Almost periodic weights and the invariant $Sd$

Written by Claude Opus 5.5 (Anthropic), September 2026, revised October 2026. The September text was spot-checked by Claude Opus 5.5 in a separate session; the October revisions are self-checked by the writing AI. The introductory conditions and amplification and subgroup descriptions were checked and corrected by GPT-6 Astra (OpenAI), Ultra, October 2026. The proof of Theorem 7.3 was drafted by GPT-6 Astra (OpenAI) in ChatGPT web, Pro mode, and checked and adapted by Claude Opus 5.5, October 2026. Public domain (CC0).

Introduction

Call a faithful semifinite normal weight $\varphi$ almost periodic when its modular operator is diagonalizable, $\Delta_\varphi=\sum_{\lambda>0}\lambda E_\lambda$. Then the modular group $\sigma^\varphi_t=\operatorname{Ad}\Delta_\varphi^{it}$ is an almost periodic function of $t$, and it extends to an action of a compact abelian group, the dual of the group generated by the eigenvalues. Compact group actions are much easier to handle than actions of $\mathbb R$: they have spectral subspaces, a conditional expectation onto the fixed points, and a Connes spectrum that is computed from eigenoperators.

For a factor $M$ this lesson studies the point modular spectrum $Sd(M)=\bigcap_{\varphi\ \text{almost periodic}}\{\text{eigenvalues of }\Delta_\varphi\}.$ The main results are these.

$Sd(M)$ is a subgroup of $\mathbb R_+^*$, it is the intersection of the Connes spectra of the extended modular actions, and it lies in the invariant $S(M)$ whenever $M$ has separable predual or is not of type III₀ (Sections 4 and 5).
On factors of type III₀ with separable predual, almost periodic states are norm dense, with eigenvalues in any prescribed dense subgroup; so $Sd$ is trivial for these factors and, more generally, for Krieger factors (Section 5).
For a full factor with separable predual that admits an almost periodic weight, one such weight suffices to compute $Sd(M)$. A weight realizing exactly $Sd(M)$ exists; infinite such weights are unique up to unitary conjugation and scaling, and a nonfinite factor has the trace-scaling crossed-product decomposition of Section 7. For every full factor with separable predual, $\overline{Sd(M)}=S(M)$, whether or not an almost periodic weight exists.
Every countable subgroup of $\mathbb R_+^*$ is $Sd$ of some full factor whose predual is separable. Hence the isomorphism classes of type III₁ factors do not form a countably separated Borel space, and the centre of $\operatorname{Out}M$ can be larger than the image of the modular group (Sections 7 and 8).

What is assumed. Tomita–Takesaki theory for weights, the cocycle derivative, crossed products by discrete groups, the invariant $S(M)$ with the type classification, and the discrete decomposition of type III factors. These are listed in "Results used from other lessons". From this course we use the lessons "Ultraproducts and the asymptotic centralizer" and "Full factors"; Section 1 restates exactly what is used from them. "Results used from other lessons" names, for each fact that the programme proves, the lesson that proves it.

Basic references are [Connes 1974] and [Connes 1973].

Conventions. "Weight" means faithful semifinite normal (f.s.n.) weight unless said otherwise. For a weight $\varphi$ on $M$: $\mathfrak n_\varphi=\{x:\varphi(x^*x)<\infty\}$, $\mathfrak m_\varphi$ is the span of $\mathfrak m_\varphi^+=\{x\ge0:\varphi(x)<\infty\}$, $(H_\varphi,\eta_\varphi)$ is the GNS construction ($\eta_\varphi:\mathfrak n_\varphi\to H_\varphi$, $\langle\eta_\varphi(x),\eta_\varphi(y)\rangle=\varphi(y^*x)$), and $M$ acts on $H_\varphi$ with $x\eta_\varphi(y)=\eta_\varphi(xy)$. $\Delta_\varphi$, $J_\varphi$, $\sigma^\varphi$ are the modular operator, conjugation and group, and $M_\varphi=\{x:\sigma^\varphi_t(x)=x\ \forall t\}$ is the centralizer. $\mathbb T$ is the unit circle, $\mathcal U(M)$ the unitary group, and $\mathbb R_+^*$ the multiplicative group of positive reals. For $h\ge0$ affiliated with $M_\varphi$ we write $\varphi_h=\varphi(h\,\cdot)$ for the weight $x\mapsto\lim_{\varepsilon\to0}\varphi(h_\varepsilon^{1/2}x h_\varepsilon^{1/2})$, $h_\varepsilon=h(1+\varepsilon h)^{-1}$. For a projection $e\in M_\varphi$, $\varphi_e$ is the restriction of $\varphi$ to $eMe$.

Results used from other lessons

(B1) Weights. A normal weight is $\sigma$-weakly lower semicontinuous, and $\varphi(x)=\sup\{\omega(x):\omega\in M_*^+,\ \omega\le\varphi\}$ for $x\ge0$. A weight is semifinite if and only if $\mathfrak n_\varphi$ is $\sigma$-weakly dense; then there is an increasing net in $\mathfrak m_\varphi^+$ converging strongly to $1$. One has $\Delta_\varphi^{it}\eta_\varphi(x)=\eta_\varphi(\sigma^\varphi_t(x))$, and $\sigma^\varphi$ fixes the centre of $M$ pointwise. If $a\in M$ is entire analytic for $\sigma^\varphi$ and $x\in\mathfrak n_\varphi$, then $xa\in\mathfrak n_\varphi$ and $\eta_\varphi(xa)=J_\varphi\,\sigma^\varphi_{i/2}(a)^*J_\varphi\,\eta_\varphi(x).\tag{0.1}$ The restriction of $\varphi$ to $M_\varphi$ is a trace, and $\varphi(ux u^*)=\varphi(x)$ for every unitary $u\in M_\varphi$. The standard form is unique up to unitary equivalence; in particular $H_\varphi$ is separable when $M_*$ is, and if $\omega$ is a faithful normal state, every normal state is a vector state $\omega_\xi$, $\xi\in H_\omega$. These facts are proved in the course Modular theory and weights: lower semicontinuity and the supremum formula in Detecting normal weights by finite observations, §NW-11; semifiniteness in General weights: finite domains, GNS spaces, and normal representations, §WG-008; the modular group on the GNS space and its action on the centre in The modular group and its analytic algebra, §MF-06; (0.1) in Fixed elements and changes of density, §CZ-04; the centralizer statements in §CZ-05 there; uniqueness of the standard form in Recovering a representation from its positive cone, §SE-11; vector states in The positive cone of a standard representation, §SF-10. Separability of $H_\varphi$ when $M_*$ is separable is shown as in Full factors, (B1). See also [Haagerup 1975].

(B2) Cocycle derivative. For weights $\varphi,\psi$, let $\theta$ be the balanced weight of (e) below. For each $t$ there is a unique unitary $u_t=(D\psi:D\varphi)_t$ in $M$ with $\sigma^\theta_t(1\otimes e_{21})=u_t\otimes e_{21}$. The family $(u_t)$ is $\sigma$-strongly continuous, $u_{s+t}=u_s\sigma^\varphi_s(u_t)$ and $\sigma^\psi_t=\operatorname{Ad}u_t\circ\sigma^\varphi_t$; these two properties alone do not determine it, since $(e^{ict}u_t)$ has them too for every $c\in\mathbb R$. Moreover:

(a) $(D\psi:D\varphi)_t(D\varphi:D\chi)_t=(D\psi:D\chi)_t$, and $(D\psi:D\varphi)_t=c^{it}$ for all $t$ (with $c>0$) if and only if $\psi=c\varphi$.
(b) For $\alpha\in\operatorname{Aut}M$: $(D(\psi\circ\alpha):D(\varphi\circ\alpha))_t=\alpha^{-1}((D\psi:D\varphi)_t)$; and $\alpha\sigma^\varphi_t\alpha^{-1}=\sigma^{\varphi\circ\alpha^{-1}}_t$. The first formula follows from the second, applied to the balanced weights.
(c) For a unitary $u$: $(D(\varphi\circ\operatorname{Ad}u):D\varphi)_t=u^*\sigma^\varphi_t(u)$, where $(\varphi\circ\operatorname{Ad}u)(x)=\varphi(uxu^*)$.
(d) For $h$ positive nonsingular affiliated with $M_\varphi$: $\varphi_h$ is a weight, and $(D\varphi_h:D\varphi)_t=h^{it}$, so $\sigma^{\varphi_h}_t=\operatorname{Ad}h^{it}\circ\sigma^\varphi_t$.
(e) Balanced weight. $\theta(\sum x_{ij}\otimes e_{ij})=\varphi(x_{11})+\psi(x_{22})$ is a weight on $M\otimes M_2(\mathbb C)$, and $\sigma^\theta_t(x\otimes e_{11})=\sigma^\varphi_t(x)\otimes e_{11}$, $\sigma^\theta_t(x\otimes e_{22})=\sigma^\psi_t(x)\otimes e_{22}$, $\sigma^\theta_t(1\otimes e_{21})=(D\psi:D\varphi)_t\otimes e_{21}$.

Consequently the class $\delta_M(t)$ of $\sigma^\varphi_t$ in $\operatorname{Out}M=\operatorname{Aut}M/ \operatorname{Int}M$ does not depend on $\varphi$ (the modular homomorphism), and it commutes with every element of $\operatorname{Out}M$. The balanced weight and the cocycle read from a matrix unit are constructed in Supported GNS representations and the balanced cocycle, §§GC-03–GC-04; continuity, the cocycle identity and $\sigma^\psi_t=\operatorname{Ad}u_t\circ\sigma^\varphi_t$ in Comparing supported weights and transporting their cuts, §WC-02; (a) in Changing reference for a weight cocycle, §CH-01 and Recognizing a weight by its fixed density, §§PT-04–PT-05; (b) in §PT-02 of the same lesson and The KMS boundary condition determines the modular group, §KM-06; (c) in Averaging a dual action and recovering its coefficients, OA-FLOW.AVG.INNER, from the balanced weight and (b); (d) in Fixed elements and changes of density, §CZ-11 and Recognizing a weight by its fixed density, §PT-04. See also [Connes 1973, Section 1.2].

(B3) Reduction, tensor products, expectations. For a nonzero projection $e\in M_\varphi$, $\varphi_e$ is a weight on $eMe$ with $\sigma^{\varphi_e}=\sigma^\varphi|_{eMe}$. For the trace $\operatorname{Tr}$ on $B(K)$, $\varphi\otimes\operatorname{Tr}$ is a weight on $M\bar\otimes B(K)$ with modular group $\sigma^\varphi_t\otimes\operatorname{id}$. If $E$ is a faithful normal conditional expectation of $M$ onto $N$ and $\chi$ a weight on $N$, then $\chi\circ E$ is a weight on $M$, $\sigma^{\chi\circ E}_t(N)=N$ and $\sigma^{\chi\circ E}_t|_N=\sigma^\chi_t$. Reduction: Fixed elements and changes of density, §CZ-09. Tensor products with the trace: Tensor operators, their full domains, and tensor weights. Expectations: Conditional expectations from modular invariance, §ME-10.

(B4) Traces. Let $N$ have an f.s.n. trace $\tau$ and centre $C$. Every f.s.n. trace on $N$ is $\tau(c\,\cdot)$ with $c$ positive nonsingular affiliated with $C$. Every $\omega\in N_*^+$ is $\tau(h\,\cdot)$ with $h\ge0$ in $L^1(N,\tau)$, $\|\omega\|=\tau(h)$, $\omega$ is faithful exactly when $h$ is nonsingular, and $\|\tau(h\,\cdot)-\tau(k\,\cdot)\|=\tau(h-k)$ when $0\le k\le h$. The restriction of a weight to its centralizer is a trace. If $C$ is countably decomposable, $C\cong L^\infty(X,\mu)$ with $\mu$ finite, and positive nonsingular operators affiliated with $C$ are the a.e. finite measurable functions $X\to\,]0,\infty[$; an automorphism of $C$ acts on them. On a semifinite factor the f.s.n. trace is unique up to a scalar, finite projections have finite trace, and automorphisms of a type I factor are inner. Traces relative to each other: Integration for a trace, Theorem 7.2 and Corollary 7.4; densities of normal functionals in $L^1(N,\tau)$: Trace densities and noncommutative integration, §§TI-06 and TI-12; the centralizer: Fixed elements and changes of density, §CZ-05; finite projections: Integration for a trace, Proposition 8.10; automorphisms of $B(H)$: The universal enveloping von Neumann algebra of a C*-algebra, and W*-algebras, Lemma 6.4 with Projections and types of von Neumann algebras, Corollary 10.4. The measure-theoretic description of $C$ is Abelian operator algebras, Corollary 3.3, applied in the GNS representation of a faithful normal state of $C$, which is cyclic; positive nonsingular operators affiliated with $L^\infty(X,\mu)$ are multiplications by a.e. finite measurable functions $X\to\,]0,\infty[$, by Unbounded decomposable operators and measurable spectral decomposition, Theorem 5.5 with one-dimensional fibres.

(B5) The invariant $S$. $S(M)$ is the intersection of the spectra $\operatorname{Sp}\Delta_\varphi$ over all weights; it is closed in $[0,\infty[$ [Connes 1973, Definition 3.1.1]. A factor is semifinite if and only if $S(M)=\{1\}$ [Connes 1973, Lemma 3.1.2]. The types are defined through $S$: a factor is of type III₀, III$_\lambda$ ($0<\lambda<1$) or III₁ according as $S(M)=\{0,1\}$, $\{\lambda^n:n\in\mathbb Z\}\cup\{0\}$ or $[0,\infty[$ [Connes 1973, opening of Chapter IV]. For a nonzero projection $e$, $S(eMe)=S(M)$ [Connes 1973, Corollary 3.2.8]. If $M_\varphi$ is a factor, then $S(M)=\operatorname{Sp}\Delta_\varphi$ [Connes 1973, Corollary 3.2.7]. Every countably decomposable III$_\lambda$ factor, $0<\lambda<1$, carries a weight $\varphi$ with $\sigma^\varphi_{T}=\operatorname{id}$ for $T=2\pi/|\log\lambda|$ and $M_\varphi$ a factor, namely a generalized trace [Connes 1973, Theorem 4.2.6, Definition 4.3.1, Theorem 4.3.2].

(B6) Projections. In a type III factor, for every nonzero projection $e$ there are partial isometries $(v_j)_{j\in J}$ with $v_j^*v_j=e$ and $\sum_jv_jv_j^*=1$; then $x\mapsto(v_i^*xv_j)_{i,j}$ maps $M$ isomorphically onto $eMe\bar\otimes B(\ell^2(J))$. A type III factor has a nonzero countably decomposable projection, and in a countably decomposable type III factor all nonzero projections are equivalent. A properly infinite countably decomposable von Neumann algebra $M$ satisfies $M\cong M\bar\otimes B(\ell^2)$ and $M\cong M\otimes M_2(\mathbb C)$. In a factor with separable predual any two infinite projections are equivalent. For a projection $e$ of a von Neumann algebra $A$, the centre of $eAe$ is $Z(A)e$. If $e$ is a projection in $A$ with central support $z$, then $z$ is the sum of orthogonal projections each equivalent in $A$ to a subprojection of $e$; and two projections with non-orthogonal central supports have nonzero equivalent subprojections. Proved in Projections and types of von Neumann algebras: the matrix decomposition by Proposition 15.3(3), which also gives $M\cong M\bar\otimes B(\ell^2)$ and hence $M\otimes M_2(\mathbb C)\cong M\bar\otimes B(\ell^2)\otimes M_2(\mathbb C)\cong M$; countably decomposable projections: the cyclic projections of Lemma 4.6 are countably decomposable (The double commutant theorem, Theorem 9.5); equivalence of infinite projections: Propositions 15.2(2) and 5.1; centres of corners: Traces on von Neumann algebras, Lemma 1.5; central supports: Lemma 5.3 and Corollary 5.4.

(B7) Compact abelian groups. Let $\Lambda$ be an abelian group with the discrete topology and $G=\widehat\Lambda$ its compact dual, with normalized Haar measure $ds$ and pairing $\langle s,\lambda\rangle$. The continuous characters of $G$ are the maps $s\mapsto\langle s,\lambda\rangle$ (Pontryagin duality); a subgroup $H\subset G$ is dense if and only if no $\lambda\neq1$ satisfies $\langle s,\lambda\rangle=1$ for all $s\in H$; in particular the elements of $G$ separate the points of $\Lambda$, and for a subgroup $\Lambda_1\subset\Lambda$ and $\lambda\notin\Lambda_1$ some $s\in G$ is trivial on $\Lambda_1$ with $\langle s,\lambda\rangle\neq1$ (apply this to $\Lambda/\Lambda_1$); the characters form an orthonormal basis of $L^2(G)$. If $W$ is a strongly continuous unitary representation of $G$, the operators $P_\lambda=\int_G\overline{\langle s,\lambda\rangle}W_s\,ds$ are orthogonal projections with $\sum_\lambda P_\lambda=1$ and $W_sP_\lambda=\langle s,\lambda\rangle P_\lambda$. If $\Lambda$ is countable, $G$ is metrizable. Proved in the course Harmonic analysis on locally compact abelian groups: The Pontryagin duality theorem, Subgroups, quotients and annihilators for density and separation, The Plancherel theorem for the orthonormal basis, and Unitary representations of abelian groups: the spectral theorem for the spectral projections.

(B8) Crossed products by discrete groups. For an action $\theta$ of a discrete group $D$ on $N\subset B(H)$, the crossed product $N\rtimes_\theta D$ is the von Neumann algebra on $\ell^2(D,H)$ generated by $(\pi(n)\xi)(g)=\theta_{g^{-1}}(n)\xi(g)$ and $(\ell_h\xi)(g)=\xi(h^{-1}g)$; up to isomorphism it does not depend on the faithful normal representation of $N$. There is a faithful normal conditional expectation $E$ onto $\pi(N)$ with $E(\pi(n)\ell_g)=0$ for $g\ne e$, and $x=0$ if $E(x\ell_g^*)=0$ for all $g$. If $D$ is abelian, the dual action $\hat\theta$ of $\widehat D$ fixes $\pi(N)$ and multiplies $\ell_g$ by $\chi(g)$; its fixed point algebra is $\pi(N)$. The crossed product and its independence of the representation: Changing the Hilbert space of a regular crossed product; the dual action and its fixed points: Averaging a dual action and recovering its coefficients; the conditional expectation: The coefficient algebra as the value space of a weight, OA-FLOW.OVW.DISCRETE, where also $E(y)=\pi(V_e^*yV_e)$ with $V_e\eta=\delta_e\otimes\eta$. Since $E(\ell_h^*y\ell_h)=\ell_h^*E(y)\ell_h$ (check it on the elements $\pi(n)\ell_g$ and use normality), the entry $V_e^*\ell_h^*x\ell_kV_e$ of $x$ is $\theta_{h^{-1}}(\pi^{-1}(E(x\ell_{hk^{-1}}^*)))$, which gives the last statement.

(B9) Structure of type III₀ factors. Suppose $M$ is a type III₀ factor whose predual is separable, and let $D=(\mathbb Z/2)^{(\mathbb N)}$, with generators $g_1,g_2,\dots$. There are a von Neumann subalgebra $N\subset M$ of type II∞ with centre $C$, a faithful normal conditional expectation $E:M\to N$, a homomorphism $\varepsilon\mapsto u_\varepsilon$ of $D$ into $\mathcal U(M)$ and projections $1=e_1\ge e_2\ge\cdots$ in $C$ such that: $N'\cap M=C$; $u_\varepsilon Nu_\varepsilon^*=N$ and $E(u_\varepsilon xu_\varepsilon^*)= u_\varepsilon E(x)u_\varepsilon^*$; $N$ and the $u_\varepsilon$ generate $M$; and, with $\theta_\varepsilon=\operatorname{Ad}u_\varepsilon|_N$, $e_{n+1}+\theta_{g_n}(e_{n+1})=e_n\qquad(n\ge1).\tag{0.2}$ [Connes 1973, Theorem 5.3.1, Corollary 5.3.6 and their proofs]. In that corollary our $e_{n+1}$ is written $e_n$, and the sequence, called increasing in its statement, decreases by relation (0.2) from its proof.

(B10) Point realizations and hyperfiniteness. A nonzero abelian von Neumann algebra $C$ whose predual is separable is $L^\infty(X,\mu)$ for a standard probability space $(X,\mu)$; every action of a countable group by automorphisms of $C$ comes from a nonsingular Borel action on $X$; and if the group is abelian, its orbit equivalence relation is, off an invariant null set, the increasing union of Borel equivalence relations $R_1\subset R_2\subset\cdots$ with finite classes. These facts are proved in Section 7.2 of this lesson (Theorem 7.3), from the lessons on Polish spaces and on commutative operator algebras in Foundations of von Neumann algebras.

(B11) The Effros Borel structure. For a separable Hilbert space $H$, the set of von Neumann algebras on $H$ carries the Effros Borel structure. A map $y\mapsto M_y$ from a standard Borel space is Borel if there are maps $y\mapsto a_n(y)\in B(H)$, $n\in\mathbb N$, Borel for the weak operator topology, with $M_y=\{a_n(y):n\in\mathbb N\}''$. Proved in The Effros Borel structure, Theorems 9.1 and 10.3. See also, e.g., [Effros 1965].

(B12) Infinite tensor products. Let $H_i$ ($i\in I$, $I$ countable) be Hilbert spaces with unit vectors $\xi_i$. If $\xi_i'$ are unit vectors with $\sum_i\|\xi_i-\xi'_i\|<\infty$, then $\bigotimes_i\xi'_i$ is a unit vector of $\bigotimes_i(H_i,\xi_i)$, and $\bigotimes_i(H_i,\xi'_i)= \bigotimes_i(H_i,\xi_i)$. Unitaries $W_i$ with $W_i\xi_i=\xi'_i$ define a unitary $\bigotimes W_i$ from $\bigotimes(H_i,\xi_i)$ onto $\bigotimes(H_i,\xi'_i)$, and a bijection of index sets defines a unitary carrying product vectors to product vectors. If $\xi_i$ is cyclic and separating for $P_i\subset B(H_i)$, then $\bigotimes\xi_i$ is cyclic and separating for the infinite tensor product algebra $\bigotimes(P_i,\xi_i)$, and the modular operator of the product state satisfies $\Delta^{it}=\bigotimes_i\Delta_i^{it}$. The Araki–Woods factors are the factors $\bigotimes_n(M_{k_n}(\mathbb C),\omega_n)$ with faithful states $\omega_n$. Every Krieger factor that is not of type III₀ and not semifinite is isomorphic to an Araki–Woods factor. The infinite tensor product statements are proved in Infinite tensor products, Theorem 3.1, Theorem 5.1, Proposition 5.2 and Theorem 6.1; the last statement is Krieger's theorem. A Krieger factor is an injective factor with separable predual (Classification of injective factors, Proposition 2.8), so in type III$_\lambda$, $0<\lambda<1$, it is isomorphic to the Powers factor $R_\lambda$, an Araki–Woods factor, by Classification of injective factors, Theorem 8.3. In type III₁ the statement is the uniqueness of the injective factor of type III₁ [Connes 1985], [Haagerup 1987], which is not proved in this programme. The Araki–Woods factors are those of [Araki–Woods 1968].

