mathrm{tr}((e-|T_i|)(e+|T_i|))\le\delta^2$, so $\|T_i-v_i\|_{2,Q}=\|v_i(|T_i|-e)\|_{2,Q}\le\delta,\qquad\|T_i^*-v_i^*\|_{2,Q}=\|T_i-v_i\|_{2,Q}\le\delta .$

Words. Let $w$ have length $L<k$, with letters $x_1,\dots,x_L\in\{u_i^{\pm1}\}$, and let $y_l\in\{v_i^{\pm1}\}$ be the corresponding letters. For $x\in\{u_i^{\pm1}\}$, $\|[x,e]\|_{\mathrm{HS}}\le\delta\|e\|_{\mathrm{HS}}$ (for $x=u_i^*$, $[u_i^*,e]=-u_i^*[u_i,e]u_i^*$). Using $ex_1x_2\cdots x_Le=(ex_1e)(ex_2\cdots x_Le)+ex_1(1-e)x_2\cdots x_Le$ and $\|ex_1(1-e)\|_{\mathrm{HS}}=\|e[x_1,e](1-e)\|_{\mathrm{HS}}\le\delta\|e\|_{\mathrm{HS}}$, induction gives $\|eu(w)e-(ex_1e)\cdots(ex_Le)\|_{2,Q}\le(L-1)\delta$. Telescoping with $\|ex_le\|\le1$ and $\|ex_le-y_l\|_{2,Q}\le\delta$ gives $\|(ex_1e)\cdots(ex_Le)-v(w)\|_{2,Q}\le L\delta$. Hence $|\mathrm{tr}(v(w))-\tau(u(w))|\le\|v(w)-eu(w)e\|_{2,Q}+\Big|\frac{\operatorname{Tr}(eu(w)e)}{\operatorname{Tr}(e)}-\tau(u(w))\Big| \le(2L-1)\delta+\delta\le2k\delta=\varepsilon .$ Identifying $Q$ with $M_m$ finishes the proof. $\square$

Theorem 7.3. Let $N$ be a II₁ factor with separable predual satisfying the Følner condition on some Hilbert space. Then there is a unital normal injective $*$-homomorphism $\pi\colon N\to R^\omega$ with $\tau_{R^\omega}\circ\pi=\tau$.

Proof. Choose unitaries $u_1,u_2,\dots$ generating $N$ as a von Neumann algebra (B1). For $k\ge1$, apply Lemma 7.2 to $u_1,\dots,u_k$, the length bound $k$ and $\varepsilon=1/k$: we get $m_k$ and unitaries $v^k_1,\dots,v^k_k\in M_{m_k}$. Every II₁ factor contains a unital copy of every matrix algebra (split $1$ into $m$ equivalent projections of trace $1/m$, B1), and on it the trace of the factor is the normalized trace; so we regard $v^k_i$ as unitaries of $R$, with $v^k_i=1$ for $i>k$. Let $v_i\in R^\omega$ be the class of the sequence $(v^k_i)_k$. Each $w\in\mathbb F_\infty$ lies in $\mathbb F_k$ and has length less than $k$ for all large $k$, so $\tau_{R^\omega}(v(w))=\lim_{k\to\omega}\tau_R(v^k(w))=\tau(u(w))\qquad(w\in\mathbb F_\infty). \tag{7.1}$ Let $A_0$ be the linear span of the $u(w)$; it is the unital $*$-algebra generated by the $u_i$ (as $u(w)^*=u(w^{-1})$ and $u(w)u(w')=u(ww')$), and it is weakly dense in $N$. If $\sum_w\lambda_wu(w)=0$ (finite sum), then by (7.1) $\Big\|\sum_w\lambda_wv(w)\Big\|_2^2=\sum_{w,w'}\overline{\lambda_w}\lambda_{w'}\tau_{R^\omega}(v(w^{-1}w'))= \sum_{w,w'}\overline{\lambda_w}\lambda_{w'}\tau(u(w^{-1}w'))=\Big\|\sum_w\lambda_wu(w)\Big\|_2^2=0,$ so $\pi_0(\sum\lambda_wu(w))=\sum\lambda_wv(w)$ is a well-defined linear map. It is a unital $*$-homomorphism (it is defined on words by a group homomorphism) and preserves traces by (7.1). Lemma 2.1 extends it to $N$. $\square$

Proposition 7.4. Let $N$ be as in Theorem 7.3. There is a unital normal injective $*$-homomorphism $\theta\colon N\bar\otimes N\to(N\bar\otimes R)^\omega$, preserving traces, such that

(a) for $x\in N$, $\theta(x\otimes1)$ is the class of the constant sequence $(x\otimes1)$;
(b) for $y\in N$, $\theta(1\otimes y)$ is the class of a sequence of the form $(1\otimes z_k)_k$ with $z_k\in R$.

Proof. Let $\pi$ be as in Theorem 7.3. For $y\in N$ let $(\pi(y)_k)_k$ be a bounded representing sequence of $\pi(y)$ and set $\pi_2(y)=$ class of $(1\otimes\pi(y)_k)_k$ in $(N\bar\otimes R)^\omega$; this does not depend on the representing sequence, since $\|1\otimes z\|_2=\|z\|_2$. The map $r\mapsto1\otimes r$ is a trace-preserving $*$-homomorphism $R\to N\bar\otimes R$, so it induces one $R^\omega\to(N\bar\otimes R)^\omega$, and $\pi_2$ is its composition with $\pi$. Let $\pi_1(x)$ be the class of the constant sequence $(x\otimes1)$. Then $\pi_1(N)$ and $\pi_2(N)$ commute, and the traces factorize: $\tau_\omega(\pi_1(x)\pi_2(y))=\lim_{k\to\omega}\tau(x)\,\tau_R(\pi(y)_k)=\tau(x)\tau(y).$ Hence $x\otimes y\mapsto\pi_1(x)\pi_2(y)$ is a unital $*$-homomorphism of $N\odot N$ preserving $\tau\otimes\tau$. The algebraic tensor product is weakly dense in $N\bar\otimes N$, and Lemma 2.1 gives $\theta$. $\square$

8. Central sequences and the McDuff property

This section proves that a II₁ factor satisfying (g) and (h) absorbs $R$: $N\cong N\bar\otimes R$. The proof has three parts. First, property Γ and the flip show that $N$ has noncommuting central sequences (Sections 8.1–8.3). Second, noncommuting central sequences give approximately central $2\times2$ matrix units (Section 8.4). Third, McDuff's theorem builds from these a tensor factor isomorphic to $R$ (Section 8.5). Basic references are [Connes 1974] and [Connes 1976, §3.2]; the theorem of Section 8.5 is due to McDuff.

Throughout, a central sequence in a II₁ factor $M$ is a bounded sequence $(x_k)$ with $\|[x_k,y]\|_2\to0$ for all $y\in M$. For a finite set $F\subset M$ and $\delta>0$, an element $a$ is $(F,\delta)$-central if $\|[a,f]\|_2<\delta$ for $f\in F$. If $(y_l)$ is $\|\cdot\|_2$-dense in the unit ball of $M$, a bounded sequence $(x_k)$ is central as soon as $\|[x_k,y_l]\|_2\to0$ for each $l$, because $\|[x,y]\|_2\le\|[x,y_l]\|_2+2\|x\|\|y-y_l\|_2$.

8.1 Property Γ

Proposition 8.1. If $N$ satisfies (g), then $N$ has property Γ.

Proof. By (g) and Proposition 5.3, $C^*(N,N')$ is isomorphic to $N\otimes_{\min}N'$, which is simple (B1), (B7). The set $C^*(N,N')\cap\mathcal K(H)$ of compact operators in it is a closed two-sided ideal. If it were nonzero it would be all of $C^*(N,N')$, and $1$ would be compact, which is false as $H$ is infinite-dimensional. So $C^*(N,N')$ contains no nonzero compact operator, and $N$ has property Γ by Theorem 2.5 of "Property Γ and the algebra generated by a factor and its commutant". $\square$

8.2 Centrally trivial automorphisms are inner

Lemma 8.2. Let $M$ be a II₁ factor with separable predual and $\theta\in\operatorname{Ct}M$. For every $\varepsilon>0$ there are a finite set $F\subset M$ and $\delta>0$ such that every $(F,\delta)$-central unitary $v$ satisfies $\|\theta(v)-v\|_2<\varepsilon$.

Proof. Otherwise there are $\varepsilon>0$ and, for each $k$, a unitary $v_k$ which is $(\{y_1,\dots,y_k\},1/k)$-central, where $(y_l)$ is $\|\cdot\|_2$-dense in the unit ball, with $\|\theta(v_k)-v_k\|_2\ge\varepsilon$. Then $(v_k)$ is a central sequence on which $\theta$ is not trivial, contradicting $\theta\in\operatorname{Ct}M$. $\square$

Lemma 8.3. Let $M$ be a II₁ factor with separable predual, $\theta\in\operatorname{Ct}M$ and $\sigma\in\overline{\operatorname{Int}}M$. Then $\theta\sigma\theta^{-1}\sigma^{-1}$ is inner.

Proof. Let $u_n$ be unitaries with $\operatorname{Ad}u_n\to\sigma$ pointwise in $\|\cdot\|_2$. For $k\ge1$, Lemma 8.2 gives $F_k$ and $\delta_k$ for $\varepsilon=2^{-k}$. Choose $n_1<n_2<\cdots$ such that $\|u_nfu_n^*-u_{n'}fu_{n'}^*\|_2<\delta_k$ for $f\in F_k$ and $n,n'\ge n_k$. Put $v_k=u_{n_k}^*u_{n_{k+1}}$. Then $\|[v_k,f]\|_2=\|v_kfv_k^*-f\|_2=\|u_{n_{k+1}}fu_{n_{k+1}}^*-u_{n_k}fu_{n_k}^*\|_2<\delta_k$ for $f\in F_k$, so $\|\theta(v_k)-v_k\|_2<2^{-k}$. Let $w_k=\theta(u_{n_k})u_{n_k}^*$. Since $u_{n_{k+1}}=u_{n_k}v_k$, $w_{k+1}=\theta(u_{n_k})\theta(v_k)v_k^*u_{n_k}^*,\qquad\|w_{k+1}-w_k\|_2=\|\theta(v_k)v_k^*-1\|_2=\|\theta(v_k)-v_k\|_2<2^{-k}.$ So $(w_k)$ is $\|\cdot\|_2$-Cauchy in the unit ball and converges to some $w\in M$ (B1). As $\|w_k^*-w^*\|_2=\|w_k-w\|_2$, products converge too, and $w^*w=\lim w_k^*w_k=1$; in a finite algebra $w$ is then unitary. For $x\in M$, $\|w_kxw_k^*-wxw^*\|_2\le2\|w_k-w\|_2\|x\|\to0$. On the other hand $\operatorname{Ad}w_k=\theta\circ\operatorname{Ad}u_{n_k}\circ\theta^{-1}\circ\operatorname{Ad}u_{n_k}^*$. Since $\operatorname{Ad}u_{n_k}^*\to\sigma^{-1}$ and $\operatorname{Ad}u_{n_k}\to\sigma$ pointwise in $\|\cdot\|_2$, all maps involved are $\|\cdot\|_2$-isometric, and $\|u(y_k)u^*-u(y)u^*\|_2=\|y_k-y\|_2$, we get $\operatorname{Ad}w_k(x)\to\theta\sigma\theta^{-1}\sigma^{-1}(x)$. Hence $\theta\sigma\theta^{-1}\sigma^{-1}= \operatorname{Ad}w$. $\square$

Lemma 8.4. Let $M_1,M_2$ be II₁ factors and $\theta_i\in\operatorname{Aut}M_i$. If $\theta_1\otimes\theta_2$ is inner on $M_1\bar\otimes M_2$, then $\theta_1$ is inner.

Proof. Let $\theta_1\otimes\theta_2=\operatorname{Ad}U$. For $x\in M_1$, $U(x\otimes1)=(\theta_1(x)\otimes1)U$. Let $\chi$ be a normal functional on $M_2$ and $V=(\mathrm{id}\otimes\chi)(U)\in M_1$, the slice map (B6) on the second factor, which is $M_1$-bimodular: $(\mathrm{id}\otimes\chi)((a\otimes1)X(b\otimes1))=a(\mathrm{id}\otimes\chi)(X)b$. Slicing the relation gives $Vx=\theta_1(x)V$ for all $x\in M_1$. Since $U\ne0$ and product functionals separate points, $\chi$ can be chosen with $V\ne0$. Now $V^*Vx=V^*\theta_1(x)V=(\theta_1(x^*)V)^*V=(Vx^*)^*V=xV^*V$, so $V^*V\in M_1\cap M_1'=\mathbb C$; similarly $VV^*\theta_1(x)=Vx V^*=\theta_1(x)VV^*$, and $\theta_1(M_1)=M_1$, so $VV^*\in\mathbb C$. With $V^*V=c>0$, $W=c^{-1/2}V$ is an isometry whose range projection is a nonzero scalar projection, hence $1$; so $W$ is unitary and $\theta_1=\operatorname{Ad}W$. $\square$

Proposition 8.5. If $N$ satisfies (h), then $\operatorname{Ct}N=\operatorname{Int}N$.

Proof. Inner automorphisms are centrally trivial: $ux_ku^*-x_k=[u,x_k]u^*$. Conversely let $\theta\in\operatorname{Ct}N$ and $M=N\bar\otimes N$, a II₁ factor with separable predual. By (L2), $\theta\otimes\mathrm{id}\in\operatorname{Ct}M$. By (h), the flip $\sigma$ is in $\overline{\operatorname{Int}}M$. By Lemma 8.3, $(\theta\otimes\mathrm{id})\sigma(\theta\otimes\mathrm{id})^{-1}\sigma^{-1}$ is inner. Evaluating on $x\otimes y$: $\sigma^{-1}(x\otimes y)=y\otimes x$, then $\theta^{-1}(y)\otimes x$, then $x\otimes\theta^{-1}(y)$, then $\theta(x)\otimes\theta^{-1}(y)$. So $\theta\otimes\theta^{-1}$ is inner, and $\theta$ is inner by Lemma 8.4. $\square$

8.3 Noncommuting central sequences

Recall that $N_\omega=N'\cap N^\omega$ is the algebra of classes of central sequences along $\omega$.