(B13) Polish groups. The quotient of a Polish group by a closed normal subgroup is a Polish group. Proved in Polish spaces and standard Borel spaces, Theorem 8.8.

1. What is used from the lessons on ultraproducts and on full factors

This section restates, without proofs, what is used from the lessons "Ultraproducts and the asymptotic centralizer" and "Full factors" of this course. Numbers in parentheses refer to the lesson "Full factors". Throughout, $M$ is a von Neumann algebra whose predual is separable.

Commutators with functionals. For $x\in M$, $\omega\in M_*$ put $(x\omega)(y)=\omega(yx)$, $(\omega x)(y)=\omega(xy)$ and $[x,\omega]=x\omega-\omega x$. A bounded sequence $(x_n)$ is centralizing if $\|[x_n,\omega]\|\to0$ for every $\omega\in M_*$, and trivial if $x_n-\lambda_n\to0$ strong* for some scalars $\lambda_n$.

The $u$-topology. $\operatorname{Aut}M$ carries the topology of pointwise convergence in norm on $M_*$: $\alpha_i\to\alpha$ if and only if $\|\omega\circ\alpha_i-\omega\circ\alpha\|\to0$ for every $\omega\in M_*$. With it $\operatorname{Aut}M$ is a Polish topological group (Proposition 2.3 there), and inversion is continuous, so also $\|\omega\circ\alpha_i^{-1}-\omega\circ\alpha^{-1}\|\to0$.

Full algebras. $M$ is full if the normal subgroup $\operatorname{Int}M$ of inner automorphisms is closed. For a factor $M$ with separable predual the following are equivalent (Definition 4.3, Theorem 4.2(a), (b) and Corollary 5.2 there):

(F1) $M$ is full;
(F2) the map $u\mapsto\operatorname{Ad}u$ induces an isomorphism of topological groups from $\mathcal U(M)/\mathbb T$ (strong topology) onto $\operatorname{Int}M$ ($u$-topology);
(F3) every centralizing sequence is trivial;
(F4) the asymptotic centralizer $M_\omega$ is $\mathbb C$ for one, or every, free ultrafilter $\omega$.

We use (F2) in the following form. If $M$ is full, $u_n\in\mathcal U(M)$, $v\in\mathcal U(M)$ and $\operatorname{Ad}u_n\to\operatorname{Ad}v$ in $\operatorname{Aut}M$, then there are $\lambda_n\in\mathbb T$ with $\lambda_nu_n\to v$ strongly. Indeed the classes converge in $\mathcal U(M)/\mathbb T$, which is metrized by $\underline d([u],[v])=\min_{\lambda\in\mathbb T}d(\lambda u,v)$ for a metric $d$ on $\mathcal U(M)$ that is invariant under multiplication by scalars, and one takes for $\lambda_n$ a minimizer. Moreover, if $M$ is full, so is $M\bar\otimes B(K)$ for every nonzero separable Hilbert space $K$ (Proposition 10.1 there).

Open mapping theorem. A continuous bijective homomorphism between Polish groups is a homeomorphism (Corollary 3.3 there).

The free Bernoulli construction. Consider a von Neumann algebra $P$ on a separable Hilbert space $H$, with a cyclic and separating unit vector $\xi_0$; put $\varphi_P=\omega_{\xi_0}|_P$, and let $a,b$ generate the free group $\mathbb F_2$. Let $N=\bigotimes_{s\in\mathbb F_2}(P,\xi_0)$ act on $\mathcal K=\bigotimes_{s\in\mathbb F_2} (H,\xi_0)$ with product vector $\eta_0$, let $\mathbb F_2$ act on $N$ by shifting the tensor factors (implemented by permutation unitaries $V_s$ of $\mathcal K$), and let $M(P,\xi_0)$ denote the von Neumann algebra on $\mathcal K\otimes\ell^2(\mathbb F_2)$ generated by $N\otimes1$ and $U_s=V_s\otimes\lambda_s$ (a model of the crossed product). Put $\zeta_0=\eta_0\otimes\delta_1$ and $\psi=\omega_{\zeta_0}$ (Construction 9.1 there). Then (Proposition 9.2 and Theorem 9.4 there):

(R1) $\psi$ is a faithful normal state and $U_s\in M_\psi$ for all $s$;
(R2) $\|T-\psi(T)\|_\psi\le20\max(\|[T,U_a]\|_\psi,\|[T,U_b]\|_\psi)$ for all $T\in M(P,\xi_0)$; hence every element commuting with $U_a$ and $U_b$ is a scalar, and $M(P,\xi_0)$ and $M(P,\xi_0)_\psi$ are factors;
(R3) $\Delta_\psi=\Delta_{\varphi_N}\otimes1$, and $\Delta_{\varphi_N}^{it}$ is the infinite tensor product of copies of $\Delta_{\varphi_P}^{it}$;
(R4) $M(P,\xi_0)$ is a full factor, and its predual is separable.

2. Actions of compact abelian groups and the Connes spectrum

A reference for this section is [Connes 1973, Chapter II].

2.1 The compact group attached to a group of eigenvalues

Let $\Lambda$ be a subgroup of $\mathbb R_+^*$, taken with the discrete topology, and let $G_\Lambda= \widehat\Lambda$ be its compact dual group. Define $\iota_\Lambda:\mathbb R\to G_\Lambda,\qquad \langle\iota_\Lambda(t),\lambda\rangle=\lambda^{it}.$ When $\Lambda=\mathbb R_+^*$, $G_\Lambda$ is the Bohr compactification of $\mathbb R$.

Lemma 2.1. $\iota_\Lambda$ is a continuous homomorphism with dense range. A character $t\mapsto c^{it}$ ($c>0$) of $\mathbb R$ is of the form $\chi\circ\iota_\Lambda$ for a continuous function $\chi$ on $G_\Lambda$ if and only if $c\in\Lambda$; then $\chi(s)=\langle s,c\rangle$.

Proof. The topology of $G_\Lambda$ is that of pointwise convergence on $\Lambda$, and $t\mapsto\lambda^{it}$ is continuous, so $\iota_\Lambda$ is continuous; it is clearly a homomorphism. If $\lambda^{it}=1$ for all $t$, then $\lambda=1$; by (B7) the range is dense. If $c\in\Lambda$, take $\chi(s)=\langle s,c\rangle$. Conversely, let $\chi$ be continuous with $\chi(\iota_\Lambda(t))=c^{it}$. For $s,s'\in G_\Lambda$ the continuous functions $\chi(s+s')$ and $\chi(s)\chi(s')$ agree on the dense set $\iota_\Lambda(\mathbb R)^2$, so $\chi$ is a continuous character. By (B7), $\chi=\langle\cdot,\lambda\rangle$ with $\lambda\in\Lambda$, and $\lambda^{it}=c^{it}$ for all $t$ gives $\lambda=c$. $\square$

The group operation of $G_\Lambda$ is written additively. We say that a map $f$ on $\mathbb R$ extends to $G_\Lambda$ if there is a continuous map $\tilde f$ on $G_\Lambda$ with $f=\tilde f\circ\iota_\Lambda$; by density, $\tilde f$ is unique.

Lemma 2.2. Let $K$ be a positive nonsingular operator on a Hilbert space $H$. Then $t\mapsto K^{it}$ extends to $G_\Lambda$ as a strongly continuous map if and only if $K=\sum_{\lambda\in\Lambda}\lambda P_\lambda$ for orthogonal projections $P_\lambda$ with sum $1$. The extension is then $W_s=\sum_\lambda\langle s,\lambda\rangle P_\lambda$, a strongly continuous unitary representation of $G_\Lambda$.

Proof. Suppose $K=\sum\lambda P_\lambda$, and define $W_s$ as stated; it is a unitary representation, and $W_{\iota_\Lambda(t)}=K^{it}$. For $\xi\in H$ and $\varepsilon>0$ choose a finite set $F\subset\Lambda$ with $\sum_{\lambda\notin F}\|P_\lambda\xi\|^2<\varepsilon$. Then $\|W_s\xi-W_{s'}\xi\|^2\le\sum_{\lambda\in F}|\langle s,\lambda\rangle-\langle s',\lambda\rangle|^2\|P_\lambda\xi\|^2 +4\varepsilon$, and the finite sum is small when $s'$ is close to $s$. So $W$ is strongly continuous.

Conversely, let $W$ be a strongly continuous extension. Multiplication is jointly strongly continuous on the unitary group, so $(s,s')\mapsto W_{s+s'}$ and $(s,s')\mapsto W_sW_{s'}$ are continuous; they agree on the dense set $\iota_\Lambda(\mathbb R)^2$, so $W$ is a unitary representation. By (B7), $H=\bigoplus_\lambda P_\lambda H$ with $W_s=\langle s,\lambda\rangle$ on $P_\lambda H$. Then $K^{it}=W_{\iota_\Lambda(t)}=\sum\lambda^{it} P_\lambda$, and by uniqueness of the generator of a one-parameter unitary group (Stone's theorem), $K=\sum\lambda P_\lambda$. $\square$

2.2 Actions, spectral subspaces and Fourier coefficients

Let $G=G_\Lambda$. By an action of $G$ on a von Neumann algebra $M$ we mean a homomorphism $\alpha:G\to \operatorname{Aut}M$ such that $s\mapsto\omega(\alpha_s(x))$ is continuous for all $x\in M$, $\omega\in M_*$. For $\lambda\in\Lambda$ put $M^\alpha(\lambda)=\{x\in M\mid\alpha_s(x)=\langle s,\lambda\rangle x\ \ \forall s\in G\},\qquad M^\alpha=M^\alpha(1).$

Lemma 2.3. Let $\alpha$ be an action of $G$ on $M$.

(a) $s\mapsto\alpha_s(x)$ is strong* continuous for every $x$.
(b) For $f\in L^\infty(G)$ and $\omega\in M_*$, the functional $x\mapsto\int_Gf(s)\omega(\alpha_s(x))\,ds$ is normal. For $\lambda\in\Lambda$, the formula $\omega(P_\lambda(x))=\int_G\overline{\langle s,\lambda\rangle}\, \omega(\alpha_s(x))\,ds$ ($\omega\in M_*$) defines a normal linear map $P_\lambda:M\to M$ of norm at most $1$, which is a projection onto $M^\alpha(\lambda)$ and commutes with every $\alpha_s$. $P_1$ is a faithful normal conditional expectation onto $M^\alpha$.
(c) $M^\alpha(\lambda)M^\alpha(\mu)\subset M^\alpha(\lambda\mu)$ and $M^\alpha(\lambda)^*=M^\alpha(\lambda^{-1})$. If $x\in M^\alpha(\lambda)$ has polar decomposition $x=v|x|$, then $v\in M^\alpha(\lambda)$ and $|x|\in M^\alpha$.
(d) Fourier uniqueness. If $P_\lambda(x)=0$ for all $\lambda$, then $x=0$. The linear span of $\bigcup_\lambda M^\alpha(\lambda)$ is $\sigma$-weakly dense in $M$.

Proof. (a) For a vector $\xi$ in a faithful normal representation, $\|\alpha_s(x)\xi-x\xi\|^2=\langle\alpha_s(x^*x)\xi,\xi\rangle-2\operatorname{Re}\langle\alpha_s(x)\xi,x\xi\rangle+ \|x\xi\|^2$, which tends to $0$ as $s\to0$ by weak continuity; the same holds for $x^*$. Continuity at other points follows by composing with $\alpha_{s_0}$.

(b) The integral defines a bounded linear functional of $\omega$, hence an element of $M=(M_*)^*$, of norm at most $\|x\|$. Invariance of Haar measure gives $\alpha_r(P_\lambda(x))=\langle r,\lambda\rangle P_\lambda(x)$ and $P_\lambda\alpha_r=\alpha_rP_\lambda$; if $x\in M^\alpha(\lambda)$, then $P_\lambda(x)=x$. Normality: for $\omega\in M_*^+$ the positive functional $\rho(x)=\int_G\omega(\alpha_s(x))\,ds$ is normal, because if $x_i\nearrow x$ the continuous functions $\omega(\alpha_s(x_i))$ increase pointwise to the continuous function $\omega(\alpha_s(x))$, hence uniformly by Dini's theorem. By the Cauchy–Schwarz inequality, $|\int f(s)\omega(\alpha_s(x))\,ds|\le\|f\|_\infty\int|\omega(\alpha_s(x))|\,ds\le\|f\|_\infty\omega(1)^{1/2} \rho(x^*x)^{1/2}$. So if $x_i\to0$ strong* and boundedly, the functional tends to $0$; being strong* continuous on bounded sets, it is normal. Since every element of $M_*$ is a combination of positive ones, this proves the first statement, and taking $f=\overline{\langle\cdot,\lambda\rangle}$ shows that each $\omega\circ P_\lambda$ is normal. For $\lambda=1$, $P_1$ is positive and unital, and if $x\ge0$ and $P_1(x)=0$, then $\int\omega(\alpha_s(x))\,ds=0$ with a continuous nonnegative integrand, so $\omega(x)=0$ for all $\omega\in M_*^+$, and $x=0$.

(c) The first two statements are immediate. If $x\in M^\alpha(\lambda)$, then $x^*x\in M^\alpha$, so $|x|\in M^\alpha$, and $\alpha_s(x)=\alpha_s(v)|x|$ with $\alpha_s(v)$ a partial isometry with initial projection $\alpha_s(v^*v)=v^*v$. By uniqueness of the polar decomposition of $\langle s,\lambda\rangle x$, $\alpha_s(v)=\langle s,\lambda\rangle v$.

(d) For $\omega\in M_*$, the continuous function $f(s)=\omega(\alpha_s(x))$ has Fourier coefficients $\omega(P_\lambda(x))$. If they all vanish, $f=0$ by (B7), and $\omega(x)=f(0)=0$. For density, if $\omega\in M_*$ vanishes on every $M^\alpha(\lambda)$, the same argument gives $\omega(x)=0$ for all $x$; by the Hahn–Banach theorem for the $\sigma$-weak topology, the span is dense. $\square$

2.3 The spectrum and the Connes spectrum

Definition 2.4. Let $\alpha$ be an action of $G_\Lambda$ on $M$. Its spectrum is $\operatorname{Sp}\alpha=\{\lambda\in\Lambda:M^\alpha(\lambda)\neq0\}$. For a nonzero projection $e\in M^\alpha$, $\alpha$ restricts to an action $\alpha_e$ on $eMe$, with $(eMe)^{\alpha_e}(\lambda)=eM^\alpha(\lambda)e$. The Connes spectrum is $\Gamma(\alpha)=\bigcap\{\operatorname{Sp}\alpha_e:\ e\ \text{a nonzero projection of }M^\alpha\}.$

Taking $e=1$ gives $\Gamma(\alpha)\subset\operatorname{Sp}\alpha$, and $1\in\Gamma(\alpha)$ because $e\in eM^\alpha e$.

Lemma 2.5. Let $\alpha$ be an action of $G_\Lambda$ on $M$ and $e,f$ nonzero projections of $M^\alpha$.

(a) If $e\le f$, then $\operatorname{Sp}\alpha_e\subset\operatorname{Sp}\alpha_f$.
(b) If $w\in M^\alpha$ is a partial isometry with $w^*w=e$ and $ww^*=f$, then $\operatorname{Sp}\alpha_e=\operatorname{Sp}\alpha_f$.
(c) If $z$ is the central support of $e$ in $M^\alpha$, then $\operatorname{Sp}\alpha_e=\operatorname{Sp} \alpha_z$. Hence $\Gamma(\alpha)$ is the intersection of the $\operatorname{Sp}\alpha_z$ over the nonzero central projections $z$ of $M^\alpha$.
(d) If $M^\alpha$ is a factor, then $\Gamma(\alpha)=\operatorname{Sp}\alpha$.

Proof. (a) $eM^\alpha(\lambda)e=e(fM^\alpha(\lambda)f)e$. (b) The maps $y\mapsto wyw^*$ and $y\mapsto w^*yw$ are mutually inverse bijections between $eM^\alpha(\lambda)e$ and $fM^\alpha(\lambda)f$, since $w$ is fixed by $\alpha$. (c) By (a), $\operatorname{Sp}\alpha_e\subset\operatorname{Sp}\alpha_z$. Write $z=\sum_if_i$ with orthogonal projections $f_i\in M^\alpha$ and partial isometries $w_i\in M^\alpha$, $w_i^*w_i=f_i$, $w_iw_i^*\le e$ (B6). Let $0\ne x\in zM^\alpha(\lambda)z$. Then $f_ixf_j\neq0$ for some $i,j$, and $y=w_ixw_j^*\in eM^\alpha(\lambda)e$ satisfies $w_i^*yw_j=f_ixf_j\neq0$. So $\lambda\in\operatorname{Sp} \alpha_e$. (d) Every central support is $1$. $\square$

Proposition 2.6. $\Gamma(\alpha)$ is a subgroup of $\Lambda$.

Proof. Inverses: $x\mapsto x^*$ maps $eM^\alpha(\lambda)e$ onto $eM^\alpha(\lambda^{-1})e$. Products: let $\lambda,\mu\in\Gamma(\alpha)$ and $e$ a nonzero projection of $M^\alpha$. Choose $0\ne x\in eM^\alpha(\lambda)e$ and let $f$ be the left support of $x$, the support of $xx^*\in M^\alpha$; so $0\ne f\le e$, $f\in M^\alpha$. Since $\mu\in\operatorname{Sp}\alpha_f$, choose $0\ne y\in fM^\alpha(\mu)f$. Then $yx\in eM^\alpha(\mu\lambda)e$, and $yx\neq0$: otherwise $y$ vanishes on the range of $x$, which is dense in $fH$, so $y=yf=0$. $\square$

The next two results relate $\Gamma$ to the fixed point algebra and to perturbations of the action.

Proposition 2.7. Let $M$ be a factor and $\alpha$ an action of $G_\Lambda$ on $M$ with $\Gamma(\alpha)=\operatorname{Sp}\alpha$. Then $M^\alpha$ is a factor. Together with Lemma 2.5(d): for an action on a factor, $M^\alpha$ is a factor if and only if $\Gamma(\alpha)=\operatorname{Sp}\alpha$.

Proof. Put $\Lambda_0=\operatorname{Sp}\alpha$, a group by Proposition 2.6. On $\mathcal M=M\bar\otimes B(\ell^2(\Lambda_0))$, with matrix units $e_{ab}$ ($a,b\in\Lambda_0$), let $\beta_s=\alpha_s\otimes \operatorname{Ad}\rho_s$, where $\rho_s\delta_c=\langle s,c\rangle\delta_c$. For $x\in M^\alpha(\lambda)$, $\beta_s(x\otimes e_{ab})=\langle s,\lambda ab^{-1}\rangle x\otimes e_{ab}$. So $1\otimes e_{aa}\in\mathcal M^\beta$, $M^\alpha\otimes e_{aa}\subset\mathcal M^\beta$, and $x\otimes e_{ab}\in\mathcal M^\beta$ whenever $x\in M^\alpha(ba^{-1})$.

Let $Z$ be a central projection of $\mathcal M^\beta$. It commutes with every $1\otimes e_{aa}$, so $Z=\sum_az_a\otimes e_{aa}$ with $z_a\in M^\alpha$; it commutes with $M^\alpha\otimes e_{aa}$, so $z_a$ is a central projection of $M^\alpha$; and it commutes with $x\otimes e_{ab}$, so $z_ax=xz_b\qquad(a,b\in\Lambda_0,\ x\in M^\alpha(ba^{-1})).\tag{2.1}$ Fix $a$ and $\lambda\in\Lambda_0$, and put $b=\lambda a$ and $w=z_a(1-z_b)$, a projection of $M^\alpha$. For $x\in M^\alpha(\lambda)$, (2.1) gives $wxw=(1-z_b)z_ax\,z_a(1-z_b)=(1-z_b)xz_bz_a(1-z_b)=0$. Since $\lambda\in\Gamma(\alpha)$, this forces $w=0$, that is $z_a\le z_b$. The same argument with $\lambda^{-1}$ and the pair $(b,a)$ gives $z_b\le z_a$. Hence all $z_a$ are equal to one projection $z$, and (2.1) says that $z$ commutes with every $M^\alpha(\lambda)$, hence with $M$ (Lemma 2.3(d)). So $z\in\{0,1\}$ and $\mathcal M^\beta$ is a factor. Finally $M^\alpha\otimes e_{11}=(1\otimes e_{11})\mathcal M^\beta(1\otimes e_{11})$ is a corner of a factor, hence a factor. $\square$

Definition. Let $\alpha$ be an action of $G_\Lambda$ on $M$. A strongly continuous map $v:G_\Lambda\to \mathcal U(M)$ is an $\alpha$-cocycle if $v_{s+s'}=v_s\alpha_s(v_{s'})$. Then $\beta_s=\operatorname{Ad}v_s \circ\alpha_s$ is again an action; $\alpha$ and $\beta$ are called exterior equivalent.

Theorem 2.8. Exterior equivalent actions have the same Connes spectrum.

Proof. Let $\beta_s=\operatorname{Ad}v_s\circ\alpha_s$. Since $\alpha_s=\operatorname{Ad}v_s^*\circ\beta_s$ and $s\mapsto v_s^*$ is a $\beta$-cocycle ($v^*_{s+s'}=\alpha_s(v_{s'})^*v_s^*=v_s^*\beta_s(v_{s'}^*)$), it suffices to prove $\Gamma(\alpha)\subset\Gamma(\beta)$. Fix a nonzero projection $p\in M^\beta$.