Lemma 8.6. Let $M$ be a II₁ factor with separable predual and suppose $M_\omega$ is commutative. For every $\varepsilon>0$ there are a finite set $F\subset M$ and $\delta>0$ such that any two $(F,\delta)$-central elements $a,b$ of the unit ball satisfy $\|[a,b]\|_2<\varepsilon$.

Proof. Otherwise, with $(y_l)$ dense as above, there are $\varepsilon>0$ and $(\{y_1,\dots,y_k\},1/k)$-central elements $a_k,b_k$ of the unit ball with $\|[a_k,b_k]\|_2\ge\varepsilon$. The sequences $(a_k)$, $(b_k)$ are central, so their classes $a,b$ lie in $M_\omega$, and $\|ab-ba\|_{2}=\lim_{k\to\omega}\|[a_k,b_k]\|_2\ge\varepsilon$. This contradicts the commutativity of $M_\omega$. $\square$

Proposition 8.7. Let $M$ be a II₁ factor with separable predual such that $M_\omega$ is commutative. Then $\overline{\operatorname{Int}}M\subset\operatorname{Ct}M$.

Proof. Let $\sigma=\lim\operatorname{Ad}u_n$ (pointwise in $\|\cdot\|_2$), let $(x_m)$ be a central sequence, which we may assume in the unit ball, and $\varepsilon>0$. Take $F,\delta$ from Lemma 8.6, and $n_0$ with $\|u_nfu_n^*-u_{n'}fu_{n'}^*\|_2<\delta$ for $f\in F$ and $n,n'\ge n_0$. Fix $n=n_0$. Choose $m_0$ such that for $m\ge m_0$, $x_m$ is $(F,\delta)$-central and $\|[u_n,x_m]\|_2<\varepsilon$. For such $m$, choose $n'\ge n_0$ with $\|\sigma(x_m)-u_{n'}x_mu_{n'}^*\|_2<\varepsilon$. The unitary $v=u_n^*u_{n'}$ is $(F,\delta)$-central (as in the proof of Lemma 8.3), so $\|[v,x_m]\|_2<\varepsilon$ by Lemma 8.6. Since $u_{n'}x_mu_{n'}^*=u_n(vx_mv^*)u_n^*$, $\|\sigma(x_m)-x_m\|_2\le\varepsilon+\|vx_mv^*-x_m\|_2+\|u_nx_mu_n^*-x_m\|_2<3\varepsilon\qquad(m\ge m_0).$ So $\|\sigma(x_m)-x_m\|_2\to0$, as required by the definition of $\operatorname{Ct}M$. $\square$

Corollary 8.8. If $N$ satisfies (g) and (h), then $N_\omega$ is not commutative.

Proof. If $N_\omega$ were commutative, Propositions 8.7 and 8.5 would give $\overline{\operatorname{Int}}N\subset \operatorname{Ct}N=\operatorname{Int}N$. By Lemma 1.4, $\operatorname{Int}N$ would be closed in the $u$-topology, so $N$ would be full and, by Theorem 6.5 of "Full factors", without property Γ. This contradicts Proposition 8.1. $\square$

8.4 Matrix units in the central sequence algebra

Lemma 8.9. A finite von Neumann algebra $A$ that is not commutative contains a nonzero partial isometry $v$ with $v^*v\perp vv^*$.

Proof. A von Neumann algebra is generated by its projections, so some projection $p\in A$ is not central. If $(1-p)Ap=0$, then also $pA(1-p)=((1-p)Ap)^*=0$, so every $x=pxp+(1-p)x(1-p)$ commutes with $p$, a contradiction. Take $y=(1-p)xp\ne0$ and its polar decomposition $y=v|y|$; then $v^*v\le p$ and $vv^*\le1-p$. $\square$

The next lemma is the reindexation trick: a finite configuration of central sequences can be moved, without changing its joint distribution, so that it commutes with, and is independent of, any given finite configuration.

Lemma 8.10. Let $N$ be a II₁ factor with separable predual, $a_1,\dots,a_r\in N_\omega$ and $S\subset N^\omega$ a finite self-adjoint set. There are $b_1,\dots,b_r\in N_\omega$ such that

(1) $b_i$ commutes with every element of $S$;
(2) $\tau_\omega(W(b))=\tau_\omega(W(a))$ for every noncommutative monomial $W$ in $r$ variables and their adjoints;
(3) $\tau_\omega(s\,W(b))=\tau_\omega(s)\,\tau_\omega(W(b))$ for every such $W$ and every $s$ in the unital $*$-algebra generated by $S$.

In particular, every $*$-polynomial relation $P(a)=0$ implies $P(b)=0$.

Proof. Take bounded representing sequences $(a_{i,k})_k$ of the $a_i$, and $(s_k)_k$ of each monomial $s$ in the elements of $S$ (products of representing sequences). Enumerate the monomials $W_1,W_2,\dots$ in the $a$'s, the monomials $s^1,s^2,\dots$ in the elements of $S$, and a $\|\cdot\|_2$-dense sequence $(y_l)$ in the unit ball of $N$. For each $n$ let $K_n$ be the set of $k\ge n$ such that, for all $i\le r$ and $l,l'\le n$,

$\|[a_{i,k},y_l]\|_2<1/n$, and $\|[a_{i,k},t_n]\|_2<1/n$ for every $t\in S$ ($t_n$ its $n$-th representative);
$|\tau(W_l(a_{\cdot,k}))-\tau_\omega(W_l(a))|<1/n$;
$|\tau(s^{l'}_nW_l(a_{\cdot,k}))-\tau(s^{l'}_n)\tau(W_l(a_{\cdot,k}))|<1/n$.

Each condition holds for all $k$ in a set belonging to $\omega$: the first because $(a_{i,k})_k$ is central along $\omega$ and $y_l$, $t_n$ are fixed elements of $N$; the second by definition of $\tau_\omega$; the third by the asymptotic independence lemma of "Full factors" (Lemma 6.3 there), applied to the bounded sequence $(W_l(a_{\cdot,k}))_k$, which is central along $\omega$, and the fixed element $x=s^{l'}_n$. Also $\{k\ge n\}\in\omega$. So $K_n\in\omega$ is nonempty; choose $k_n\in K_n$ and let $b_i$ be the class of $(a_{i,k_n})_n$.

The first condition makes $(a_{i,k_n})_n$ central along the cofinite filter, hence along $\omega$, so $b_i\in N_\omega$; and $\|[b_i,t]\|_2=\lim_{n\to\omega}\|[a_{i,k_n},t_n]\|_2=0$ for $t\in S$, which is (1). The second and third conditions give (2) and (3) in the limit $n\to\omega$. Finally, for a $*$-polynomial $P$, $\|P(b)\|_2^2=\tau_\omega(P(b)^*P(b))$ is a combination of moments, so it equals $\|P(a)\|_2^2$ by (2), and $\tau_\omega$ is faithful. $\square$

Proposition 8.11. Let $N$ be a II₁ factor with separable predual such that $N_\omega$ is not commutative. Then $N_\omega$ contains a unital copy of $M_2$: elements $E_{11},E_{21}$ with $E_{21}^*E_{21}=E_{11}$ and $E_{21}E_{21}^*=1-E_{11}$, and $E_{11}$ a projection.