For $\mu\in\Lambda$ let $Y(\mu)=\{y\in M:v_s\alpha_s(y)=\langle s,\mu\rangle y\ \forall s\}$. On $Q=M\otimes M_2(\mathbb C)$, the map $W_s=1\otimes e_{11}+v_s\otimes e_{22}$ is an $(\alpha\otimes \operatorname{id})$-cocycle, so $\gamma_s=\operatorname{Ad}W_s\circ(\alpha_s\otimes\operatorname{id})$ is an action, with $\gamma_s(x\otimes e_{11})=\alpha_s(x)\otimes e_{11},\quad\gamma_s(x\otimes e_{22})=\beta_s(x)\otimes e_{22},\quad \gamma_s(y\otimes e_{21})=v_s\alpha_s(y)\otimes e_{21}.$ Hence $(1\otimes e_{22})Q^\gamma(\mu)(1\otimes e_{11})=Y(\mu)\otimes e_{21}$. If $pY(\mu)=0$ for all $\mu$, then $(p\otimes e_{22})Q^\gamma(\mu)(1\otimes e_{11})=0$ for all $\mu$, and by Lemma 2.3(d) $(p\otimes e_{22})Q(1\otimes e_{11})=0$, which is false since it contains $p\otimes e_{21}$. So there are $\mu$ and $y\in Y(\mu)$ with $py\neq0$.

Replace $v$ by $v'_s=\overline{\langle s,\mu\rangle}v_s$. This is again an $\alpha$-cocycle with $\operatorname{Ad}v'_s\circ\alpha_s=\beta_s$, and $y$ satisfies $v'_s\alpha_s(y)=y$. Let $\gamma'$ be the action built from $v'$. Then $w=py\otimes e_{21}$ is fixed by $\gamma'$, because $v'_s\alpha_s(py)=\beta_s(p)v'_s\alpha_s(y)=py$. Let $w=w_0|w|$ be its polar decomposition; $w_0\in Q^{\gamma'}$, and $E=w_0^*w_0$, $F=w_0w_0^*$ are nonzero projections of $Q^{\gamma'}$ with $E\le1\otimes e_{11}$ and $F\le p\otimes e_{22}$. Write $E=e\otimes e_{11}$ with $e$ a nonzero projection of $M^\alpha$. By Lemma 2.5, $\Gamma(\alpha)\subset\operatorname{Sp}\alpha_e=\operatorname{Sp}\gamma'_E=\operatorname{Sp}\gamma'_F\subset \operatorname{Sp}\gamma'_{p\otimes e_{22}}=\operatorname{Sp}\beta_p .$ As $p$ was arbitrary, $\Gamma(\alpha)\subset\Gamma(\beta)$. $\square$

2.4 Crossed products recognized from eigenunitaries

Lemma 2.9. Let $\alpha$ be an action of $G_\Lambda$ on $P$, with $N=P^\alpha$, and let $\lambda\mapsto U_\lambda$ be a homomorphism of $\Lambda$ into $\mathcal U(P)$ with $U_\lambda\in P^\alpha(\lambda)$ for every $\lambda\in\Lambda$. Then $P^\alpha(\lambda)=NU_\lambda$, $V_\lambda=\operatorname{Ad}U_\lambda|_N$ is an action of $\Lambda$ on $N$, and there is an isomorphism of $P$ onto $N\rtimes_V\Lambda$ with $n\mapsto\pi(n)$ and $U_\lambda\mapsto\ell_\lambda$. It carries $P_1$ to the canonical conditional expectation.

Proof. If $x\in P^\alpha(\lambda)$, then $xU_\lambda^*\in P^\alpha(1)=N$, so $x\in NU_\lambda$; conversely $NU_\lambda\subset P^\alpha(\lambda)$. Also $U_\lambda NU_\lambda^*\subset P^\alpha(1)=N$. By Lemma 2.3(d), $N$ and the $U_\lambda$ generate $P$.

Let $P\subset B(H)$ be a faithful normal representation and $G=G_\Lambda$. On $L^2(G,H)=L^2(G)\otimes H$ define $(\delta(x)\xi)(s)=\alpha_s(x)\xi(s)$. This is a $*$-homomorphism; it is injective, since $\delta(x)=0$ forces $\alpha_s(x)=0$ for almost all $s$, hence for all $s$ by continuity. It is normal: if $\xi,\xi'$ are simple functions $\sum_k1_{A_k}h_k$, $\sum_l1_{A_l'}h'_l$, then $\langle\delta(x)\xi,\xi'\rangle=\sum_{k,l} \int1_{A_k\cap A'_l}(s)\langle\alpha_s(x)h_k,h'_l\rangle ds$ is normal by Lemma 2.3(b); for general $\xi,\xi'$ the functional $x\mapsto\langle\delta(x)\xi,\xi'\rangle$ is a norm limit of such functionals, hence normal. Its range $\delta(P)$ is a von Neumann algebra generated by $\delta(N)=1\otimes N$ and $\delta(U_\lambda)=m_\lambda\otimes U_\lambda$, where $m_\lambda$ is multiplication by $\langle\cdot,\lambda\rangle$. The Fourier transform $\mathcal F:L^2(G)\to\ell^2(\Lambda)$, which sends the character $\langle\cdot,\mu\rangle$ to $\delta_\mu$ (B7), satisfies $\mathcal Fm_\lambda\mathcal F^*=\ell_\lambda$. So $P\cong\{1\otimes N,\ \ell_\lambda\otimes U_\lambda\}''$ on $\ell^2(\Lambda,H)$. Finally the unitary $(W\xi)(\mu)=U_\mu\xi(\mu)$ satisfies $W\pi(n)W^*=1\otimes n$ and $W\ell_\lambda W^*=\ell_\lambda\otimes U_\lambda$ (direct check: $(W\ell_\lambda W^*\xi)(\mu)=U_\mu U_{\lambda^{-1}\mu}^*\xi(\lambda^{-1}\mu)=U_\lambda \xi(\lambda^{-1}\mu)$). Composing gives the isomorphism. Under it $P^\alpha(\lambda)=NU_\lambda$ goes to $\pi(N)\ell_\lambda$, so $P_1$ goes to the canonical expectation. $\square$

3. Almost periodic weights

A reference for this section is [Connes 1974]; almost periodic states were introduced in [Connes 1973, Definition 3.7.1].

3.1 Definitions and the extension of the modular group

Definition 3.1. Let $\varphi$ be a weight on $M$.

For $\lambda>0$, the eigenoperators of $\varphi$ with eigenvalue $\lambda$ form the space $M^\varphi(\lambda)=\{x\in M:\sigma^\varphi_t(x)=\lambda^{it}x\ \forall t\in\mathbb R\}$.
$\operatorname{Sp}_p(\varphi)$ is the set of eigenvalues (the point spectrum) of $\Delta_\varphi$.
$\varphi$ is almost periodic if $H_\varphi$ is the closed span of eigenvectors of $\Delta_\varphi$. For a subgroup $\Lambda\subset\mathbb R_+^*$, $\varphi$ is $\Lambda$-almost periodic if it is almost periodic and $\operatorname{Sp}_p(\varphi)\subset\Lambda$.
$\varphi$ is strictly semifinite if its restriction to $M_\varphi$ is semifinite.

An almost periodic weight is $\Lambda$-almost periodic for the group $\Lambda$ generated by $\operatorname{Sp}_p(\varphi)$. The spaces $M^\varphi(\lambda)$ have the algebraic properties of Lemma 2.3(c), with the same proofs, and $M^\varphi(1)=M_\varphi$.

Lemma 3.2. Let $\varphi$ be a weight on $M$.

(a) $\varphi$ is strictly semifinite if and only if there are orthogonal projections $e_i\in M_\varphi$ with $\sum_ie_i=1$ and $\varphi(e_i)<\infty$.
(b) In that case let $\varphi_i=\varphi(e_i\,\cdot\,e_i)\in M_*^+$. Each $\varphi_i$ is $\sigma^\varphi$-invariant, and on bounded subsets of $M$ the strong* topology is given by the seminorms $p_i(x)=(\varphi_i(x^*x)+\varphi_i(xx^*))^{1/2}$.
(c) In that case the finite sums of the $e_i$ form an increasing net in $\mathfrak n_\varphi\cap M_\varphi$ converging strongly to $1$.

Proof. (a) If $\tau=\varphi|_{M_\varphi}$ is semifinite, it is a semifinite trace (B4), so every nonzero projection of $M_\varphi$ majorizes a nonzero projection of finite trace, and a maximal orthogonal family of projections of finite trace has sum $1$. Conversely, given the $e_i$, let $q$ be a nonzero projection of $M_\varphi$. For some $i$, $qe_i\neq0$, so $e_iqe_i\neq0$, and for small $\varepsilon>0$ the spectral projection $r=1_{[\varepsilon,1]} (e_iqe_i)\in M_\varphi$ is nonzero with $\tau(r)\le\varepsilon^{-1}\tau(e_iqe_i)\le\varepsilon^{-1}\tau(e_i)<\infty$. The projection $r$ lies under the right support of $qe_i$, which is equivalent in $M_\varphi$ to its left support, a subprojection of $q$. So $q$ majorizes a nonzero projection of finite trace, and the trace $\tau$ is semifinite.

(b) $\varphi_i(\sigma^\varphi_t(x))=\varphi(\sigma^\varphi_t(e_ixe_i))=\varphi_i(x)$. In $H_\varphi$ put $\xi_i=\eta_\varphi(e_i)$, so $\varphi_i=\omega_{\xi_i}|_M$. The projection onto $[M'\xi_i]$ is the support of $\varphi_i$, which is $e_i$ because $\varphi$ is faithful. So $\sum_i[M'\xi_i]=1$. If $(x_k)$ is a bounded net with $p_i(x_k)\to0$ for all $i$, then $x_ky'\xi_i=y'x_k\xi_i\to0$ and $x_k^*y'\xi_i\to0$ for $y'\in M'$, so $x_k\to0$ strong* on the dense span of the $M'\xi_i$, hence everywhere by boundedness. The converse is clear.

Theorem 3.3. Let $\Lambda$ be a subgroup of $\mathbb R_+^*$, $G=G_\Lambda$, $\iota=\iota_\Lambda$, and $\varphi$ a weight on $M$. The following are equivalent.

(a) $\varphi$ is $\Lambda$-almost periodic.
(b) There is an action $\sigma^{\varphi,\Lambda}$ of $G$ on $M$ with $\sigma^{\varphi,\Lambda}_{\iota(t)}= \sigma^\varphi_t$ for all $t$. (It is unique, since $\iota(\mathbb R)$ is dense.)
(c) $\varphi$ is strictly semifinite, and some set $\mathcal S\subset M$ with $M=(\mathcal S\cup\mathcal S^*)''$ has this property: for each $x\in\mathcal S$ the map $t\mapsto\sigma^\varphi_t(x)$ extends to a strong* continuous map $G\to M$.

When they hold:

$\varphi\circ\sigma^{\varphi,\Lambda}_s=\varphi$; the spectral subspaces of $\sigma^{\varphi,\Lambda}$ are the $M^\varphi(\lambda)$, $\lambda\in\Lambda$; the fixed point algebra is $M_\varphi$; $M^\varphi(\lambda)=0$ for $\lambda\notin\Lambda$; and $\operatorname{Sp}_p(\varphi)=\operatorname{Sp}\sigma^{\varphi,\Lambda}=\{\lambda>0:M^\varphi(\lambda)\neq0\}.\tag{3.1}$
The eigenspace of $\Delta_\varphi$ for $\lambda$ is the closed span of the vectors $\eta_\varphi(xa)$, $x\in M^\varphi(\lambda)$, $a\in\mathfrak n_\varphi\cap M_\varphi$.
With $\Delta_\varphi=\sum_\lambda\lambda E_\lambda$, the unitaries $D_s=\sum_\lambda\langle s,\lambda\rangle E_\lambda$ form a strongly continuous representation of $G$ with $\sigma^{\varphi,\Lambda}_s=\operatorname{Ad} D_s|_M$, and $s\mapsto\sigma^{\varphi,\Lambda}_s$ is continuous from $G$ to $\operatorname{Aut}M$ with the $u$-topology.

Proof. (a)⇒(b). By Lemma 2.2 applied to $K=\Delta_\varphi$, $D$ is a strongly continuous representation of $G$ with $D_{\iota(t)}=\Delta_\varphi^{it}$. For $s\in G$ choose a net $t_k$ with $\iota(t_k)\to s$. Then $\sigma^\varphi_{t_k}(x)=\Delta_\varphi^{it_k}x\Delta_\varphi^{-it_k}\to D_sxD_s^*$ strongly for $x\in M$, so $D_sMD_s^*\subset M$, and likewise $D_s^*MD_s\subset M$. Thus $\sigma^{\varphi,\Lambda}_s=\operatorname{Ad} D_s|_M$ is an action with the required property. This also proves the first half of item 3.

(b)⇒(a). Write $\alpha=\sigma^{\varphi,\Lambda}$. Invariance. For $x\ge0$ and a net $\iota(t_k)\to s$, $\sigma^\varphi_{t_k}(x)\to\alpha_s(x)$ $\sigma$-weakly, so lower semicontinuity (B1) gives $\varphi(\alpha_s(x))\le\liminf\varphi(\sigma^\varphi_{t_k}(x))=\varphi(x)$. Applying this to $-s$ and $\alpha_s(x)$ gives equality.

Spectral subspaces. For $\lambda\in\Lambda$, $M^\alpha(\lambda)\subset M^\varphi(\lambda)$ by restriction to $\iota(\mathbb R)$; conversely if $x\in M^\varphi(\lambda)$, the continuous maps $s\mapsto\alpha_s(x)$ and $s\mapsto\langle s,\lambda\rangle x$ agree on a dense set. If $\lambda\notin\Lambda$ and $x\in M^\varphi(\lambda)$, then for $\mu\in\Lambda$, $P_\mu(x)\in M^\varphi(\mu)$ and $\sigma^\varphi_t(P_\mu(x))=P_\mu(\sigma^\varphi_t(x))= \lambda^{it}P_\mu(x)$; so $(\lambda^{it}-\mu^{it})P_\mu(x)=0$ for all $t$, $P_\mu(x)=0$, and $x=0$ by Lemma 2.3(d). The fixed point algebra is $M^\varphi(1)=M_\varphi$.

Strict semifiniteness. Let $E=P_1$, a normal conditional expectation onto $M_\varphi$. For $x\in \mathfrak m_\varphi^+$ and $\omega\in M_*^+$ with $\omega\le\varphi$, $\omega(E(x))=\int\omega(\alpha_s(x))\,ds\le\int\varphi(\alpha_s(x))\,ds=\varphi(x)$; by (B1), $\varphi(E(x))\le\varphi(x)<\infty$. So $E(\mathfrak m_\varphi)\subset\mathfrak m_\varphi\cap M_\varphi$, and since $\mathfrak m_\varphi$ is $\sigma$-weakly dense and $E$ is normal and onto $M_\varphi$, $\varphi|_{M_\varphi}$ is semifinite. Let $(a_k)$ be the net of Lemma 3.2(c).

Eigenvectors. If $x\in M^\varphi(\lambda)$ and $a\in\mathfrak n_\varphi\cap M_\varphi$, then $xa\in\mathfrak n_\varphi$ and $\Delta_\varphi^{it}\eta_\varphi(xa)=\eta_\varphi(\sigma^\varphi_t(xa))=\lambda^{it}\eta_\varphi(xa)$, so $\eta_\varphi(xa)$ is an eigenvector of $\Delta_\varphi$ for $\lambda$ (or zero). Let $K$ be the closed span of these vectors for all $\lambda\in\Lambda$. For $y\in\mathfrak n_\varphi$, (0.1) with $a_k\in M_\varphi$ (so $\sigma^\varphi_{i/2}(a_k)=a_k$) gives $\eta_\varphi(ya_k)=J_\varphi a_kJ_\varphi\eta_\varphi(y)\to\eta_\varphi(y)$. And $\eta_\varphi(ya_k)=y\,\eta_\varphi(a_k)$, where $y\mapsto y\,\eta_\varphi(a_k)$ is continuous from the $\sigma$-weak to the weak topology; by Lemma 2.3(d), $y$ is a $\sigma$-weak limit of finite sums of eigenoperators, so $y\,\eta_\varphi(a_k)$ lies in the weak closure of $K$, which is $K$. Hence $K\supset\eta_\varphi(\mathfrak n_\varphi)$, so $K=H_\varphi$, and $\Delta_\varphi$ is diagonal with eigenvalues in $\Lambda$. Vectors for different eigenvalues are orthogonal, so the eigenspace for $\lambda$ is the closed span described in item 2. This gives $\operatorname{Sp}_p(\varphi)\subset\{\lambda:M^\varphi(\lambda)\neq0\}$. Conversely, if $0\ne x\in M^\varphi(\lambda)$, then $xa_k\to x$ strongly, so some $xa_k\neq0$ and $\|\eta_\varphi(xa_k)\|^2= \varphi(a_kx^*xa_k)>0$: $\lambda$ is an eigenvalue. This proves item 1.

(b)⇒(c). Strict semifiniteness was just proved; take $\mathcal S=M$ and Lemma 2.3(a).

(c)⇒(b). Let $e_i,\varphi_i,p_i$ be as in Lemma 3.2. Each $p_i$ is $\sigma^\varphi$-invariant, so $p_i(\sigma^\varphi_t(x)-\sigma^\varphi_{t'}(x))=p_i(\sigma^\varphi_{t-t'}(x)-x)$. Let $\mathcal A$ be the set of $x\in M$ such that $p_i(\sigma^\varphi_t(x)-x)\to0$ whenever $\iota(t)\to0$ (along nets in $\mathbb R$), for every $i$. On bounded sets the $p_i$ define the strong* topology, for which multiplication is jointly continuous on bounded sets; hence $\mathcal A$ is a unital $*$-subalgebra. If $x$ is a bounded strong* limit of $x_k\in\mathcal A$, then $p_i(\sigma^\varphi_t(x)-x)\le2p_i(x-x_k)+p_i(\sigma^\varphi_t(x_k)-x_k)$, so $x\in\mathcal A$. If $x\in \mathcal S$, the extension $\tilde x$ of $t\mapsto\sigma^\varphi_t(x)$ is strong* continuous at $0$, so $x\in\mathcal A$. By the Kaplansky density theorem, $\mathcal A=M$.

Now fix $x\in M$ and $s\in G$, and let $\iota(t_k)\to s$. The net $y_k=\sigma^\varphi_{t_k}(x)$ is bounded and Cauchy for each $p_i$, since $\iota(t_k-t_l)\to0$. In $H_\varphi$ (notation of Lemma 3.2(b)) this means that $y_ky'\xi_i=y'y_k\xi_i$ and $y_k^*y'\xi_i$ converge for $y'\in M'$; by boundedness $y_k$ and $y_k^*$ converge strongly on all of $H_\varphi$, so $y_k$ converges strong* to an element of $M$. We may therefore define $\alpha_s(x)=\lim_k\sigma^\varphi_{t_k}(x)$, independently of the net. Each $\alpha_s$ is a unital $*$-endomorphism, being a pointwise bounded strong* limit of automorphisms, and $p_i(\alpha_s(x))=p_i(x)$. The same estimates show $\alpha_s(\alpha_{s'}(x))=\alpha_{s+s'}(x)$ and that $s\mapsto\alpha_s(x)$ is strong* continuous; since $\alpha_0=\operatorname{id}$, each $\alpha_s$ is an automorphism. So $\alpha$ is the required action.

Item 3, second half. Every $\omega\in M_*$ has the form $\omega(x)=\sum_n\langle x\xi_n,\xi'_n\rangle$ with $\xi_n,\xi'_n\in H_\varphi$ and $\sum\|\xi_n\|\|\xi'_n\|<\infty$. Then $\|\omega\circ\alpha_s-\omega\circ\alpha_{s'}\|\le\sum_n(\|D_s^*\xi_n-D_{s'}^*\xi_n\|\|\xi'_n\|+ \|\xi_n\|\|D_s^*\xi'_n-D_{s'}^*\xi'_n\|)$, which tends to $0$ as $s'\to s$ by dominated convergence (over $n$). $\square$

Remark 3.4. Formula (3.1) shows that $\operatorname{Sp}_p(\varphi)$ and the spaces $M^\varphi(\lambda)$ do not depend on $\Lambda$. The action $\sigma^{\varphi,\Lambda}$ does depend on $\Lambda$, but only through the choice of compact group: if $\Lambda\subset\Lambda'$, then $\sigma^{\varphi,\Lambda'}$ is $\sigma^{\varphi,\Lambda}$ composed with the restriction map $G_{\Lambda'}\to G_\Lambda$.

3.2 Constructions

Proposition 3.5. Let $\varphi$ be a weight on $M$ that is $\Lambda$-almost periodic.

(a) For a nonzero projection $e\in M_\varphi$, $\varphi_e$ is $\Lambda$-almost periodic on $eMe$, and $\operatorname{Sp}_p(\varphi_e)=\{\lambda:eM^\varphi(\lambda)e\neq0\}$.
(b) For every Hilbert space $K\neq0$, $\varphi\otimes\operatorname{Tr}$ is $\Lambda$-almost periodic on $M\bar\otimes B(K)$, with the same point spectrum as $\varphi$.
(c) Let $b=\sum_{\lambda\in\Lambda}\lambda f_\lambda$, where the $f_\lambda$ are orthogonal projections of $M_\varphi$ with sum $1$. Then $\varphi_b$ is $\Lambda$-almost periodic, and $M_{\varphi_b}$ contains every element of $M_\varphi$ commuting with all $f_\lambda$.
(d) If $\kappa:M\to N$ is an isomorphism, $\varphi\circ\kappa^{-1}$ is $\Lambda$-almost periodic on $N$ with the same point spectrum. In particular $\varphi\circ\operatorname{Ad}u$ is, for $u\in\mathcal U(M)$.

Proof. (a) $\sigma^{\varphi,\Lambda}_s(e)=e$, so $\sigma^{\varphi,\Lambda}$ restricts to an action on $eMe$ extending $\sigma^{\varphi_e}=\sigma^\varphi|_{eMe}$ (B3). Apply Theorem 3.3(b)⇒(a) and (3.1), noting $(eMe)^{\varphi_e}(\lambda)=eM^\varphi(\lambda)e$.

(b) By (B3), $\sigma^{\varphi\otimes\operatorname{Tr}}_t=\sigma^\varphi_t\otimes\operatorname{id}$. The unitaries $D_s\otimes1$ implement automorphisms of $M\bar\otimes B(K)$ (they map $x\otimes y$ to $\sigma^{\varphi,\Lambda}_s(x)\otimes y$), giving an action which extends $\sigma^\varphi_t\otimes\operatorname{id}$. For its spectrum: $x\otimes e_{11}$ is an eigenoperator when $x$ is; conversely the matrix entries $(1\otimes e_{1i})X(1\otimes e_{j1})$ of an eigenoperator $X\neq0$ are eigenoperators for the same eigenvalue, and one of them is nonzero. Conclude with (3.1).

(c) Put $b^{(s)}=\sum_\lambda\langle s,\lambda\rangle f_\lambda$, a strongly continuous unitary representation of $G$ (Lemma 2.2) with values in $M_\varphi$, hence fixed by $\sigma^{\varphi,\Lambda}$. So $\beta_s=\operatorname{Ad}b^{(s)}\circ\sigma^{\varphi,\Lambda}_s$ is an action, and $\beta_{\iota(t)}= \operatorname{Ad}b^{it}\circ\sigma^\varphi_t=\sigma^{\varphi_b}_t$ by (B2d). By Theorem 3.3, $\varphi_b$ is $\Lambda$-almost periodic. An element of $M_\varphi$ commuting with the $f_\lambda$ is fixed by every $\beta_s$.

(d) is clear. $\square$

3.3 The relative commutant of the centralizer

Lemma 3.6. Let $\varphi$ be a weight, $\lambda>0$, and $v\in M^\varphi(\lambda)$ a partial isometry with $f=v^*v\in M_\varphi$. Then $\varphi(vcv^*)=\lambda\varphi(c)$ for every $c\ge0$ in $fM_\varphi f$ with $\varphi(c)<\infty$.

Proof. $v^*$ is entire analytic with $\sigma^\varphi_z(v^*)=\lambda^{-iz}v^*$, so $\sigma^\varphi_{i/2}(v^*)^*=\lambda^{1/2}v$. Let $y=c^{1/2}\in\mathfrak n_\varphi$. By (0.1), $\eta_\varphi(yv^*)=\lambda^{1/2}J_\varphi vJ_\varphi\eta_\varphi(y)$. Since $y=yf$ and $f\in M_\varphi$, (0.1) also gives $\eta_\varphi(y)=J_\varphi fJ_\varphi\eta_\varphi(y)$, so $J_\varphi\eta_\varphi(y)$ lies in the range of $f=v^*v$, on which $v$ is isometric. Hence $\varphi(vcv^*)=\|\eta_\varphi(yv^*)\|^2=\lambda \|v\,J_\varphi\eta_\varphi(y)\|^2=\lambda\|\eta_\varphi(y)\|^2=\lambda\varphi(c)$. $\square$

Proposition 3.7. For every almost periodic weight $\varphi$ on $M$, $M_\varphi'\cap M\subset M_\varphi$; that is, $M_\varphi'\cap M$ is the centre of $M_\varphi$.

Proof. Let $\Lambda$ be the group generated by $\operatorname{Sp}_p(\varphi)$ and $\alpha= \sigma^{\varphi,\Lambda}$. The algebra $R=M_\varphi'\cap M$ is $\alpha$-invariant, because $\alpha_s$ fixes $M_\varphi$ pointwise; so the Fourier coefficients $P_\lambda(x)$ of $x\in R$ lie in $R$. By Lemma 2.3(d) it suffices to show $R\cap M^\varphi(\lambda)=0$ for $\lambda\neq1$.

Let $x\in R\cap M^\varphi(\lambda)$, $x\neq0$, with polar decomposition $x=v|x|$. Then $v\in R$ (as $R$ is a von Neumann algebra) and $v\in M^\varphi(\lambda)$ (Lemma 2.3(c)). So $f=v^*v$ and $g=vv^*$ lie in $R\cap M_\varphi$, the centre of $M_\varphi$. Since $v$ commutes with $f$ and $g$, we get $g=vfv^*=fvv^*=fg$ and $f=v^*gv=gv^*v=gf$, so $f=g\neq0$. By strict semifiniteness (Theorem 3.3) there is $a\in\mathfrak m_\varphi^+\cap M_\varphi$ with $c=faf\neq0$; then $c\in fM_\varphi f$ and $\varphi(c)=\varphi(a^{1/2}fa^{1/2}) \le\varphi(a)<\infty$, since $\varphi|_{M_\varphi}$ is a trace, and $\varphi(c)>0$. As $v$ commutes with $c\in M_\varphi$, $vcv^*=cvv^*=cf=c$. Lemma 3.6 gives $\varphi(c)=\lambda\varphi(c)$, so $\lambda=1$. $\square$

3.4 Examples

Example 3.8 (type I). On $B(H)$, every weight is $\varphi=\operatorname{Tr}(K\,\cdot)$ with $K$ positive nonsingular, $H_\varphi$ is the Hilbert–Schmidt space and $\Delta_\varphi\xi=K\xi K^{-1}$. If $K=\sum_ak_aq_a$ is diagonal (with distinct eigenvalues $k_a$ and spectral projections $q_a$), the operators $\xi$ with $\xi=q_a\xi q_b$ are eigenvectors for $k_a/k_b$, so $\varphi$ is almost periodic with $\operatorname{Sp}_p(\varphi)=\{k_a/k_b\}$. If $K$ has a nonzero continuous spectral part, the eigenvectors of $\Delta_\varphi$ do not span. Indeed, let $q$ be the projection onto the closed span of the eigenvectors of $K$. A Hilbert–Schmidt operator $\xi$ with $\Delta_\varphi^{it}\xi=\mu^{it}\xi$ satisfies $K^{it}\xi^*\xi K^{-it}= \xi^*\xi$; the eigenspaces of the compact operator $\xi^*\xi$ for nonzero eigenvalues are finite dimensional and invariant under $K^{it}$, hence spanned by eigenvectors of $K$. So $\xi(1-q)=0$, and all eigenvectors of $\Delta_\varphi$ lie in the proper closed subspace $\{\xi:\xi(1-q)=0\}$. The trace $\operatorname{Tr}$ is almost periodic with $\operatorname{Sp}_p=\{1\}$. The eigenvalues of $\Delta_\varphi$ for an almost periodic weight need not form a group: $K=\operatorname{diag}(1,2)$ on $\mathbb C^2$ gives $\{1/2,1,2\}$.

Example 3.9 (product states). Let $\omega_n=\operatorname{Tr}(\rho_n\,\cdot)$ be faithful states on $M_{k_n}(\mathbb C)$, with $\rho_n$ of eigenvalues $p_{n,1},\dots,p_{n,k_n}$, and $R=\bigotimes_n(M_{k_n}, \omega_n)$ with product state $\omega$. The modular operator of $\omega_n$ on $M_{k_n}$ (as a Hilbert–Schmidt space) is $\xi\mapsto\rho_n\xi\rho_n^{-1}$, diagonal with eigenvalues $p_{n,i}/p_{n,j}$ and eigenvector $\rho_n^{1/2}$ for $1$. By (B12), $\Delta_\omega^{it}=\bigotimes\Delta_{\omega_n}^{it}$; the product vectors built from eigenvectors, differing from the reference vector in finitely many places, form an orthonormal basis of eigenvectors. So $\omega$ is almost periodic, and $\operatorname{Sp}_p(\omega)$ is the set of products $\prod_nr_n$, where $r_n$ is a ratio $p_{n,i}/p_{n,j}$ and $r_n=1$ for all but finitely many $n$. It is contained in the group generated by all the ratios, and equal to it when every ratio occurs for infinitely many $n$. For $k_n=2$ and $\rho_n=\operatorname{diag} (1,\lambda)/(1+\lambda)$, $R$ is the Powers factor $R_\lambda$ and $\operatorname{Sp}_p(\omega)=\lambda^{\mathbb Z}$.

Example 3.10 (dual weights). Take a von Neumann algebra $N$ with an f.s.n. trace $\tau$, a countable subgroup $\Lambda$ of $\mathbb R_+^*$, and an action $V$ of $\Lambda$ on $N$ with $\tau\circ V_\lambda=\lambda\tau$. In $P=N\rtimes_V\Lambda$ with expectation $E$, put $\varphi=\tau\circ E$. By (B3), $\sigma^\varphi_t$ is the identity on $N$, and $\varphi\circ\operatorname{Ad}\ell_\lambda=\tau\circ V_\lambda\circ E=\lambda\varphi$, so $\ell_\lambda^*\sigma^\varphi_t(\ell_\lambda)=\lambda^{it}$ by (B2a, c), i.e. $\ell_\lambda\in P^\varphi(\lambda)$. By Theorem 3.3(c) with $\mathcal S=N\cup\{\ell_\lambda\}$ ($\varphi$ is strictly semifinite since $N\subset P_\varphi$ and $\varphi|_N=\tau$), $\varphi$ is $\Lambda$-almost periodic. The dual action of $G_\Lambda$ (B8) agrees with $\sigma^\varphi_t$ at $\iota(t)$ on $N$ and on the $\ell_\lambda$, so it is $\sigma^{\varphi,\Lambda}$; by Lemma 2.9, $P^\varphi(\lambda)=N\ell_\lambda$, so $P_\varphi=N$ and $\operatorname{Sp}_p(\varphi)=\Lambda$.

4. The point modular spectrum

A reference for this section is [Connes 1974].

Definition 4.1. For a factor $M$, define the point modular spectrum by $Sd(M)=\bigcap\{\operatorname{Sp}_p(\varphi):\ \varphi\ \text{an almost periodic weight on }M\}\subset\mathbb R_+^*,$ with the convention $Sd(M)=\mathbb R_+^*$ if $M$ has no almost periodic weight. For an almost periodic weight $\varphi$ put $\Gamma(\varphi)=\bigcap\{\operatorname{Sp}_p(\varphi_e):\ e\ \text{a nonzero projection of }M_\varphi\}.$

By Theorem 3.3 and Proposition 3.5(a), $\operatorname{Sp}_p(\varphi_e)=\operatorname{Sp}(\sigma^{\varphi,\Lambda})_e$ for every group $\Lambda\supset\operatorname{Sp}_p(\varphi)$, and the projections of $M_\varphi$ are those of the fixed point algebra. So $\Gamma(\varphi)=\Gamma(\sigma^{\varphi,\Lambda})$ is the Connes spectrum of the extended modular action, for any such $\Lambda$, including $\Lambda=\mathbb R_+^*$. From Section 2:

$\Gamma(\varphi)$ is a subgroup of $\mathbb R_+^*$ contained in $\operatorname{Sp}_p(\varphi)$ (Proposition 2.6);
if $M_\varphi$ is a factor, then $\Gamma(\varphi)=\operatorname{Sp}_p(\varphi)$ (Lemma 2.5(d)), and conversely, for a factor $M$, $\Gamma(\varphi)=\operatorname{Sp}_p(\varphi)$ implies that $M_\varphi$ is a factor (Proposition 2.7).

Also $1\in\operatorname{Sp}_p(\varphi)$ for every almost periodic $\varphi$, since $1\in\Gamma(\varphi)$. The invariant $Sd$ is an isomorphism invariant: an isomorphism transports almost periodic weights with their point spectra (Proposition 3.5(d)).

Lemma 4.2 (amplification). Let $M$ be a type III factor and $e$ a nonzero projection of $M$. For every almost periodic weight $\chi$ on $eMe$, some almost periodic weight $\varphi$ on $M$ has $\operatorname{Sp}_p(\varphi)=\operatorname{Sp}_p(\chi)$. Consequently $Sd(M)\subset Sd(eMe)$.

Proof. By (B6), $M\cong eMe\bar\otimes B(\ell^2(J))$ for a nonempty set $J$. Transport $\chi\otimes \operatorname{Tr}$ to $M$ and apply Proposition 3.5(b), (d). $\square$

Theorem 4.3. For every factor $M$ one has $Sd(M)=\bigcap\{\Gamma(\varphi):\varphi\ \text{almost periodic on }M\},\tag{4.1}$ and $Sd(M)$ is a subgroup of $\mathbb R_+^*$. Moreover, $Sd(M)=\{1\}$ if $M$ is semifinite.

Proof. Since $\Gamma(\varphi)\subset\operatorname{Sp}_p(\varphi)$, the right side of (4.1) is contained in $Sd(M)$. Both sides equal $\mathbb R_+^*$ if there is no almost periodic weight.

Semifinite case. A trace $\tau$ is almost periodic with $\Delta_\tau=1$, so $Sd(M)\subset\{1\}$ and $\Gamma(\tau)=\{1\}$; since $1\in\Gamma(\varphi)\subset\operatorname{Sp}_p(\varphi)$ for all $\varphi$, both sides of (4.1) are $\{1\}$.

Type III case. Let $\varphi$ be almost periodic and $e$ a nonzero projection of $M_\varphi$. By Proposition 3.5(a), $\varphi_e$ is almost periodic on $eMe$; by Lemma 4.2, $M$ carries an almost periodic weight whose point spectrum is $\operatorname{Sp}_p(\varphi_e)$. Hence $Sd(M)\subset\operatorname{Sp}_p(\varphi_e)$ for all $e$, that is $Sd(M)\subset\Gamma(\varphi)$. This proves (4.1). As an intersection of subgroups, $Sd(M)$ is a subgroup. $\square$

Remark 4.4. If $M_*$ is separable and $Sd(M)\neq\mathbb R_+^*$, then $Sd(M)$ is countable. Indeed $M$ then has an almost periodic weight $\varphi$, $Sd(M)\subset\operatorname{Sp}_p(\varphi)$, and $\Delta_\varphi=\sum \lambda E_\lambda$ with pairwise orthogonal nonzero projections $E_\lambda$ on the separable space $H_\varphi$ (B1), so there are countably many eigenvalues.

The formula (4.1) is not practical in general, because it involves all almost periodic weights. Section 6 shows that on full factors a single weight suffices. On non-full factors this fails (Example 6.4).

5. Factors of type III₀: almost periodic states are dense

A reference for this section is [Connes 1974]; the structure theory used is in [Connes 1973, Theorem 5.3.1 and Corollary 5.3.6].

Throughout this section $M$ is a type III₀ factor whose predual is separable, and $N,C,E,D,u,e_n$ are as in (B9). Write $\theta_\varepsilon=\operatorname{Ad}u_\varepsilon|_N$, $\theta_n=\theta_{g_n}$, and let $F_n$ be the subgroup of $D$ generated by $g_1,\dots,g_n$ ($F_0=\{0\}$). Every $\theta_\varepsilon$ is an involution. A positive nonsingular operator $c$ affiliated with $C$ is $\Lambda$-valued if $c=\sum_{\lambda\in\Lambda}\lambda f_\lambda$ with orthogonal projections $f_\lambda\in C$ of sum $1$. Products and inverses of $\Lambda$-valued operators and their images under automorphisms of $C$ are $\Lambda$-valued, and $c$ is $\Lambda$-valued if $cf_i$ is $\Lambda$-valued on $f_iC$ for some partition of unity $(f_i)$ of $C$.

5.1 The structure as a crossed product

Lemma 5.1.

(a) For each $n\ge0$ the projections $\theta_\varepsilon(e_{n+1})$, $\varepsilon\in F_n$, are pairwise orthogonal with sum $1$; and $\theta_n(e_n)=e_n$ for $n\ge1$.
(b) $E(u_\varepsilon)=0$ for $\varepsilon\neq0$.
(c) There is an isomorphism of $M$ onto $N\rtimes_\theta D$ with $n\mapsto\pi(n)$, $u_\varepsilon\mapsto \ell_\varepsilon$, carrying $E$ to the canonical expectation.
(d) Denote by $N_k$ the von Neumann subalgebra generated by $N$ and $\{u_\varepsilon:\varepsilon\in F_k\}$. There is a faithful normal conditional expectation $E_k$ of $M$ onto $N_k$ with $E\circ E_k=E$, and $\bigcup_kN_k$ is a $\sigma$-weakly dense $*$-subalgebra of $M$.

Proof. (a) Induction on $n$; the case $n=0$ is $e_1=1$. Since $F_n=F_{n-1}\sqcup(g_n+F_{n-1})$, (0.2) gives $\sum_{\varepsilon\in F_n}\theta_\varepsilon(e_{n+1})=\sum_{\varepsilon\in F_{n-1}}\theta_\varepsilon(e_{n+1}+ \theta_n(e_{n+1}))=\sum_{\varepsilon\in F_{n-1}}\theta_\varepsilon(e_n)=1$. Projections with sum $1$ are pairwise orthogonal. Finally $\theta_n(e_n)=\theta_n(e_{n+1})+e_{n+1}=e_n$.

(b) Let $a=E(u_\varepsilon)\in N$ and $c\in C$. From $u_\varepsilon c=\theta_\varepsilon(c)u_\varepsilon$ and the bimodule property of $E$, $ac=\theta_\varepsilon(c)a$; as $c$ is central in $N$, $ca=\theta_\varepsilon(c)a$. Choose $k$ with $\varepsilon\in F_k$, and put $c=\theta_\delta(e_{k+1})$ for $\delta\in F_k$. Then $\theta_\delta(e_{k+1})a=\theta_\delta(e_{k+1})\theta_{\delta+\varepsilon}(e_{k+1})a=0$ by (a), since $\delta+\varepsilon\neq\delta$. Summing over $\delta\in F_k$ gives $a=0$.

(c) $M$ has separable predual, so $N$ has a faithful normal state $\chi$; put $\varphi_0=\chi\circ E$. The span $\mathcal A_0$ of the elements $nu_\varepsilon$ is a $\sigma$-weakly dense $*$-algebra, since $u_\varepsilon n=\theta_\varepsilon(n)u_\varepsilon$ and $u_\varepsilon u_\delta=u_{\varepsilon+\delta}$. By (b), $E(nu_\varepsilon u_\delta^*)=0$ unless $\varepsilon=\delta$, so an element $x=\sum n_\varepsilon u_\varepsilon$ of $\mathcal A_0$ determines its coefficients $n_\delta=E(xu_\delta^*)$, and $\varphi_0\Bigl(\bigl(\textstyle\sum_\delta m_\delta u_\delta\bigr)^*\bigl(\sum_\varepsilon n_\varepsilon u_\varepsilon\bigr)\Bigr)=\sum_\varepsilon\chi\bigl(\theta_\varepsilon(m_\varepsilon^*n_\varepsilon)\bigr).$ The same formula holds in $N\rtimes_\theta D$ for the state $\chi\circ E_R$, $E_R$ the canonical expectation, and the elements $\sum n_\varepsilon\ell_\varepsilon$ (B8). So $\eta_{\varphi_0}(x)\mapsto\eta_{\chi\circ E_R}(\Phi(x))$, $\Phi(\sum n_\varepsilon u_\varepsilon)=\sum\pi(n_\varepsilon)\ell_\varepsilon$, extends to a unitary $W$ between the GNS spaces (both images are dense, the algebras being $\sigma$-weakly dense and the states faithful), and $Wx W^*=\Phi(x)$ on $\mathcal A_0$. Hence $\operatorname{Ad}W$ is an isomorphism of the generated von Neumann algebras, extending $\Phi$. It carries $E$ to $E_R$, as both are normal and agree on $\mathcal A_0$.

(d) Use (c) and the dual action $\hat\theta$ of the compact group $\widehat D$. Let $F_k^\perp\subset\widehat D$ be the annihilator of $F_k$ and $E_k=\int_{F_k^\perp}\hat\theta_\chi\,d\chi$, a faithful normal conditional expectation onto the fixed points of $F_k^\perp$ (the proof of Lemma 2.3(b) works for any compact abelian group). These fixed points contain $N$ and $u_\varepsilon$, $\varepsilon\in F_k$. Conversely, if $x$ is fixed, its coefficients $x_\delta=E(xu_\delta^*)$ satisfy $x_\delta=\overline{\chi(\delta)}x_\delta$ for $\chi\in F_k^\perp$; for $\delta\notin F_k$ some $\chi\in F_k^\perp$ has $\chi(\delta)\ne1$ (B7), so $x_\delta=0$, and $x-\sum_{\delta\in F_k}x_\delta u_\delta$ has all coefficients zero, hence vanishes (B8). So the fixed points are $N_k$. The map $\int_{\widehat D}\hat\theta_\chi\,d\chi$ is normal and agrees with $E$ on the elements $nu_\varepsilon$ (it kills $nu_\varepsilon$ for $\varepsilon\neq0$, by orthogonality of characters), which span a $\sigma$-weakly dense subspace; so $E=\int_{\widehat D}\hat\theta_\chi\,d\chi$, and since Haar measure is invariant, $E\circ E_k=E$. The union of the increasing algebras $N_k$ contains $N$ and all $u_\varepsilon$. $\square$

5.2 Trace-scaling derivatives

Let $\tau$ be an f.s.n. trace on $N$. For $\varepsilon\in D$, $\tau\circ\theta_\varepsilon$ is an f.s.n. trace, so $\tau\circ\theta_\varepsilon=\tau(c_\varepsilon\,\cdot)$ with $c_\varepsilon$ positive nonsingular affiliated with $C$ (B4). We call $c_\varepsilon=c_\varepsilon(\tau)$ the derivative of $\theta_\varepsilon$ with respect to $\tau$.

Lemma 5.2.

(a) $c_{\varepsilon+\delta}=c_\delta\,\theta_\delta(c_\varepsilon)$; in particular $\theta_\varepsilon(c_\varepsilon)= c_\varepsilon^{-1}$.
(b) If $\rho\in C$ is positive and invertible, the derivatives for $\tau_\rho=\tau(\rho\,\cdot)$ are $c_\varepsilon(\tau_\rho)=c_\varepsilon\,\theta_\varepsilon(\rho)\rho^{-1}$.
(c) For $\varphi=\tau\circ E$: $\sigma^\varphi_t(n)=n$ for $n\in N$, and $\sigma^\varphi_t(u_\varepsilon)= u_\varepsilon c_\varepsilon^{it}$. If all $c_\varepsilon$ are $\Lambda$-valued, $\varphi$ is $\Lambda$-almost periodic and $N\subset M_\varphi$.

Proof. (a) $\tau(\theta_\varepsilon(\theta_\delta(x)))=\tau(c_\varepsilon\theta_\delta(x))=\tau(\theta_\delta( \theta_\delta(c_\varepsilon)x))=\tau(c_\delta\theta_\delta(c_\varepsilon)x)$, using $\theta_\delta^{-1}=\theta_\delta$. With $\delta=\varepsilon$: $1=c_0=c_\varepsilon\theta_\varepsilon(c_\varepsilon)$.

(b) $\tau(\rho\theta_\varepsilon(x))=\tau(\theta_\varepsilon(\theta_\varepsilon(\rho)x))=\tau(c_\varepsilon \theta_\varepsilon(\rho)x)=\tau_\rho(\rho^{-1}c_\varepsilon\theta_\varepsilon(\rho)x)$.