Proof. $N_\omega$ is a finite von Neumann algebra (from "Ultraproducts and the asymptotic centralizer"). By Lemma 8.9 it contains $v\ne0$ with $v^*v\perp vv^*$. Put $e_{11}=v^*v$, $e_{21}=v$, $e_{12}=v^*$, $e_{22}=vv^*$, a system of $2\times2$ matrix units with unit $q=e_{11}+e_{22}$, and $t=\tau_\omega(e_{11})\in(0,\tfrac12]$. If $q=1$ we are done. Otherwise we build copies $e^{(1)}=e,e^{(2)},e^{(3)},\dots$ of this system inductively: $e^{(k)}$ is given by Lemma 8.10 applied to $a=(e_{11},e_{21})$ and $S=\{e^{(j)}_{cd}:j<k,\ c,d\le2\}$. So $e^{(k)}$ is a system of matrix units in $N_\omega$ with unit $q_k=e^{(k)}_{11}+e^{(k)}_{22}$, commuting with all earlier systems, and $\tau_\omega(sq_k)= 2t\,\tau_\omega(s)$ for $s$ in the algebra generated by the earlier systems.

The projections $q_j$ commute. Put $p_1=1$ and $p_k=\prod_{j<k}(1-q_j)$. Then $p_k$ commutes with $e^{(k)}$, and $\tau_\omega(p_{k+1})=\tau_\omega(p_k)-\tau_\omega(p_kq_k)=(1-2t)\tau_\omega(p_k)$, so $\tau_\omega(p_k)=(1-2t)^{k-1}$. Let $f^{(k)}_{cd}=p_ke^{(k)}_{cd}$. For $j<k$, $e^{(j)}_{cd}p_k=e^{(j)}_{cd}q_jp_k=0$ because $q_jp_k=0$, and $p_ke^{(k)}_{ab}$ commutes with $e^{(j)}$; hence $f^{(j)}_{cd}f^{(k)}_{ab}=0=f^{(k)}_{ab}f^{(j)}_{cd}$. Within one index, $f^{(k)}_{ab}f^{(k)}_{cd}=\delta_{bc}f^{(k)}_{ad}$. So $F^K_{cd}=\sum_{k\le K}f^{(k)}_{cd}$ is a system of matrix units with unit $\sum_{k\le K}p_kq_k=1-p_{K+1}$. For $K<K'$, $\|F^{K'}_{cd}-F^K_{cd}\|_2^2=\sum_{K<k\le K'}\tau_\omega\big(p_ke^{(k)}_{dd}\big)=\tfrac12\sum_{K<k\le K'}\tau_\omega(p_kq_k) \le\tau_\omega(p_{K+1}),$ because the blocks are orthogonal and $p_ke^{(k)}_{11}$, $p_ke^{(k)}_{22}$ are equivalent projections with sum $p_kq_k$; and $\sum_{K<k\le K'}\tau_\omega(p_kq_k)=\tau_\omega(p_{K+1})-\tau_\omega(p_{K'+1})$. The bound tends to $0$ (if $t=\frac12$, then $q=1$). The unit ball of $N_\omega$ is complete for $\|\cdot\|_2$, so $F^K_{cd}\to E_{cd}$, and the matrix unit relations pass to the limit (products of bounded $\|\cdot\|_2$-convergent sequences converge). The unit is $E_{11}+E_{22}=\lim(1-p_{K+1})=1$. $\square$

8.5 McDuff's theorem

Proof. All limits are along $\omega$, and "small" means tending to $0$ in $\|\cdot\|_2$ along $\omega$; we use the inequalities of (B1) freely, together with the uniform bound on the norms.

A projection of trace $\frac12$. Let $a'_n=(a_n+a_n^*)/2$; then $a'_n-a_n$ and $a'^2_n-a'_n$ are small. Let $e_n=\chi_{[1/2,\infty)}(a'_n)$. For real $s$, $|\chi_{[1/2,\infty)}(s)-s|\le2|s^2-s|$ (for $s\ge\frac12$ because $2s\ge1$, for $s<\frac12$ because $2|s-1|>1$), so $\|e_n-a'_n\|_2\le2\|a'^2_n-a'_n\|_2$ is small, and $\tau(e_n)\to\frac12$. By (B1) choose a projection $p_n$ with $\tau(p_n)=\frac12$ and $p_n\le e_n$ or $p_n\ge e_n$; then $\|p_n-e_n\|_2^2=|\tau(e_n)-\frac12|$ is small. So $p_n-a_n$ is small.

Cutting $b_n$. Let $c_n=(1-p_n)b_np_n$. Now $b_n-c_n=p_nb_n+(1-p_n)b_n(1-p_n)$. Since $b_nb_n^*-(1-p_n)$ is small, $\|p_nb_n\|_2^2=\tau(p_nb_nb_n^*p_n)$ differs from $\tau(p_n(1-p_n)p_n)=0$ by a small amount; likewise $\|b_n(1-p_n)\|_2^2=\tau((1-p_n)b_n^*b_n(1-p_n))$ is small. So $b_n-c_n$ is small. Also $c_n^*c_n-p_n=(p_nb_n^*b_np_n-p_n)-p_nb_n^*p_nb_np_n$ is small.

Polar decomposition. Let $c_n=w_n|c_n|$, with $s_n=w_n^*w_n\le p_n$ the support of $|c_n|$ and $w_nw_n^*\le1-p_n$. In $p_nPp_n$, $|t-1|\le|t^2-1|$ for $t\ge0$ gives $\||c_n|-p_n\|_2\le\|c_n^*c_n-p_n\|_2$, so $\|c_n-w_n\|_2=\|w_n(|c_n|-p_n)\|_2$ is small (note $w_np_n=w_n$). Hence $\tau(s_n)=\|w_n\|_2^2\to\lim\|c_n\|_2^2= \lim\tau(c_n^*c_n)=\frac12$, and $\tau(p_n-s_n)\to0$.

Completing the partial isometry. The projections $p_n-s_n$ and $(1-p_n)-w_nw_n^*$ have the same trace, so they are equivalent (B1): let $r_n$ be a partial isometry with $r_n^*r_n=p_n-s_n$, $r_nr_n^*=(1-p_n)-w_nw_n^*$, and $v_n=w_n+r_n$. The initial and final projections of $w_n$ and $r_n$ are orthogonal, so $v_n^*v_n=s_n+(p_n-s_n)=p_n$ and $v_nv_n^*=1-p_n$. Finally $\|v_n-b_n\|_2\le\|r_n\|_2+\|w_n-c_n\|_2+\|c_n-b_n\|_2$, and $\|r_n\|_2^2=\tau(p_n-s_n)$. $\square$

Lemma 8.13. Let $N$ be a II₁ factor with separable predual such that $N_\omega$ is not commutative, and let $A\subset N$ be a unital subfactor isomorphic to a matrix algebra $M_m$, with matrix units $\varepsilon_{cd}$, and $P=A'\cap N$. Then $P$ is a II₁ factor, and for every finite set $G\subset P$ and $\delta>0$ there are a projection $p$ and a partial isometry $v$ in $P$ with $v^*v=p$, $vv^*=1-p$, such that $p$ and $v$ are $(G,\delta)$-central.