(c) By (B3), $\sigma^\varphi|_N=\sigma^\tau=\operatorname{id}$, so $N\subset M_\varphi$. By the bimodule property, $\varphi(u_\varepsilon xu_\varepsilon^*)=\tau(\theta_\varepsilon(E(x)))=\tau(c_\varepsilon E(x))=\varphi_{c_\varepsilon} (x)$, with $c_\varepsilon$ affiliated with $C\subset M_\varphi$. By (B2c, d), $u_\varepsilon^*\sigma^\varphi_t(u_\varepsilon)=c_\varepsilon^{it}$. If $c_\varepsilon=\sum\lambda f_\lambda$ is $\Lambda$-valued, $t\mapsto u_\varepsilon c_\varepsilon^{it}$ extends to the strongly continuous unitary-valued map $s\mapsto u_\varepsilon\sum_\lambda\langle s,\lambda\rangle f_\lambda$ (Lemma 2.2), which is strong* continuous. Moreover $\varphi$ is strictly semifinite: $\tau$ gives orthogonal projections of $N\subset M_\varphi$ with finite weight and sum $1$ (Lemma 3.2(a)). Theorem 3.3(c) with $\mathcal S=N\cup\{u_\varepsilon\}$ applies. $\square$

Lemma 5.3 (one involution). Consider a von Neumann algebra $Q$ with f.s.n. trace $\tau$ and countably decomposable centre $Z$, an automorphism $\theta$ of $Q$ with $\theta^2=\operatorname{id}$, and a projection $e\in Z$ with $e+\theta(e)=1$. Let $\Lambda$ be a countable dense subgroup of $\mathbb R_+^*$ and $\delta>0$. Then there is $r\in Z$ with $\exp(-\delta)\le r\le\exp(\delta)$ and $re=e$ such that the derivative of $\theta$ with respect to $\tau(r\,\cdot)$ is $\Lambda$-valued.

Proof. Let $c$ be the derivative of $\theta$ for $\tau$; by Lemma 5.2(a) (whose proof only used that $\theta$ is an involution), $\theta(c)=c^{-1}$. Enumerate $\Lambda=\{\lambda_1,\lambda_2,\dots\}$ and let $f(x)=\lambda_m$ for the least $m$ with $|\log\lambda_m-\log x|<\delta$; $f$ is a Borel map $]0,\infty[\to\Lambda$, and $g(x)=x/f(x)$ takes values in $]\exp(-\delta),\exp(\delta)[$. Put $a=f(c)$, which is $\Lambda$-valued, and $r=e+g(c)(1-e)$. By Lemma 5.2(b) the new derivative is $c'=c\,\theta(r)r^{-1}$. On $1-e=\theta(e)$: $\theta(r)(1-e)=\theta(re)=\theta(e)=1-e$ and $r^{-1}(1-e)=ac^{-1}(1-e)$, so $c'(1-e)=a(1-e)$. On $e$: $\theta(r)e=\theta(r(1-e))=\theta(g(c))e=c^{-1}\theta(a)^{-1}e$, since $\theta(g(c))=g(\theta(c))=\theta(c)/f(\theta(c)) =c^{-1}\theta(a)^{-1}$; and $r^{-1}e=e$. So $c'e=\theta(a)^{-1}e$. Both pieces are $\Lambda$-valued. $\square$

Proposition 5.4. Let $\Lambda$ be a countable dense subgroup of $\mathbb R_+^*$ and $\tau$ an f.s.n. trace on $N$. There is $\rho\in C$ with $\exp(-1)\le\rho\le\exp(1)$ such that all derivatives $c'_\varepsilon$, $\varepsilon\in D$, with respect to $\tau'=\tau(\rho\,\cdot)$ are $\Lambda$-valued. Consequently $\varphi'=\tau'\circ E$ is a $\Lambda$-almost periodic weight on $M$ with $N\subset M_{\varphi'}$.

Proof. By Lemma 5.2(a) it suffices that the derivatives of the generators $\theta_n$ be $\Lambda$-valued. We build $\rho_n\in C$, $n\ge1$, such that:

(i) $\exp(-2^{-n})\le\rho_n\le\exp(2^{-n})$;
(ii) $\theta_\varepsilon(\rho_n)=\rho_n$ for $\varepsilon\in F_{n-1}$;
(iii) with $\rho^{(n)}=\rho_1\cdots\rho_n$, the derivatives of $\theta_1,\dots,\theta_n$ with respect to $\tau(\rho^{(n)}\,\cdot)$ are $\Lambda$-valued.

Suppose $\rho_1,\dots,\rho_{n-1}$ are built (nothing for $n=1$), and let $c'_\varepsilon$ be the derivatives for $\tau(\rho^{(n-1)}\,\cdot)$. Apply Lemma 5.3 to $Q=Ne_n$, the trace $\tau(\rho^{(n-1)}\,\cdot)$ restricted to it, the involution $\theta_n|_{Ne_n}$ (recall $\theta_n(e_n)=e_n$), the projection $e_{n+1}$ (by (0.2), $e_{n+1}+\theta_n(e_{n+1})=e_n$) and $\delta=2^{-n}$. We get $r\in Ce_n$ such that $c'_{g_n}\theta_n(r)r^{-1}$ is $\Lambda$-valued on $e_n$. Put $\rho_n=\sum_{\varepsilon\in F_{n-1}}\theta_\varepsilon(r)$, using that the projections $\theta_\varepsilon(e_n)$, $\varepsilon\in F_{n-1}$, are pairwise orthogonal with sum $1$ (Lemma 5.1(a)). Then (i) and (ii) hold, and $\rho_ne_n=r$.

Check (iii). For $j<n$, $g_j\in F_{n-1}$, so by (ii) and Lemma 5.2(b) the derivative of $\theta_j$ is unchanged: still $\Lambda$-valued. Hence, by Lemma 5.2(a), the new derivatives $c''_\varepsilon$ are $\Lambda$-valued for all $\varepsilon\in F_{n-1}$. For $\theta_n$: since $\theta_n(e_n)=e_n$, $\theta_n(\rho_n)e_n=\theta_n(\rho_ne_n)= \theta_n(r)$, so $c''_{g_n}e_n=c'_{g_n}\theta_n(r)r^{-1}e_n$ is $\Lambda$-valued. For $\delta\in F_{n-1}$, Lemma 5.2(a) gives $c''_\delta\theta_\delta(c''_{g_n})=c''_{g_n+\delta}=c''_{g_n}\theta_n(c''_\delta)$, so $c''_{g_n}\theta_\delta(e_n)=c''_\delta\,\theta_n(c''_\delta)^{-1}\,\theta_\delta\bigl(c''_{g_n}e_n\bigr),$ a product of $\Lambda$-valued operators. Summing over $\delta\in F_{n-1}$, $c''_{g_n}$ is $\Lambda$-valued.

Now $\rho=\prod_n\rho_n$ converges in norm (the $\log\rho_n$ commute and $\sum\|\log\rho_n\|\le1$), and $\exp(-1)\le\rho\le\exp(1)$. For each $n$, $\prod_{m>n}\rho_m$ is $\theta_n$-invariant by (ii), so by Lemma 5.2(b) the derivative of $\theta_n$ for $\tau(\rho\,\cdot)$ equals the one for $\tau(\rho^{(n)}\,\cdot)$, which is $\Lambda$-valued. The last statement is Lemma 5.2(c). $\square$

5.3 Density of almost periodic states

Theorem 5.5. Let $M$ be a type III₀ factor whose predual is separable, and $\Lambda$ a dense subgroup of $\mathbb R_+^*$. Then every normal state on $M$ can be approximated in norm by faithful normal states that are $\Lambda$-almost periodic.

Reference: [Connes 1974, Theorem 1.5] states this for every countably decomposable factor of type III₀; the proof uses the structure (B9), which [Connes 1973, Corollary 5.3.6] gives for factors with separable predual, so we assume that here.

Proof. Replacing $\Lambda$ by a countable dense subgroup, we may assume $\Lambda$ countable. Let $\tau'$ and $\varphi'$ be as in Proposition 5.4, with derivatives $c'_\varepsilon$.

Step 1: a trace on $N_k$ giving an almost periodic weight. Fix $k$ and put $\tau_k(x)=\sum_{\varepsilon\in F_k}\tau'\bigl(e_{k+1}\theta_\varepsilon(x)\bigr)=\tau'(g_kx),\qquad g_k=\sum_{\varepsilon\in F_k}c'_\varepsilon\,\theta_\varepsilon(e_{k+1}),$ where we used $\tau'(e_{k+1}\theta_\varepsilon(x))=\tau'(\theta_\varepsilon(\theta_\varepsilon(e_{k+1})x))= \tau'(c'_\varepsilon\theta_\varepsilon(e_{k+1})x)$. By Lemma 5.1(a), $g_k$ is $\Lambda$-valued, so $\tau_k$ is an f.s.n. trace on $N$; and $\tau_k\circ\theta_\delta=\tau_k$ for $\delta\in F_k$, by reindexing the sum. By Lemma 5.2(c) applied to $\tau_k$, $\sigma^{\tau_k\circ E}$ fixes $N$ and every $u_\delta$, $\delta\in F_k$, so $N_k\subset M_{\tau_k\circ E}$. Hence $T_k=(\tau_k\circ E)|_{N_k}$ is a faithful normal trace on $N_k$ (B4), semifinite because $\tau_k$ gives orthogonal projections of $N$ of finite trace with sum $1$ (argument of Lemma 3.2(a)). Since $E\circ E_k=E$, $T_k\circ E_k=\tau_k\circ E=\varphi'_{g_k},$ which Proposition 3.5(c) shows to be $\Lambda$-almost periodic, with $N_k$ in its centralizer.

Step 2: almost periodic states from $N_k$. Let $b=\kappa\sum_{n\in\mathbb Z}\lambda^np_n$ with $\kappa>0$, $\lambda\in\Lambda$, orthogonal projections $p_n\in N_k$ of sum $1$, and $T_k(b)=1$. Then $T_k(b\,\cdot)\circ E_k=(T_k\circ E_k)_b$ is a faithful normal state. It is $\Lambda$-almost periodic: $(T_k\circ E_k)_b=\kappa(T_k\circ E_k)_{b/\kappa}$, a positive multiple has the same modular operator, and Proposition 3.5(c) applies to $b/\kappa$ because the $p_n$ lie in $N_k$, inside the centralizer of $T_k\circ E_k$.

Step 3: such states are dense among the states $\chi\circ E_k$. Take a normal state $\chi$ on $N_k$. Replacing $\chi$ by $(1-\epsilon)\chi+\epsilon\chi_0$ with $\chi_0$ faithful, we may assume $\chi$ faithful, so $\chi=T_k(h\,\cdot)$ with $h\in L^1(N_k,T_k)$ positive nonsingular and $T_k(h)=1$ (B4). Let $\lambda\in\Lambda$, $\lambda>1$, write $p_n$ for the spectral projection of $h$ for $]\lambda^n,\lambda^{n+1}]$, and $b_0=\sum_n \lambda^np_n$. Then $\lambda^{-1}h\le b_0\le h$, so $\|\chi-T_k(b_0\,\cdot)\|=T_k(h-b_0)\le1-\lambda^{-1}$, and $b=b_0/T_k(b_0)$ satisfies $\|T_k(b\,\cdot)-T_k(b_0\,\cdot)\|=1-T_k(b_0)\le1-\lambda^{-1}$. Hence $\|\chi-T_k(b\,\cdot)\|\le2(1-\lambda^{-1})$, where $b$ is as in Step 2. As $\Lambda$ is dense, $\lambda$ can be taken arbitrarily close to $1$. Since $E_k$ is unital and completely positive, $\|\chi\circ E_k-T_k(b\,\cdot)\circ E_k\|\le\|\chi-T_k(b\,\cdot)\|$.

Step 4: states $\chi\circ E_k$ are dense. Fix a faithful normal state $\chi_0$ of $N$ and put $\varphi_0=\chi_0\circ E$; then $\varphi_0\circ E_k=\varphi_0$. Given a normal state $\psi$ on $M$, (B1) writes it as $\psi=\omega_\xi$ for a unit vector $\xi\in H_{\varphi_0}$. Since $\bigcup_kN_k$ is a $\sigma$-weakly dense $*$-algebra and $\eta_{\varphi_0}(1)$ is cyclic, there are $x\in\bigcup_kN_k$ with $x\eta_{\varphi_0}(1)$ as close to $\xi$ as we like, and $\|\omega_\xi-\omega_{\xi'}\|\le\|\xi-\xi'\|(\|\xi\|+\|\xi'\|)$. Normalizing, we can approximate $\psi$ in norm by states $y\mapsto\varphi_0(x^*yx)$ with $x\in N_k$ for some $k$. For such $x$, $\varphi_0(x^*yx)=\varphi_0(E_k(x^*yx))=\varphi_0(x^*E_k(y)x)=\chi(E_k(y))$, where $\chi=\varphi_0(x^*\,\cdot\,x)|_{N_k}$.

Steps 1–4 together prove the theorem. $\square$

5.4 Consequences for $Sd$

Corollary 5.6. Let $M$ be a factor. If $M$ has separable predual, or is not of type III₀, then $Sd(M)\subset S(M)$.

Proof. If $M$ is semifinite, $Sd(M)=\{1\}=S(M)$ (Theorem 4.3, (B5)).

Type III₀. By hypothesis $M$ has separable predual. Apply Theorem 5.5 with $\Lambda_1=\{\exp(q):q\in\mathbb Q\}$ and $\Lambda_2=\{\exp(q\sqrt2):q\in\mathbb Q\}$. There are almost periodic states $\varphi_1,\varphi_2$ with $\operatorname{Sp}_p(\varphi_j)\subset\Lambda_j$, so $Sd(M)\subset\Lambda_1\cap\Lambda_2=\{1\}\subset S(M)$.

In the remaining cases $M$ is of type III$_\lambda$ with $0<\lambda\le1$. Let $e$ be a nonzero countably decomposable projection (B6). By Lemma 4.2 and (B5), $Sd(M)\subset Sd(eMe)$ and $S(eMe)=S(M)$; so we may assume $M$ countably decomposable.

Type III$_\lambda$, $0<\lambda<1$. Let $\varphi$ be as in (B5), with $\sigma^\varphi_T=\operatorname{id}$, $T=2\pi/|\log\lambda|$. Then $\Delta_\varphi^{iT}\eta_\varphi(x)=\eta_\varphi(x)$, so $\Delta_\varphi^{iT}=1$: the spectral measure of $\log\Delta_\varphi$ lives on the discrete set $(2\pi/T)\mathbb Z=|\log\lambda|\mathbb Z$. Hence $\Delta_\varphi=\sum_n\lambda^nE_n$ is diagonal, and $Sd(M)\subset\operatorname{Sp}_p(\varphi)\subset\lambda^{\mathbb Z}\subset S(M)$.

Type III₁. $S(M)=[0,\infty[$. $\square$

Lemma 5.7. Let $R=\bigotimes_n(M_{k_n}(\mathbb C),\omega_n)$ be an Araki–Woods factor and $\mu\neq1$. There is a faithful normal almost periodic state $\omega'$ on $R$ with $\mu\notin\operatorname{Sp}_p(\omega')$.

Proof. Let $\omega_n=\operatorname{Tr}(\rho_n\,\cdot)$, and diagonalize $\rho_n$ with eigenvalues $p_{n,1},\dots,p_{n,k_n}$ in a basis that we keep fixed. $R$ acts on $\mathcal K=\bigotimes_n(\mathrm{HS}_{k_n}, \rho_n^{1/2})$, where $\mathrm{HS}_{k}$ is $M_k(\mathbb C)$ with the Hilbert–Schmidt inner product and $M_k$ acts by left multiplication. Choose real numbers $t_{n,j}$ ($n\ge1$, $2\le j\le k_n$) one after the other, each in a small interval around $\log(p_{n,j}/p_{n,1})$ and outside the countable $\mathbb Q$-linear span of $\log\mu$ and the previously chosen ones. Then $\{\log\mu\}\cup\{t_{n,j}\}$ is linearly independent over $\mathbb Q$ (recall $\log\mu\neq0$). Let $\rho'_n$ be the diagonal density with eigenvalue ratios $p'_{n,j}/p'_{n,1}=e^{t_{n,j}}$. If the intervals are small enough, $\|\rho_n'^{1/2}-\rho_n^{1/2}\|_{\mathrm{HS}}\le 2^{-n}$. By (B12), $\xi'=\bigotimes\rho_n'^{1/2}$ is a unit vector of $\mathcal K$, and $\mathcal K=\bigotimes_n (\mathrm{HS}_{k_n},\rho_n'^{1/2})$, so $R$ is also the infinite tensor product algebra built on these reference vectors, and $\xi'$ is cyclic and separating. The state $\omega'=\omega_{\xi'}$ is the product state $\bigotimes\operatorname{Tr}(\rho'_n\,\cdot)$. By Example 3.9 it is almost periodic and $\operatorname{Sp}_p(\omega')$ is contained in the group generated by the ratios $p'_{n,i}/p'_{n,j}=e^{t_{n,i}-t_{n,j}}$ ($t_{n,1}=0$), that is, in $\exp$ of the $\mathbb Z$-span of the $t_{n,j}$, which does not contain $\log\mu$. $\square$

Corollary 5.8. If $M$ is a Krieger factor, then $Sd(M)=\{1\}$.

Proof. A Krieger factor has separable predual. If it is semifinite, apply Theorem 4.3; if it is of type III₀, apply the proof of Corollary 5.6. Otherwise it is an Araki–Woods factor (B12), and by Lemma 5.7 every $\mu\neq1$ lies outside $Sd(M)$. $\square$

So $Sd$ carries no information on Krieger factors. It becomes useful on full factors, where it can be computed from a single weight.

6. On a full factor one weight computes $Sd$

A reference for this section is [Connes 1974].

The key point is that on a full factor the extended modular actions of two almost periodic weights are exterior equivalent. For $\mathbb R$ this is the cocycle theorem; the content is that the cocycle, corrected by a scalar character, extends to the compact group.

Theorem 6.1. Consider a full factor $M$ whose predual is separable, a countable subgroup $\Lambda$ of $\mathbb R_+^*$, $G=G_\Lambda$, $\iota=\iota_\Lambda$, and two $\Lambda$-almost periodic weights $\varphi_1,\varphi_2$ on $M$; put $u_t=(D\varphi_2:D\varphi_1)_t$ and $\alpha^j=\sigma^{\varphi_j,\Lambda}$.

(a) For some $\mu>0$, the map $t\mapsto\mu^{-it}u_t$ has a strongly continuous extension $v:G\to\mathcal U(M)$.
(b) $v$ is an $\alpha^1$-cocycle and $\alpha^2_s=\operatorname{Ad}v_s\circ\alpha^1_s$ for all $s\in G$.
(c) The balanced weight $\theta=\mu\varphi_1\oplus\varphi_2$ on $M\otimes M_2(\mathbb C)$ is $\Lambda$-almost periodic.
(d) A number $\mu'>0$ has the property (a) if and only if $\mu'/\mu\in\Lambda$.

Proof. Step 1: the automorphisms $\operatorname{Ad}u_t$ extend to $G$. By Theorem 3.3(3), $s\mapsto\alpha^j_s$ is continuous into $\operatorname{Aut}M$, a topological group; so $A_s=\alpha^2_s(\alpha^1_s)^{-1}$ is continuous in $s$, and $A_{\iota(t)}=\sigma^{\varphi_2}_t\sigma^{\varphi_1}_{-t}=\operatorname{Ad}u_t$ (B2). Since $\operatorname{Int}M$ is closed and $\iota(\mathbb R)$ is dense, $A_s=\operatorname{Ad}w_s$ with $w_s\in \mathcal U(M)$, for every $s\in G$; $w_s$ is unique up to a scalar.

Step 2: a one-parameter group. On $H=H_{\varphi_1}$, with $\Delta=\Delta_{\varphi_1}$, put $V_t=u_t\Delta^{it}$. From $\Delta^{it}x\Delta^{-it}=\sigma^{\varphi_1}_t(x)$ and the cocycle identity, $V_tV_{t'}=u_t\sigma^{\varphi_1}_t(u_{t'})\Delta^{i(t+t')}=V_{t+t'}$; so $V$ is a strongly continuous unitary group and $V_t=K^{it}$ for a positive nonsingular $K$ (Stone's theorem). Let $D_s$ be the representation of $G$ of Theorem 3.3(3) for $\varphi_1$.

Step 3: $\operatorname{Ad}V_t$ extends to $G$ on compact operators. We use: if $X_n\to X$ and $Y_n\to Y$ strongly with $\sup\|X_n\|,\sup\|Y_n\|<\infty$ and $T$ is compact, then $X_nTY_n^*\to XTY^*$ in norm. (Indeed $\|(X_n-X)T\|\to0$, as one sees for finite rank $T$ first, and $\|T(Y_n-Y)^*\|=\|(Y_n-Y)T^*\|\to0$.) Let $s\in G$ and $t_n\in\mathbb R$ with $\iota(t_n)\to s$ ($G$ is metrizable). Then $\operatorname{Ad}u_{t_n}=A_{\iota(t_n)} \to\operatorname{Ad}w_s$, so by Section 1 there are $\lambda_n\in\mathbb T$ with $\lambda_nu_{t_n}\to w_s$ strongly; and $\Delta^{it_n}=D_{\iota(t_n)}\to D_s$ strongly. Applying the fact twice, for every compact $T$, $V_{t_n}TV_{t_n}^*=(\lambda_nu_{t_n})\,\Delta^{it_n}T\Delta^{-it_n}\,(\lambda_nu_{t_n})^*\longrightarrow \beta_s(T):=w_sD_sTD_s^*w_s^*\quad\text{in norm.}\tag{6.1}$ Thus $\beta_s$ is a $*$-automorphism of the algebra $\mathcal K(H)$ of compact operators, determined by $s$, and (6.1) holds for every sequence with $\iota(t_n)\to s$.

Step 4: $\beta$ is a norm continuous action. If $\iota(t_n)\to s$ and $\iota(t'_n)\to s'$, then $V_{t_n+t'_n}TV_{t_n+t'_n}^*=V_{t_n}(V_{t'_n}TV_{t'_n}^*)V_{t_n}^*$; the inner term tends to $\beta_{s'}(T)$ in norm, so by (6.1) the whole tends both to $\beta_s\beta_{s'}(T)$ and to $\beta_{s+s'}(T)$. For continuity, let $s_k\to s$ and choose $t_k$ with $\iota(t_k)$ within $1/k$ of $s_k$ (for a fixed metric) and $\|V_{t_k}TV_{t_k}^*-\beta_{s_k}(T)\|<1/k$, which (6.1) allows. Then $\iota(t_k)\to s$, so $V_{t_k}TV_{t_k}^*\to \beta_s(T)$, and $\beta_{s_k}(T)\to\beta_s(T)$.