Proof. $P$ is a II₁ factor. The map $y\mapsto y\varepsilon_{11}$ is a $*$-isomorphism of $P$ onto $\varepsilon_{11}N\varepsilon_{11}$ (injective since $y\varepsilon_{cc}=\varepsilon_{c1}(y\varepsilon_{11}) \varepsilon_{1c}$; surjective since for $z\in\varepsilon_{11}N\varepsilon_{11}$ the element $\sum_c\varepsilon_{c1}z \varepsilon_{1c}$ lies in $P$), and $\varepsilon_{11}N\varepsilon_{11}$ is a II₁ factor (B1). Its trace is the restriction of $\tau$.

The expectation onto $P$. Let $E(x)=\frac1m\sum_{c,d}\varepsilon_{dc}x\varepsilon_{cd}$. One checks on matrix units that $\varepsilon_{kl}E(x)=\frac1m\sum_c\varepsilon_{kc}x\varepsilon_{cl}=E(x)\varepsilon_{kl}$, so $E(x)\in P$; $E$ is the identity on $P$ (as $\sum_{c,d}\varepsilon_{dc}\varepsilon_{cd}=m$), preserves $\tau$, and, since $x=\frac1m\sum_{c,d}\varepsilon_{dc}\varepsilon_{cd}x$, $\|E(x)-x\|_2\le\frac1m\sum_{c,d}\|\varepsilon_{dc}(x\varepsilon_{cd}-\varepsilon_{cd}x)\|_2\le\frac1m\sum_{c,d}\|[x,\varepsilon_{cd}]\|_2 . \tag{8.1}$

Lifting. By Proposition 8.11, let $E_{11},E_{21}\in N_\omega$ be unital $2\times2$ matrix units, with bounded representing sequences $(a_n),(b_n)$. The hypotheses of Lemma 8.12 hold in $N$ (the quantities are $\|\cdot\|_2$-norms of representatives of $0$, and $\tau_\omega(E_{11})=\frac12$ because $E_{11}\sim E_{22}$), so we get exact matrix units $f^{(n)}_{11}=p_n$, $f^{(n)}_{21}=v_n$ in $N$ which still represent $E_{11},E_{21}$. Hence $\|[f^{(n)}_{cd},y]\|_2\to0$ along $\omega$ for all $y\in N$. By (8.1), $a'_n=E(p_n)$ and $b'_n=E(v_n)$ satisfy $\|a'_n-p_n\|_2\to0$, $\|b'_n-v_n\|_2\to0$ along $\omega$; they lie in $P$, are bounded by $1$, and $\tau(a'_n)=\frac12$. So Lemma 8.12, applied in $P$, gives exact matrix units $p'_n,v'_n$ in $P$ with $\|p'_n-p_n\|_2+\|v'_n-v_n\|_2\to0$ along $\omega$. Then $\|[p'_n,g]\|_2\le\|[p_n,g]\|_2+2\|g\|\|p'_n-p_n\|_2\to0$ along $\omega$ for $g\in G$, and likewise for $v'_n$. Some $n$ gives the required $p=p'_n$, $v=v'_n$. $\square$

Theorem 8.14 (McDuff). Let $N$ be a II₁ factor with separable predual such that $N_\omega$ is not commutative. Then $N\cong N\bar\otimes R$.

Reference: due to McDuff; Connes's extension is stated in [Connes 1976, §3.2.2].

Proof. Let $(y_j)_{j\ge1}$ be $\|\cdot\|_2$-dense in the unit ball of $N$, and $\delta_k=8^{-k}$. We construct mutually commuting unital subfactors $Q_1,Q_2,\dots$ of $N$, each isomorphic to $M_2$, and finite sets $G_k$. Put $A_0=\mathbb C$, $A_k$ the algebra generated by $Q_1,\dots,Q_k$ and $P_k=A_k'\cap N$. Since $Q_{k+1}\subset P_k$ commutes with $A_k$, the map $a\otimes q\mapsto aq$ is a unital $*$-homomorphism of the simple algebra $A_k\otimes Q_{k+1}$, hence injective, so $A_{k+1}\cong A_k\otimes M_2$ and $A_k\cong M_2^{\otimes k}$ compatibly. Also $P_0=N$ and $P_{k+1}=Q_{k+1}'\cap P_k$. Let $E_k$ be the expectation of (8.1) onto $P_k$.

For $x\in N$ and matrix units $\varepsilon_{cd}$ of $A_k$ ($c,d\le2^k$), the elements $x_{cd}=\sum_e\varepsilon_{ec}x\varepsilon_{de}$ lie in $P_k$, have norm at most $\|x\|$, and $x=\sum_{c,d}\varepsilon_{cd}x_{cd}, \tag{8.2}$ as one checks from the matrix unit relations. Given $Q_1,\dots,Q_k$, let $G_k=\{(y_j)_{cd}:j\le k,\ c,d\le2^k\}\cup\{E_k(g):g\in G_i,\ i<k\}\subset P_k ,$ and choose, by Lemma 8.13 applied to $A=A_k$ and the finite set $G_k\cup G_k^*$, a unital copy $Q_{k+1}$ of $M_2$ in $P_k$ with matrix units $\phi_{11}=p$, $\phi_{21}=v$, $\phi_{12}=v^*$, $\phi_{22}=1-p$, where $p,v$ are $(G_k\cup G_k^*,\delta_k)$-central. Since $\|[v^*,g]\|_2=\|[g^*,v]\|_2$ and $[1-p,g]=-[p,g]$, each $\phi_{cd}$ then $\delta_k$-commutes with every $g\in G_k$.

Step 1: each $g\in G_i$ is close to $P=\bigcap_kP_k$. For $h\in P_k$, the expectation onto $P_{k+1}=Q_{k+1}'\cap P_k$ is $h\mapsto\frac12\sum_{c,d}\phi_{dc}h\phi_{cd}$, and it agrees with $E_{k+1}(h)$: both are trace-preserving and $P_{k+1}$-bimodular maps of $P_k$ onto $P_{k+1}$ that fix $P_{k+1}$ (B4). By (8.1), for $h\in G_k$, $\|E_{k+1}(h)-h\|_2\le\frac12\cdot4\delta_k=2\delta_k$. Let $g\in G_i$. For $k\ge i$, $E_k(g)\in G_k$ (by construction for $k>i$, and $E_i(g)=g$), and $E_{k+1}(g)=E_{k+1}(E_k(g))$. Hence $\|E_{k+1}(g)-E_k(g)\|_2\le2\delta_k$, and $(E_k(g))_{k\ge i}$ is $\|\cdot\|_2$-Cauchy and bounded by $\|g\|$. Its limit $z_g$ lies in $N$ (B1), and in each $P_l$ (for $k\ge l$, $E_k(g)\in P_k\subset P_l$, and the ball of $P_l$ is $\|\cdot\|_2$-closed); so $z_g\in P$ and $\|g-z_g\|_2\le\sum_{k\ge i}2\delta_k<3\cdot8^{-i}.$

Step 2: $N$ is generated by $P$ and $A_\infty=\bigcup_kA_k$. Let $L$ be the von Neumann algebra generated by $P\cup A_\infty$, and fix $j$. For $i\ge j$, (8.2) gives $y_j=\sum_{c,d}\varepsilon_{cd}(y_j)_{cd}$ with $(y_j)_{cd}\in G_i$, so $z=\sum_{c,d}\varepsilon_{cd}z_{(y_j)_{cd}}\in L$ satisfies $\|y_j-z\|_2\le4^i\cdot3\cdot8^{-i}=3\cdot2^{-i}$. So $\hat y_j$ lies in the closure of $\hat L$ in $L^2(N)$, hence $y_j\in L$ (B4). The $y_j$ are dense in the unit ball of $N$ and $L$ is strongly closed, so $L=N$.

Step 3: $N\cong P\bar\otimes R$. The von Neumann algebra $R_0$ generated by $A_\infty$ is isomorphic to $R$ by Lemma 2.2(a). For $y\in P$ and matrix units $\varepsilon_{cd}$ of $A_k$: $\tau(y\varepsilon_{cd})=0$ for $c\ne d$ (since $\tau(y\varepsilon_{cd})=\tau(y\varepsilon_{cd}\varepsilon_{dd})=\tau(\varepsilon_{dd}y\varepsilon_{cd})= \tau(y\varepsilon_{dd}\varepsilon_{cd})$), and $\tau(y\varepsilon_{cc})=\tau(y\varepsilon_{c1}\varepsilon_{1c})= \tau(\varepsilon_{1c}y\varepsilon_{c1})=\tau(y\varepsilon_{11})$ for all $c$, so $\tau(y\varepsilon_{cc})=2^{-k}\tau(y)$. Hence $\tau(yr)=\tau(y)\tau(r)$ for $y\in P$, $r\in A_\infty$. So $y\otimes r\mapsto yr$ is a unital $*$-homomorphism from $P\odot A_\infty$ (weakly dense in $P\bar\otimes R_0$) into $N$ preserving the traces; by Lemma 2.1 it extends to an isomorphism of $P\bar\otimes R_0$ onto the von Neumann algebra generated by $P\cup A_\infty$, which is $N$ by Step 2.

Conclusion. By Lemma 2.2(b), $N\bar\otimes R\cong P\bar\otimes R\bar\otimes R\cong P\bar\otimes R\cong N$. $\square$

Corollary 8.15. If $N$ satisfies (g) and (h), then $N\cong N\bar\otimes R$.

Proof. Corollary 8.8 and Theorem 8.14. $\square$

9. End of the proof

A reference for this section is [Connes 1976, §4.2]; see also [Anantharaman–Popa, Chapter 11].

Proposition 9.1 ((e), (g), (h) ⇒ (i)). Let $N$ satisfy (e), (g) and (h). Then $N$ satisfies (i).

Proof. $N\cong N\bar\otimes R$ by Corollary 8.15. Let $x_1,\dots,x_n\in N$ and $\varepsilon>0$. By (h) there is a unitary $v\in N\bar\otimes N$ with $\|x_j\otimes1-v(1\otimes x_j)v^*\|_2\le\varepsilon/2$ for all $j$ (as $\sigma(1\otimes x_j)=x_j\otimes1$). Let $\theta\colon N\bar\otimes N\to(N\bar\otimes R)^\omega$ be as in Proposition 7.4 (which uses (e)). As $\theta$ is a trace-preserving $*$-homomorphism, it preserves $\|\cdot\|_2$, so $\|\theta(x_j\otimes1)-\theta(v)\theta(1\otimes x_j)\theta(v)^*\|_2\le\varepsilon/2 .$ By Lemma 2.3, $\theta(v)$ has a representing sequence of unitaries $X_k\in N\bar\otimes R$, and by Proposition 7.4, $\theta(x_j\otimes1)$ and $\theta(1\otimes x_j)$ are represented by $(x_j\otimes1)_k$ and $(1\otimes z_{j,k})_k$ with $z_{j,k}\in R$. The norm $\|\cdot\|_2$ in the ultrapower is the limit along $\omega$ of the norms of representatives, so $\lim_{k\to\omega}\|x_j\otimes1-X_k(1\otimes z_{j,k})X_k^*\|_2\le\varepsilon/2\qquad(j=1,\dots,n).$ The set of $k$ for which all $n$ norms are at most $\varepsilon$ belongs to $\omega$, so it is nonempty; any such $k$ gives $X=X_k$ and $z_j=z_{j,k}$. $\square$

Proposition 9.2 ((i)⇒(a)). If $N$ satisfies (i), then $N\cong R$.

Proof. We verify the Murray–von Neumann criterion (B11). Fix an isomorphism $\Psi\colon N\bar\otimes R\to N$; it preserves traces, hence $\|\cdot\|_2$. Let $x_1,\dots,x_n\in N$ and $\varepsilon>0$, and put $y_j=\Psi^{-1}(x_j)\in N\bar\otimes R$.

Approximation by finitely many coordinates. The algebra $N\odot\bigcup_kA_k$ is weakly dense in $N\bar\otimes R$, so by (B5) there are $k$ and elements $y'_j=\sum_{c,d}a^{(j)}_{cd}\otimes\varepsilon_{cd}\in N\otimes A_k$ with $\|y_j-y'_j\|_2\le\varepsilon/3$, where $\varepsilon_{cd}$ ($c,d\le2^k$) are matrix units of $A_k$ and $a^{(j)}_{cd}\in N$.

Using (i) on a second copy of $R$. By Lemma 2.2(b), $R=A_k\bar\otimes R_{>k}$ with $R_{>k}\cong R$, so $N\bar\otimes R=N\bar\otimes A_k\bar\otimes R_{>k}$. Condition (i), transported by an isomorphism $R\cong R_{>k}$, applied to the finitely many $a^{(j)}_{cd}$ with $\eta=\varepsilon/(3\cdot4^k)$, gives a unitary $X\in N\bar\otimes R_{>k}$ and $z^{(j)}_{cd}\in R_{>k}$ with $\|a^{(j)}_{cd}\otimes1-X(1\otimes z^{(j)}_{cd})X^*\|_2\le\eta$. Let $Y\in N\bar\otimes A_k \bar\otimes R_{>k}$ be $X$ placed on the first and third tensor factors; $Y$ commutes with $1\otimes A_k\otimes1$. Then, with $w_j=\sum_{c,d}\varepsilon_{cd}\otimes z^{(j)}_{cd}\in A_k\bar\otimes R_{>k}=R$, $\Big\|y'_j-Y(1\otimes w_j)Y^*\Big\|_2=\Big\|\sum_{c,d}\big(a^{(j)}_{cd}\otimes\varepsilon_{cd}\otimes1- Y(1\otimes\varepsilon_{cd}\otimes z^{(j)}_{cd})Y^*\big)\Big\|_2\le\sum_{c,d}\eta=\varepsilon/3 ,$ because the $(c,d)$ term equals $(a^{(j)}_{cd}\otimes1\otimes1-Y(1\otimes1\otimes z^{(j)}_{cd})Y^*)(1\otimes\varepsilon_{cd} \otimes1)$, whose $\|\cdot\|_2$ is at most that of the first factor, which is computed in $N\bar\otimes R_{>k}$.