Step 5: $K$ is diagonal. Let $q$ be the projection onto the closed span of the eigenvectors of $K$; it commutes with $V_t$, so by (6.1) $\beta_s((1-q)T(1-q))=(1-q)\beta_s(T)(1-q)$. Suppose $q\neq1$ and take a compact $T\neq0$ with $T=(1-q)T(1-q)$. The norm-continuous function $s\mapsto\beta_s(T)$ has Fourier coefficients $T_\lambda=\int_G\overline{\langle s,\lambda\rangle}\beta_s(T)\,ds$ ($\lambda\in\Lambda$); they are compact, satisfy $T_\lambda=(1-q)T_\lambda(1-q)$ and $\beta_s(T_\lambda)=\langle s,\lambda\rangle T_\lambda$, and not all vanish (for a bounded functional $f$, the continuous function $f(\beta_s(T))$ has Fourier coefficients $f(T_\lambda)$, (B7)). Take $T_\lambda\neq0$ and $S=T_\lambda^*T_\lambda$. Then $\beta_s(S)=S$, so $S$ commutes with all $V_t=K^{it}$. An eigenspace of $S$ for a nonzero eigenvalue is finite dimensional, contained in $(1-q)H$, and invariant under $K^{it}$, so it contains an eigenvector of $K$. This contradicts the definition of $q$. Hence $q=1$.

Step 6: ratios of eigenvalues lie in $\Lambda$. Let $K\xi=\mu\xi$ and $K\xi'=\mu'\xi'$ with unit vectors, and $T=\langle\cdot,\xi\rangle\xi'$. Then $V_tTV_t^*=(\mu'/\mu)^{it}T$. By (6.1), $\beta_s(T)=\chi(s)T$ with $\chi$ continuous, and $\chi$ is a character by Step 4, with $\chi(\iota(t))=(\mu'/\mu)^{it}$. By Lemma 2.1, $\mu'/\mu\in\Lambda$.

Step 7: proof of (a). Fix an eigenvalue $\mu$ of $K$. By Steps 5 and 6, $\mu^{-1}K=\sum_{\lambda\in\Lambda} \lambda P_\lambda$, so by Lemma 2.2 the group $\mu^{-it}V_t$ extends to a strongly continuous representation $Y$ of $G$. Then $\mu^{-it}u_t=\mu^{-it}V_t\Delta^{-it}=Y_{\iota(t)}D_{\iota(t)}^*$ extends to the strongly continuous unitary map $v_s=Y_sD_s^*$. Each $v_s$ is a strong limit of elements $\mu^{-it_n}u_{t_n}$ of $M$, so $v_s\in\mathcal U(M)$.

Step 8: proof of (c) and (b). By (B2a), $(D\varphi_2:D(\mu\varphi_1))_t=\mu^{-it}u_t$. Let $Q=M\otimes M_2(\mathbb C)$. By (B2e), $\sigma^\theta$ acts on the corners $M\otimes e_{11}$ and $M\otimes e_{22}$ by $\sigma^{\varphi_1}$ and $\sigma^{\varphi_2}$, and $\sigma^\theta_t(1\otimes e_{21})=\mu^{-it}u_t\otimes e_{21}$. The three maps extend strong* continuously to $G$, by $\alpha^1$, $\alpha^2$ and $v$ (unitary valued strongly continuous maps are strong* continuous). The projections of Lemma 3.2(a) for $\varphi_1$ and $\varphi_2$, placed in the two corners, are orthogonal projections of $Q_\theta$ with finite weight and sum $1$, so $\theta$ is strictly semifinite. The set $(M\otimes e_{11})\cup(M\otimes e_{22})\cup\{1\otimes e_{21}\}$ generates $Q$. By Theorem 3.3(c), $\theta$ is $\Lambda$-almost periodic; this is (c). Let $\beta=\sigma^{\theta,\Lambda}$. It fixes $1\otimes e_{11}$ and $1\otimes e_{22}$, restricts to $\alpha^1$ and $\alpha^2$ on the corners (by uniqueness of extensions), and $\beta_s(1\otimes e_{21})=v_s\otimes e_{21}$ (both sides are continuous and agree on $\iota(\mathbb R)$). Then $\alpha^2_s(x)\otimes e_{22}=\beta_s\bigl((1\otimes e_{21})(x\otimes e_{11})(1\otimes e_{12})\bigr)=v_s\alpha^1_s(x)v_s^* \otimes e_{22},$ and $v_{s+s'}\otimes e_{21}=\beta_s(\beta_{s'}(1\otimes e_{21}))=\beta_s\bigl((1\otimes e_{21})(v_{s'}\otimes e_{11})\bigr)= v_s\alpha^1_s(v_{s'})\otimes e_{21}$. This is (b).

(d) If $\mu'$ also has property (a), the quotient $(\mu/\mu')^{it}=(\mu'^{-it}u_t)(\mu^{-it}u_t)^*$ extends continuously to $G$, so $\mu/\mu'\in\Lambda$ by Lemma 2.1. The converse is clear. $\square$

Remark 6.2. The cocycle $(D\varphi_2:D\varphi_1)$ itself need not extend. On the full factor $B(H)$ take $\varphi_1=\operatorname{Tr}$, $\varphi_2=2\operatorname{Tr}$ and $\Lambda=\{1\}$. Then $G_\Lambda$ is a point, and $u_t=2^{it}$ is not constant; the admissible $\mu$ of Theorem 6.1 is $\mu=2$.

Theorem 6.3. Consider a full factor $M$ whose predual is separable and any almost periodic weight $\varphi$ on $M$. Then $Sd(M)=\Gamma(\varphi)=\bigcap\{\operatorname{Sp}_p(\varphi_e):e\ \text{a nonzero projection of }M_\varphi\}.$

Proof. Take a second almost periodic weight $\psi$, and let $\Lambda$ be the group generated by $\operatorname{Sp}_p(\varphi)\cup\operatorname{Sp}_p(\psi)$, which is countable (Remark 4.4). By Theorem 6.1(b), $\sigma^{\psi,\Lambda}$ and $\sigma^{\varphi,\Lambda}$ are exterior equivalent, so $\Gamma(\psi)=\Gamma(\varphi)$ by Theorem 2.8 and Section 4. By (4.1), $Sd(M)=\Gamma(\varphi)$. $\square$

Example 6.4 (fullness cannot be dropped). Let $M=R_\lambda$ be the Powers factor, $0<\lambda<1$, a Krieger factor of type III$_\lambda$, and $\varphi$ the weight of (B5), with $\sigma^\varphi_T=\operatorname{id}$ and $M_\varphi$ a factor. By the proof of Corollary 5.6, $\varphi$ is almost periodic with $\operatorname{Sp}_p(\varphi)\subset\lambda^{\mathbb Z}$, and by (B5) $\operatorname{Sp}\Delta_\varphi=S(M)= \lambda^{\mathbb Z}\cup\{0\}$. The spectrum of a diagonal operator is the closure of its eigenvalues, and the points of $\lambda^{\mathbb Z}$ are isolated, so $\operatorname{Sp}_p(\varphi)=\lambda^{\mathbb Z}$. As $M_\varphi$ is a factor, $\Gamma(\varphi)=\lambda^{\mathbb Z}$, while $Sd(M)=\{1\}$ by Corollary 5.8. So the formula of Theorem 6.3 fails for $R_\lambda$, which is therefore not full.

7. Almost periodic weights on full factors

A reference for this section is [Connes 1974].

7.1 Characterizations of the good weights

Lemma 7.1. Suppose $M$ is a full factor whose predual is separable and $Sd(M)=\Lambda_0\neq\mathbb R_+^*$. For an almost periodic weight $\varphi$ on $M$, the following are equivalent.

(a) $\varphi$ is $\Lambda_0$-almost periodic.
(b) $\operatorname{Sp}_p(\varphi)=\Lambda_0$.
(c) $M_\varphi'\cap M=\mathbb C$.
(d) The centralizer $M_\varphi$ is a factor.
(e) If $\psi$ is a weight on $M$ with $M_\varphi\subset M_\psi$, then $\psi=c\varphi$ for some $c>0$.

Proof. (a)⇔(b): $Sd(M)\subset\operatorname{Sp}_p(\varphi)$ always. (b)⇒(d): by Theorem 6.3, $\Gamma(\varphi)=\Lambda_0=\operatorname{Sp}_p(\varphi)$, so $M_\varphi$ is a factor by Proposition 2.7 (Section 4). (d)⇒(b): $\operatorname{Sp}_p(\varphi)=\Gamma(\varphi)=Sd(M)$ by Lemma 2.5(d) and Theorem 6.3. (c)⇔(d): by Proposition 3.7, $M_\varphi'\cap M$ is the centre of $M_\varphi$.

(c)⇒(e). Let $u_t=(D\psi:D\varphi)_t$ and $w$ a unitary of $M_\varphi\subset M_\psi$. Then $\varphi\circ\operatorname{Ad}w=\varphi$ and $\psi\circ\operatorname{Ad}w=\psi$ (B1), so by (B2b) $u_t=w^*u_tw$. Hence $u_t\in M_\varphi'\cap M=\mathbb C$. The cocycle identity becomes $u_{s+t}=u_su_t$, so $u_t=c^{it}$ for some $c>0$, and $\psi=c\varphi$ by (B2a).

(e)⇒(d). Let $h$ be a self-adjoint element of the centre of $M_\varphi$ with $\frac12\le h\le1$. By (B2d), $\sigma^{\varphi_h}_t=\operatorname{Ad}h^{it}\circ\sigma^\varphi_t$ fixes $M_\varphi$ pointwise, so $M_\varphi\subset M_{\varphi_h}$ and $\varphi_h=c\varphi$. Then $h^{it}=(D\varphi_h:D\varphi)_t=c^{it}$, so $h=c$. Every self-adjoint central element is an affine function of such an $h$, so the centre is $\mathbb C$. $\square$

7.2 Crossed products and non-full algebras

Lemma 7.2. Consider a von Neumann algebra $M$, a countable subgroup $\Lambda$ of $\mathbb R_+^*$, and a weight $\varphi$ on $M$ that is $\Lambda$-almost periodic. On $\ell^2(\Lambda)$ let $B\delta_\gamma=\gamma\delta_\gamma$, $L_\gamma\delta_\mu=\delta_{\gamma\mu}$, and $\omega=\operatorname{Tr}(B\,\cdot)$. Put $P=M\bar\otimes B(\ell^2(\Lambda))$, $\psi=\varphi\otimes\omega$ and $U_\gamma=1\otimes L_\gamma$. Then:

(a) $\psi$ is $\Lambda$-almost periodic and $U_\gamma\in P^\psi(\gamma)$;
(b) $V_\gamma=\operatorname{Ad}U_\gamma|_{P_\psi}$ is an action of $\Lambda$ on $P_\psi$, and $P\cong P_\psi\rtimes_V\Lambda$;
(c) $\tau=\psi|_{P_\psi}$ is an f.s.n. trace with $\tau\circ V_\gamma=\gamma\tau$; in particular $V_\gamma$ is outer for $\gamma\neq1$;
(d) $P_\psi'\cap P\subset P_\psi$.

Proof. (a) $\psi=(\varphi\otimes\operatorname{Tr})_{1\otimes B}$, and $1\otimes B=\sum_\gamma\gamma(1\otimes E_\gamma)$, $E_\gamma$ the projection onto $\mathbb C\delta_\gamma$, with $1\otimes E_\gamma$ in the centralizer of $\varphi\otimes\operatorname{Tr}$ (whose modular group is $\sigma^\varphi\otimes\operatorname{id}$). By Proposition 3.5(b), (c), $\psi$ is $\Lambda$-almost periodic, and $\sigma^\psi_t=\operatorname{Ad}(1\otimes B^{it})\circ (\sigma^\varphi_t\otimes\operatorname{id})$. Since $B^{it}L_\gamma B^{-it}\delta_\mu=(\gamma\mu)^{it}\mu^{-it} \delta_{\gamma\mu}$, $\sigma^\psi_t(U_\gamma)=\gamma^{it}U_\gamma$.

(b) Lemma 2.9 applied to $\sigma^{\psi,\Lambda}$, whose fixed point algebra is $P_\psi$.

(c) $\tau$ is a faithful normal trace (B4), semifinite since $\psi$ is strictly semifinite. By (B2c), $(D(\psi\circ \operatorname{Ad}U_\gamma):D\psi)_t=U_\gamma^*\sigma^\psi_t(U_\gamma)=\gamma^{it}$, so $\psi\circ\operatorname{Ad} U_\gamma=\gamma\psi$ by (B2a), and $\tau\circ V_\gamma=\gamma\tau$. Inner automorphisms of $P_\psi$ preserve the trace $\tau$.

(d) Proposition 3.7. $\square$

Point realizations and hyperfinite orbit relations. Lemma 7.4 below turns the centre of a crossed product into a measure space on which the acting group moves points, and it needs the orbits to be exhausted by finite pieces. Theorem 7.3 provides both. Part (b) is a special case of the theorem of Connes, Feldman and Weiss that the orbit relation of a nonsingular action of a countable amenable group is hyperfinite off a null set; for countable abelian groups, Gao and Jackson proved that the orbit relation of every Borel action is hyperfinite, with no measure at all. Both theorems are stated in [Kechris 2025, Theorems 9.21 and 8.34]. Steps 4 and 5 of the proof below follow [Marks 2023, Lemma 3.1 and Theorem 3.2]; Steps 6 and 7 treat finitely generated abelian groups directly.

A standard probability space $(X,\mu)$ is a standard Borel space $X$ with a Borel probability measure $\mu$; $L^\infty(X,\mu)$ is formed with the completion of $\mu$. A Borel automorphism $T$ of $X$ is nonsingular if $\mu(T^{-1}A)=0$ exactly when $\mu(A)=0$. For an equivalence relation $R$ on $X$ and $A\subseteq X$, the saturation is $[A]_R=\{y:(x,y)\in R\text{ for some }x\in A\}$, and $A$ is $R$-invariant if $[A]_R=A$. A Borel equivalence relation is finite if all its classes are finite.

Theorem 7.3.

(a) Let $C\neq\{0\}$ be an abelian von Neumann algebra whose predual is separable. Then $C\cong L^\infty(X,\mu)$ for a standard probability space $(X,\mu)$, and $C$ is diffuse if and only if $\mu$ is nonatomic. Every $*$-automorphism $V$ of $L^\infty(X,\mu)$ is $V(f)=f\circ T^{-1}$ for a nonsingular Borel automorphism $T$ of $X$, unique up to equality almost everywhere. If $\gamma\mapsto V_\gamma$ is an action of a countable group $\Lambda$, the $T_\gamma$ can be chosen with $T_e=\operatorname{id}_X$ and $T_{\gamma\delta}=T_\gamma T_\delta$ everywhere on $X$.
(b) Let a countable abelian group $\Lambda$ act on a standard probability space $(X,\mu)$ by nonsingular Borel automorphisms $T_\gamma$, and let $E=\{(x,T_\gamma x):x\in X,\ \gamma\in\Lambda\}$ be the orbit relation. There are an $E$-invariant conull Borel set $X_0$ and finite Borel equivalence relations $R_1\subseteq R_2\subseteq\cdots$ on $X_0$ with $\bigcup_nR_n=E|_{X_0}$.
(c) If in (b) the measure $\mu$ is nonatomic, then every $R_n$-invariant Borel set $A\subseteq X_0$ with $\mu(A)>0$ is the union of two disjoint $R_n$-invariant Borel sets of positive measure.

The zero algebra is excluded in (a), because $L^\infty(X,\mu)\neq\{0\}$ for every probability measure. The group in (a) need not be abelian.

Proof. Step 1: a standard model. By Theorem 9.4(b) of the lesson on the predual of $B(H)$, $C=(C_*)^*$, and the weak* topology is the $\sigma$-weak topology. So the closed unit ball of $C$ is $\sigma$-weakly compact, and metrizable because $C_*$ is separable (Weak topologies, Theorem 3.1 and Proposition 3.2). It therefore has a countable $\sigma$-weakly dense subset, which generates $C$, since a von Neumann algebra is $\sigma$-weakly closed. Let $(\varphi_j)_{j\ge1}$ be norm dense in the set of normal states and $\varphi=\sum_j2^{-j}\varphi_j$. If $a\in C_+$ and $\varphi(a)=0$, then every $\varphi_j$, hence every normal state, vanishes at $a$; vector states included, so $a=0$. Thus $\varphi$ is a faithful normal state, and $C$ is $\sigma$-finite: the values of $\varphi$ on pairwise orthogonal nonzero projections are positive with sum at most $1$. By Commutative operator algebras, Theorem 8.2, $C$ is generated by one self-adjoint element $a$. Step 1 of the proof of Theorem 8.4 in that lesson uses only commutativity and $\sigma$-finiteness; it gives a $*$-isomorphism of $C$ onto a von Neumann algebra $C_e$ generated by the image $b$ of $a$, with a cyclic unit vector $\xi$. Let $K=\sigma(b)\subseteq\mathbb R$, and let $\mu$ be the Radon probability measure on $K$ with $\int g\,d\mu=\langle g(b)\xi,\xi\rangle$ for $g\in C(K)$. The vector $\xi$ is cyclic for the C*-algebra of the $g(b)$, which is strongly dense in $C_e$. By Theorem 3.1(2), (3) of the same lesson there is a unitary $U$ from $L^2(K,\mu)$ onto the Hilbert space of $C_e$ with $UM_gU^*=g(b)$, and $U$ carries the multiplication algebra $L^\infty(K,\mu)$, the von Neumann algebra generated by the $M_g$, onto $C_e$. The compact set $K$ is a standard Borel space. A projection of $L^\infty(K,\mu)$ is $1_A$ with $A$ Borel, and it is minimal exactly when $A$ is an atom of $\mu$; so $C$ is diffuse exactly when $\mu$ is nonatomic.

Step 2: one automorphism. Let $X$ be a standard Borel space with a Borel probability measure $\mu$, let $(A_j)_{j\ge1}$ be Borel sets that separate the points of $X$, and let $\iota(x)=(1_{A_j}(x))_j\in\mathcal C=\{0,1\}^{\mathbb N}$. The map $\iota$ is Borel and injective, so $\iota(X)$ is Borel and $\iota$ is a Borel isomorphism onto it (Polish spaces and standard Borel spaces, Theorem 4.3(5)). Let $V$ be a $*$-automorphism of $L^\infty(X,\mu)$; it preserves suprema of bounded increasing sequences (Commutative operator algebras, Proposition 7.3). Choose Borel sets $B_j$ with $V(1_{A_j})=1_{B_j}$, and put $p(x)=(1_{B_j}(x))_j$. Then $V(1_{\iota^{-1}(D)})=1_{p^{-1}(D)}\qquad\text{for every Borel set }D\subseteq\mathcal C.\tag{7.1}$ Indeed, (7.1) holds when $D$ prescribes finitely many coordinates, because $V$ preserves products and $e\mapsto 1-e$; and the sets $D$ satisfying (7.1) are closed under complements, finite intersections and increasing countable unions, so they form a $\sigma$-algebra. By (7.1) for $D=\mathcal C\setminus\iota(X)$, the Borel set $X'=p^{-1}(\iota(X))$ is conull. Fix $x_*\in X$, and let $S=\iota^{-1}\circ p$ on $X'$ and $S=x_*$ off $X'$. Then $S$ is Borel, and (7.1) for $D=\iota(A)$ gives $V(1_A)=1_{S^{-1}(A)}$ for Borel $A$. Since $V$ is injective, $\mu(S^{-1}(A))=0$ exactly when $\mu(A)=0$; so $f\mapsto f\circ S$ is well defined on $L^\infty(X,\mu)$, and uniform approximation by Borel simple functions gives $V(f)=f\circ S$. In the same way $V^{-1}(f)=f\circ Q$ for a Borel map $Q$ with the same property of null sets. Then $f=f\circ S\circ Q=f\circ Q\circ S$ almost everywhere for every $f$, and the functions $1_{A_j}$ show that $S\circ Q=Q\circ S=\operatorname{id}$ on a conull Borel set $D_0$. Let $Y$ be the intersection of the sets $w^{-1}(D_0)$ over the countably many finite composites $w$ of $S$ and $Q$, the identity included. It is a conull Borel set with $S(Y)\cup Q(Y)\subseteq Y$, and on $Y$ the maps $S$ and $Q$ are mutually inverse bijections. Put $T=Q$ on $Y$ and $T=\operatorname{id}$ off $Y$. Then $T$ is a Borel automorphism, its inverse is $S$ on $Y$ and $\operatorname{id}$ off $Y$, both are nonsingular, and $V(f)=f\circ T^{-1}$. If $T'$ also implements $V$, then $f\circ T=V^{-1}(f)=f\circ T'$ almost everywhere; with $f=1_{A_j}$ for all $j$, the points $T(x)$ and $T'(x)$ lie in the same sets $A_j$ for almost every $x$, so $T=T'$ almost everywhere.

Step 3: a whole group. For an action $\gamma\mapsto V_\gamma$ of a countable group $\Lambda$, Step 2 gives nonsingular Borel automorphisms $\tilde T_\gamma$ with $V_\gamma(f)=f\circ\tilde T_\gamma^{-1}$. Since $V_\gamma V_\delta(f)=f\circ(\tilde T_\gamma\tilde T_\delta)^{-1}$, uniqueness gives $\tilde T_{\gamma\delta}=\tilde T_\gamma\tilde T_\delta$ and $\tilde T_e=\operatorname{id}$ almost everywhere, all on one conull Borel set $D_1$. The group $H$ generated by the $\tilde T_\gamma$ is countable and consists of nonsingular Borel automorphisms, so $Y=\bigcap_{h\in H}h^{-1}(D_1)$ is an $H$-invariant conull Borel set on which these identities hold everywhere. Let $T_\gamma=\tilde T_\gamma$ on $Y$ and $T_\gamma=\operatorname{id}$ off $Y$. This proves (a).

Step 4: a criterion. Let a countable group $\Lambda$ act on $(X,\mu)$ by nonsingular Borel automorphisms $T_\gamma$, with orbit relation $E$. It is Borel, as the countable union of the graphs of the $T_\gamma$ (Polish spaces and standard Borel spaces, Proposition 4.1); the saturation $[A]_E=\bigcup_\gamma T_\gamma(A)$ of a Borel set is Borel, and null if $A$ is null. Suppose that for every finite $Q\subseteq\Lambda$ and $\varepsilon>0$ there is a finite Borel equivalence relation $F\subseteq E$ with $\mu\{x:(x,T_\gamma x)\notin F\text{ for some }\gamma\in Q\}<\varepsilon.\tag{7.2}$ Then the conclusion of (b) holds for $E$. Indeed, enumerate $\Lambda=\{\gamma_1,\gamma_2,\dots\}$, choose $F_k$ for $Q=\{\gamma_1,\dots,\gamma_k\}$ and $\varepsilon=2^{-k}$, and let $B_k$ be the set in (7.2) for $F_k$. The Borel set $B=\bigcap_m\bigcup_{k\ge m}B_k$ is null, since $\sum_k\mu(B_k)<\infty$. Put $X_0=X\setminus[B]_E$ and $R_n=\big(\bigcap_{k\ge n}F_k\big)|_{X_0}$. These are Borel equivalence relations, increasing in $n$, with finite classes because $R_n\subseteq F_n$. For $x\in X_0$ there is $k_0$ with $x\notin B_k$ for $k\ge k_0$; so $(x,T_{\gamma_i}x)\in F_k$ whenever $k\ge\max(i,k_0)$, that is, $(x,T_{\gamma_i}x)\in R_n$ for $n=\max(i,k_0)$. Hence $E|_{X_0}=\bigcup_nR_n$.