Approximation inside $R$. By (B5) there are $m$ and $q_j\in A_m\subset R$ with $\|w_j-q_j\|_2\le\varepsilon/3$. Then $Q=\Psi(Y(1\otimes A_m)Y^*)$ is a unital subfactor of $N$ isomorphic to $M_{2^m}$, and $\|x_j-\Psi(Y(1\otimes q_j)Y^*)\|_2=\|y_j-Y(1\otimes q_j)Y^*\|_2\le\|y_j-y'_j\|_2+\|y'_j-Y(1\otimes w_j)Y^*\|_2+\|w_j-q_j\|_2 \le\varepsilon .$ By (B11), $N\cong R$. $\square$

Proof of Theorem 1.5. (a)⇒(c): Proposition 3.1. (c)⇒(b): Proposition 3.2. (b)⇒(d): Proposition 3.3. (d)⇒(e): Theorem 4.1. (e)⇒(f): Proposition 5.1. (f)⇒(g): Proposition 5.3. (g)⇒(b): Proposition 5.4. So (b)–(g) are equivalent and follow from (a). (g)⇒(h): Theorem 6.2; (h)⇒(g): Proposition 5.6. If these hold, then (e), (g) and (h) hold, so (i) holds by Proposition 9.1, and (i)⇒(a) is Proposition 9.2. $\square$

Proof of Theorem 1.6. Let $N$ be an injective II₁ factor on a separable Hilbert space. Then $N$ has separable predual. Injectivity does not depend on the representation (Corollary 2.2(a) of "Injective von Neumann algebras"), so $N$ is injective on $L^2(N,\tau)$, and Theorem 1.5 (b)⇒(a) gives $N\cong R$. $\square$

Proof of Corollary 1.7. Let $P\subset R$ be a subfactor. Then $P$ is finite, as a von Neumann subalgebra of a finite algebra, so it is a matrix algebra or a II₁ factor (B1); it has separable predual. Suppose it is a II₁ factor. By Theorem 1.5, $R$ is injective on $L^2(R)$: there is a projection of norm one $\Phi\colon B(L^2(R))\to R$. Let $E_P\colon R\to P$ be the trace-preserving conditional expectation (B4), which has norm one. Then $E_P\circ\Phi$ is a projection of norm one of $B(L^2(R))$ onto $P$, so $P$ is injective (in this faithful representation), and $P\cong R$ by Theorem 1.6. $\square$

Corollary 1.7 and Proposition 4.6 together say that no infinite-dimensional factor other than $R$ has a hypertrace; and every II₁ factor contains a copy of $R$, whereas $R$ contains no other infinite-dimensional factor.

10. Amenability

A reference for this section is [Connes 1976, §2.2–§2.3]; the notion of amenability is from [Johnson–Kadison–Ringrose 1972].

Definition 10.1. Let $M$ be a von Neumann algebra. A dual normal $M$-bimodule is a Banach space $X$ which is the dual of a Banach space $X_*$, with a unital bimodule structure $x\psi$, $\psi x$ ($x\in M$, $\psi\in X$) such that $\|x\psi\|,\|\psi x\|\le\|x\|\|\psi\|$, the maps $\psi\mapsto x\psi$ and $\psi\mapsto\psi x$ are weak* continuous, and for fixed $\psi$ and $\omega\in X_*$ the functionals $x\mapsto\langle x\psi,\omega\rangle$ and $x\mapsto\langle\psi x,\omega\rangle$ are normal. A derivation is a bounded linear map $D\colon M\to X$ with $D(xy)=xD(y)+D(x)y$; it is inner if $D(x)=x\psi_0-\psi_0x$ for some $\psi_0\in X$. $M$ is amenable if every derivation of $M$ into every dual normal $M$-bimodule is inner.

Theorem 10.2. For a II₁ factor $N$ with separable predual, the following are equivalent: (1) $N$ is amenable; (2) $N$ has a hypertrace; (3) $N\cong R$.

Proof. (2)⇒(3) is Theorem 1.5, and (3)⇒(1) is the Johnson–Kadison–Ringrose theorem (B12). We prove (1)⇒(2). Let $N$ act on $H=L^2(N,\tau)$.

The module. Let $X$ consist of the linear functionals $\psi$ on $B(H)$ with the following two properties:

(i) there is $C<\infty$ with $|\psi(aTb)|\le C\|a\|_2\|T\|\|b\|_2$ for all $a,b\in N$, $T\in B(H)$; the least such $C$ is $\|\psi\|_X$;
(ii) $\psi(x)=0$ for $x\in N$.

With $(x\psi)(T)=\psi(Tx)$ and $(\psi x)(T)=\psi(xT)$, $X$ is an $N$-bimodule: $x\psi$ satisfies (i) with constant $\|x\|\|\psi\|_X$ since $\|bx\|_2\le\|b\|_2\|x\|$, and (ii) since $Nx\subset N$; likewise for $\psi x$.

$X$ is a dual normal bimodule. For $\psi\in X$, the trilinear form $(a,T,b)\mapsto\psi(aTb)$ on $N\times B(H)\times N$ is bounded for $\|\cdot\|_2\times\|\cdot\|\times\|\cdot\|_2$, so it extends uniquely to a bounded trilinear form $\Psi$ on $H\times B(H)\times H$ with $\|\Psi\|=\|\psi\|_X$. It satisfies $\Psi(\widehat{ac},T,\beta)=\Psi(\hat a,cT,\beta),\quad\Psi(\alpha,Tc,\hat b)=\Psi(\alpha,T,\widehat{cb}),\quad \Psi(\hat1,c,\hat1)=0\qquad(a,b,c\in N) \tag{10.1}$ (the first two hold by continuity for all $\alpha,\beta\in H$ once they hold on $N\hat1$). Conversely, a bounded trilinear form $\Psi$ satisfying (10.1) defines $\psi(T)=\Psi(\hat1,T,\hat1)$, and (10.1) gives $\psi(aTb)= \Psi(\hat a,T,\hat b)$, so $\psi\in X$ with $\|\psi\|_X=\|\Psi\|$. The bounded trilinear forms form the dual of the projective tensor product $H\hat\otimes B(H)\hat\otimes H$ (B9), and the conditions (10.1) are weak* closed. So $X$ is isometric to a weak* closed subspace of a dual space, hence a dual space. In terms of $\Psi$, $x\psi$ corresponds to $\Psi(\alpha,T,\beta\cdot x)$, where $\beta\cdot x=Jx^*J\beta$ is right multiplication, a bounded operator on $H$; so $\psi\mapsto x\psi$ is the transpose of a bounded map on the predual and is weak* continuous. For fixed $\psi$ and an elementary tensor $\omega=\alpha\otimes T\otimes\beta$, $x\mapsto\Psi(\alpha,T,Jx^*J\beta)$ is a vector functional of $x\mapsto Jx^*J$ composed with a bounded linear functional on $H$, hence normal; for general $\omega$ it is a norm limit of normal functionals, hence normal. The left action is treated in the same way.

The derivation. Define $D(x)(T)=\langle(xT-Tx)\hat1,\hat1\rangle$. Then $D(x)\in X$: for $a,b\in N$, $D(x)(aTb)=\langle Tb\hat1,(xa)^*\hat1\rangle-\langle T\,bx\hat1,a^*\hat1\rangle ,$ so $|D(x)(aTb)|\le2\|x\|\|a\|_2\|T\|\|b\|_2$, and $D(x)(c)=\tau(xc-cx)=0$ for $c\in N$. The identity $D(xy)=xD(y)+D(x)y$ follows from $(xD(y))(T)=D(y)(Tx)=\langle(yTx-Txy)\hat1,\hat1\rangle$ and $(D(x)y)(T)=D(x)(yT)=\langle(xyT-yTx)\hat1,\hat1\rangle$. So $D$ is a bounded derivation (it is also normal, in the sense that $x\mapsto\langle D(x),\omega\rangle$ is normal, by the same argument as above).

A hypertrace from an inner derivation. By amenability, $D(x)=x\psi_0-\psi_0x$ for some $\psi_0\in X$. Put $\varphi(T)=\langle T\hat1,\hat1\rangle+\psi_0(T)$, a bounded functional (by (i) with $a=b=1$). Then $\varphi(xT)-\varphi(Tx)=\langle(xT-Tx)\hat1,\hat1\rangle+\psi_0(xT)-\psi_0(Tx)=D(x)(T)-(x\psi_0-\psi_0x)(T)=0,$ and $\varphi(1)=1$. The functional $\varphi^*(T)=\overline{\varphi(T^*)}$ has the same properties, so $h=(\varphi+\varphi^*)/2$ is a self-adjoint functional with $h\circ\operatorname{Ad}u=h$ for $u\in\mathcal U(N)$ and $h(1)=1$. Let $h=h_+-h_-$ be its Jordan decomposition (B9). Since $\operatorname{Ad}u$ is an isometric positive automorphism of $B(H)$, $h_+\circ\operatorname{Ad}u-h_-\circ\operatorname{Ad}u$ is again a Jordan decomposition of $h$, so by uniqueness $h_+\circ\operatorname{Ad}u=h_+$. As $h_+(1)\ge h(1)=1$, $h_+/h_+(1)$ is a hypertrace. $\square$

Conversely, a hypertrace $\varphi$ makes $D$ inner, with $\psi_0=\varphi-\langle\,\cdot\,\hat1,\hat1\rangle$: by the Cauchy–Schwarz inequality $|\varphi(aTb)|\le\varphi(aTT^*a^*)^{1/2}\varphi(b^*b)^{1/2}\le\|T\|\|a^*\|_2\|b\|_2$, since $\varphi|_N=\tau$, so $\psi_0$ satisfies (i), and (ii) holds as $\varphi|_N=\tau$. So for this particular derivation, being inner amounts to having a hypertrace.

11. Exercises

Exercise 1. Let $N$ be a II₁ factor. Show that the flip $\sigma$ of $N\bar\otimes N$ is not inner. (For $N=R$ it is approximately inner by Proposition 3.4, so $\overline{\operatorname{Int}}(R\bar\otimes R)\ne \operatorname{Int}(R\bar\otimes R)$.)

Solution. Suppose $\sigma=\operatorname{Ad}U$ with $U$ a unitary of $M=N\bar\otimes N$, which acts standardly on $H\otimes H$ with trace vector $\zeta=\hat1\otimes\hat1$ and conjugation $J\otimes J$ (B3). Put $\hat U=U\zeta\ne0$. From $U(x\otimes1)=\sigma(x\otimes1)U=(1\otimes x)U$, applied to $\zeta$, we get $(Jx^*J\otimes1)\hat U=(1\otimes x)\hat U$ for all $x\in N$: the left side is $U(x\otimes1)\zeta$, written with the right action of $M$ on $L^2(M)$ (B3). Now use the unitary $V\colon H\otimes H\to\mathrm{HS}(H)$ sending $\xi\otimes\eta$ to the rank-one operator $\alpha\mapsto\langle\alpha,J\eta\rangle\xi$. One checks on elementary tensors that $V(A\otimes B)V^*T=A\,T\,JB^*J$. Hence $T=V\hat U$ is a nonzero Hilbert–Schmidt operator with $Jx^*J\,T=T\,Jx^*J$ for all $x\in N$, that is, $T\in(JNJ)'=N$. But a II₁ factor on an infinite-dimensional space contains no nonzero compact operator: if $T\ne0$ were compact, a spectral projection of $T^*T$ for some $(t,\infty)$, $t>0$, would be a nonzero finite-rank projection $p\in N$, and $pNp$ would be a II₁ factor acting on the finite-dimensional space $pH$, which is impossible. So $\sigma$ is outer.

Exercise 2. Let $\mathbb F_2$ be the free group on two generators and $L(\mathbb F_2)$ its von Neumann algebra, a II₁ factor. Show that $L(\mathbb F_2)$ is not injective and has no hypertrace on any Hilbert space.

Solution. By "Full factors" (Example 6.6 there), $L(\mathbb F_2)$ is full, so by Theorem 6.5 there it does not have property Γ. If it were injective, or had a hypertrace on some space, it would satisfy (g) by Theorem 1.5 and Remark 5.2, and then it would have property Γ by Proposition 8.1. (Compare Example 4.4: for amenable groups the Følner sets produce a hypertrace; the free group has no Følner sets.)

Exercise 3. Let $p$ be a nonzero projection of $R$. Show that $pRp\cong R$. Deduce that every factor $P\subset R$ whose unit is a projection $p$ of $R$ (a nonunital subfactor) is finite-dimensional or isomorphic to $R$.

Solution. $pRp$ is a II₁ factor with separable predual (B1), acting faithfully on $pL^2(R)$. Let $\Phi\colon B(L^2(R))\to R$ be a projection of norm one (Theorem 1.5). For $S\in B(pL^2(R))$ let $\tilde S=pSp$, viewed in $B(L^2(R))$, and $\Psi(S)=p\Phi(\tilde S)p\in pRp$. Then $\|\Psi\|\le1$ and $\Psi(y)=pyp=y$ for $y\in pRp$. So $pRp$ is injective, and $pRp\cong R$ by Theorem 1.6. If $P\subset R$ is a factor with unit $p$, then $P$ is a unital subfactor of $pRp\cong R$, and Corollary 1.7 applies.

Exercise 4. Let $N_1,N_2$ be injective II₁ factors with separable predual. Show that $N_1\bar\otimes N_2\cong R$. In particular $R\bar\otimes R\cong R$, which gives a second proof of part of Lemma 2.2.

Solution. By Proposition 3.3, $N_i$ has a hypertrace on $L^2(N_i)$. By Lemma 6.1, $N_1\bar\otimes N_2$ has a hypertrace on $L^2(N_1)\otimes L^2(N_2)=L^2(N_1\bar\otimes N_2)$ (B3). It is a II₁ factor with separable predual, so Theorem 1.5 (d)⇒(a) applies. (The second proof is not independent of Lemma 2.2, which is used in Section 8; it only illustrates the theorem.)

Exercise 5. Let $N$ be a II₁ factor and $e$ a nonzero finite-rank projection on $L^2(N)$. Show that $\|[u,e]\|_{\mathrm{HS}}$ cannot be $0$ for all unitaries $u$ of $N$; thus the Følner condition can only hold approximately.

Solution. If $ue=eu$ for all $u\in\mathcal U(N)$, then $e\in N'$, a II₁ factor on an infinite-dimensional space, which contains no nonzero finite-rank projection (see the solution of Exercise 1).

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