Step 5: increasing unions. In the situation of Step 4, let $E_1\subseteq E_2\subseteq\cdots$ be Borel equivalence relations with union $E$, each satisfying the conclusion of (b) on a conull Borel set $Y_k$, with relations $F_{k,1}\subseteq F_{k,2}\subseteq\cdots$. Then $E$ satisfies the criterion of Step 4. Indeed, $Y=X\setminus[\bigcup_k(X\setminus Y_k)]_E$ is an $E$-invariant conull Borel set contained in every $Y_k$. Given a finite $Q\subseteq\Lambda$ and $\varepsilon>0$, the Borel sets $\{x\in Y:(x,T_\gamma x)\in E_k\text{ for all }\gamma\in Q\}$ increase to $Y$ as $k\to\infty$; choose $k$ for which this set has measure $>1-\varepsilon/2$. The sets $\{x\in Y:(x,T_\gamma x)\in F_{k,m}\text{ for all }\gamma\in Q\}$ increase in $m$ to that set; choose $m$ for which the measure is $>1-\varepsilon$. Then $F_{k,m}|_Y$, extended by equality off $Y$, satisfies (7.2).

Step 6: separated sets. Let $G$ be an abelian group generated by $g_1,\dots,g_d$, $d\ge1$ (the trivial group with $g_1=e$), acting on $X$ by Borel automorphisms $T_g$. Put $q(v)=g_1^{v_1}\cdots g_d^{v_d}$ for $v\in\mathbb Z^d$, a surjective homomorphism, and on each orbit let $\rho(x,y)=\min\{\|v\|_\infty:T_{q(v)}x=y\}$, with $\rho=\infty$ between different orbits. This is a metric on each orbit; the ball $B_r(x)=\{y:\rho(x,y)\le r\}=\{T_{q(v)}x:\|v\|_\infty\le r\}$ has at most $(2r+1)^d$ points; and the relation $\rho\le r$ is a finite union of graphs of Borel maps, hence Borel. No freeness of the action is assumed. Fix an integer $r\ge1$. We construct a Borel set $S\subseteq X$ with $\rho(s,s')>r\quad(s\neq s'\text{ in }S),\qquad B_r(x)\cap S\neq\varnothing\quad(x\in X).\tag{7.3}$ Fix an injective Borel map $c:X\to\mathcal C$, as in Step 2, and let $m(x)$ be the least $m\ge1$ such that the first $m$ coordinates of $c(x)$ differ from those of $c(y)$ for every $y\in B_r(x)\setminus\{x\}$. The condition $m(x)\le m$ is a finite conjunction of Borel conditions, one for each $v$ with $\|v\|_\infty\le r$; so $\kappa(x)=(m(x),c(x)|_{m(x)})$ is a Borel map into a countable set, and $\kappa(x)\neq\kappa(y)$ whenever $0<\rho(x,y)\le r$. Enumerate the classes of $\kappa$ as $C_1,C_2,\dots$, and put $S_1=C_1,\qquad S_j=C_j\setminus\bigcup_{\|v\|_\infty\le r}T_{q(v)}\Big(\bigcup_{i<j}S_i\Big),\qquad S=\bigcup_jS_j.$ Points of one $S_j$ lie in one class of $\kappa$, so they are more than $r$ apart; the recursion keeps later points more than $r$ away from earlier ones; and a point of $C_j\setminus S_j$ lies within $r$ of $\bigcup_{i<j}S_i$. This proves (7.3). Moreover, $|S\cap B_{3r}(x)|\le7^d.\tag{7.4}$ Write each $s\in S\cap B_{3r}(x)$ as $T_{q(v_s)}x$ with $\|v_s\|_\infty\le3r$. Cutting $[-3r,3r]$ into the six intervals $[jr,(j+1)r[$, $-3\le j\le2$, and the point $\{3r\}$ cuts the cube into $7^d$ boxes of $\|\cdot\|_\infty$-diameter at most $r$. If $v_s$ and $v_{s'}$ lie in one box, then $s=T_{q(v_s-v_{s'})}s'$, so $\rho(s,s')\le r$ and $s=s'$ by (7.3).

Step 7: finitely generated abelian groups. Let $G$ be as in Step 6, acting by nonsingular Borel automorphisms; we verify the criterion of Step 4. Fix a Borel linear order $\prec$ on $X$, for instance from a Borel injection into $[0,1]$ (Polish spaces and standard Borel spaces, Theorem 5.2). For $t\in[r,2r]$ let $f_t(x)$ be the $\prec$-least element of $S\cap\{s:\rho(x,s)\le t\}$, a nonempty finite set by (7.3), and write $x\,F_t\,y$ when $f_t(x)=f_t(y)$. The map $(t,x)\mapsto f_t(x)$ is Borel: its value is one of the finitely many points $T_{q(v)}x$, $\|v\|_\infty\le2r$, selected by Borel conditions. So $F_t$ is a Borel equivalence relation contained in the orbit relation, and its classes are finite, since $f_t(x)=s$ forces $x\in B_{2r}(s)$. Let $\rho(x,y)\le a$ with $1\le a\le r$. If $f_t(x)\neq f_t(y)$, some $s\in S$ is within distance $t$ of exactly one of $x$ and $y$; then $|\rho(x,s)-t|\le a$ by the triangle inequality, and $\rho(x,s)\le t+a\le3r$. By (7.4) such $t$ lie in at most $7^d$ intervals of length $2a$, so $\frac1r\int_r^{2r}1_{\{(x,y)\notin F_t\}}\,dt\le\frac{2a\cdot7^d}{r}.\tag{7.5}$ Let $Q\subseteq G$ be finite and nonempty, write each $\gamma\in Q$ as $q(v_\gamma)$, and put $a=\max(1,\max_\gamma\|v_\gamma\|_\infty)$, so that $\rho(x,T_\gamma x)\le a$. Integrating (7.5) over $x$ (Tonelli) and summing over $\gamma\in Q$ gives $\frac1r\int_r^{2r}\mu\{x:(x,T_\gamma x)\notin F_t\text{ for some }\gamma\in Q\}\,dt\le\frac{2|Q|\,a\,7^d}{r},$ which is less than a given $\varepsilon>0$ when $r\ge a$ is large. Some $t$ then satisfies (7.2) with $F=F_t$. No invariance of $\mu$ is used: the average is over the radius $t$.

Step 8: proof of (b). Write $\Lambda=\{\gamma_1,\gamma_2,\dots\}$ and $\Lambda_k=\langle\gamma_1,\dots,\gamma_k\rangle$. By Steps 7 and 4, the orbit relation $E_k$ of $\Lambda_k$ satisfies the conclusion of (b) on a conull Borel set. The $E_k$ increase to $E$, so Steps 5 and 4 prove (b).

Step 9: proof of (c). Let $\mu$ be nonatomic, and let $F$ be a finite Borel equivalence relation contained in $E|_{X_0}$, for instance $R_n$. Fix a Borel injection $\beta:X_0\to[0,1]$. The set $D=\bigcap_{\gamma\in\Lambda}\{x\in X_0:(x,T_\gamma x)\notin F\text{ or }\beta(x)\le\beta(T_\gamma x)\}$ is Borel, and it meets each class of $F$ in exactly one point, the point where $\beta$ is least. For a Borel set $B\subseteq X_0$, $[B]_F=\bigcup_\gamma T_\gamma(B\cap\{x:(x,T_\gamma x)\in F\})$ is Borel, and null if $B$ is null. Let $A\subseteq X_0$ be an $F$-invariant Borel set with $\mu(A)>0$. Since $A=[A\cap D]_F$, the set $A\cap D$ has positive measure; split it into disjoint Borel sets $B$ and $B'$ of positive measure. Then $[B]_F$ and $[B']_F$ are disjoint, since a class meeting both would contain two points of $D$; they are $F$-invariant Borel sets of positive measure, and their union is $A$. $\square$

A factor $P$ has the Pukánszky property L if there are unitaries $u_n\in P$ with $u_n\to0$ weakly and $u_nxu_n^*-x\to0$ strongly for every $x\in P$.

Lemma 7.4. Suppose the von Neumann algebra $N\neq\{0\}$ has separable predual and diffuse centre $C$; let $\Lambda$ be a countable abelian group, $V$ an action of $\Lambda$ on $N$, and $P=N\rtimes_V\Lambda$, with canonical unitaries $U_\gamma$ and expectation $E$. There are self-adjoint unitaries $u_n\in C$ such that $u_n\to0$ weakly, $(u_n)$ is centralizing in $P$, and $u_nxu_n-x\to0$ strongly for all $x\in P$. Consequently, if $P$ is a factor, it is not full and has property L.

Proof. By Theorem 7.3(a), $C=L^\infty(X,\mu)$ with $\mu$ a nonatomic probability measure (as $C$ is nonzero and diffuse), and $V_\gamma(f)=f\circ T_\gamma^{-1}$ for a nonsingular Borel action $T$ of $\Lambda$ on $X$. By Theorem 7.3(b), after an invariant null set is discarded, the orbit relation is $\bigcup_nR_n$ with $R_1\subseteq R_2\subseteq\cdots$ Borel equivalence relations with finite classes.

Invariant sets are plentiful. Let $\mathfrak B_n$ be the $\sigma$-algebra of $R_n$-invariant Borel sets. By Theorem 7.3(c), $\mu$ is nonatomic on $\mathfrak B_n$. Consequently, for $g_0,\dots,g_n\in L^1(X,\mu)$ there is $F\in\mathfrak B_n$ with $\int_Fg_m\,d\mu=\frac12\int g_m\,d\mu$ for $0\le m\le n$. Indeed, let $L^1_n$ and $L^\infty_n$ be formed from $\mathfrak B_n$-measurable functions with the restriction of $\mu$ to $\mathfrak B_n$, so that $L^\infty_n=(L^1_n)^*$ (Measure and Hilbert space tools for Haar integration, Theorem 4.2). By the complex Radon–Nikodym theorem there are $h_m\in L^1_n$ with $\int_Fg_m\,d\mu=\int_Fh_m\,d\mu$ for $F\in\mathfrak B_n$. The set $K$ of $f\in L^\infty_n$ with $0\le f\le1$ and $\int fh_m\,d\mu=\frac12\int g_m\,d\mu$ for $0\le m\le n$ contains the constant $\frac12$; it is convex, and weak* compact as a weak*-closed subset of the unit ball (Weak topologies, Theorem 3.1). By the Krein–Milman theorem it has an extreme point $f$, which we take $\mathfrak B_n$-measurable with values in $[0,1]$. If $\mu\{0<f<1\}>0$, then $E'=\{\varepsilon\le f\le1-\varepsilon\}$ has positive measure for some $\varepsilon>0$; splitting repeatedly, $E'$ is the disjoint union of $2n+3$ sets $E_j\in\mathfrak B_n$ of positive measure. The $2n+2$ real linear conditions $\operatorname{Re}\int hh_m\,d\mu=\operatorname{Im}\int hh_m\,d\mu=0$ have a nonzero solution $h=\sum_jc_j1_{E_j}$ with real $|c_j|\le\varepsilon$; then $f\pm h\in K$, which contradicts extremality. So $f=1_F$ almost everywhere, with $F=\{f=1\}\in\mathfrak B_n$, and $\int_Fg_m\,d\mu=\int fh_m\,d\mu=\frac12\int g_m\,d\mu$.

Choice of $u_n$. Fix a faithful normal state $\chi_0$ of $N$, put $\psi_0=\chi_0\circ E$, let $g_0\in L^1(X,\mu)$ be the density of $\chi_0|_C$, and let $(g_m)_{m\ge1}$ be a dense sequence in $L^1(X,\mu)$. By the preceding paragraph there is $F_n\in\mathfrak B_n$ with $\int_{F_n}g_m\,d\mu=\frac12\int g_m\,d\mu$ for $0\le m\le n$. Put $u_n=2\cdot1_{F_n}-1$, a self-adjoint unitary of $C$, constant on the classes of $R_n$.

Properties. (i) $\int u_ng_m\,d\mu=0$ for $m\le n$; by density of $(g_m)$, $u_n\to0$ in the weak* topology of $L^\infty$, i.e. weakly in $P$; and $\psi_0(u_n)=\chi_0(u_n)=0$.

(ii) Let $D_{n,\gamma}=\{x:(T_\gamma^{-1}x,x)\in R_n\}$, which increases to $X$. On $D_{n,\gamma}$, $V_\gamma(u_n)=u_n$; so $|V_\gamma(u_n)-u_n|\le2\cdot1_{X\setminus D_{n,\gamma}}\to0$ strongly. For $x\in N$, $[u_n,xU_\gamma]=x(u_n-V_\gamma(u_n))U_\gamma\to0$ strongly, and since $(xU_\gamma)^*=V_{\gamma^{-1}}(x^*)U_{\gamma^{-1}}$ is of the same form and $u_n=u_n^*$, also strong*. Hence $[u_n,y]\to0$ strong* for $y$ in the $\sigma$-weakly dense $*$-algebra $\mathcal A$ spanned by the $xU_\gamma$.

(iii) $u_n$ lies in the centre of $N$, which the modular group of $\chi_0$ fixes pointwise; by (B3), $u_n\in P_{\psi_0}$. Write $\|z\|_{\psi_0}=\psi_0(z^*z)^{1/2}$. Then $\|zu_n\|_{\psi_0}=\|z\|_{\psi_0}$ (B1).

(iv) Centralizing. For $y\in\mathcal A$ and $\omega=y\psi_0$ (that is, $\omega(z)=\psi_0(zy)$): since $\psi_0(u_nzy)=\psi_0(zyu_n)$, $[u_n,\omega](z)=\psi_0(z[u_n,y])$, so $\|[u_n,\omega]\|\le\|[u_n,y]\|_{\psi_0}\to0$. The functionals $y\psi_0$, $y\in P$, are norm dense in $P_*$ (if $x\in P$ annihilates them, $\psi_0(xx^*)=0$), and $\|y\psi_0-y'\psi_0\|\le\|y-y'\|_{\psi_0}$, which can be made small with $y'\in\mathcal A$. As $\|u_n\|=1$, $(u_n)$ is centralizing.

(v) Asymptotic commutation. For $y\in P$ and $y'\in\mathcal A$, by (iii), $\|[u_n,y]\|_{\psi_0}\le2\|y-y'\|_{\psi_0}+\|[u_n,y']\|_{\psi_0}$, and the same for adjoints. So $[u_n,y]\to0$ strong*, and $u_nyu_n-y=[u_n,y]u_n$ has $\|u_nyu_n-y\|_{\psi_0}=\|[u_n,y]\|_{\psi_0}\to0$: $u_nyu_n-y\to0$ strongly.

Finally, if $P$ is a factor, $(u_n)$ is not trivial: if $u_n-\lambda_n\to0$ strong*, then $|\lambda_n|=|\psi_0(\lambda_n-u_n)|\to0$, so $u_n\to0$ strongly, which is impossible for unitaries. By (F3), $P$ is not full. $\square$

7.3 Existence and uniqueness

Theorem 7.5. Consider a full factor $M$ whose predual is separable, with $Sd(M)=\Lambda_0\neq\mathbb R_+^*$.

(1) Some almost periodic weight $\varphi$ on $M$ has $\operatorname{Sp}_p(\varphi)=Sd(M)$; equivalently (Lemma 7.1), $M_\varphi$ is a factor.
(2) If $\varphi_1,\varphi_2$ are $\Lambda_0$-almost periodic weights with $\varphi_1(1)=\varphi_2(1)=+\infty$, there are $u\in\mathcal U(M)$ and $c>0$ with $\varphi_2(x)=c\,\varphi_1(u^*xu)$ for all $x\in M^+$.

Proof of (1). If $M$ is semifinite, a trace has $\operatorname{Sp}_p=\{1\}=Sd(M)$. Let $M$ be of type III and $\varphi$ any almost periodic weight (one exists since $Sd(M)\ne\mathbb R_+^*$).

The centre of $M_\varphi$ has an atom. Suppose not. Let $\Lambda$ be the countable group generated by $\operatorname{Sp}_p(\varphi)$, and $P,\psi,U,V$ as in Lemma 7.2. The projections $f_a=1\otimes E_a$ ($a\in\Lambda$) lie in $P_\psi$ and have sum $1$; and $f_aP_\psi f_a=M_\varphi\otimes E_a$, since $x\otimes e_{aa}$ is fixed by $\sigma^\psi$ exactly when $x\in M_\varphi$. The centre of $f_aP_\psi f_a$ is (centre of $P_\psi$)$f_a$ (B6), and it is isomorphic to the centre of $M_\varphi$, which is diffuse. If $z$ were an atom of the centre of $P_\psi$, some $zf_a$ would be nonzero, hence an atom of (centre of $P_\psi$)$f_a$, which is impossible. So the centre of $P_\psi$ is diffuse. $P$ is a factor with separable predual, and $P\cong P_\psi\rtimes_V\Lambda$ (Lemma 7.2(b)), so by Lemma 7.4 $P$ is not full. But $M$ is full, hence so is $P=M\bar\otimes B(\ell^2(\Lambda))$ (Section 1). This contradiction shows that the centre of $M_\varphi$ has an atom $e$.

Then $M_{\varphi_e}=eM_\varphi e$ (B3) has centre $\mathbb Ce$: it is a factor. Since $M$ is countably decomposable of type III, there is $v\in M$ with $v^*v=1$, $vv^*=e$ (B6), and the formula $\varphi'(x)=\varphi_e(vxv^*)$ defines an almost periodic weight on $M$ (Proposition 3.5(a), (d)) whose centralizer $v^*(eM_\varphi e)v$ is a factor. By Lemma 7.1, $\operatorname{Sp}_p(\varphi')=Sd(M)$.

Proof of (2). If $M$ is semifinite, $\Lambda_0=\{1\}$ (Theorem 4.3), so $\sigma^{\varphi_j}$ is trivial and $\varphi_1,\varphi_2$ are traces (B4); they are proportional, and $u=1$ works. Let $M$ be of type III. By Theorem 6.1 with $\Lambda=\Lambda_0$, there is $\mu>0$ such that $\theta=\mu\varphi_1\oplus\varphi_2$ is $\Lambda_0$-almost periodic on $Q=M\otimes M_2(\mathbb C)$. As $Q\cong M$ (B6), $Q$ is a full factor with $Sd(Q)=\Lambda_0$, and $Q_\theta$ is a factor by Lemma 7.1. The restriction of $\theta$ to $Q_\theta$ is an f.s.n. trace, and $f_1=1\otimes e_{11}$, $f_2=1\otimes e_{22}$ are projections of $Q_\theta$ with $\theta(f_1)=\mu\varphi_1(1)=\infty=\theta(f_2)$. Finite projections of a semifinite factor have finite trace (B4), so $f_1,f_2$ are infinite, hence equivalent in $Q_\theta$, which has separable predual (B6). A partial isometry $w\in Q_\theta$ with $w^*w=f_1$, $ww^*=f_2$ has the form $w=u\otimes e_{21}$ with $u\in\mathcal U(M)$. The unitary $W=w+w^*$ lies in $Q_\theta$, so $\theta\circ\operatorname{Ad}W=\theta$ (B1), and for $x\in M^+$ $\varphi_2(uxu^*)=\theta\bigl(W(x\otimes e_{11})W^*\bigr)=\theta(x\otimes e_{11})=\mu\varphi_1(x).$ So $\varphi_2(y)=\mu\varphi_1(u^*yu)$. $\square$

7.4 Consequences

Corollary 7.6. For a full factor $M$ whose predual is separable, $\overline{Sd(M)}=S(M)$, the closure being taken in $[0,\infty[$.

Proof. If $Sd(M)=\mathbb R_+^*$, then $S(M)\supset\mathbb R_+^*$ by Corollary 5.6, and $S(M)=[0,\infty[$ since it is closed. Otherwise take $\varphi$ as in Theorem 7.5(1). Since $M_\varphi$ is a factor, $S(M)= \operatorname{Sp}\Delta_\varphi$ (B5), and the spectrum of the diagonal operator $\Delta_\varphi$ is the closure of $\operatorname{Sp}_p(\varphi)=Sd(M)$. $\square$

In particular, a full type III factor with separable predual has $Sd(M)\neq\{1\}$, and a full factor whose predual is separable and whose $Sd$ is $\{1\}$ is semifinite.

Corollary 7.7. Consider a full factor $M$ whose predual is separable, with $Sd(M)=\Lambda_0\neq\mathbb R_+^*$, and assume $M$ is not finite.

(1) $M\cong N\rtimes_V\Lambda_0$, where $N$ is an infinite semifinite factor with an f.s.n. trace $\tau$ and $\tau\circ V_\gamma=\gamma\tau$ for $\gamma\in\Lambda_0$. If $\Lambda_0\neq\{1\}$, $N$ is of type II∞. If $\Lambda_0=\{1\}$, $M$ is semifinite and $N=M$, which can be of type I∞.
(2) In such a decomposition the isomorphism class of $N$ is determined by $M$, and so are the classes of the $V_\gamma$ in $\operatorname{Out}N$, in the following sense: for two decompositions $(N_1,V^1)$ and $(N_2,V^2)$ there is an isomorphism $\kappa:N_1\to N_2$ with $\kappa V^1_\gamma\kappa^{-1}V^{2}_{\gamma^{-1}}\in \operatorname{Int}N_2$ for every $\gamma\in\Lambda_0$.

Reference: [Connes 1974, Corollary 4.12] asserts that $N$ is always of type II∞; this fails for $M=B(H)$, which is full with $Sd=\{1\}$, so we allow type I∞ when $\Lambda_0=\{1\}$.

Proof. (1) If $\Lambda_0=\{1\}$, $M$ is semifinite (Corollary 7.6) and infinite, and $M=M\rtimes\{1\}$. Let $\Lambda_0\neq\{1\}$; then $\Lambda_0$ is infinite and $M$ is of type III. Take $\varphi$ as in Theorem 7.5(1) and $P,\psi$ as in Lemma 7.2 with $\Lambda=\Lambda_0$. Since $M$ is properly infinite and countably decomposable, $P\cong M$ (B6), so $P$ is full with $Sd(P)=\Lambda_0$, and $\psi$ is $\Lambda_0$-almost periodic; by Lemma 7.1, $N=P_\psi$ is a factor. By Lemma 7.2, $P\cong N\rtimes_V\Lambda_0$ with $\tau\circ V_\gamma=\gamma\tau$. $N$ is semifinite with $\tau(1)=\psi(1)\ge\varphi(e)\,\omega(1)=\infty$ for any projection $e\in M_\varphi$ with $0<\varphi(e)<\infty$, since $\omega(1)=\sum_\gamma\gamma=\infty$; so $N$ is infinite (B4). It is not of type I, since automorphisms of a type I factor are inner and preserve its trace, while $V_\gamma$ scales $\tau$ by $\gamma\neq1$.

(2) Realize both decompositions inside $M$: $N_j\subset M$, unitaries $U^j_\gamma$, expectations $E_j$. By Example 3.10, $\varphi_j=\tau_j\circ E_j$ is $\Lambda_0$-almost periodic, $M_{\varphi_j}=N_j$ and $M^{\varphi_j}(\gamma)=N_jU^j_\gamma$; and $\varphi_j(1)=\tau_j(1)=\infty$ because $N_j$ is infinite. By Theorem 7.5(2), $\varphi_2=c\,\varphi_1\circ\operatorname{Ad}u^*$ for a unitary $u$, so $\sigma^{\varphi_2}_t= \operatorname{Ad}u\circ\sigma^{\varphi_1}_t\circ\operatorname{Ad}u^*$ (B2b), and $\operatorname{Ad}u$ maps $M^{\varphi_1}(\gamma)$ onto $M^{\varphi_2}(\gamma)$. So $\kappa=\operatorname{Ad}u|_{N_1}$ is an isomorphism of $N_1$ onto $N_2$, and $uU^1_\gamma u^*=w_\gamma U^2_\gamma$ with $w_\gamma\in N_2$, necessarily unitary. For $n\in N_2$, $\kappa V^1_\gamma\kappa^{-1}(n)=(uU^1_\gamma u^*)n(uU^1_\gamma u^*)^*=w_\gamma V^2_\gamma(n)w_\gamma^*$. $\square$

Corollary 7.8. Let $M$ be a full type III₁ factor with separable predual that has an almost periodic weight. Then the modular homomorphism $\delta_M:\mathbb R\to\operatorname{Out}M$ is injective, and the centre of $\operatorname{Out}M$ is strictly larger than $\delta_M(\mathbb R)$.

Proof. $\Lambda_0=Sd(M)$ is countable (Remark 4.4) and dense in $\mathbb R_+^*$, since its closure is $S(M)=[0,\infty[$ (Corollary 7.6). Let $\varphi$ be as in Theorem 7.5(1); then $M_\varphi'\cap M=\mathbb C$ (Lemma 7.1). As $\operatorname{Int}M$ is closed, $\operatorname{Out}M$ is a Polish group (B13), in particular Hausdorff.

Injectivity. If $\sigma^\varphi_t=\operatorname{Ad}w$, then $w$ commutes with $M_\varphi$, so $w\in\mathbb C$ and $\sigma^\varphi_t=\operatorname{id}$. Each $\lambda\in\Lambda_0=\operatorname{Sp}_p(\varphi)$ has a nonzero eigenoperator (3.1), so $\lambda^{it}=1$ for all $\lambda\in\Lambda_0$; since $\log\Lambda_0$ is dense in $\mathbb R$, $t=0$.

The compact group $\bar\delta(G)$. Let $G=G_{\Lambda_0}$, compact metrizable, and $\bar\delta(s)$ the class of $\sigma^{\varphi,\Lambda_0}_s$ in $\operatorname{Out}M$. By Theorem 3.3(3), $\bar\delta$ is a continuous homomorphism, with $\bar\delta\circ\iota=\delta_M$. Its image is a compact subgroup, the closure of $\delta_M(\mathbb R)$. By (B2), $\delta_M(\mathbb R)$ lies in the centre of $\operatorname{Out}M$, which is closed; so $\bar\delta(G)$ lies in the centre.

Conclusion. Suppose $\bar\delta(G)\subset\delta_M(\mathbb R)$. Then $\delta_M$ is a homomorphism from $\mathbb R$ onto the compact metrizable group $\bar\delta(G)$ that is continuous and bijective. Both are Polish groups, so by the open mapping theorem (Section 1) $\delta_M$ is a homeomorphism, and $\mathbb R$ would be compact. Hence some central element $\bar\delta(s)$ is not in $\delta_M(\mathbb R)$. $\square$

8. Every countable subgroup of $\mathbb R_+^*$ is $Sd$ of a full factor

A reference for this section is [Connes 1974].

For $\lambda\in\,]0,1]$ let $\rho_\lambda=(1+\lambda)^{-1}\operatorname{diag}(1,\lambda)$, $\omega_\lambda= \operatorname{Tr}(\rho_\lambda\,\cdot)$ on $M_2(\mathbb C)$, and $\xi(\lambda)=\rho_\lambda^{1/2}$ in the Hilbert–Schmidt space $\mathrm{HS}_2\cong\mathbb C^4$. For a countable family $(\lambda_i)_{i\in I}$ in $]0,1]$, the infinite tensor product $P=\bigotimes_{i\in I}(M_2(\mathbb C),\omega_{\lambda_i})$ acts on $\bigotimes_i (\mathrm{HS}_2,\xi(\lambda_i))$ with the cyclic and separating product vector $\xi_P$ (B12). By Example 3.9, the state $\varphi_P=\omega_{\xi_P}$ is almost periodic and $\operatorname{Sp}_p(\varphi_P)$ is the set of products $\prod_i\lambda_i^{n_i}$ with $n_i\in\{-1,0,1\}$, almost all zero.

Corollary 8.1. For every countable subgroup $\Lambda$ of $\mathbb R_+^*$ there is a full factor $M$ whose predual is separable and whose $Sd(M)$ equals $\Lambda$. It is of type III₁ if $\Lambda$ is dense, of type III$_\lambda$ if $\Lambda=\lambda^{\mathbb Z}$ ($0<\lambda<1$), and semifinite if $\Lambda=\{1\}$.

Proof. Let $(\lambda_i)_{i\in I}$ be a family in $\Lambda\cap\,]0,1]$ in which every element of $\Lambda\cap\,]0,1]$ occurs (if $\Lambda=\{1\}$, all $\lambda_i=1$), and $P,\xi_P$ as above. Every $\lambda\in\Lambda$ is $\lambda_i$ or $\lambda_i^{-1}$ for some $i$, and products of elements of $\Lambda$ lie in $\Lambda$; so $\operatorname{Sp}_p(\varphi_P)=\Lambda$. Let $M=M(P,\xi_P)$ be the free Bernoulli construction of Section 1, with state $\psi$. By (R4), $M$ is a full factor and its predual is separable. By (R3), $\Delta_\psi=\Delta_{\varphi_N}\otimes1$ and $\Delta_{\varphi_N}^{it}=\bigotimes_{s\in\mathbb F_2}\Delta_{\varphi_P}^{it}$; as in Example 3.9, the product vectors of eigenvectors of $\Delta_{\varphi_P}$, equal to $\xi_P$ at all but finitely many places, form an orthonormal basis of eigenvectors of $\Delta_{\varphi_N}$, with eigenvalues the finite products of elements of $\Lambda$. So $\psi$ is almost periodic with $\operatorname{Sp}_p(\psi)=\Lambda$. By (R2), $M_\psi$ is a factor, so $\Gamma(\psi)=\operatorname{Sp}_p(\psi)=\Lambda$ (Section 4), and $Sd(M)=\Lambda$ by Theorem 6.3. The type follows from Corollary 7.6 and (B5). $\square$

Lemma 8.2. Let $A$ be the countable group of affine maps $t\mapsto at+b$ of $\mathbb R$ with $a\in\mathbb Q\setminus\{0\}$, $b\in\mathbb Q$.

(a) $s$ and $t$ are in the same $A$-orbit if and only if $\mathbb Qs+\mathbb Q=\mathbb Qt+\mathbb Q$.
(b) Every $A$-invariant Borel set is Lebesgue null or conull.
(c) There is no sequence of $A$-invariant Borel sets separating the $A$-orbits.

Proof. (a) If $s=at+b$, then $\mathbb Qs+\mathbb Q=\mathbb Qt+\mathbb Q$. Conversely, if the two sets agree and $t\in\mathbb Q$, then $s\in\mathbb Q$ and $s=t+(s-t)$; if $t\notin\mathbb Q$, then $s=at+b$ with $a,b\in\mathbb Q$ and $a\neq0$, since $s\notin\mathbb Q$.

(b) Let $B$ be invariant and $f=1_B$, which is $1$-periodic. Its Fourier coefficients $c_n=\int_0^1f(x)e^{-2\pi inx}dx$ satisfy $c_n=c_ne^{2\pi inq}$ for all $q\in\mathbb Q$, by invariance under $x\mapsto x+q$. So $c_n=0$ for $n\neq0$, $f$ is a.e. constant on $[0,1[$, and by periodicity $B$ is null or conull.

(c) Let $B_n$ be invariant Borel sets and $B'_n=B_n$ if $B_n$ is conull, $B'_n=\mathbb R\setminus B_n$ otherwise. Then $\bigcap_nB'_n$ is conull, hence uncountable, while orbits are countable; so it contains two points of different orbits, and no $B_n$ separates them. $\square$

Let $H_0$ be a separable infinite dimensional Hilbert space, $\mathfrak F$ the set of factors of type III₁ on $H_0$ with the relative Effros Borel structure (B11), and $\mathfrak F/{\cong}$ the quotient by isomorphism with the quotient Borel structure: a subset of $\mathfrak F/{\cong}$ is Borel when its preimage in $\mathfrak F$ is Borel.

Corollary 8.3. The quotient Borel space $\mathfrak F/{\cong}$ of type III₁ factors on a separable Hilbert space, up to isomorphism, admits no countable separating family of Borel sets.

Proof. The family. For $t\in\mathbb R$ and $i=(a,b)\in I=\mathbb Q^2$ put $\lambda_i(t)=e^{-|a+bt|}\in\,]0,1]$, and let $P_t=\bigotimes_{i\in I}(M_2(\mathbb C),\omega_{\lambda_i(t)})$ with its product vector, and $M_t=M(P_t,\xi_{P_t})$. The numbers $\lambda_i(t)^{\pm1}$ are exactly the numbers $e^{a+bt}$, $a,b\in\mathbb Q$, which form the dense countable group $\Lambda_t=\exp(\mathbb Q+\mathbb Qt)$. As in the proof of Corollary 8.1, $M_t$ is a full factor of type III₁ with $Sd(M_t)=\Lambda_t$.

Isomorphisms. If $s=\alpha t+\beta$ with $\alpha\in\mathbb Q\setminus\{0\}$, $\beta\in\mathbb Q$, the bijection $\pi(a,b)=(a+b\beta,b\alpha)$ of $I$ satisfies $\lambda_i(s)=\lambda_{\pi(i)}(t)$. Permuting the tensor factors (B12) gives a unitary carrying $P_s$ onto $P_t$ and the product vector to the product vector. It induces a unitary between the spaces of the free Bernoulli constructions, commuting with the shifts, which carries $M_s$ onto $M_t$. If $s,t$ are not in the same $A$-orbit, then $\Lambda_s\neq\Lambda_t$ by Lemma 8.2(a) (take logarithms), so $M_s\not\cong M_t$ since $Sd$ differs.

A Borel realization. Let $\varepsilon_{jk}$ denote the matrix units of $M_2(\mathbb C)$, viewed as an orthonormal basis of $\mathrm{HS}_2$, and $W(\lambda)$ the unitary of $\mathrm{HS}_2$ that rotates the plane spanned by $\varepsilon_{11},\varepsilon_{22}$ so that $W(\lambda)\varepsilon_{11}=\xi(\lambda)$ and fixes $\varepsilon_{12}, \varepsilon_{21}$; it depends continuously on $\lambda$. Let $\mathcal K_0=\bigotimes_{i\in I}(\mathrm{HS}_2, \varepsilon_{11})$. By (B12), $\bigotimes_iW(\lambda_i(t))$ is a unitary from $\mathcal K_0$ onto the space of $P_t$ carrying $\bigotimes\varepsilon_{11}$ to $\xi_{P_t}$. Hence $(P_t,\xi_{P_t})$ is spatially isomorphic to the algebra $\tilde P_t$ on $\mathcal K_0$ generated by the operators $A_i(t,x)$, acting in the tensor factor $i$ as $W(\lambda_i(t))^*L_xW(\lambda_i(t))$ ($L_x$ left multiplication by $x\in M_2(\mathbb C)$), with the vector $\bigotimes\varepsilon_{11}$. Running the free Bernoulli construction with $(\tilde P_t,\bigotimes\varepsilon_{11})$, the algebra $M_t$ is realized on $H_0=\Bigl(\bigotimes_{(s,i)\in\mathbb F_2\times I}(\mathrm{HS}_2,\varepsilon_{11})\Bigr)\otimes\ell^2(\mathbb F_2),$ which does not depend on $t$. It is generated by the countably many operators $A_{s,i}(t,\varepsilon_{jk})\otimes1$ (the operator $A_i(t,\varepsilon_{jk})$ placed in the tensor factor $(s,i)$), which are norm continuous in $t$, and the unitaries $U_g=V_g\otimes\lambda_g$, $g\in\mathbb F_2$, which do not depend on $t$. By (B11), $t\mapsto M_t$ is a Borel map $\mathbb R\to\mathfrak F$.

Conclusion. Suppose Borel sets $\mathcal S_n\subset\mathfrak F$, saturated for $\cong$, separate the isomorphism classes. The Borel sets $B_n=\{t:M_t\in\mathcal S_n\}$ are $A$-invariant by the isomorphisms above, and they separate $A$-orbits because factors from different orbits are not isomorphic. This contradicts Lemma 8.2(c). $\square$

Corollary 8.4. For some factors $M$ of type III₁ with separable predual, the centre of $\operatorname{Out}M$ is strictly larger than $\delta_M(\mathbb R)$.

Proof. Take $M$ as in Corollary 8.1 with $\Lambda$ dense, for instance $\Lambda=\exp\mathbb Q$, and apply Corollary 7.8. $\square$

9. Exercises

Exercise 1. Let $K=\sum_ak_aq_a$ be a diagonal positive nonsingular operator on $H$ and $\varphi=\operatorname{Tr}(K\,\cdot)$ on $B(H)$. Show directly from the definition that $\Gamma(\varphi)=\{1\}$, even when $\operatorname{Sp}_p(\varphi)$ is large.

Solution. Let $\xi$ be a unit eigenvector of $K$ and $e$ the projection onto $\mathbb C\xi$. Then $e$ commutes with $K$, so $\sigma^\varphi_t(e)=K^{it}eK^{-it}=e$ and $e\in M_\varphi$. Since $eB(H)e=\mathbb Ce$, the weight $\varphi_e$ is a multiple of the unique state on $\mathbb Ce$, whose modular operator is $1$; so $\operatorname{Sp}_p(\varphi_e)=\{1\}$ and $\Gamma(\varphi)=\{1\}$, while $\operatorname{Sp}_p(\varphi)= \{k_a/k_b\}$ (Example 3.8). This agrees with Theorem 4.3, which gives $Sd(B(H))=\{1\}$.

Exercise 2. Let $\Lambda$ be a subgroup of $\mathbb R_+^*$ and $c>0$. Fix a $\Lambda$-almost periodic weight $\varphi$, and put $\psi=c\varphi$. Show that $t\mapsto(D\psi:D\varphi)_t$ extends to $G_\Lambda$ if and only if $c\in\Lambda$. Which $\mu$ are admissible in Theorem 6.1 for $\varphi_1=\varphi$, $\varphi_2=c\varphi$?

Solution. $(D(c\varphi):D\varphi)_t=c^{it}$ by (B2a), and $t\mapsto c^{it}$ extends continuously to $G_\Lambda$ exactly when $c\in\Lambda$ (Lemma 2.1). In Theorem 6.1, $\mu^{-it}c^{it}=(c/\mu)^{it}$ must extend, so the admissible $\mu$ are those with $c/\mu\in\Lambda$, that is $\mu\in c\Lambda$, in agreement with Theorem 6.1(d).

Exercise 3. Let $M$ be a factor and $\varphi$ an almost periodic weight whose centralizer $M_\varphi$ is a factor. Show that $\operatorname{Sp}_p(\varphi)$ is a subgroup of $\mathbb R_+^*$. Show that the hypothesis on $M_\varphi$ cannot be dropped.

Solution. By Lemma 2.5(d) (Section 4), $\operatorname{Sp}_p(\varphi)=\Gamma(\varphi)$, a group by Proposition 2.6. For $M=M_2(\mathbb C)$ and $\varphi=\operatorname{Tr}(K\,\cdot)$ with $K=\operatorname{diag}(1,2)$, $\operatorname{Sp}_p(\varphi)=\{\frac12,1,2\}$ is not a group (Example 3.8); here $M_\varphi$ is the diagonal algebra, not a factor.

Exercise 4. Let $N$ be a semifinite factor with separable predual and an f.s.n. trace $\tau$, $\Lambda$ a countable subgroup of $\mathbb R_+^*$ and $V$ an action of $\Lambda$ on $N$ with $\tau\circ V_\lambda=\lambda\tau$. Let $P=N\rtimes_V \Lambda$ and $\varphi=\tau\circ E$. Show that $\Gamma(\varphi)=\Lambda$ and that $P_\varphi'\cap P=\mathbb C$. If moreover $P$ is full, show that $Sd(P)=\Lambda$ and $S(P)=\overline\Lambda$.

Solution. By Example 3.10, $\varphi$ is $\Lambda$-almost periodic with $P_\varphi=N$ and $\operatorname{Sp}_p(\varphi)=\Lambda$. As $N$ is a factor, $\Gamma(\varphi)=\operatorname{Sp}_p(\varphi)=\Lambda$ (Lemma 2.5(d)), and $P_\varphi'\cap P$ is the centre of $N$ by Proposition 3.7, that is $\mathbb C$; in particular the centre of $P$, which lies in $P_\varphi'\cap P$, is trivial. $P$ has separable predual, as $\Lambda$ is countable. If $P$ is full, Theorem 6.3 gives $Sd(P)=\Gamma(\varphi)=\Lambda$, and Corollary 7.6 gives $S(P)=\overline\Lambda$.

Exercise 5. Let $M$ be a type III factor with separable predual. Show that $Sd(M\bar\otimes B(K))=Sd(M)$ for every separable Hilbert space $K\neq0$.

Solution. $M$ is countably decomposable and properly infinite, so $M\bar\otimes B(K)\cong M$ (B6; for finite dimensional $K$ use $M\otimes M_n(\mathbb C)\cong M$, obtained in the same way from $n$ orthogonal projections equivalent to $1$). $Sd$ is an isomorphism invariant. The inclusion $Sd(M\bar\otimes B(K))\subset Sd(M)$ can also be seen directly: by Proposition 3.5(b), every almost periodic weight $\varphi$ on $M$ gives the almost periodic weight $\varphi\otimes\operatorname{Tr}$ with the same point spectrum. The reverse inclusion is where the isomorphism is needed.

References

[Connes 1974] A. Connes, Almost periodic states and factors of type III₁, J. Functional Analysis 16 (1974), 415–445. Free at https://doi.org/10.1016/0022-1236(74)90059-7
[Connes 1973] A. Connes, Une classification des facteurs de type III, Ann. Sci. École Norm. Sup. (4) 6 (1973), no. 2, 133–252. https://doi.org/10.24033/asens.1247 (open access: http://www.numdam.org/item?id=ASENS_1973_4_6_2_133_0). Free at https://alainconnes.org/wp-content/uploads/classificationfacteurs.pdf
[Araki–Woods 1968] H. Araki and E. J. Woods, A classification of factors, Publ. Res. Inst. Math. Sci. Ser. A 4 (1968), 51–130. https://doi.org/10.2977/prims/1195195263
[Effros 1965] E. G. Effros, The Borel space of von Neumann algebras on a separable Hilbert space, Pacific J. Math. 15 (1965), 1153–1164. https://doi.org/10.2140/pjm.1965.15.1153
[Haagerup 1975] U. Haagerup, The standard form of von Neumann algebras, Math. Scand. 37 (1975), 271–283. https://doi.org/10.7146/math.scand.a-11606
[Kechris 2025] A. S. Kechris, The Theory of Countable Borel Equivalence Relations, Cambridge University Press, 2025; pre-publication version free from the author: https://www.pma.caltech.edu/documents/7948/CBER.pdf
[Marks 2023] A. Marks, A short proof of the Connes–Feldman–Weiss theorem, note of 20 November 2017, updated 20 March 2023. Free from the author: https://math.berkeley.edu/~marks/notes/cfw.pdf
[Haagerup 1987] U. Haagerup, Connes' bicentralizer problem and uniqueness of the injective factor of type III₁, Acta Math. 158 (1987), 95–148. Free at https://doi.org/10.1007/BF02392257
[Connes 1985] A. Connes, Factors of type III₁, property $L'_\lambda$ and closure of inner automorphisms, J. Operator Theory 14 (1985), 189–211. Free at https://www.theta.ro/jot/archive/1985-014-001/1985-014-001-011.html

Almost periodic weights and the invariant \(Sd\)

Introduction

Results used from other lessons

1. What is used from the lessons on ultraproducts and on full factors

2. Actions of compact abelian groups and the Connes spectrum

2.1 The compact group attached to a group of eigenvalues

2.2 Actions, spectral subspaces and Fourier coefficients

2.3 The spectrum and the Connes spectrum

2.4 Crossed products recognized from eigenunitaries

3. Almost periodic weights

3.1 Definitions and the extension of the modular group

3.2 Constructions

3.3 The relative commutant of the centralizer

3.4 Examples

4. The point modular spectrum

5. Factors of type III₀: almost periodic states are dense

5.1 The structure as a crossed product

5.2 Trace-scaling derivatives

5.3 Density of almost periodic states

5.4 Consequences for \(Sd\)

6. On a full factor one weight computes \(Sd\)

7. Almost periodic weights on full factors

7.1 Characterizations of the good weights

7.2 Crossed products and non-full algebras

7.3 Existence and uniqueness

7.4 Consequences

8. Every countable subgroup of \(\mathbb R_+^*\) is \(Sd\) of a full factor

9. Exercises

References